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Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

The number of observations that have determined the least trimmed squares estimator, scalar.h is an integer greater or equal thanp but smaller thenn. Generally if the purpose is outlier detectionh=[0.5*(n+p+1)] (default value).h can be smaller than this threshold if the purpose is to find subgroups of homogeneous observations. In this function the LTS/LMS estimator is used just to initialize the search.

Number of subsamples which will be extracted to find the robust estimator, scalar. Ifnsamp=0 all subsets will be extracted. They will be(n choose p). If the number of all possible subset is<1000 the default is to extract all subsets otherwise just 1000.

Criterion to use to find the initial subset to initialize the search (LMS, LTS with concentration steps, LTS without concentration steps or subset supplied directly by the user). The default value is 1 (Least Median of Squares is computed to initialize the search). On the other hand, if the user wants to initialze the search with LTS with all the default options for concentration steps then lms=2. If the user wants to use LTS without concentration steps, lms can be a scalar different from 1 or 2. If lms is a list it is possible to control a series of options for concentration steps (for more details see optionlms insideLXS_control). If, on the other hand, the user wants to initialize the search with a prespecified set of units there are two possibilities:

1. lms can be a vector with length greater than 1 which contains the list of units forming the initial subset. For example, if the user wants to initialize the search with units 4, 6 and 10 thenlms=c(4, 6, 10);

2. lms is a struct which contains a field named bsb which contains the list of units to initialize the search. For example, in the case of simple regression through the origin with just one explanatory variable, if the user wants to initialize the search with unit 3 thenlms=list(bsb=3).

Search initialization, scalar. It specifies the initial subset size to start monitoring exceedances of minimum deletion residual, if init is not specified it set equal to:p+1, if the sample size is smaller than 40 ormin(3*p+1,floor(0.5*(n+p+1))), otherwise. For example, ifinit=100, the procedure starts monitoring from stepm=100.

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

Option to be used if the distribution of the data is strongly non normal and, thus, the general signal detection rule based on consecutive exceedances cannot be used. In this case bonflev can be:

1. a scalar smaller than 1 which specifies the confidence level for a signal and a stopping rule based on the comparison of the minimum MD with a Bonferroni bound. For example if bonflev=0.99 the procedure stops when the trajectory exceeds for the first time the 99% bonferroni bound.

2. A scalar value greater than 1. In this case the procedure stops when the residual trajectory exceeds for the first time this value.

Default value is ”, which means to rely on general rules based on consecutive exceedances.

Controls whether to display or not messages on the screen Ifmsg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

How to behave in case subset at step m (say bsbm) produces a singular X. In other words, this options controls what to do whenrank(X[bsbm, ]) is smaller then number of explanatory variables. Ifbsbmfullrank=1 (default) these units (whose number is say mnofullrank) are constrained to enter the search in the final n-mnofullrank steps else the search continues using as estimate of beta at step m the estimate of beta found in the previous step.

Plot on the screen. Scalar. Ifplot=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. Ifplot=2 the user can also monitor the intermediate plots based on envelope superimposition. Ifplot=FALSE (default) no plot is produced.

Wheather to superimpose bivariate least square lines on the plot (ifplot=TRUE.This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default isbivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. Ifbivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. Ifbivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. Ifbivarfit='i1' orbivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default ismultivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). Ifmultivarfit=2, same action as withmultivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

Add outlier labels in plot. Iflabeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value islabeladd=FALSE, i.e. no label is added.

Add variable labels in plot. A vector of strings of lengthp containing the labels of the variables of the regression dataset. If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

Controly scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Controlx scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Search initialization, scalar. It specifies the initial subset size to start monitoring exceedances of minimum deletion residual, if init is not specified it set equal to:p+1, if the sample size is smaller than 40 ormin(3*p+1,floor(0.5*(n+p+1))), otherwise. For example, ifinit=100, the procedure starts monitoring from stepm=100.

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

The kind of t-statistics which have to be monitored.tstat="trad" implies monitoring of traditional t statistics (out$Tols). In this case the estimate of s2 at step m is based on s2m (notice that s2m<<s2 when m/n is small)tstat="scal" (default) implies monitoring of rescaled t statistics. In this case the estimate of s2 at step m is based on s2m/vartruncnorm(m/n) where vartruncnorm(m/n) is the variance of the truncated normal distribution.

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975.

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Criterion to use to find the initial subset to initialize the search (LMS, LTS with concentration steps, LTS without concentration steps or subset supplied directly by the user). The default value is 1 (Least Median of Squares is computed to initialize the search). On the other hand, if the user wants to initialze the search with LTS with all the default options for concentration steps then lms=2. If the user wants to use LTS without concentration steps, lms can be a scalar different from 1 or 2. If lms is a list it is possible to control a series of options for concentration steps (for more details see optionlms insideLXS_control). If, on the other hand, the user wants to initialize the search with a prespecified set of units there are two possibilities:

1. lms can be a vector with length greater than 1 which contains the list of units forming the initial subset. For example, if the user wants to initialize the search with units 4, 6 and 10 thenlms=c(4, 6, 10);

2. lms is a struct which contains a field named bsb which contains the list of units to initialize the search. For example, in the case of simple regression through the origin with just one explanatory variable, if the user wants to initialize the search with unit 3 thenlms=list(bsb=3).

The number of observations that have determined the least trimmed squares estimator, scalar.h is an integer greater or equal thanp but smaller thenn. Generally if the purpose is outlier detectionh=[0.5*(n+p+1)] (default value).h can be smaller than this threshold if the purpose is to find subgroups of homogeneous observations. In this function the LTS/LMS estimator is used just to initialize the search.

Breakdown point. It measures the fraction of outliers the algorithm should resist. In this case any value greater than 0 but smaller or equal than 0.5 will do fine. If on the other hand the purpose is subgroups detection then bdp can be greater than 0.5. In any case however n*(1-bdp) must be greater than p. If this condition is not fulfilled an error will be given. Please specify h or bdp not both.

Number of subsamples which will be extracted to find the robust estimator, scalar. Ifnsamp=0 all subsets will be extracted. They will be(n choose p). If the number of all possible subset is<1000 the default is to extract all subsets otherwise just 1000.

LXS reweighted - if rew=1 the reweighted version of LTS (LMS) is used and the output quantities refer to the reweighted version else no reweighting is performed (default).

Confidence level which is used to declare units as outliers, usuallyconflev=0.95, 0.975, 0.99(individual alpha) or 1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975.

Controls whether to display or not messages on the screen Ifmsg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

It controls whether to display or not on the screen messages about estimated time to compute LMS (LTS). If nomes is equal to 1 no message about estimated time to compute LMS (LTS) is displayed, else if nomes is equal to 0 (default), a message about estimated time is displayed.

Plot on the screen. Scalar. Ifplots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. Ifplots=2 the user can also monitor the intermediate plots based on envelope superimposition. Ifplots=FALSE (default) no plot is produced.

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Starting values of the MM-estimator, a list with the fiollowingelements:loc, a $p x 1$ vector, location vector estimate andscale, a scaler, estimate of the scale. If empty (default) the program will use S estimators. In this last case it is possible to specify the options given in function Sreg.

Scalar defining nominal efficiency (i.e. a number between 0.5 and 0.99). The default value is 0.95.

Location or scale efficiency. Ifeffshape=1 efficiency refers to shape efficiency else (default) efficiency refers to location efficiency.

Specifies the rho function which must be used to weight the residuals. Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel'.

1. 'bisquare' uses Tukey's rho and psi functions. See TBrho and TBpsi.

2. 'optimal' uses optimal rho and psi functions. See OPTrho and OPTpsi.

3. 'hyperbolic' uses hyperbolic rho and psi functions. See HYPrho and HYPpsi.

4. 'hampel' uses Hampel rho and psi functions. See HArho and HApsi.

The default is 'bisquare'.

Additional parameters for the specified rho function. For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Number of refining iterations in each subsample (default isrefsteps=3).refsteps = 0 means "raw-subsampling" without iterations.

Scalar controlling tolerance in the MM loop. The default value istol=1e-6

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

Controls whether to display or not messages on the screen IfSmsg==TRUE (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

Plot on the screen. Scalar. Ifplots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. Ifplots=2 the user can also monitor the intermediate plots based on envelope superimposition. Ifplots=FALSE (default) no plot is produced.

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Starting values of the MM-estimator, a list with the fiollowingelements:loc, a $p x 1$ vector, location vector estimate andscale, a scaler, estimate of the scale. If empty (default) the program will use S estimators. In this last case it is possible to specify the options given in function Sreg.

Options to pass to Sreg, anSreg_control object. The options are: Srhofunc, Snsamp, Srefsteps, Sreftol, Srefstepsbestr, Sreftolbestr, Sminsctol, Sbestr.See functionSreg_control for more details on these options.

It is necessary to add to the S options the letter S at the beginning. For example, if you want to use the optimal rho function the supplied option is 'Srhofunc','optimal'. For example, if you want to use 3000 subsets, the supplied option is 'Snsamp',3000

Vector defining nominal efficiency (i.e. a series of numbers between 0.5 and 0.99). The default value is the sequenceseq(0.5, 0.99, 0.01)

Location or scale efficiency. Ifeffshape=1 efficiency refers to shape efficiency else (default) efficiency refers to location efficiency.

Specifies the rho function which must be used to weight the residuals. Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel'.

1. 'bisquare' uses Tukey's rho and psi functions. See TBrho and TBpsi.

2. 'optimal' uses optimal rho and psi functions. See OPTrho and OPTpsi.

3. 'hyperbolic' uses hyperbolic rho and psi functions. See HYPrho and HYPpsi.

4. 'hampel' uses Hampel rho and psi functions. See HArho and HApsi.

The default is 'bisquare'.

Additional parameters for the specified rho function. For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Number of refining iterations in each subsample (default isrefsteps=3).refsteps = 0 means "raw-subsampling" without iterations.

Scalar controlling tolerance in the MM loop. The default value istol=1e-6.

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

Plot on the screen. Scalar. Ifplots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. Ifplots=2 the user can also monitor the intermediate plots based on envelope superimposition. Ifplots=FALSE (default) no plot is produced.

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Breakdown point. It measures the fraction of outliers the algorithm should resist. In this case any value greater than 0 but smaller or equal than 0.5 will do fine.

Note that given bdp nominal efficiency is automatically determined.

Specifies the rho function which must be used to weight the residuals. Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel'.

1. 'bisquare' uses Tukey's rho and psi functions. See TBrho and TBpsi.

2. 'optimal' uses optimal rho and psi functions. See OPTrho and OPTpsi.

3. 'hyperbolic' uses hyperbolic rho and psi functions. See HYPrho and HYPpsi.

4. 'hampel' uses Hampel rho and psi functions. See HArho and HApsi.

The default is 'bisquare'.

Additional parameters for the specified rho function. For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Number of subsamples which will be extracted to find the robust estimator, scalar. Ifnsamp=0 all subsets will be extracted. They will be(n choose p). If the number of all possible subset is<1000 the default is to extract all subsets otherwise just 1000.

Number of refining iterations in each subsample (default isrefsteps=3).refsteps = 0 means "raw-subsampling" without iterations.

Tolerance for the refining steps. The default value is 1e-6

Scalar defining number of refining iterations for each best subset (default = 50).

Tolerance for the refining steps for each of the best subsets. The default value isreftolbestr=1e-8.

Value of tolerance for the iterative procedure for finding the minimum value of the scale for each subset and each of the best subsets (It is used by subroutineminscale.m).The default value isminsctol=1e-7.

Defins the number of "best betas" to remember from the subsamples. These will be later iterated until convergence (default isbestr=5).

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Controls whether to display or not messages on the screen Ifmsg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

Plot on the screen. Scalar. Ifplots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. Ifplots=2 the user can also monitor the intermediate plots based on envelope superimposition. Ifplots=FALSE (default) no plot is produced.

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

Breakdown point. It measures the fraction of outliers the algorithm should resist. In this case any value greater than 0 but smaller or equal than 0.5 will do fine.

The default value of bdp is a sequence from 0.5 to 0.01 with step 0.01

Specifies the rho function which must be used to weight the residuals. Possible values are 'bisquare' 'optimal' 'hyperbolic' 'hampel'.

1. 'bisquare' uses Tukey's rho and psi functions. See TBrho and TBpsi.

2. 'optimal' uses optimal rho and psi functions. See OPTrho and OPTpsi.

3. 'hyperbolic' uses hyperbolic rho and psi functions. See HYPrho and HYPpsi.

4. 'hampel' uses Hampel rho and psi functions. See HArho and HApsi.

The default is 'bisquare'.

Additional parameters for the specified rho function. For hyperbolic rho function it is possible to set up the value of k = sup CVC (the default value of k is 4.5).

For Hampel rho function it is possible to define parameters a, b and c (the default values are a=2, b=4, c=8)

Number of subsamples which will be extracted to find the robust estimator, scalar. Ifnsamp=0 all subsets will be extracted. They will be(n choose p). If the number of all possible subset is<1000 the default is to extract all subsets otherwise just 1000.

Number of refining iterations in each subsample (default isrefsteps=3).refsteps = 0 means "raw-subsampling" without iterations.

Tolerance for the refining steps. The default value is 1e-6

Scalar defining number of refining iterations for each best subset (default = 50).

Tolerance for the refining steps for each of the best subsets. The default value isreftolbestr=1e-8.

Value of tolerance for the iterative procedure for finding the minimum value of the scale for each subset and each of the best subsets (It is used by subroutineminscale.m).The default value isminsctol=1e-7.

Defins the number of "best betas" to remember from the subsamples. These will be later iterated until convergence (default isbestr=5).

Confidence level which is used to declare units as outliers. Usually conflev=0.95, 0.975, 0.99 (individual alpha) or conflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Controls whether to display or not messages on the screen Ifmsg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

Check input arguments, scalar. Ifnocheck=TRUE no check is performed on matrixy and matrixX. Notice thaty andX are left unchanged. In other words the additional column of ones for the intercept is not added. As defaultnocheck=FALSE.

Plot on the screen. Scalar. Ifplots=TRUE the plot of minimum deletion residual with envelopes based on n observations and the scatterplot matrix with the outliers highlighted is produced. Ifplots=2 the user can also monitor the intermediate plots based on envelope superimposition. Ifplots=FALSE (default) no plot is produced.

An S3 object of classtclusticsol.object,(output oftclustICsol) containing the relevant solutions.

Wheather to include or not spurious solutions. By default spuriossolutions are not included into the plot.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An object of S3 class returned by one of the estimation functions with themonitoring option selected (monitoring=TRUE):fsreda.object,sregeda.object,mmregeda.object,fsmeda.object,smulteda.object ormmmulteda.object.This is a list containing the monitoring of minimum Mahalanobis distance in case ofmultivariate analysis or the monitoring of residuals in case of regression.

The needed elements ofout are

1. MAL: matrix containing the squared Mahalanobis distances monitored in eachstep of the forward search. Every row is associated with a unit (row of data matrix Y).This matrix can be created using one of the functionsfsmult,smult ormmmult with the monitoring option selected, i.e.monitoring=TRUE.

