This function can be used to perform the functional boxplot of univariate ormultivariate functional data.
fbplot(Data, Depths="MBD", Fvalue=1.5, adjust=FALSE, display=TRUE, xlab=NULL, ylab=NULL, main=NULL,...)# S3 method for fDatafbplot(Data, Depths="MBD", Fvalue=1.5, adjust=FALSE, display=TRUE, xlab=NULL, ylab=NULL, main=NULL,...)# S3 method for mfDatafbplot(Data, Depths=list(def="MBD", weights="uniform"), Fvalue=1.5, adjust=FALSE, display=TRUE, xlab=NULL, ylab=NULL, main=NULL,...)the univariate or multivariate functional dataset whosefunctional boxplot must be determined, in form offData ormfData object.
either a vector containing the depths for each element of thedataset, or:
univariate case: a string containing the name of the method youwant to use to compute it. The default is'MBD'.
multivariate case: a list with elementsdef, containing thename of the depth notion to be used to compute depths (BD orMBD), andweights, containing the value of parameterweights to be passed to the depth function. Default islist(def = 'MBD', weights = 'uniform').
In both cases the name of the functions to compute depths must be availablein the caller's environment.
the value of the inflation factor \(F\), default isF =1.5.
eitherFALSE if you would like the default value for theinflation factor, \(F = 1.5\), to be used, or (for nowonly in theunivariate functional case) a list specifying the parameters required bythe adjustment:
N_trials: the number of repetitions of the adjustment procedurebased on the simulation of a gaussian population of functional data, eachone producing an adjusted value of \(F\), which will lead to the averagedadjusted value \(\bar{F}\). Default is 20.
trial_size: the number of elements in the gaussian population offunctional data that will be simulated at each repetition of the adjustmentprocedure. Default is 8 *Data$N.
TPR: the True Positive Rate of outliers, i.e. the proportion ofobservations in a dataset without amplitude outliers that have to beconsidered outliers. Default is2 * pnorm(4 * qnorm(0.25)).
F_min: the minimum value of \(F\), defining the left boundaryfor the optimization problem aimed at finding, for a given dataset ofsimulated gaussian data associated toData, the optimal value of\(F\). Default is 0.5.
F_max: the maximum value of \(F\), defining the right boundaryfor the optimization problem aimed at finding, for a given dataset ofsimulated gaussian data associated toData, the optimal value of\(F\). Default is 5.
tol: the tolerance to be used in the optimization problem aimed atfinding, for a given dataset of simulated gaussian data associated toData, the optimal value of \(F\). Default is1e-3.
maxiter: the maximum number of iterations to solve theoptimization problem aimed at finding, for a given dataset of simulatedgaussian data associated toData, the optimal value of \(F\).Default is100.
VERBOSE: a parameter controlling the verbosity of the adjustmentprocess.
either a logical value indicating whether you want thefunctional boxplot to be displayed, or the number of the graphical devicewhere you want the functional boxplot to be displayed.
the label to use on the x axis when displaying the functionalboxplot.
the label (or list of labels for the multivariate functionalcase) to use on the y axis when displaying the functional boxplot.
the main title (or list of titles for the multivariate functionalcase) to be used when displaying the functional boxplot.
additional graphical parameters to be used in plotting functions.
Even when used in graphical way to plot the functional boxplot, thefunction returns a list of three elements:
Depths: contains the depths of each element of the functionaldataset.
Fvalue: is the value of F used to obtain the outliers.
ID_out: contains the vector of indices of dataset elements flaggedas outliers (if any).
In theunivariate functional case, when the adjustment option isselected, the value of \(F\) is optimized for the univariate functionaldataset provided withData.
In practice, a numberadjust$N_trials of times a synthetic population(of sizeadjust$tiral_size with the same covariance (robustlyestimated from data) and centerline asfData is simulated withoutoutliers and each time an optimized value \(F_i\) is computed so that agiven proportion (adjust$TPR) of observations is flagged as outliers.The final value ofF for the functional boxplot is determined as anaverage of \(F_1, F_2, \dots, F_{N_{trials}}\). At each time step theoptimization problem is solved usingstats::uniroot (Brent's method).
Sun, Y., & Genton, M. G. (2012). Functional boxplots. Journal ofComputational and Graphical Statistics.
Sun, Y., & Genton, M. G. (2012). Adjusted functional boxplots forspatio-temporal data visualization and outlier detection. Environmetrics,23(1), 54-64.