2. RES: matrix containing the residuals monitored in eachstep of the forward search. Every row is associated with a unit (row of data matrix Y).This matrix can be created using the functionfsreg withthe monitoring option selected, i.e.monitoring=TRUE .

3. bdp: a vector containing breakdown point values that have been used, in case of S estimates.

4. eff: a vector containing efficiency values that have been used, in case of MM estimates.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An object of S3 classfsmeda.object returned byfsmult withmonitoring=TRUE -a list containing the monitoring of minimum Mahalanobis distance.

The needed elements ofout are

1. S2cov: matrix containing the monitoring of the elementsof the covariance matrix in each step of the forward search:

2. Un: matrix containing the order of entry of each unit(necessary if datatooltip or databrush is selected).

3. X: The data matrix.

Controls thex scale in the plot.xlim is a vector with two elements controllingminimum and maximum on thex-axis. Default is to use automatic scale.

Controls they scale in the plot.ylim is a vector with two elements controllingminimum and maximum on they-axis. Default is to use automatic scale.

A title for the x axis

A title for the y axis

An overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed,3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid","dashed", "dotted", "dotdash", "longdash", or "twodash".The latter two are not supported by Matlab.

Colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Numeric vector containing the subset size with length equal to the number of columns ofmatrix of mahalanobis distances. The default value of subsize is(nrow(MAL) - ncol(MAL) + 1):nrow(MAL)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifiesthe highlighted trajectories.Iflength(fg.thresh) == 1 the highlighted trajectories are those of units that throughtoutthe search had at leat once a mahalanobis distance greater thanfg.thresh.The default value isfg.thresh=2.5. Iflength(fg.thresh) == 2 the highlightedtrajectories are those of units that throughtout the search had a mahalanobis distance atleast once bigger thanfg.thresh[2] or smaller thanfg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted.Iffg.unit is supplied,fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels forthe highlighted trajectories (units). The default is to put the labels at theinitial and final steps of the search.fg.labstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types caneither be specified as an integer (1=solid (default), 2=dashed,3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid","dashed", "dotted", "dotdash", "longdash", or "twodash".The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols.iffg.mark==TRUE each line is plotted using a differentsymbol else no marker is used (default).

Controls the font size of the labels of the trajectories in foreground. Iffg.cex=0 no labels will be shown - equivalent tofg.labstop="".

MATLAB-style arguments - appearance of the plot in terms of xlim, ylim, axes labelsand their font size style, color of the lines, etc.

MATLAB-style arguments - for the trajectories in foregroud.

Plot handle. String which identifies the handle of the plot which is about to be created.The default istag='pl_mmd'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to haveinformation about the unit selected, the step in which the unit enters the search andthe associated label. If datatooltip is a list, it is possible to control the aspectof the data cursor (see MATLAB functiondatacursormode() for more details orsee the examples below). The default options areDisplayStyle="Window" andSnapToDataVertex="on".

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

Ifrew=TRUE all subsequent output refers to reweighted else no reweighting is done.

p-by-1 vector containing the estimated regression parameters.

p x 1 vector containing the units forming subset associated with bLMS (bLTS).

residuals.

scale estimate of the residuals.

Vector like y containing weights. The elements of this vector are 0 or 1. These weights identify the h observations which are used to compute the final LTS (LMS) estimate. sum(weights)=h if there is not a perfect fit otherwise sum(weights) can be greater than h

The number of observations that have determined the LTS (LMS) estimator, i.e. the value of h.

vector containing the list of the units declared as outliers using confidence level specified in input scalar conflev.

confidence level which is used to declare outliers. Remark:conflev will be used to draw the horizontal lines (confidence bands) in the plots. Default value is 0.975

Number of subsets wihtout full rank. Notice that if this number is greater than 0.1*(number of subsamples) a warning is produced

the data matrix X

the response vector y

Ifrew=TRUE all subsequent output refers to reweighted else no reweighting is done.

p-by-1 vector containing the estimated regression parameters.

p x 1 vector containing the units forming subset associated with bLMS (bLTS).

residuals.

scale estimate of the residuals.

Vector like y containing weights. The elements of this vector are 0 or 1. These weights identify the h observations which are used to compute the final LTS (LMS) estimate. sum(weights)=h if there is not a perfect fit otherwise sum(weights) can be greater than h

The number of observations that have determined the LTS (LMS) estimator, i.e. the value of h.

vector containing the list of the units declared as outliers using confidence level specified in input scalar conflev.

confidence level which is used to declare outliers. Remark:conflev will be used to draw the horizontal lines (confidence bands) in the plots. Default value is 0.975

Number of subsets wihtout full rank. Notice that if this number is greater than 0.1*(number of subsamples) a warning is produced

the data matrix X

the response vector y

An n x p data matrix (n observations and p variables).Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

Plots the random starts minimum Mahalanobis distance with 1Ifplot=FALSE (default) orplot=0 no plot is produced.The scale (ylim) for the y axis is defined as follows:

• ylim[2] is the maximum between the values ofmmd in steps[n*0.2 n] and the finalvalue of the 99 per cent envelope multiplied by 1.1.

• ylim[1] is the minimum between the values ofmmd in steps[n*0.2 n] and the 1 per cent envelope multiplied by 0.9.

Remark: the plot which is produced is very simple. In order to control a series of optionsin this plot (including the y scale) and in order to connect it dynamically to the otherforward plots it is necessary to use functionmmdrsplot.

Point where to start monitoring required diagnostics.Ifinit is not specified it will be set equal to(p+1).

A vector which specifies for which steps of the forwardsearch it is necessary to save the units forming subset for eachrandom start. ifbsbsteps = 0 for each random start we storethe units forming subset in all steps. The default is store theunits forming subset in all steps ifn <= 500 else to storethe units forming subset at step init and steps which are multipleof 100. For example, ifn = 753 andinit = 6,units forming subset are stored form=init, 100, 200, 300, 400, 500 and 600.

REMARK: The vector bsbsteps must contain numbers from init to n.ifmin(bsbsteps) < init a warning message will be issued.

Number of random starts. Default value isnsimul=200.

It controls whether to perform checks on matrix Y. Ifnocheck=TRUE,no check is performed.

Ifnumpool > 1, the routine automatically checks if theParallel Computing Toolbox is installed and distributes the random starts overnumpool parallel processes. Ifnumpool <= 1, the random starts are runsequentially. By default, numpool is set equal to the number of physical coresavailable in the CPU (this choice may be inconvenient if other applicationsare running concurrently). The same happens if the numpool value chosen bythe user exceeds the available number of cores.

REMARK: up to R2013b, there was a limitation on the maximum number of cores thatcould be addressed by the parallel processing toolbox (8 and, more recently, 12).From R2014a, it is possible to run a local cluster of more than 12 workers.

REMARK: Unless you adjust the cluster profile, the default maximum number ofworkers is the same as the number of computational (physical) cores on the machine.

REMARK: In modern computers the number of logical cores is larger than the numberof physical cores. By default, MATLAB is not using all logical cores because,normally, hyper-threading is enabled and some cores are reserved to this feature.

REMARK: It is because of Remarks 3 that we have chosen as default value fornumpool the number of physical cores rather than the number of logical ones.The user can increase the number of parallel pool workers allocated to themultiple start monitoring by:

• setting the NumWorkers option in the local cluster profile settings tothe number of logical cores (Remark 2). To do so go on the menuHome|Parallel|Manage Cluster Profile and set the desired"Number of workers to start on your local machine".

• setting numpool to the desired number of workers

Therefore, *if a parallel pool is not already open*, UserOption numpool (if set)overwrites the number of workers set in the local/current profile. Similarly,the number of workers in the local/current profile overwrites default value ofnumpool obtained as feature('numCores') (i.e. the number of physical cores).

Set cleanpoolcleanpool=TRUE if the parallel pool has to be cleanedafter the execution of the random starts. Otherwise (default)cleanpool=FALSE.Clearly this option has an effect just if previous optionnumpool > 1.

Level of output to sidplay. It controls whether to display or not messagesabout random start progress. More precisely, if previous optionnumpool > 1,then a progress bar is displayed, on the other hand a message will be displayed onthe screen when 10

REMARK: in order to create the progress bar whennparpool > 1 the programwrites on a temporary .txt file in the folder where the user is working.Therefore it is necessary to work in a folder where the user has writepermission. If this is not the case and the user (say) is working withoutwrite permission in folder C:/Program Files/MATLAB the following messagewill appear on the screen:

Error using ProgressBar (line 57) Do you have write permissions for C:/Program Files/MATLAB?"

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An n x p data matrix (n observations and p variables).Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

List of units forming the initial subset or size of the initial subset.Ifmonitoring=FALSE the default is to start the search withp+1units, containing those observationswhich are not outlying on any scatterplot, found as the intersection of all points lyingwithin a robust contour containing a specified portion of the data (Riani and Zani 1997)and inside the univariate boxplot.

Remark: if bsb is a vector, the option crit is ignored.

Wheather to perform monitoring of Mahalanobis distances and other specific quantities

If specified, the criterion to be used to initialize the search.

• Ifcrit="md" the units which form initial subset are those which have thesmallestm0 pseudo Mahalanobis distances computed using procedureunibiv() (bivariate robust ellipses).

• Ifcrit="biv" sorting is done first in terms of times units fell outside robust bivariate ellipsesand then in terms of pseudoMD. In other words, the units forming initial subset are chosen firstamong the set of those which never fell outside robust bivariate ellipses then among those whichfell only once outside bivariate ellipses ... up to reachm0.

• Ifcrit="uni" sorting is done first in terms of times units fell outside univariate boxplots andthen in terms of pseudoMD. In other words, the units forming initial subset are chosen first amongthe set of those which never fell outside univariate boxplots then among those which fell onlyonce outside univariate boxplots... up to reachm0.

Remark: as the user can see the starting point of the search is not going to affect at all the resultsof the analysis. The user can explore this point with his own datasets.

Remark: ifcrit="biv" the user can also supply in scalar rf (see below) the confidence level ofthe bivariate ellipses.

Confidence level for bivariate ellipses. The default is 0.95. This option is useful only ifcrit='biv'.

Point where to start monitoring required diagnostics. Note that if a vectorm0 issupplied,init >= length(m0). Ifinit is not specified it will be set equal tofloor(n*0.6).

Plots the minimum Mahalanobis distance. Ifplot=FALSE (default) orplot=0 no plot is produced.Ifplot=TRUE the plot of minimum MD with envelopes based on n observations and thescatterplot matrix with the outliers highlighted is produced.Ifplot=2 the additional plots of envelope resuperimposition are produced.If plot is a list it may contain the following fields:

• ylim vector with two elements controlling minimum and maximum on the y axis.Default value is ” (automatic scale)

• xlim vector with two elements controlling minimum and maximum on the x axis.Default value is ” (automatic scale)

• resuper vector which specifies for which steps it is necessary to show the plotsof resuperimposed envelopes if resuper is not supplied a plot of each step in whichthe envelope is resuperimposed is shown. Example: ifresuper = c(85 87)plots of resuperimposedenvelopes are shown at stepsm=85 andm=87

• ncoord Ifncoord=1 plots are shown in normal coordinates else (default)plots are shown in traditional mmd coordinates

• labeladd Iflabeladd=1, the outliers in the spm are labelled with the unitrow index. The default value islabeladd="", i.e. no label is added

• nameY character vector containing the labels of the variables. As default value,the labels which are added are Y1, ...Yp.

• lwd controls line width of the curve which contains the monitoring of minimumMahalanobis distance. Default islwd=2.

• lwdenv Controls linewidth of the envelopes. Default islwdenv=2.

Option that might be used to identify extreme outliers when the distribution ofthe data is strongly non normal. In these circumstances, the general signal detection rulebased on consecutive exceedances cannot be used. In this casebonflev can be:

1. a scalar smaller than 1, which specifies the confidence level for a signal and a stopping rulebased on the comparison of the minimum deletion residual with a Bonferroni bound.For example ifbonflev=0.99 the procedure stops when the trajectory exceedsfor the first time the 99 per cent bonferroni bound.

2. a scalar value greater than 1. In this case the procedure stops when theresidual trajectory exceeds for the first time this value.

Default value is empty, which means to rely on general rules based on consecutive exceedances.

It controls whether to display or not messages on the screen. Ifmsg=TRUE(default) messages about the progression of the search are displayed on the screenotherwise only error messages will be displayed.

It controls whether to perform checks on matrix Y. Ifnocheck=TRUE, no check is performed.

Controls whether to monitor scaled Mahalanobis distances (only ifmonitoring=TRUE). Ifscaled=TRUEMahalanobis distances monitored during the search are scaled using ratio of determinant.Ifscaled=2 Mahalanobis distances monitored during the search are scaled usingasymptotic consistency factor. The default isscaled=FALSE, that is Mahalanobis distances are not scaled.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

p-by-1 vector containing location of the data.

p-by-p robust estimate of covariance matrix.

n-by-1 vector containing the estimates of the robust Mahalanobis distances (in squared units). This vector contains the distances of each observation from the location of the data, relative to the scatter matrix cov.

(n-init)-by-2 matrix. 1st col is the forward search index; 2nd col is the value of minimum Mahalanobis Distance in each step of the fwd search.

(n-init)-by-11 matrix which contains the unit(s) includedin the subset at each step of the fwd search.REMARK: in every step the new subset is compared with theold subset. Un contains the unit(s) present in the newsubset but not in the old one. Un[1 ,2] for example contains the unit included in step init+1.Un[end, 2] contains the units included in the final stepof the search.

2 x 5 matrix containing the number of times mdr went outof particular quantiles.First row contains quantiles 1 99 99.9 99.99 99.999.Second row contains the frequency distribution.It is NULL if bonflev threshold is used.

This output is produced only if the search found at acertain step is a non singular matrix X. In this case thesearch run in a constrained mode, that is including theunits which produced a singular matrix in the last n-constrsteps. out.constr is a vector which contains the list ofunits which produced a singular X matrix.

the data matrix X

p-by-1 vector containing the estimated regression parameters (in step n-k).

scalar containing the estimate of the scale (sigma).

residuals.

fitted values.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

(n-init) x 2 matrix 1st col = fwd search index, 2nd col = value of minimum deletion residual in each step of the fwd search

(n-init) x 11 matrix which contains the unit(s) includedin the subset at each step of the fwd search.REMARK: in every step the new subset is compared with theold subset. Un contains the unit(s) present in the newsubset but not in the old one.Un(1,2) for example contains the unit included in stepinit+1.Un(end,2) contains the units included in the final stepof the search.

2 x 5 matrix containing the number of times mdr went outof particular quantiles.First row contains quantiles 1 99 99.9 99.99 99.999.Second row contains the frequency distribution.

This output is produced only if the search found at acertain step is a non singular matrix X. In this case thesearch run in a constrained mode, that is including theunits which produced a singular matrix in the last n-constrsteps. out.constr is a vector which contains the list ofunits which produced a singular X matrix.

the data matrix X

the response vector y

aformula of the formy ~ x1 + x2 + ....

data frame from which variables specified informula are to be taken.

an optional vector specifying a subset of observationsto be used in the fitting process.

an optional vector of weights to be usedin the fitting process.NOT USED YET.

a function which indicates what should happenwhen the data containNAs. The default is set bythena.action setting ofoptions, and isna.fail if that is unset. The “factory-fresh”default isna.omit. Another possible value isNULL, no action. Valuena.exclude can be useful.

logicals indicating if themodel frame, the model matrix and the response are to be returned,respectively.

an optional list. See thecontrasts.argofmodel.matrix.default.

this can be used to specify ana prioriknown component to be included in the linear predictorduring fitting. Anoffset term can be included in theformula instead or as well, and if both are specified their sum is used.

Predictor variables. Matrix. Matrix of explanatoryvariables (also called 'regressors') of dimension n x (p-1)where p denotes the number of explanatory variablesincluding the intercept.Rows of X represent observations, and columns representvariables. By default, there is a constant term in themodel, unless you explicitly remove it using input optionintercept=FALSE, so do not include a column of 1s in X. Missingvalues (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite valueswill automatically be excluded from the computations.

Response variable. Vector. Response variable, specified asa vector of length n, where n is the number ofobservations. Each entry in y is the response for thecorresponding row of X.Missing values (NA's) and infinite values (Inf's) areallowed, since observations (rows) with missing or infinitevalues will automatically be excluded from thecomputations.

Initial subset - vector of indices. Ifbsb=0 (default) then the procedure starts with p units randomly chosen. If bsb is not 0 the search will start withm0=length(bsb).

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

wheather to perform monitoring for several quantities in each step of the forward search. Deafault ismonitoring=FALSE.

A control object (S3) containing estimation options, as returned byFSR_control. Use the functionFSR_control and see its help page. If the control object is supplied, the parameters from itwill be used. If parameters are passed also in the invocation statement, they will override the corresponding elements of the control object.

Whether to print intermediate results. Default istrace=FALSE.

Potential further optional arguments, see the help of the functionFSR_control.

n x (n-init+1) matrix containing the monitoring of scaled residuals: the first row is the residual for the first unit, ..., n-th row is the residual for the n-th unit.

(n+1) x (n-init+1) matrix containing the monitoring of leverage: the first row is the leverage for the first unit, ..., n-th row is the leverage for the n-th unit.

n x (n-init+1) matrix containing the information about the units belonging to the subset at each step of the forward search: first col contains indexes of the units forming subset in the initial step; ...; last column contains units forming subset in the final step (all units).

n-init x 3 matrix which contains the monitoring of minimum deletion residual or (m+1)-ordered residual at each step of the forward search: first col is the fwd search index (from init to n-1);2nd col = minimum deletion residual; 3rd col = (m+1)-ordered residual.

Remark: these quantities are stored with sign, that is the min deletion residual is stored with negative sign if it corresponds to a negative residual.

n-init+1 x 3 matrix which contains the monitoring of maximum studentized residual or m-th ordered residual: first col is the fwd search index (from init to n); 2nd col = maximum studentized residual; 3rd col = (m)-ordered studentized residual.

(n-init+1) x 4 matrix containing the monitoring of normality test in each step of the forward search: first col = fwd search index (from init to n); 2nd col = Asymmetry test; 3rd col = Kurtosis test; 4th col = Normality test.

(n-init+1) x (p+1) matrix containing the monitoring of estimated beta coefficients in each step of the forward search.

(n-init+1) x 5 matrix containing the monitoring of S2 or R2 and F-test in each step of the forward search:

1. 1st col = fwd search index (from init to n);

2. 2nd col = monitoring of S2;

3. 3rd col = monitoring of R2;

4. 4th col = monitoring of rescaled S2. In this case theestimated of sigma^2 at step m is divided by theconsistency factor (to make the estimate asymptoticallyunbiased)

5. 5th col = monitoring of F test. Note that an asymptoticunbiased estimate of sigma^2 is used.

In this case the estimated of s2 at step m is divided by the consistency factor (to make the estimate asymptotically unbiased).

(n-init+1) x 3 matrix containing the monitoring of Cook or modified Cook distance in each step of the forward search:

1. 1st col = fwd search index (from init to n);

2. 2nd col = monitoring of Cook distance;

3. 3rd col = monitoring of modified Cook distance.

(n-init+1) x (p+1) matrix containing the monitoring of estimated t-statistics (as specified in option input 'tstat') in each step of the forward search

(n-init) x 11 matrix which contains the unit(s) included in the subset at each step of the fwd search.

REMARK: in every step the new subset is compared with the old subset. Un contains the unit(s) present in the new subset but not in the old one Un(1,2) for example contains the unit included in step init+1 Un(end,2) contains the units included in the final step of the search.

Confidence intervals for the elements of β. betaINT is a (n-init+1)-by-2*length(confint)-by-p 3D array. Each third dimension refers to an element of beta:

1. betaINT[,,1] is associated with first element of beta;

2. ...;

3. betaINT[,,p] is associated with last element of beta.

The first two columns contain the lower and upper confidence limits associated with conflev(1). Columns three and four contain the lower and upper confidence limits associated with conflev(2); ...;The last two columns contain the lower and upper confidence limits associated with conflev(end).For examplebetaINT[,3:4,5] contain the lower and upper confidence limits for the fifth element of beta using confidence level specified in the second element of input optionconflev.

confidence interval for s2.

1. 1st col = fwd search index;

2. 2nd col = lower confidence limit based on conflev(1);

3. 3rd col = upper confidence limit based on conflev(1);

4. 4th col = lower confidence limit based on conflev(2);

5. 5th col = upper confidence limit based on conflev(2);

6. ...

7. penultimate col = lower confidence limit based on conflev(end);

8. last col = upper confidence limit based on conflev(end).

the data matrix X

the response vector y

aformula of the formy ~ x1 + x2 + ....

data frame from which variables specified informula are to be taken.

an optional vector specifying a subset of observationsto be used in the fitting process.

an optional vector of weights to be usedin the fitting process.NOT USED YET.

a function which indicates what should happenwhen the data containNAs. The default is set bythena.action setting ofoptions, and isna.fail if that is unset. The “factory-fresh”default isna.omit. Another possible value isNULL, no action. Valuena.exclude can be useful.

logicals indicating if themodel frame, the model matrix and the response are to be returned,respectively.

an optional list. See thecontrasts.argofmodel.matrix.default.

this can be used to specify ana prioriknown component to be included in the linear predictorduring fitting. Anoffset term can be included in theformula instead or as well, and if both are specified their sum is used.

family of robust regression models, can be 'homo' for homoscedastic (same variance) regression model, 'hetero' for heteroskedastic regression model or 'bayes' Bayesian linear regression. The default isfamily='homo' for homoscedastic regression model.

robust regression estimation model, can be 'FS' for Forward search, 'S' for S regression, 'MM' for MM regression, 'LMS' or 'LTS'. The default ismethod='FS' for forward search estimation.

wheather to perform monitoring for several quantities in each step of the forward search or for series of values of the breakdown point in case of S estimates or for series of values of the efficiency in case of MM estimates. Deafault ismonitoring=FALSE.

Response variable. Vector. Response variable, specified asa vector of length n, where n is the number ofobservations. Each entry in y is the response for thecorresponding row of X.Missing values (NA's) and infinite values (Inf's) areallowed, since observations (rows) with missing or infinitevalues will automatically be excluded from thecomputations.

Predictor variables. Matrix. Matrix of explanatoryvariables (also called 'regressors') of dimension n x (p-1)where p denotes the number of explanatory variablesincluding the intercept.Rows of X represent observations, and columns representvariables. By default, there is a constant term in themodel, unless you explicitly remove it using input optionintercept=FALSE, so do not include a column of 1s in X. Missingvalues (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite valueswill automatically be excluded from the computations.

Initial subset - vector of indices. Ifbsb=0 (default) then the procedure starts with p units randomly chosen. If bsb is not 0 the search will start withm0=length(bsb).

Indicator for constant term. Scalar. Ifintercept=TRUE, a model with constant term will be fitted (default), else, no constant term will be included.

A control object (S3) containing estimation options. If the control object is supplied, the parameters from itwill be used. If parameters are passed also in the invocation statement, they will override the corresponding elements of the control object.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

Ann x p data matrix (n observations andp variables).Rows ofx represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

potential further arguments passed to lower level functions.

aformula of the formy ~ x1 + x2 + ....

data frame from which variables specified informula are to be taken.

an optional vector specifying a subset of observationsto be used in the fitting process.

an optional vector of weights to be usedNOT USED YET.

a function which indicates what should happenwhen the data containNAs. The default is set bythena.action setting ofoptions, and isna.fail if that is unset. The “factory-fresh”default isna.omit. Another possible value isNULL, no action. Valuena.exclude can be useful.

logical indicating if themodel frame, is to be returned.

logical indicating if thethe model matrixis to be returned.

logical indicating if theresponse is to be returned.

an optional list. See thecontrasts.argofmodel.matrix.default.

this can be used to specify ana prioriknown component to be included in the linear predictorduring fitting. Anoffset term can be included in the

Response variable. A vector withn elements thatcontains the response variable.

wheather to use constant term (default isintercept=TRUE

Ifplot=FALSE (default) no plot is produced.Ifplot=TRUE a fan plot is shown.

string which identifies the family of transformations which must be used. Possible values arec('BoxCox', 'YJ', 'YJpn', 'YJall'). Default is'BoxCox'. The Box-Cox family of powertransformations equals(y^{\lambda}-1)/\lambda for\lambda not equal to zero, and\log(y)if\lambda = 0.The Yeo-Johnson (YJ) transformation is the Box-Cox transformation ofy+1 for nonnegative values, and of|y|+1 with parameter2-\lambda fory negative. Remember that BoxCox can be used onlyif input y is positive. Yeo-Johnson family of transformations does not have this limitation.Iffamily='YJpn' Yeo-Johnson family is applied but in this case it is also possibleto monitor (in the output argumentsScorep andScoren) the score test forpositive and negative observations respectively. Iffamily='YJall', it is alsopossible to monitor the joint F test for the presence of the two constructed variablesfor positive and negative observations.

values of the transformation parameter for which it is necessaryto compute the score test. Default value of lambda isla=c(-1, -0.5, 0, 0.5, 1), i.e., the five most common values of lambda.

how to find the initlal subset to initialize the search. Iflms=1 (default)Least Median of Squares (LMS) is computed, else Least Trimmed Squares (LTS) is computed.If,lms is matrix of sizep - 1 + intercept X length(la) it contains in columnj=1,..., lenght(la) the list of units forming the initial subset for the searchassociated withla(j). In this case the input optionnsamp is ignored.

the percentage (roughly) of squared residuals whose sum will be minimized,by defaultalpha=0.5. In general, alpha must between 0.5 and 1.

The number of observations that have determined the least trimmed squaresestimator, scalar.h is an integer greater or equal thanp but smallerthenn. Generallyh=[0.5*(n+p+1)] (default value).

Search initialization. It specifies the initial subset size to startmonitoring the value of the score test. Ifinit is not specified it willbe set equal to:p+1, if the sample size is smaller than 40 ormin(3 * p + 1, floor(0.5 * (n+p+1))), otherwise.

Controls whether to display or not messages on the screen. Ifmsg==TRUEmessages are displayed on the screen. Ifmsg=2, detailed messages are displayed,for example the information at iteration level.

Whether to check input arguments. Ifnocheck=TRUE no check is performedon matrixy and matrixX. Notice thaty andXare left unchanged. In other words the additional column of ones for theintercept is not added. The default isnocheck=FALSE.

number of subsamples which will be extracted to find the robust estimator. Ifnsamp=0all subsets will be extracted. They will ben choose p.

Remark: if the number of all possible subset is <1000 the default is to extract all subsetsotherwise just 1000. Ifnsamp is a matrix of sizer-by-p, it contains in the rowsthe subsets which sill have to be extracted. For example, ifp=3 andnsamp=c(2,4,9; 23, 45, 49; 90, 34, 1)the first subset is made up of unitsc(2, 4, 9), the second subset of unitsc(23, 45, 49)and the third subset of unitsc(90 34 1).

Confidence level for the bands (default is 0.99, that is,we plot two horizontal lines corresponding to values -2.58 and 2.58).

A label for the X-axis, default is 'Subset size m'

A label for the Y-axis, default is 'Score test statistic'

A label for the title, default is 'Fan plot'

Minimum and maximum for the X-axis

Minimum and maximum for the Y-axis

The line width of the curves which contain the score test, a positive number, default islwd=2

The line width of the lines associated with the envelopes, a positive number, default islwd.env=1

Whether to print intermediate results. Default istrace=FALSE.

a vector specifying the colors for the lines, eachone corresponding to ala value. iflength(col) < length(la),the colors will be recycled.

a vector specifying the line types for the lines, eachone corresponding to ala value. iflength(col) < length(la),the colors will be recycled.

An object containing monitoring of leverage,fsreda.object.

The needed elements ofout are

1. LEV: matrix containing the leverage monitored in each step of the forward search. Every row is associated with a unit.This matrix can be created using functionfsreg() withmethod="FS", monitoring=TRUE.

2. Un: (for FSR only) - matrix containing the order of entry in the subset of each unit (required only when datatooltip is true or databrush is not empty).

3. y: a vector containing the response (required only when option databrush is requested).

4. X: a matrix containing the explanatory variables (required only when option databrush is requested).

5. Bols: (n-init+1) x (p+1) matrix containing the estimated beta coefficients monitored in each step of the robust procedure (required only when option databrush is requested and suboption multivarfit is requested).

Controly scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Controlx scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

a title for the x axis

a title for the y axis

an overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

values for the x axis. Numeric vector ofncol(RES) controlling the x axis coordinates. The default value of xvalues is(nrow(RES) - ncol(RES) + 1):nrow(RES)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifies the highlighted trajectories.Iflength(fthresh) == 1 the highlighted trajectories are those of units that throughtout the search had at leat once a residual greater (in absolute value) than thresh. The default value isfg.thresh=2.5. Iflength(fthresh) == 2 the highlighted trajectories are those of units that throughtout the search had a residual at leat once bigger thanfg.thresh[2] or smaller thanfg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted. Iffg.unit is supplied,fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels for the highlighted trajectories (units). The default is to put the labels at the initial and final steps of the search.flabstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols.iffg.mark==TRUE each line is plotted using a different symbol else no marker is used (default).

controls the font size of the labels of the trajectories in foreground.

numeric vector of length 1 or 2 which specifies how to define the unimmportant trajectories.Unimmportant trajectories will be plotted using a colormap, in greysh or will be hidden.Iflength(thresh) == 1 the irrelevant units are those which always had a residual smaller (in absolute value) than thresh.Iflength(bthresh) == 2 the irrelevant units are those which always had a residual greater than bthresh(1) and smaller than bthresh(2). The default is:bg.thresh=2.5 ifn > 100 andbg.thresh=-Inf ifn <= 100 i.e. to treat all trajectories as important ifn <= 100 and, ifn > 100, to reduce emphasis only to trajectories having in all steps of the search a value of scaled residual smaller than 2.5.

specifies how to plot the unimportant trajectories as defined in option bthresh.

1. bg.style="faint": unimportant trajectories are plotted using a colormap.

2. bg.style="hide": unimportant trajectories are hidden.

3. bg.style="greyish": unimportant trajectories are displayed in a faint grey.

Whenn>100 the default option isbg.style='faint'. Whenn <= 100 andbg.thresh == -Inf option bstyle is ignored.Remark: bground=” is equivalent to -Inf that is all trajectories are considered relevant.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is to use tag 'pl_resfwd'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

trajectories to highlight in connection with resfwdplot. Ifxground="lev" (default), the levfwdplot trajectories are put in foreground or in background depending on the leverage values. Ifxground="res", the levfwdplot trajectories are put in foreground or in background depending on the residual values. See optionsbg.thresh andfg.thresh.

Interactive clicking. It is inactive if this parameter is missing or empty. The default isdatatooltip=TRUE, i.e. the user can select with the mouse an individual residual trajectory in order to have information about the corresponding unit. The information displayed depends on the estimator in use.

For example for classfsreda.object the information concerns the label and the step of the search in which the unit enters the subset. If datatooltip is a list it may contain the following fields:

1. DisplayStyle determines how the data cursor displays. Possible values are'datatip' and'window' (default).'datatip' displays data cursor information in a small yellow text box attached to a black square marker at a data point you interactively select.'window' displays data cursor information for the data point you interactively select in a floating window within the figure.

2. SnapToDataVertex: specifies whether the data cursor snaps to the nearest data value or is located at the actual pointer position. Possible values areSnapToDataVertex='on' (default) andSnapToDataVertex='off'.

3. LineColor: controls the color of the trajectory selected with the mouse. It can be an RGB triplet of values between 0 and 1, or character vector indicating a color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation.

4. SubsetLinesColor: enables to control the color of the trajectories of the units that are in the subset at a given step of the search (iflevfwdplot() is applied to an object of classfsreda.object) or have a weight greater than 0.9 (iflevfwdplot() is applied to an object of classsregeda.object ormmregeda.object). This can be done (repeatedly) with a left mouse click in proximity of the step of interest. A right mouse click will terminate the selection by marking with a up-arrow the step corresponding to the highlighted lines. The highlighted lines by default are in red, but a different color can be specified as RGB triplet or character of color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation. By defaultSubsetLinesColor="", i.e. the modality is not active.Any initialization forSubsetLinesColor which cannot be interpreted as RGB vector will be converted to blue, i.e.SubsetLinesColor will be forced to be [0 0 1].IfSubsetLinesColor is not empty the previous optionLineColor is ignored.

Character vector containing the labels of the units (optional argument used whendatatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

Add variable labels in plot. A vector of strings of lengthp containing the labels of the variables of the regression dataset. If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

Controls whether to display or not messages on the screen Ifmsg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is a scalar or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the y|X plot, i.e. a matrix of scatter plots of y against each column of X, grouped according to the selection(s) done by brushing. If the plot y|X does not exist it is automatically created.In addition, brushed units are automatically highlighted in the minimum deletion residual plot if it is already open. The extension to the following plots will be available in future versions of the toolbox:

1. monitoring leverage plot;

2. maximum studentized residual;

3. s^2 and R^2;

4. Cook distance and modified Cook distance;

5. deletion t statistics.

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments of functionselectdataFS() and the following optional argument:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value ispersist="", that is brushing is allowed only once. Ifpersist="on" orpersis="off" brushing can be done as many time as the user requires. Ifpersist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. Ifpersist='off' every time a new brush is performed units previously brushed are removed.

2. bivarfit. Wheather to superimpose bivariate least square lines on the plot (ifplot=TRUE.This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default isbivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. Ifbivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. Ifbivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. Ifbivarfit='i1' orbivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

3. multivarfit. Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default ismultivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). Ifmultivarfit=2, same action as withmultivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

4. labeladd. Add outlier labels in plot. Iflabeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value islabeladd=FALSE, i.e. no label is added.

(MATLAB-style arguments) appearance of the plot in terms of xlim, ylim, axes labels and their font size style, color of the lines, etc.

MATLAB-style arguments for the fground trajectories in foregroud.

MATLAB-style arguments for the fground trajectories in backgroud.

potential further arguments passed to lower level functions.

An object of S3 classfsmeda.object returned byfsmult withmonitoring=TRUE -a list containing the monitoring of minimum Mahalanobis distance.

The needed elements ofout are

1. MAL: matrix containing the squared Mahalanobis distances monitored in eachstep of the forward search. Every row is associated with a unit (row of data matrix X).

2. Un: matrix containing the order of entry of each unit(necessary if datatooltip or databrush is selected).

3. X: The data matrix.

Controls thex scale in the plot.xlim is a vector with two elements controllingminimum and maximum on thex-axis. Default is to use automatic scale.

Controls they scale in the plot.ylim is a vector with two elements controllingminimum and maximum on they-axis. Default is to use automatic scale.

A title for the x axis

A title for the y axis, deafult is "Squared Mahalanobis distances".

An overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed,3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid","dashed", "dotted", "dotdash", "longdash", or "twodash".The latter two are not supported by Matlab.

Colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Numeric vector containing the subset size with length equal to the number of columns ofmatrix of mahalanobis distances. The default value of subsize is(nrow(MAL) - ncol(MAL) + 1):nrow(MAL)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifiesthe highlighted trajectories.Iflength(fg.thresh) == 1 the highlighted trajectories are those of units that throughtoutthe search had at leat once a mahalanobis distance greater thanfg.thresh.The default value isfg.thresh=2.5. Iflength(fg.thresh) == 2 the highlightedtrajectories are those of units that throughtout the search had a mahalanobis distance atleast once bigger thanfg.thresh[2] or smaller thanfg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted.Iffg.unit is supplied,fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels forthe highlighted trajectories (units). The default is to put the labels at theinitial and final steps of the search.fg.labstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types caneither be specified as an integer (1=solid (default), 2=dashed,3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid","dashed", "dotted", "dotdash", "longdash", or "twodash".The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols.iffg.mark==TRUE each line is plotted using a differentsymbol else no marker is used (default).

Controls the font size of the labels of the trajectories in foreground. Iffg.cex=0 no labels will be shown - equivalent tofg.labstop="".

Numeric vector of length 1 or 2 which specifies how to define theunimmportant trajectories.Unimmportant trajectories will be plotted using a colormap, in greysh or will be hidden.Iflength(bg.thresh) == 1 the irrelevant units are those which alwayshad a mahalanobis distance smaller thanbg.thresh.Iflength(bg.thresh) == 2 the irrelevant units are those which alwayshad a mahalanobis distance greater thanbg.thresh[1] and smaller thanbg.thresh[2].The default isbg.thresh=2.5 ifn > 100 andbg.thresh=-Inf ifn <= 100i.e. to treat all trajectories as important ifn <= 100 and, ifn > 100,to reduce emphasis only to trajectories having in all steps of the search a valueof mahalanobis distance smaller than 2.5.

Specifies how to plot the unimportant trajectories as defined in option bg.thresh.

1. bg.style="faint": unimportant trajectories are plotted using a colormap.

2. bg.style="hide": unimportant trajectories are hidden.

3. bg.style="greyish": unimportant trajectories are displayed in a faint grey.

Whenn > 100 the default option isbg.style='faint'. Whenn <= 100andbg.thresh == -Inf option bg.style is ignored.Remark: bground=” is equivalent to -Inf that is all trajectories are considered relevant.

MATLAB-style arguments - appearance of the plot in terms of xlim, ylim, axes labelsand their font size style, color of the lines, etc.

MATLAB-style arguments - for the trajectories in foregroud.

MATLAB-style arguments - for the trajectories in backgroud.

Plot handle. String which identifies the handle of the plot which is about to be created.The default is to use tag 'pl_resfwd'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

Interactive clicking. It is inactive if this parameter is set to FALSE.The default isdatatooltip=TRUE, the user can select with the mouse an individualmahalanobis distance trajectory in order to have information about the corresponding unit, the associatedlabel and the step of the search in which the unit enters the subset.If datatooltip is a list it may contain the following fields:

1. DisplayStyle determines how the data cursor displays. Possible values are'datatip' and'window' (default).'datatip' displaysdata cursor information in a small yellow text box attached to a blacksquare marker at a data point you interactively select.'window'displays data cursor information for the data point you interactivelyselect in a floating window within the figure.

2. SnapToDataVertex: specifies whether the data cursor snaps to the nearest data value oris located at the actual pointer position.Possible values areSnapToDataVertex='on' (default) andSnapToDataVertex='off'.

3. LineColor: controls the color of the trajectory selected with the mouse. It can be an RGB tripletof values between 0 and 1, or character vector indicating a color name. Note that a RGB vectorcan be conveniently chosen with our MATLAB class FSColor, see documentation.

4. SubsetLinesColor: enables to control the color of the trajectories of the units that arein the subset at a given step of the search (ifresfwdplot() is applied to anobject of classfsreda.object) or have a weight greater than 0.9 (ifresfwdplot() is applied to an object of classsregeda.objectormmregeda.object). This can be done (repeatedly) with a left mouseclick in proximity of the step of interest. A right mouse click will terminate theselection by marking with a up-arrow the step corresponding to the highlightedlines. The highlighted lines by default are in red, but a different color can bespecified as RGB triplet or character of color name. Note that a RGB vector canbe conveniently chosen with our MATLAB class FSColor, see documentation.By defaultSubsetLinesColor="", i.e. the modality is not active.Any initialization forSubsetLinesColor which cannot be interpreted asRGB vector will be converted to blue, i.e.SubsetLinesColor will be forced to be [0 0 1].IfSubsetLinesColor is not empty the previous optionLineColor is ignored.

Character vector containing the labels of the units (optional argument used whendatatooltip=TRUE. If this field is not present labelsrow1, ..., rown will be automatically created and includedin the pop up datatooltip window).

Add variable labels in plot. A vector of strings of lengthpcontaining the labels of the variables in the dataset.If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush isTRUE or a list) enables the user to selecta set of trajectories in the current plot and to see them highlighted in the scatterplot matrix.If the scatterplot matrix does not exist it is automatically created.In addition, brushed units can be highlighted in the monitoring mahalanobis distance plot.Note that the window style of the other figures is set equal to that which contains the monitoring mahalanobis distance plot.In other words, if the monitoring mahalanobis distance plot is docked all the other figures will be docked too.

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it ispossible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments ofthe MATLAB functionselectdataFS() and the following optional arguments:

• persist. Persist is an empty value or a character containing 'on' or 'off'.The default value ispersist="", that is brushing is allowed only once.Ifpersist="on" orpersis="off" brushing can be done as many time asthe user requires. Ifpersist='on' then the unit(s) currently brushed areadded to those previously brushed. It is possible, every time a new brushing isdone, to use a different color for the brushed units. Ifpersist='off'every time a new brush is performed units previously brushed are removed.

• label: add labels of brushed units in the monitoring plot.

• labeladd: add labels of brushed units in the scatterplot matrix.If this option is '1', we label the units of the last selected group withthe unit row index in the matrix X. The default value is labeladd=”,i.e. no label is added.

confidence interval for the horizontal bands. It can be a vector ofdifferent confidence level values, e.g. c(0.95, 0.99, 0.999).The confidence interval is based on the $chi^2$ distribution.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

a numeric vector or an object of S3 class (one offsmult.object,smult.object ormmmult.object) returned byone of the functionsfsmult orsmult ormmmult -a list containing the monitoring of minimum Mahalanobis distance

Ifout is a vector, p is the number of variables of theoriginal data matrix which have been used to compute md.

A title for the x axis

A title for the y axis

An overall title for the plot

Add variable labels in the plot. A vector of strings of lengthpcontaining the labels of the variables of the original data matrixX.If it is empty (default) the sequenceX1, ..., Xp will be created automatically

confidence interval for the horizontal bands. It can be a vector ofdifferent confidence level values, e.g. c(0.95, 0.99, 0.999).The confidence interval is based on the chi^2 distribution.

Number of points to be labeled in the plot. Ifnumlab is asingle number, e.g.numlab]k, the units with thek largestmd are labelled in the plots. Ifnumlab is a vector, the units indexedby the vector are labelled in the plot. Default isnumlab=5, i.e. the 5 unitsunits with the largest md are labelled. Usenumlab="" for no labelling.

Tag of the figure which will host the malindexplot. The default tag istag="pl_malindex".

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An object returned by FSReda() (seeFSReda_control).

The needed elements ofout are

1. mdr: Minimum deletion residual. A matrix containing the monitoring of minimum deletion residual in each step of the forward search. The first column of mdr must contain the fwd search index.

2. Un: (for FSR only) - matrix containing the order of entry in the subset of each unit (required only when datatooltip is true or databrush is not empty).

3. y: a vector containing the response (required only when option databrush is requested).

4. X: a matrix containing the explanatory variables (required only when option databrush is requested).

5. Bols: (n-init+1) x (p+1) matrix containing the estimated beta coefficients monitored in each step of the robust procedure (required only when option databrush is requested and suboption multivarfit is requested).

Quantiles for which envelopes have to be computed. The default is to produce 1%, 50% and 99% envelopes. In other words the default isquant=c(0.01, 0.5, 0.99).

Wheather to use MDR with sign: ifsign=TRUE (default) we distinguish steps for which minimum deletion residual was associated with positive or negative value of the residual. Steps associated with positive values of mdr are plotted in black, while other steps are plotted in red.

Wheather to plot the (m+1)-th order statistic. Specifies if it is necessary to plot the curve associated with (m+1)-th order statistic.

Sample size for drawing enevlopes. Specifies the size of the sample which is used to superimpose the envelope. The default is to add an envelope based on all the observations (size n envelope).

Controly scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Controlx scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

a title for the x axis

a title for the y axis

an overall title for the plot

Controls the width of the lines associated with the envelopes, default islvdenv=1.

Controls the linewidth of the curve which contains the monitoring of minimum deletion residual.

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Plot handle. String which identifies the handle of the plot which is about to be created. The default is to use tag 'pl_mdr'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to haveinformation about the unit selected, the step in which the unit enters the search andthe associated label. If datatooltip is a list, it is possible to control the aspectof the data cursor (see MATLAB functiondatacursormode() for more details orsee the examples below). The default options areDisplayStyle="Window" andSnapToDataVertex="on".

Character vector containing the labels of the units (optional argument used whendatatooltip=TRUE. If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

Add variable labels in plot. A vector of strings of lengthp containing the labels of the variables of the regression dataset. If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is a scalar or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the y|X plot, i.e. a matrix of scatter plots of y against each column of X, grouped according to the selection(s) done by brushing. If the plot y|X does not exist it is automatically created.In addition, brushed units are automatically highlighted in the minimum deletion residual plot if it is already open. The extension to the following plots will be available in future versions of the toolbox:

1. monitoring leverage plot;

2. maximum studentized residual;

3. s^2 and R^2;

4. Cook distance and modified Cook distance;

5. deletion t statistics.

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments of functionselectdataFS() and the following optional argument:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value ispersist="", that is brushing is allowed only once. Ifpersist="on" orpersis="off" brushing can be done as many time as the user requires. Ifpersist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. Ifpersist='off' every time a new brush is performed units previously brushed are removed.

2. bivarfit. Wheather to superimpose bivariate least square lines on the plot (ifplot=TRUE.This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default isbivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. Ifbivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. Ifbivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. Ifbivarfit='i1' orbivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

3. multivarfit. Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default ismultivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). Ifmultivarfit=2, same action as withmultivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

4. labeladd. Add outlier labels in plot. Iflabeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value islabeladd=FALSE, i.e. no label is added.

potential further arguments passed to lower level functions.

An object of S3 classfsmeda.object returned byfsmult withmonitoring=TRUE -a list containing the monitoring of minimum Mahalanobis distance

Quantiles for which envelopes have to be computed.The default is to produce 1%, 50% and 99% envelopes. In otherwords the default isquant=c(0.01, 0.5, 0.99).

Wheather to plot the (m+1)-th order statistic.

Sample size for drawing enevlopes. Specifies the size of the sample which isused to superimpose the envelope. The default is to add an envelope based onall the observations (sizen envelope).

Controls the line width of the curve which contains the monitoringof minimum deletion residual.

Controls the width of the lines associated with the envelopes. Default islwdenv=1

Control the x scale in plot. Vector with two elements controllingminimum and maximum on the x axis. Default is to use automatic scale.

Control the y scale in plot. Vector with two elements controllingminimum and maximum on the y axis. Default is to use automatic scale.

Plot handle. String which identifies the handle of the plot which is about to be created.The default istag='pl_mmd'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to haveinformation about the unit selected, the step in which the unit enters the search andthe associated label. If datatooltip is a list, it is possible to control the aspectof the data cursor (see MATLAB functiondatacursormode() for more details orsee the examples below). The default options areDisplayStyle="Window" andSnapToDataVertex="on".

Row labels. Character vector containing the labels of the units (optional argument usedwhendatatooltip=TRUE. If this field is not present labelsrow1, ..., rown will be automatically created and included in the pop up datatooltip window).

A title for the x axis

A title for the y axis

An overall title for the plot

Add variable labels in the plot. A vector of strings of lengthpcontaining the labels of the variables of the original data matrixX.If it is empty (default) the sequenceX1, ..., Xp will be created automatically

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is TRUE or a list) enables the user to selecta set of trajectories in the current plot and to see them highlighted in the scatterplot matrix.If the scatterplot matrix does not exist it is automatically created.In addition, brushed units can be highlighted in the monitoring MD plot. Note that the windowstyle of the other figures is set equal to that which contains the monitoring residual plot.In other words, if the monitoring residual plot is docked all the other figures will be docked too.

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it ispossible to brush only once (that is persist=”).

Note that the window style of the other figures is set equal to that which contains themonitoring residual plot. In other words, if the monitoring residual plot is docked allthe other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush andit is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments ofthe MATLAB functionselectdataFS() and the following optional arguments:

• persist: This option can be an empty value or a character containing 'on' or 'off'.The default value ispersist="", that is brushing is allowed only once.Ifpersist="on" orpersis="off" brushing can be done as many time asthe user requires. Ifpersist='on' then the unit(s) currently brushed areadded to those previously brushed. It is possible, every time a new brushing isdone, to use a different color for the brushed units. Ifpersist='off'every time a new brush is performed units previously brushed are removed.

• labeladd: add labels of brushed units in the scatterplot matrix.If this option is '1', we label the units of the last selected group withthe unit row index in the matrix X. The default value is labeladd=”,i.e. no label is added.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An object of S3 classfsmmmdrs.object returned byfsmmmdrs -a list containing the following elements:

• mmdrs = a matrix of size (n-ninit)-by-(nsimul+1) containing the monitoringof minimum Mahalanobis distance in each step of the forward search for eachof the nsimul random starts. The first column of mmdrs must contain the forward search index.This matrix can be created using functionfsmmmdrs.

• BBrs = 3D array of size n-by-n-(init)-by-nsimul containingunits forming subset for rach random start. This field is necessaryif datatooltip is true or databrush is not empty.

• X = n-by-v matrix containing the original data matrix. Thisfield is necessary if datatooltip is true or databrush is not empty.

Quantiles for which envelopes have to be computed.The default is to produce 1%, 50% and 99% envelopes. In otherwords the default isquant=c(0.01, 0.5, 0.99).

Sample size for drawing enevlopes. Specifies the size of the sample which isused to superimpose the envelope. The default is to add an envelope based onall the observations (sizen envelope).

Controls the linewidth of the curve which contains the monitoringof minimum deletion residual.

line width: a scalar which controls the width of the lines associatedwith the envelopes. Default islwdenv=1

Control the x scale in plot. Vector with two elements controllingminimum and maximum on the x axis. Default is to use automatic scale.

Control the y scale in plot. Vector with two elements controllingminimum and maximum on the y axis. Default is to use automatic scale.

Plot handle. String which identifies the handle of the plot which is about to be created.The default istag='pl_mmdrs'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to haveinformation about the unit selected, the step in which the unit enters the search andthe associated label. If datatooltip is a list, it is possible to control the aspectof the data cursor (see MATLAB functiondatacursormode() for more details orsee the examples below). The default options areDisplayStyle="Window" andSnapToDataVertex="on".

Row labels. Character vector containing the labels of the units (optional argument usedwhendatatooltip=TRUE. If this field is not present labelsrow1, ..., rown will be automatically created and included in the pop up datatooltip window).

A title for the x axis

A title for the y axis

wheather to label the envelopes. Ifenvlab is true (default)labels of the confidence envelopes which are used are added on the y axis.

An overall title for the plot

Add variable labels in the plot. A vector of strings of lengthpcontaining the labels of the variables of the original data matrixX.If it is empty (default) the sequenceX1, ..., Xp will be created automatically

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is TRUE or a list) enables the user to selecta set of trajectories in the current plot and to see them highlighted in the scatterplot matrix.If the scatterplot matrix does not exist it is automatically created.In addition, brushed units can be highlighted in the monitoring MD plot. Note that the windowstyle of the other figures is set equal to that which contains the monitoring residual plot.In other words, if the monitoring residual plot is docked all the other figures will be docked too.

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it ispossible to brush only once (that is persist=”).

Note that the window style of the other figures is set equal to that which contains themonitoring residual plot. In other words, if the monitoring residual plot is docked allthe other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush andit is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments ofthe MATLAB functionselectdataFS() and the following optional arguments:

1. persist. Persist is an empty value or a character containing 'on' or 'off'.The default value ispersist="", that is brushing is allowed only once.Ifpersist="on" orpersis="off" brushing can be done as many time asthe user requires. Ifpersist='on' then the unit(s) currently brushed areadded to those previously brushed. It is possible, every time a new brushing isdone, to use a different color for the brushed units. Ifpersist='off'every time a new brush is performed units previously brushed are removed.

Wheather to use scaled or unscaled envelopes. Ifscaled=TRUEthe envelopes are produced for scaled Mahalanobis distances (no consistency factoris applied) else the traditional consistency factor is applied.Default isscaled=FALSE

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An n x p data matrix (n observations and p variables).Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

Wheather to perform monitoring of Mahalanobis distances and other specific quantities

Plots the Mahalanobis distances against index number. Ifplot=FALSE(default) orplot=0 no plot is produced. The confidencelevel used to draw the confidence bands for the MD is given by the input option conflev.If conflev is not specified a nominal 0.975 confidence interval will be used.Ifplot=2 a scatter plot matrix with the outliers highlighted is produced.If plot is a list it may contain the following fields:

• labeladd Iflabeladd=1, the outliers in the spm are labelled with the unitrow index. The default value islabeladd="", i.e. no label is added

• nameY character vector containing the labels of the variables. As default value,the labels which are added are Y1, ...Yp.

Defining the nominal efficiency (i.e. a number between 0.5 and 0.99). The default value iseff=0.95.

Confidence level which is used to declare units as outliers (scalar).Usuallyconflev=0.95,conflev=0.975 orconflev=0.99 (individual alpha)conflev=1-0.05/n,conflev=1-0.025/n orconflev=1-0.01/n (simultaneous alpha).Default value isconvlev=0.975.

It controls whether to perform checks on matrix Y. Ifnocheck=TRUE, no check is performed.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

p-by-1 vector containing MM estimate of location.

p-by-p matrix with MM estimate of the shape matrix.

matrix with MM estimate of the covariance matrix.Remark:covariance = auxscale^2 * shape.

A vector containing the estimates of the weights.

A vector containing the list of the units declared as outliers using confidence level specified in input scalarconflev.

A vector with S estimate of location.

A matrix with S estimate of the shape matrix.

A matrix with S estimate of the covariance matrix.

S estimate of the scale.

n-by-1 vector containing the estimates of the robust Mahalanobis distances (in squared units).

Confidence level that was used to declare outliers.

the data matrix X

length(eff)-by-p matrix containing MM estimate of location for each value ofeff.

p-by-p-by-length(eff) 3D array containing robust estimate of the shape for each value of eff. Remark: det|shape|=1.

length(eff) vector containing robust estimate of the scalefor each value of eff.

p-by-p-by-length(eff) 3D array containing robust estimate of covariance matrix for each value ofeff.Note thatscale(i)^2 * shape[,,i] = robust estimate of covariance matrix.

(p+1)-by-length(eff) matrix containing the units forming best subset for each value of eff.

n-by-length(eff) matrix containing the estimates of the robust Mahalanobis distances (in squared units) for each value of eff.

n-by-length(eff) matrix containing flags for the outliers. Boolean matrix containing the list of the units declared as outliers for each value of eff using confidence level specified in input scalarconflev

n x length(eff) matrix containing the weights for each value of eff.

Confidence level that was used to declare outliers.

Number of subsets without full rank. Notice thatsingsub > 0.1*(number of subsamples) produces a warning.

vector which contains the values of eff which have been used.

the data matrix X.

p-by-1 vector containing the MM estimate of regression coefficients.

scalar, S estimate of the scale (or supplied external estimate of scale, if option InitialEst is not empty).

residuals.

fitted values.

n x 1 vector. Weights assigned to each observation.

p x 1 vector containing S estimate of regression coefficients (or supplied initial external estimate of regression coefficients, if option InitialEst is not empty)

Number of subsets without full rank in the S preliminary part. Notice that out.singsub > 0.1*(number of subsamples) produces a warning.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

Confidence level which is used to declare units as outliers. Usuallyconflev=0.95, 0.975, 0.99 (individual alpha) orconflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Specifies the rho function which has been used to weight the residuals. If a different rho function is specified for S and MM loop then insted ofrhofunc we will haverhofuncS andrhofuncMM.

Vector which contains the additional parameters for the specified rho function which has been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c. If a different rho function is specified for S and MM loop then insted ofrhofuncparam we will haverhofuncparamS andrhofuncparamMM.

the data matrix X

the response vector y

scalar, S estimate of the scale (or supplied external estimate of scale, if option InitialEst is not empty).

p x length(eff) matrix containing MM estimate of regression coefficients for each value of eff.

n x length(eff) matrix containing the monitoring of scaled residuals for each value ofeff.

n x length(eff) matrix containing the estimates of the weights for each value of eff

Boolean matrix containing the list of the units declared as outliers for each value of eff using confidence level specified in input scalar conflev.

Confidence level which is used to declare units as outliers. Remark: conflev will be used to draw the horizontal line (confidence band) in the plot.

Number of subsets wihtout full rank. Notice that Notice that singsub > 0.1*(number of subsamples) produces a warning

string identifying the rho function which has been used.

vector which contains the additional parameters for the specified rho function which have been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c.

vector containing the value of eff which have been used.

the data matrix X

the response vector y

a single value, interpreted as an integer

optional vector containing scaled residuals or Mahalanobisdistances for then units of the sample. If not provided,rho, psi, psider, psix and weights are not computed

number of variables (p=1 for regression)

psi function class. One of TB, HA, HU, HP, OPT or PD.

requested breakdown point

requested asymptotic efficiency

tuning constant c

additional parameters

whether to print intermediate results. Default istrace=FALSE.

responce variable or an object containing the responce,the predictors and possibly other variable resulting from monitoringof regression.

Ify is a vector, a data matrixX must be present as an argumentIfy is a list containing justy andX, the call is equivallent toregspmplot(y, X). Otherwisey must be an an object of S3 classfsreda.object returned byfsreg withmonitoring=TRUE - a list containingthe monitoring along a search

Predictor variables. Data matrix of explanatory variables (also called 'regressors') ofdimensionn byp if the argumenty is a vector. The rows ofXrepresent observations, and the columns represent variables.

grouping variable. Vector withn elements. Specifiesa grouping variable defined as a categorical variable (factor), numeric, or array of strings,or string matrix, and it must have the same number of rows asX.This grouping variable determines the marker and color assigned to each point.Remark: ifgroup is used to distinguish a set of outliers froma set of good units, the id number for the outliers should be the larger(see optional fieldlabeladd of parameterplo for details).

This option controls the names which are displayed in the marginsof the scatterplot matrix as well as the labels of the legend.Ifplot=FALSE, then namey, nameX and labeladd are both set tothe empty string (default), and no label and no name is added to the plot.Ifplot=TRUE the names y, and X1,..., Xp are added to the margins ofthe the scatter plot matrix else nothing is shown.Ifplot is a list, it is possible to control not only the names but also,point labels, colors and symbols. More precisely listplot may contain the following elements:

1. labeladd - see parameterlabeladd

2. namey - a character string containing the response variable name. See parameternamey.

3. nameX - a vector of character strings containing the labels ofthe explanatory variables. As default value, the labels which areadded areY1, ..., Yp. See parameternameX.

4. clr - see parametercol

5. sym - see parameterpch

6. siz - see parametercex

7. doleg - see parameterlegend

8. xlimx - see parameterxlim

9. ylimy - see parameterylim

a character string with the name of the responce variable

a vector of character strings with the names of the explanatory variables

color specification for the data point. Can be different for each group.By default, the order of the colors isblue,red,black,magenta,green,cyan andyelow.

the size of the symbols used for plotting. By defaultcex=1the symbol size depends on the number of plots and the size of thefigure window. Values larger than 1 will increase the size andvalues smaller than 1 will decrease the size.

specification of the symbols to use. For example, ifthere are three groups, andpch=c(1, 3, 4), the first group will beplotted with a circle, the second with a plus, and the third with a 'x' (see?pch or?points fora list of symbols. NOTE: not all symbols available in R can be mapped to the symbols in MATLAB.

logical, controls wheather the elements belonging to the last groupin the scatterplot matrix are labelled with their unit row indexor their rowname. The rowname is taken from the parameterlabelor if it is missing, from the sequence1:n. The default value islabeladd=FALSE, i.e. no label is added.

logical, controls where a legend is shown or not.

x limits. A vector with two elements controlling minimumand maximum on the x axis. By defaul automatic scale is used.

y limits. A vector with two elements controlling minimumand maximum on the y axis. By defaul automatic scale is used.

Plot handle. String which identifies the handle of the plot which is about to be created.The default istag='pl_mmd'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to haveinformation about the unit selected, the step in which the unit enters the search andthe associated label. If datatooltip is a list, it is possible to control the aspectof the data cursor (see MATLAB functiondatacursormode() for more details orsee the examples below). The default options areDisplayStyle="Window" andSnapToDataVertex="on".

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is TRUE or a list) enables the user to selecta set of trajectories in the current plot and to see them highlighted in the scatterplot matrix.If the scatterplot matrix does not exist it is automatically created.In addition, brushed units can be highlighted in the monitoring MD plot. Note that the windowstyle of the other figures is set equal to that which contains the monitoring residual plot.In other words, if the monitoring residual plot is docked all the other figures will be docked too.

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it ispossible to brush only once (that is persist=”).

Note that the window style of the other figures is set equal to that which contains themonitoring residual plot. In other words, if the monitoring residual plot is docked allthe other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush andit is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments ofthe MATLAB functionselectdataFS() and the following optional arguments:

• persist: This option can be an empty value or a character containing 'on' or 'off'.The default value ispersist="", that is brushing is allowed only once.Ifpersist="on" orpersis="off" brushing can be done as many time asthe user requires. Ifpersist='on' then the unit(s) currently brushed areadded to those previously brushed. It is possible, every time a new brushing isdone, to use a different color for the brushed units. Ifpersist='off'every time a new brush is performed units previously brushed are removed.

• labeladd: add labels of brushed units in the scatterplot matrix.If this option is '1', we label the units of the last selected group withthe unit row index in the matrix X. The default value is labeladd=”,i.e. no label is added.

x axis control, a numeric vector containing the subset sizewith length equal to the number of columns of matrix residuals. If it isnot specified it will be set equal to(nrow(residuals) - ncol(residuals) + 1) : nrow(residuals).

Text shown in selected steps, a numeric vector which specifiesfor which steps of the forward search textlabels are added in the monitoringresidual plot after a brushing action in the yXplot. The default is towrite the labels at the initial and final step. The default isselstep=c(m0, n) wherem0 andn are respectivelythe first and final step of the search.

Unit labelling. A vector of strings, a string, or a numericvector for labelling units. If out is an object the threshold is associatedwith the trajectories of the residuals monitored along the search else itrefers to the values of the response variable. If it is a vector of strings,only the lines associated with the units that in at least one step of thesearch had a residual smaller thanselunit[1] or greater thansellunit[2] will have a textbox. If it is a string it specifiesthe threshold above which labels have to be put. For exampleselunit='2.6'means that the text labels are written only for the units which havein at least one step of the search a value of the scaled residualgreater than 2.6 in absolute value. If it is a numeric vector itcontains the list of the units for which it is necessary to putthe text labels. The default value of selunit is string'2.5'ify is an object else it is an empty value.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An object returned by one of the monitoring functions (seeFSReda_control,Sregeda_control andMMregeda_control). The object is one offsreda.object,sregeda.object ormmregeda.object.

The needed elements ofout are

1. RES: matrix containing the residuals monitored in each step of the forward search or any other robust procedure. Every row is associated with a residual (unit). This matrix can be created using function FSReda, Sregeda, MMregeda.

2. Un: (for FSR only) - matrix containing the order of entry in the subset of each unit (required only when datatooltip is true or databrush is not empty).

3. bdp: (for Sreg only) - vector containing a sequence of breakdown point values to monitor on.

4. eff: (for MMreg only) - vector containing a sequence of efficiency values to monitor on.

5. y: a vector containing the response (required only when option databrush is requested).

6. X: a matrix containing the explanatory variables (required only when option databrush is requested).

7. Bols: (n-init+1) x (p+1) matrix containing the estimated beta coefficients monitored in each step of the robust procedure (required only when option databrush is requested and suboption multivarfit is requested).

Controly scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

Controlx scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

a title for the x axis

a title for the y axis

an overall title for the plot

The line width, a positive number, defaulting to 1

The line type. Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units

The magnification to be used for x and y labels relative to the current setting of cex

The magnification to be used for axis annotation relative to the current setting of cex

values for the x axis. Numeric vector ofncol(RES) controlling the x axis coordinates. The default value of xvalues is(nrow(RES) - ncol(RES) + 1):nrow(RES)

(alternative to fg.unit) numeric vector of length 1 or 2 which specifies the highlighted trajectories.Iflength(fthresh) == 1 the highlighted trajectories are those of units that throughtout the search had at leat once a residual greater (in absolute value) than thresh. The default value isfg.thresh=2.5. Iflength(fthresh) == 2 the highlighted trajectories are those of units that throughtout the search had a residual at leat once bigger thanfg.thresh[2] or smaller thanfg.thresh[1].

(alternative to fg.thresh), vector containing the list of the units to be highlighted. Iffg.unit is supplied,fg.thresh is ignored.

numeric vector which specifies the steps of the search where to put labels for the highlighted trajectories (units). The default is to put the labels at the initial and final steps of the search.fg.labstep='' means no label.

The line width for the highlighted trajectories (units). Default is 1.

The line type for the highlighted trajectories (units). Line types can either be specified as an integer (1=solid (default), 2=dashed, 3=dotted, 4=dotdash, 5=longdash, 6=twodash) or as one of the character strings "solid", "dashed", "dotted", "dotdash", "longdash", or "twodash". The latter two are not supported by Matlab.

colors to be used for the highlighted units.

Controlls whether to plot highlighted trajectories as symbols.iffg.mark==TRUE each line is plotted using a different symbol else no marker is used (default).

controls the font size of the labels of the trajectories in foreground.

numeric vector of length 1 or 2 which specifies how to define the unimmportant trajectories.Unimmportant trajectories will be plotted using a colormap, in greysh or will be hidden.Iflength(thresh) == 1 the irrelevant units are those which always had a residual smaller (in absolute value) than thresh.Iflength(bthresh) == 2 the irrelevant units are those which always had a residual greater than bthresh(1) and smaller than bthresh(2). The default is:bg.thresh=2.5 ifn > 100 andbg.thresh=-Inf ifn <= 100 i.e. to treat all trajectories as important ifn <= 100 and, ifn > 100, to reduce emphasis only to trajectories having in all steps of the search a value of scaled residual smaller than 2.5.

specifies how to plot the unimportant trajectories as defined in option bthresh.

1. bg.style="faint": unimportant trajectories are plotted using a colormap.

2. bg.style="hide": unimportant trajectories are hidden.

3. bg.style="greyish": unimportant trajectories are displayed in a faint grey.

Whenn>100 the default option isbg.style='faint'. Whenn <= 100 andbg.thresh == -Inf option bstyle is ignored.Remark: bground=” is equivalent to -Inf that is all trajectories are considered relevant.

Plot handle. String which identifies the handle of the plot which is about to be created. The default is to use tag 'pl_resfwd'. Notice that if the program finds a plot which has a tag equal to the one specified by the user, then the output of the new plot overwrites the existing one in the same window else a new window is created.

Interactive clicking. It is inactive if this parameter is missing or empty. The default isdatatooltip=TRUE, i.e. the user can select with the mouse an individual residual trajectory in order to have information about the corresponding unit. The information displayed depends on the estimator in use.

For example for classfsreda.object the information concerns the label and the step of the search in which the unit enters the subset. If datatooltip is a list it may contain the following fields:

1. DisplayStyle determines how the data cursor displays. Possible values are'datatip' and'window' (default).'datatip' displays data cursor information in a small yellow text box attached to a black square marker at a data point you interactively select.'window' displays data cursor information for the data point you interactively select in a floating window within the figure.

2. SnapToDataVertex: specifies whether the data cursor snaps to the nearest data value or is located at the actual pointer position. Possible values areSnapToDataVertex='on' (default) andSnapToDataVertex='off'.

3. LineColor: controls the color of the trajectory selected with the mouse. It can be an RGB triplet of values between 0 and 1, or character vector indicating a color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation.

4. SubsetLinesColor: enables to control the color of the trajectories of the units that are in the subset at a given step of the search (ifresfwdplot() is applied to an object of classfsreda.object) or have a weight greater than 0.9 (ifresfwdplot() is applied to an object of classsregeda.object ormmregeda.object). This can be done (repeatedly) with a left mouse click in proximity of the step of interest. A right mouse click will terminate the selection by marking with a up-arrow the step corresponding to the highlighted lines. The highlighted lines by default are in red, but a different color can be specified as RGB triplet or character of color name. Note that a RGB vector can be conveniently chosen with our MATLAB class FSColor, see documentation. By defaultSubsetLinesColor="", i.e. the modality is not active.Any initialization forSubsetLinesColor which cannot be interpreted as RGB vector will be converted to blue, i.e.SubsetLinesColor will be forced to be [0 0 1].IfSubsetLinesColor is not empty the previous optionLineColor is ignored.

Character vector containing the labels of the units (optional argument used whendatatooltip=TRUE). If this field is not present labels row1, ..., rown will be automatically created and included in the pop up datatooltip window).

Add variable labels in plot. A vector of strings of lengthp containing the labels of the variables of the regression dataset. If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

Controls whether to display or not messages on the screen Ifmsg==1 (default) messages are displayed on the screen about step in which signal took place else no message is displayed on the screen.

interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is a scalar or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the y|X plot, i.e. a matrix of scatter plots of y against each column of X, grouped according to the selection(s) done by brushing. If the plot y|X does not exist it is automatically created.In addition, brushed units are automatically highlighted in the minimum deletion residual plot if it is already open. The extension to the following plots will be available in future versions of the toolbox:

1. monitoring leverage plot;

2. maximum studentized residual;

3. s^2 and R^2;

4. Cook distance and modified Cook distance;

5. deletion t statistics.

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments of functionselectdataFS() and the following optional argument:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value ispersist="", that is brushing is allowed only once. Ifpersist="on" orpersis="off" brushing can be done as many time as the user requires. Ifpersist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. Ifpersist='off' every time a new brush is performed units previously brushed are removed.

2. bivarfit. Wheather to superimpose bivariate least square lines on the plot (ifplot=TRUE.This option adds one or more least squares lines, based on SIMPLE REGRESSION of y on Xi, to the plots of y|Xi. The default isbivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. Ifbivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. Ifbivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. Ifbivarfit='i1' orbivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

3. multivarfit. Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default ismultivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). Ifmultivarfit=2, same action as withmultivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

4. labeladd. Add outlier labels in plot. Iflabeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value islabeladd=FALSE, i.e. no label is added.

(MATLAB-style arguments) appearance of the plot in terms of xlim, ylim, axes labels and their font size style, color of the lines, etc.

MATLAB-style arguments for the fground trajectories in foregroud.

MATLAB-style arguments for the fground trajectories in backgroud.

potential further arguments passed to lower level functions.

A vector containing the residuals from a regression analysis or an object returned by one of the regression functions (seeFSR_control,LXS_control,Sreg_control andMMreg_control). The object is one offsr.object,fsdalts.object,fsdalms.object,sreg.object ormmreg.object. The needed elements ofout are at leastresiduals, but if the optiondatabrush is used, alsoX amdy will be needed.

The vector to be plotted on the x-axis. As default the sequence 1:length(residuals) will be used

Controlx scale in plot. Vector with two elements controlling minimum and maximum on the x axis. Default is to use automatic scale.

Controly scale in plot. Vector with two elements controlling minimum and maximum on the y axis. Default is to use automatic scale.

a title for the x axis

a title for the y axis

an overall title for the plot

Number of points to be identified in plots (see alsoindlab) . By default the five points with largest values will be identified.If numlab is a single number containing scalar k, the units with the k largest residuals are labelled in the plots.If numlab is a vector, the units inside vector numlab are labelled in the plots. The default value ofnumlab=5 and the units with the 5 largest residuals will be labelled. Ifnumlib=0 ornumlib=NULL no labelling will be done.

Which points to be identified in plots (see alsonumlab) - the units with indexes in the vector indlab are labelled in the plots.

Confidence interval for the horizontal bands (a numeric vector). It can be a vector of different confidence level values.

Remark: confidence interval is based on the chi^2 distribution

The magnification to be used for axis annotation relative to the current setting of cex

The magnification to be used for x and y labels relative to the current setting of cex

The line width, a positive number, defaulting to 1

Figure tag (character). Tag of the figure which will host theresindexplot. The default tag ispl_resindex.

Fill color for markers that are closed shapes (circle, square, diamond, pentagram, hexagram, and the four triangles).Can be'none' or'auto' or color name(string) or RGB triplet.

Size of the point symbols. The magnification to be used relative to the current setting of cex.

Add variable labels in plot. A vector of strings of lengthp containing the labels of the variables of the regression dataset. If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Add response label. A string containing the label of the response

Interactive mouse brushing. If databrush is missing or empty (default) ordatabrush=FALSE, no brushing is done.The activation of this option (databrush is a scalar or a list) enables the user to select a set of trajectories in the current plot and to see them highlighted in the y|X plot, i.e. a matrix of scatter plots of y against each column of X, grouped according to the selection(s) done by brushing. If the plot y|X does not exist it is automatically created.In addition, brushed units are automatically highlighted in the minimum deletion residual plot if it is already open. The extension to the following plots will be available in future versions of the package:

1. monitoring leverage plot;

2. maximum studentized residual;

3. s^2 and R^2;

4. Cook distance and modified Cook distance;

5. deletion t statistics.

Note that the window style of the other figures is set equal to that which contains the monitoring residual plot. In other words, if the monitoring residual plot is docked all the other figures will be docked too

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it is possible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments of functionselectdataFS() and the following optional argument:

1. persist. Persist is an empty value or a character containing 'on' or 'off'. The default value ispersist="", that is brushing is allowed only once. Ifpersist="on" orpersis="off" brushing can be done as many time as the user requires. Ifpersist='on' then the unit(s) currently brushed are added to those previously brushed. It is possible, every time a new brushing is done, to use a different color for the brushed units. Ifpersist='off' every time a new brush is performed units previously brushed are removed.

2. bivarfit. This option adds one or more least square lines based on SIMPLE REGRESSION to the plots of y|X, depending on the selected groups.The default isbivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. Ifbivarfit=2, two OLS lines are fitted: one to all points and another to the group of the genuine observations. The group of the potential outliers is not fitted. Ifbivarfit=0 one OLS line is fitted to each group. This is useful for the purpose of fitting mixtures of regression lines. Ifbivarfit='i1' orbivarfit='i2', etc. an OLS line is fitted to a specific group, the one with index 'i' equal to 1, 2, 3 etc. Again, useful in case of mixtures.

3. multivarfit. Wheather to superimpose multivariate least square lines. This option adds one or more least square lines, based on MULTIVARIATE REGRESSION of y on X, to the plots of y|Xi. The default ismultivarfit=FALSE: no line is fitted. Ifbivarfit=1, a single OLS line is fitted to all points of each bivariate plot in the scatter matrix y|X. The line added to the scatter plot y|Xi is avconst + Ci*Xi, where Ci is the coefficient of Xi in the multivariate regression and avconst is the effect of all the other explanatory variables different from Xi evaluated at their centroid (that is overline(y)'C)). Ifmultivarfit=2, same action as withmultivarfit=1 but this time we also add the line based on the group of unselected observations (i.e. the normal units).

4. labeladd. Add outlier labels in plot. Iflabeladd=TRUE, we label the outliers with the unit row index in matrices X and y. The default value islabeladd=FALSE, i.e. no label is added.

potential further arguments passed to lower level functions.

Ann x p data matrix (n observations andp variables).Rows ofx represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

potential further arguments passed to lower level functions.

aformula of the formy ~ x1 + x2 + ....

data frame from which variables specified informula are to be taken.

an optional vector specifying a subset of observationsto be used in the fitting process.

an optional vector of weights to be usedNOT USED YET.

a function which indicates what should happenwhen the data containNAs. The default is set bythena.action setting ofoptions, and isna.fail if that is unset. The “factory-fresh”default isna.omit. Another possible value isNULL, no action. Valuena.exclude can be useful.

logical indicating if themodel frame, is to be returned.

an optional list. See thecontrasts.argofmodel.matrix.default.

this can be used to specify ana prioriknown component to be included in the linear predictorduring fitting. Anoffset term can be included in the

Response variable. A vector withn elements thatcontains the response variable.

wheather to use constant term (default isintercept=TRUE

values of the transformation parameter for which it is necessaryto compute the score test. Default value of lambda isla=c(-1, -0.5, 0, 0.5, 1), i.e., the five most common values of lambda.

likelihood for the augmented model. If true the value of the likelihoodfor the augmented model will be calculated and returend otherwise (default) onlythe value of the score test will be given

Whether to check input arguments. Ifnocheck=TRUE no check is performedon matrixy and matrixX. Notice thaty andXare left unchanged. In other words the additional column of ones for theintercept is not added. The default isnocheck=FALSE.

Whether to print intermediate results. Default istrace=FALSE.

An n x p data matrix (n observations and p variables).Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

Wheather to perform monitoring of Mahalanobis distances and other specific quantities

Plots the Mahalanobis distances against index number. Ifplot=FALSE(default) orplot=0 no plot is produced. The confidencelevel used to draw the confidence bands for the MD is given by the input option conflev.If conflev is not specified a nominal 0.975 confidence interval will be used.Ifplot=2 a scatter plot matrix with the outliers highlighted is produced.If plot is a list it may contain the following fields:

• labeladd Iflabeladd=1, the outliers in the spm are labelled with the unitrow index. The default value islabeladd="", i.e. no label is added

• nameY character vector containing the labels of the variables. As default value,the labels which are added are Y1, ...Yp.

Measures the fraction of outliers the algorithm should resist.In this case any value greater than 0 but smaller or equal than 0.5 will do fine (default isbdp=0.5).Note that given bdp nominal efficiency is automatically determined.

Number of subsamples which will be extracted to find the robust estimator.Ifnsamp=0 all subsets will be extracted. They will be(n choose p). If the number of all possible subset is<1000the default is to extract all subsets otherwise just 1000.

Confidence level which is used to declare units as outliers (scalar).Usuallyconflev=0.95,conflev=0.975 orconflev=0.99 (individual alpha)conflev=1-0.05/n,conflev=1-0.025/n orconflev=1-0.01/n (simultaneous alpha).Default value isconvlev=0.975.

It controls whether to perform checks on matrix Y. Ifnocheck=TRUE, no check is performed.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

p-by-1 vector containing S estimate of location.

p-by-p matrix containing robust estimate of the shapematrix. Remark: det|shape|=1.

robust estimate of the scale.

scale^2 * shape: robust estimate of covariance matrix.

a (p+1) vector containing the units forming best subset associated with S estimate of location.

n-by-1 vector containing the estimates of the robust Mahalanobis distances (in squared units). This vector contains the distances of each observation from the location of the data, relative to the scatter matrix cov.

A vector containing the list of the units declared as outliers using confidence level specified in input scalarconflev.

Confidence level that was used to declare outliers.

Number of subsets without full rank. Notice thatsingsub > 0.1*(number of subsamples) produces a warning.

n x 1 vector containing the estimates of the weights.

the data matrix X

length(bdp)-by-p matrix containing S estimate of location for each value ofbdp.

p-by-p-by-length(bdp) 3D array containing robust estimate of the shape for each value of bdp. Remark: det|shape|=1.

length(bdp) vector containing robust estimate of the scalefor each value of bdp.

p-by-p-by-length(bdp) 3D array containing robust estimate of covariance matrix for each value ofbdp.Note thatscale(i)^2 * shape[,,i] = robust estimate of covariance matrix.

(p+1)-by-length(bdp) matrix containing the units forming best subset for each value of bdp.

n-by-length(bdp) matrix containing the estimates of the robust Mahalanobis distances (in squared units) for each value of bdp.

n-by-length(bdp) matrix containing flags for the outliers. Boolean matrix containing the list of the units declared as outliers for each value of bdp using confidence level specified in input scalarconflev

n x length(bdp) matrix containing the weights for each value of bdp.

Confidence level that was used to declare outliers.

Number of subsets without full rank. Notice thatsingsub > 0.1*(number of subsamples) produces a warning.

vector which contains the values of bdp which have been used.

the data matrix X.

data matrix (2D array) containingn observations onp variablesor an object of S3 classfsmeda.object returned byfsmult withmonitoring=TRUE - a list containingthe monitoring of minimum Mahalanobis distance

grouping variable. Vector withn elements. Specifiesa grouping variable defined as a categorical variable (factor), numeric, or array of strings,or string matrix, and it must have the same number of rows asX.This grouping variable determines the marker and color assigned to each point.Remark: ifgroup is used to distinguish a set of outliers froma set of good units, the id number for the outliers should be the larger(see optional fieldlabeladd of parameterplot for details).

controls the names which are displayed in the margins of thescatter-plot matrix, the labels of the legend the colors and the symbols.Ifplot isempty (plot=FALSE orplot=0 orplot=c()orplot=NULL) empty strings are displayed and no label and no name is addedto the plot. Ifplot=TRUE orplot=1, the namesY1,..., Ypare added to the margins of the the scatter plot matrix else nothing is added.Ifplot is a list, it is possible to control not only the names but also,point labels, colors and symbols. More precisely listplot may contain the following elements:

1. labeladd - see parameterlabeladd

2. nameY - a character string containing the labels ofthe variables. As default value, the labels which areadded areY1, ..., Yp. See parametervariables.

3. clr - see parametercol

4. sym - see parameterpch

5. siz - see parametercex

6. doleg - see parameterlegend

7. label - see parameterlabel

a character string with the names of the variables

color specification for the data point. Can be different for each group.By default, the order of the colors isblue,red,black,magenta,green,cyan andyelow.

the size of the symbols used for plotting. By defaultcex=1the symbol size depends on the number of plots and the size of thefigure window. Values larger than 1 will increase the size andvalues smaller than 1 will decrease the size.

specification of the symbols to use. For example, ifthere are three groups, andpch=c(1, 3, 4), the first group will beplotted with a circle, the second with a plus, and the third with a 'x' (see?pch or?points fora list of symbols. NOTE: not all symbols available in R can be mapped to the symbols in MATLAB.

logical, controls wheather the elements belonging to the last groupin the scatterplot matrix are labelled with their unit row indexor their rowname. The rowname is taken from the parameterlabelor if it is missing, from the sequence1:n. The default value islabeladd=FALSE, i.e. no label is added.

a character vector of lengthn (the number of rows in the data matrix)containing the labels of the units. If this field is emptythe sequence1:n will be used to label the units.

logical, controls where a legend is shown or not.

controls how to fill the diagonals in the plot (main diagonal ofthe scatter plot matrix). Setdispopt='hist' (default) to plot histograms,ordispopt='box' to plot boxplots. The style which is used for univariateboxplots is traditional, if the number of groups is less or equal 5, else it is'compact' (plot boxes using a smaller box style designed for plots with many groups).

Plot handle. String which identifies the handle of the plot which is about to be created.The default istag='pl_mmd'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

If datatooltip is not empty the user can use the mouse in order to haveinformation about the unit selected, the step in which the unit enters the search andthe associated label. If datatooltip is a list, it is possible to control the aspectof the data cursor (see MATLAB functiondatacursormode() for more details orsee the examples below). The default options areDisplayStyle="Window" andSnapToDataVertex="on".

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush is TRUE or a list) enables the user to selecta set of trajectories in the current plot and to see them highlighted in the scatterplot matrix.If the scatterplot matrix does not exist it is automatically created.In addition, brushed units can be highlighted in the monitoring MD plot.Note that the windowstyle of the other figures is set equal to that which contains the monitoring residual plot.In other words, if the monitoring residual plot is docked all the other figures will be docked too.

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it ispossible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments ofthe MATLAB functionselectdataFS() and the following optional arguments:

• persist: This option can be an empty value or a character containing 'on' or 'off'.The default value ispersist="", that is brushing is allowed only once.Ifpersist="on" orpersis="off" brushing can be done as many time asthe user requires. Ifpersist='on' then the unit(s) currently brushed areadded to those previously brushed. It is possible, every time a new brushing isdone, to use a different color for the brushed units. Ifpersist='off'every time a new brush is performed units previously brushed are removed.

• labeladd: add labels of brushed units in the scatterplot matrix.If this option is '1', we label the units of the last selected group withthe unit row index in the matrix X. The default value is labeladd=”,i.e. no label is added.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

p-by-1 vector containing the estimated regression parameters (in step n-k).

scalar containing the estimate of the scale (sigma).

p x 1 vector containing the units forming best subset associated with S estimate of regression coefficient.

residuals.

fitted values.

kx1 vector containing the list of the k units declared as outliers or NULL if the sample is homogeneous.

Confidence level which is used to declare units as outliers. Usuallyconflev=0.95, 0.975, 0.99 (individual alpha) orconflev=1-0.05/n, 1-0.025/n, 1-0.01/n (simultaneous alpha). Default value is 0.975

Number of subsets wihtout full rank. Notice thatsingsub > 0.1*(number of subsamples) produces a warning

n x 1 vector containing the estimates of the weights

Specifies the rho function which has been used to weight the residuals.

Vector which contains the additional parameters for the specified rho function which has been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c.

the data matrix X

the response vector y

matrix containing the S estimator of regression coefficients for each value of bdp.

vector containing the estimate of the scale (sigma) for each value of bdp. This is the value of the objective function.

p x 1 vector containing the units forming best subset associated with S estimate of regression coefficient.

n x length(bdp) matrix containing the monitoring of scaled residuals for each value ofbdp.

n x length(bdp) matrix containing the estimates of the weights for each value of bdp

Boolean matrix containing the list of the units declared as outliers for each value of bdp using confidence level specified in input scalar conflev.

Confidence level which is used to declare units as outliers. Remark: conflev will be used to draw the horizontal line (confidence band) in the plot.

Number of subsets wihtout full rank. Notice that singsub[bdp[jj]] > 0.1*(number of subsamples) produces a warning

Specifies the rho function which has been used to weight the residuals.

Vector which contains the additional parameters for the specified rho function which has been used. For hyperbolic rho function the value of k =sup CVC. For Hampel rho function the parameters a, b and c.

the data matrix X

the response vector y

an object of class"fsdalms" (or"summary.fsdalms");usually, a result of a call tofsreg.

logical; ifTRUE, the correlation matrix of the estimated parameters is returned and printed.

the number of significant digits to use when printing.

logical indicating if “significance stars”should be printer, seeprintCoefmat.

further arguments passed to or from other methods.

an object of class"fsdalts" (or"summary.fsdalts");usually, a result of a call tofsreg.

logical; ifTRUE, the correlation matrix of the estimated parameters is returned and printed.

the number of significant digits to use when printing.

logical indicating if “significance stars”should be printer, seeprintCoefmat.

further arguments passed to or from other methods.

an object of class"fsr" (or"summary.fsr");usually, a result of a call tofsreg.

logical; ifTRUE, the correlation matrix of the estimated parameters is returned and printed.

the number of significant digits to use when printing.

logical indicating if “significance stars”should be printer, seeprintCoefmat.

further arguments passed to or from other methods.

An n x p data matrix (n observations and p variables).Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

an integer vector specifying the number of mixture components (clusters) for which the BIC is to be calculated. By defaultkk=1:5.

an vector specifying the values of the restriction factor which have to be considered. By defaultcc=c(1, 2, 4, 8, 16, 32, 64, 128).

Global trimming level. A scalar between 0 and 0.5 or an integer specifying the number ofobservations which have to be trimmed. Ifalpha=0 all observations are considered. By defaultalpha=0.

More in detail, if0 < alpha < 1 clustering is based onh = fix(n * (1-alpha))observations, else if alpha is an integer greater than 1 clustering is based onh = n - floor(alpha).

A character value which specifies which information criteria must be computedfor eachk (number of groups) and each value of the restriction factorc. Possible values forwhichIC are:

• "MIXMIX": a mixture model is fitted and for computing the information criterionthe mixture likelihood is used. This option corresponds to the use of the BayesianInformation criterion (BIC). In output just the matrixMIXMIX is given.

• "MIXCLA": a mixture model is fitted but to compute the information criterionthe classification likelihood is used. This option corresponds to the use of theIntegrated Complete Likelihood (ICL). In the output just the matrixMIXCLA is given.

• "CLACLA": everything is based on the classification likelihood. This informationcriterion will be called CLA. In the output just the matrixCLACLA is given.

• "ALL": both classification and mixture likelihood are used. In this case allthree information criteria CLA, ICL and BIC are computed. In the output allthree matricesMIXMIX,MIXCLA andCLACLA are given.

If a scalar, it contains the number of subsamples which will be extracted.Ifnsamp = 0 all subsets will be extracted. Remark - if the number of all possiblesubset is greater than 300 the default is to extract all subsets, otherwise just 300.Ifnsamp is a matrix it contains in the rows the indexes of the subsets whichhave to be extracted.nsamp in this case can be conveniently generated byfunctionsubsets().nsamp can havek columns ork * (p + 1)columns. Ifnsamp hask columns thek initial centroids eachiteration i are given byX[nsamp[i,] ,] and the covariance matrices are equalto the identity.

Ifnsamp hask * (p + 1) columns, the initial centroids and covariancematrices in iterationi are computed as follows:

• X1 <- X[nsamp[i ,] ,]

• mean(X1[1:p + 1, ]) contains the initial centroid for group 1

• cov(X1[1:p + 1, ]) contains the initial cov matrix for group 1

• mean(X1[(p + 2):(2*p + 2), ]) contains the initial centroid for group 2

• cov(X1[(p + 2):(2*p + 2), ]) contains the initial cov matrix for group 2

• ...

• mean(X1[(k-1)*p+1):(k*(p+1), ]) contains the initial centroids for group k

• cov(X1[(k-1)*p+1):(k*(p+1), ]) contains the initial cov matrix for group k.

REMARK: Ifnsamp is not a scalar, the optionstartv1 given below is ignored.More precisely, ifnsamp hask columnsstartv1 = 0 else ifnsamp hask*(p+1) columns optionstartv1=1.

Number of refining iterations in each subsample. Default isrefsteps=15.refsteps = 0 means "raw-subsampling" without iterations.

Tolerance of the refining steps. The default value is 1e-14

A logical specifying wheather cluster weights in the concentrationand assignment steps shall be considered. Ifequalweights=TRUE we are (ideally)assuming equally sized groups, else ifequalweights = false (default) we allow fordifferent group weights. Please, check in the given references which functionsare maximized in both cases.

Controls whether to display or not messages on the screen Ifmsg==TRUE (default)messages are displayed on the screen. Ifmsg=2, detailed messages are displayed,for example the information at iteration level.

Check input arguments. Ifnocheck=TRUE no check is performedon matrixX. The defaultnocheck=FALSE.

Ifplot=TRUE, a plot of the BIC (MIXMIX), ICL (MIXCLA) curveand CLACLA is shown on the screen. The plots which are shown depend onthe input optionwhichIC.

How to initialize centroids and covariance matrices. Scalar.Ifstartv1=1 then initial centroids and covariance matrices are basedon(p+1) observations randomly chosen, else each centroid is initializedtaking a random row of input data matrix and covariance matrices are initializedwith identity matrices. The default value isstartv1=1.

Remark 1: in order to start with a routine which is in the required parameter space,eigenvalue restrictions are immediately applied.

Remark 2 - optionstartv1 is used just ifnsamp is a scalar(see for more details the help associated withnsamp).

Type of restriction to be applied on the cluster scatter matrices.Valid values are'eigen' (default), or'deter'."eigen" implies restriction on the eigenvalues while"deter"implies restriction on the determinants.

List of the units which must (whenever possible) havea particular label. For exampleUnitsSameGroup=c(20, 26), means thatgroup which contains unit 20 is always labelled with number 1. Similarly,the group which contains unit 26 is always labelled with number 2, (unlessit is found that unit 26 already belongs to group 1).In general, group which contains unitUnitsSameGroup(r) wherer=2, ...length(kk)-1is labelled with numberr (unless it is found that unitUnitsSameGroup(r)has already been assigned to groups1, 2, ..., r-1.

The number of parallel sessions to open. If numpool is not defined,then it is set equal to the number of physical cores in the computer.

Logical, indicating if the open pool must be closed or not.It is useful to leave it open if there are subsequent parallel sessions to execute,so that to save the time required to open a new pool.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An S3 object of classtclustic.object(output oftclustIC) containing the valuesof the information criteria BIC (MIXMIX), ICL (MIXCLA) or CLA (CLACLA),for different values of k (number of groups) and differentvalues of c (restriction factor), for a prespecified level of trimming.

Specifies the information criterion to use for the plot.SeetclustIC() for the possible values of whichIC.

plot handle. String which identifies the handle of the plot which is about to be created.The default is to use tag 'pl_IC'. Notice that if the program finds a plot which hasa tag equal to the one specified by the user, then the output of the new plot overwritesthe existing one in the same window else a new window is created.

Interactive clicking. It is inactive if this parameter is set to FALSE.Ifdatatooltip=TRUE, the user can select with the mouse a solutionin order to have the following information:

• 1) value ofk which has been selected

• 2) value ofc which has been selected

• 3) values of the information criterion

• 4) frequency distribution of the associated classification.

If datatooltip is a list it may contain the following fields:

1. DisplayStyle determines how the data cursor displays. Possible values are'datatip' and'window' (default).'datatip' displaysdata cursor information in a small yellow text box attached to a blacksquare marker at a data point you interactively select.'window'displays data cursor information for the data point you interactivelyselect in a floating window within the figure.

2. SnapToDataVertex: specifies whether the data cursor snaps to the nearest data value oris located at the actual pointer position.Possible values areSnapToDataVertex='on' (default) andSnapToDataVertex='off'.

Interactive mouse brushing. If databrush is missing or empty (default), no brushing is done.The activation of this option (databrush isTRUE or a list) enables the user to selecta set of values of IC in the current plot and to see thecorresponding classification highlighted in the scatterplot matrix.If the scatterplot matrix does not exist it is automatically created.Note that the window style of the other figures is set equal to that which contains the IC plot.In other words, if the IC plot is docked all the other figures will be docked too.

Ifdatabrush=TRUE the default selection tool is a rectangular brush and it ispossible to brush only once (that is persist=”).

Ifdatabrush=list(...), it is possible to use all optional arguments ofthe MATLAB functionselectdataFS() and the following optional arguments:

• persist: Persist is an empty value or a character containing 'on' or 'off'.The default value ispersist="", that is brushing is allowed only once.Ifpersist="on" orpersis="off" brushing can be done as many time asthe user requires. Ifpersist='on' then the unit(s) currently brushed areadded to those previously brushed. It is possible, every time a new brushing isdone, to use a different color for the brushed units. Ifpersist='off'every time a new brush is performed units previously brushed are removed.

• label: add labels of brushed units in the monitoring plot.

• dispopt: controls how to fill the diagonals in the scatterplot matrixof the brushed solutions. Setdispopt="hist" (default) to plot histograms,ordispopt="box" to plot boxplots.

Add variable labels in plot. A vector of strings of lengthpcontaining the labels of the variables in the dataset.If it is empty (default) the sequenceX1, ..., Xp will be created automatically

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

An S3 object of classtclustic.object(output oftclustIC) containing the valuesof the information criteria BIC (MIXMIX), ICL (MIXCLA) or CLA (CLACLA),for different values of k (number of groups) and differentvalues of c (restriction factor), for a prespecified level of trimming.

Number of best solutions to extract from BIC/ICL matrix.The default value of NumberOfBestSolutions is 5

Threshold to identify spurious solutions - the thresholdof the adjusted Rand index to use to consider two solutions as equivalent.The default value of ThreshRandIndex is 0.7

Specifies the information criterion to use to extract best solutions.Possible values for whichIC are:

• CLACLA = in this case best solutions are referred to the classification likelihood.

• MIXMIX = in this case in this case best solutions are referred to the mixture likelihood (BIC).

• MIXCLA = in this case in this case best solutions are referred to ICL.

• ALL = in this case best solutions both three solutions using classificationand mixture likelihood are produced. In the output classout all thethree matricesMIXMIXbs,CLACLAbs andMIXCLAbs are given.

The default value iswhichIC="ALL".

Index to use to compare partitions. IfRand=TRUE (default) the adjusted Randindex is used, else the adjusted Fowlkes and Mallows index is used.

It controls whether to display or not messages (from MATLAB) on the screen. Ifmsg=TRUE(default) messages about the progression of the search are displayed on the screenotherwise only error messages will be displayed.

Ifplot=TRUE, the best solutions which have been found are shown on the screen.

Whether to print intermediate results. Default istrace=FALSE.

potential further arguments passed to lower level functions.

the matched call

number of groups

trimming level

restriction factor

ann-by-length(alpha) vector containing assignment of each unit to each of thekgroups. Cluster names are integer numbers from 1 to k.0 indicates trimmed observations. The first column refers toalpha[1], the second columnrefers toalpha[2], ..., the last column refers toalpha[length(alpha)].

a 3 dimensional array of size k-by-p-by-length(alpha) containing the monitoringof the centroid for each value of alpha.MU[,,1], refers toalpha[1] ...,MU(,,length(alpha)] refers toalpha[length(alpha)].The first row in each slice refers to group 1, second row refers to group 2, etc.

A list of lengthlength(alpha) containing in elementj,withj=1, 2, ..., length(alpha), the 3D array of size p-by-p-by-k containingthek (constrained) estimated covariance matrices associated withalpha[j].

Amon stands for alpha monitoring. Matrix of size(length(alpha)-1)-by-4 whichcontains for two consecutive values of alpha the monitoring of three quantities(change in classification, change in centroid location, change in covariance location).

• 1st col = value of alpha.

• 2nd col = ARI index.

• 3rd col = squared Euclidean distance between centroids.

• 4th col = squared Euclidean distance between covariance matrices.

An n x p data matrix (n observations and p variables).Rows of x represent observations, and columns represent variables.

Missing values (NA's) and infinite values (Inf's) are allowed,since observations (rows) with missing or infinite values willautomatically be excluded from the computations.

Number of groups.

A scalar between 0 and 0.5 or an integer specifying the number ofobservations which have to be trimmed. Ifalpha=0,tclust reduces totraditional model based or mixture clustering (mclust): see for example theMatlab functiongmdistribution.

More in detail, if0 < alpha < 1 clustering is based onh = floor(n * (1-alpha))observations, else if alpha is an integer greater than 1 clustering is based onh = n - floor(alpha).Ifmonitoring=TRUE,alpha is a vector which specifies the values oftrimming levels which have to be considered - contains decresing elements whichlie in the interval 0 and 0.5.For example ifalpha=c(0.1, 0.05, 0),tclust() considers these 3 values of trimming level.The default for alpha is vectoralpha=c(0.1, 0.05, 0). The sequence isforced to be monotonically decreasing.

Positive scalar which constrains the allowed differences among group scatters.Larger values imply larger differences of group scatters. On the other handa value of 1 specifies the strongest restriction forcing alleigenvalues/determinants to be equal and so the method looksfor similarly scattered (respectively spherical) clusters.The default is to applyrestrfactor to eigenvalues. In order toapplyrestrfactor to determinants it is necessary to use theoptional input argumentrestrtype.

Ifmonitoring=TRUE TCLUST is performed for a series of valuesof the trimming factoralpha givenk (number of groups) and givenc (restriction factor).

Ifplot=FALSE (default) orplot=0 no plot is produced.Ifplot=TRUE andmonitoring=FALSE a plot with the classificationis shown (using the spmplot function). The plot can be:

• forp = 1, a histogram of the univariate data,

• forp = 2, a bivariate scatterplot,

• forp > 2, a scatterplot matrix generated by the MATLAB functionspmplot().

Whenp >= 2 the following additional features are offered(forp = 1 the behaviour is forced to be as forplots=TRUE):

• plot = 'contourf' adds in the background of the bivariate scatterplots afilled contour plot. The colormap of the filled contour is based on grey levels as default.This argument may also be inserted in a field named 'type' of a list. In the latter caseit is possible to specify the additional field 'cmap', which changes the default colorsof the color map used. The field 'cmap' may be a three-column matrix of values in therange [0,1] where each row is an RGB triplet that defines one color. Check the colormapfunction for additional informations.

• plot = 'contour' adds in the background of the bivariate scatterplots a contour plot.The colormap of the contour is based on grey levels as default. This argument may also beinserted in a field namedtype of a list. In the latter case it is possible to specifythe additional fieldcmap, which changes the default colors of the color map used. The fieldcmap may be a three-column matrix of values in the range [0,1] where each row is an RGBtriplet that defines one color. Check thecolormap() (MATLAB) function for additional information.

• plot = 'ellipse' superimposes confidence ellipses to each group in the bivariatescatterplots. The size of the ellipse isqchisq(0.95, 2), i.e. the confidence level usedby default is 95 percent. This argument may also be inserted in a field namedtype ofa list. In the latter case it is possible to specify the additional fieldconflev,which specifies the confidence level to use and it is a value between 0 and 1.

• plot = 'boxplotb' superimposes on the bivariate scatterplots the bivariate boxplotsfor each group, using the boxplotb function. This argument may also be inserted in a fieldnamedtype of a list.

The parameterplot can be also a list and in this case its elements are:

• type - specifies the type of plot as when plot option is a character.Therefore, plots$type can be one of 'contourf', 'contour', 'ellipse' or 'boxplotb'.

• cmap - used to set a colormap for the plot type (MATLAB style). For example, plot$cmap = 'autumn'.See the MATLAB help of colormap for a list of colormap possiblilites.

• conflev - this is the confidence level for the confidence ellipses.It must me a scalar between 0 and 1.

Ifplot=TRUE andmonitoring=TRUE two plots are shown.The first plot (monitor plot) shows three panels with the monitoring between twoconsecutive values of alpha: (i) the change in classification using ARI index (top panel),(ii) the change in centroids using squared euclidean distances (central panel) and(iii) the change in covariance matrices using squared euclidean distance (bottom panel).

The second plot (gscatter plot) shows a series of subplots which monitor the classificationfor each value ofalpha. In order to make sure that consistent labels are used for thegroups, between two consecutive values ofalpha, we assign labelr to a group ifthis group shows the smallest distance with groupr for the previous value ofalpha.The type of plot which is used to monitor the stability of the classification depends on thedata dimensionalityp.

• forp = 1, a histogram of the univariate data (the MATLAB functionhistFS() is called),

• forp = 2, a bivariate scatterplot (the MATLAB functiongscatter() is called),

• forp > 2, a scatterplot of the first two principal components (functiongscatter()is called and we show on the axes titles the percentage of variance explained by the firsttwo principal components).

Also in the case ofmonitoring=TRUE the parameterplot can be a list and its elements are:

• name: character vector which enables to specify which plot to display.name = "gscatter" produces a figure with a series of subplots which show theclassification for each value ofalpha.name = "monitor" shows a figurewith three panels which monitor between two consecutive values of alpha the change inclassification using ARI index (top panel), the change in centroids using squaredeuclidean distances (central panel), the change in covariance matrices using squaredeuclidean distance (bottom panel). If this field is not specified, by defaultname=c("gscatter", "monitor") and both figures will be shown.

• alphasel: a numeric vector which specifies for which values of alpha itis possible to see the classification. For example ifalphasel = c(0.05, 0.02),the classification will be shown just foralpha=0.05 andalpha=0.02.If this field is not specifiedalphasel=alpha and therefore the classificationis shown for each value of alpha.

If a scalar, it contains the number of subsamples which will be extracted.Ifnsamp = 0 all subsets will be extracted. Remark - if the number of all possiblesubset is greater than 300 the default is to extract all subsets, otherwise just 300.Ifnsamp is a matrix it contains in the rows the indexes of the subsets whichhave to be extracted.nsamp in this case can be conveniently generated byfunctionsubsets().nsamp can havek columns ork * (p + 1)columns. Ifnsamp hask columns thek initial centroids eachiteration i are given byX[nsamp[i,] ,] and the covariance matrices are equalto the identity.

Ifnsamp hask * (p + 1) columns, the initial centroids and covariancematrices in iterationi are computed as follows:

• X1 <- X[nsamp[i ,] ,]

• mean(X1[1:p + 1, ]) contains the initial centroid for group 1

• cov(X1[1:p + 1, ]) contains the initial cov matrix for group 1

• mean(X1[(p + 2):(2*p + 2), ]) contains the initial centroid for group 2

• cov(X1[(p + 2):(2*p + 2), ]) contains the initial cov matrix for group 2

• ...

• mean(X1[(k-1)*p+1):(k*(p+1), ]) contains the initial centroids for group k

• cov(X1[(k-1)*p+1):(k*(p+1), ]) contains the initial cov matrix for group k.

REMARK: Ifnsamp is not a scalar, the optionstartv1 given below is ignored.More precisely, ifnsamp hask columnsstartv1 = 0 else ifnsamp hask*(p+1) columns optionstartv1=1.

Number of refining iterations in each subsample. Default isrefsteps=15.refsteps = 0 means "raw-subsampling" without iterations.

Tolerance of the refining steps. The default value is 1e-14

A logical specifying wheather cluster weights in the concentrationand assignment steps shall be considered. Ifequalweights=TRUE we are (ideally)assuming equally sized groups, else ifequalweights = false (default) we allow fordifferent group weights. Please, check in the given references which functionsare maximized in both cases.