# UNIVARIATE FUNCTIONAL BOXPLOT - NO ADJUSTMENTset.seed(1)N=2*100+1P=2e2grid=seq(0,1, length.out=P)D=10*matrix(sin(2*pi*grid), nrow=N, ncol=P, byrow=TRUE)D=D+rexp(N, rate=0.05)# c( 0, 1 : (( N - 1 )/2), -( ( ( N - 1 ) / 2 ) : 1 ) )^4fD=fData(grid,D)dev.new()oldpar<-par(mfrow=c(1,1))par(mfrow=c(1,3))plot(fD, lwd=2, main='Functional dataset', xlab='time', ylab='values')fbplot(fD, main='Functional boxplot', xlab='time', ylab='values', Fvalue=1.5)#> $Depth#> [1] 0.506666667 0.446517413 0.189054726 0.181044776 0.435621891 0.067860697#> [7] 0.435621891 0.471194030 0.496268657 0.196965174 0.372935323 0.507263682#> [13] 0.423830846 0.019850746 0.468457711 0.473830846 0.256666667 0.496268657#> [19] 0.372935323 0.489502488 0.130895522 0.494726368 0.334278607 0.483383085#> [25] 0.139502488 0.086368159 0.487562189 0.029651741 0.449950249 0.487562189#> [31] 0.362388060 0.039353234 0.362388060 0.388009950 0.227611940 0.483383085#> [37] 0.351442786 0.503432836 0.506218905 0.277412935 0.465621891 0.481144279#> [43] 0.406716418 0.415472637 0.476368159 0.345820896 0.402189055 0.489502488#> [49] 0.459651741 0.227611940 0.423830846 0.181044776 0.048955224 0.478805970#> [55] 0.491343284 0.494726368 0.242338308 0.356965174 0.462686567 0.507263682#> [61] 0.113383085 0.459651741 0.297263682 0.328358209 0.009950249 0.427860697#> [67] 0.095472637 0.456517413 0.504278607 0.501442786 0.284129353 0.148009950#> [73] 0.058457711 0.328358209 0.415472637 0.048955224 0.388009950 0.334278607#> [79] 0.503432836 0.086368159 0.411144279 0.485522388 0.502487562 0.077164179#> [85] 0.164726368 0.506218905 0.316218905 0.427860697 0.383084577 0.212487562#> [91] 0.476368159 0.446517413 0.471194030 0.310000000 0.497711443 0.322338308#> [97] 0.442985075 0.249552239 0.172935323 0.383084577 0.249552239 0.462686567#> [103] 0.449950249 0.406716418 0.419701493 0.104477612 0.220099502 0.156417910#> [109] 0.481144279 0.507412935 0.322338308 0.270597015 0.029651741 0.164726368#> [115] 0.367711443 0.501442786 0.431791045 0.095472637 0.256666667 0.340099502#> [121] 0.284129353 0.505024876 0.297263682 0.491343284 0.009950249 0.212487562#> [127] 0.039353234 0.378059701 0.122189055 0.439353234 0.316218905 0.411144279#> [133] 0.402189055 0.397562189 0.148009950 0.500298507 0.204776119 0.130895522#> [139] 0.220099502 0.507014925 0.468457711 0.478805970 0.172935323 0.442985075#> [145] 0.263681592 0.505671642 0.485522388 0.419701493 0.303681592 0.204776119#> [151] 0.456517413 0.290746269 0.290746269 0.077164179 0.242338308 0.502487562#> [157] 0.263681592 0.500298507 0.058457711 0.019850746 0.345820896 0.493084577#> [163] 0.310000000 0.356965174 0.351442786 0.270597015 0.113383085 0.196965174#> [169] 0.378059701 0.122189055 0.497711443 0.277412935 0.493084577 0.189054726#> [175] 0.397562189 0.340099502 0.235024876 0.499054726 0.439353234 0.507014925#> [181] 0.505671642 0.303681592 0.367711443 0.499054726 0.392835821 0.473830846#> [187] 0.507462687 0.465621891 0.235024876 0.139502488 0.453283582 0.506666667#> [193] 0.156417910 0.431791045 0.392835821 0.507412935 0.505024876 0.453283582#> [199] 0.504278607 0.067860697 0.104477612#>#> $Fvalue#> [1] 1.5#>#> $ID_outliers#> [1] 6 14 28 53 65 73 127 154#>boxplot(fD$values[,1], ylim=range(fD$values), main='Boxplot of functional dataset at t_0 ')par(oldpar)# UNIVARIATE FUNCTIONAL BOXPLOT - WITH ADJUSTMENTset.seed(161803)P=2e2grid=seq(0,1, length.out=P)N=1e2# Generating a univariate synthetic gaussian datasetData=generate_gauss_fdata(N, centerline=sin(2*pi*grid), Cov=exp_cov_function(grid, alpha=0.3, beta=0.4))fD=fData(grid,Data)dev.new()# \donttest{fbplot(fD, adjust=list( N_trials=10, trial_size=5*N, VERBOSE=TRUE), xlab='time', ylab='Values', main='My adjusted functional boxplot')#> * * * Iteration 1 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 2 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 3 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 4 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 5 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 6 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 7 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 8 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 9 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> * * * Iteration 10 / 10#> * * * * beginning optimization#> * * * * optimization finished.#> $Depth#> [1] 0.30070101 0.44452323 0.36620606 0.45850303 0.37377980 0.48135354#> [7] 0.38181818 0.02659596 0.13369899 0.42945859 0.06214949 0.47510707#> [13] 0.30211111 0.23325253 0.17038384 0.12049495 0.42031313 0.49717778#> [19] 0.33230101 0.44418990 0.48692525 0.48365657 0.35160404 0.40534141#> [25] 0.39773131 0.42371515 0.22158586 0.36634141 0.14679192 0.39269293#> [31] 0.46537778 0.23329495 0.34899596 0.36024646 0.16920606 0.44801616#> [37] 0.11143434 0.46201818 0.28396162 0.46948687 0.08282020 0.13537576#> [43] 0.36103636 0.38312323 0.24703030 0.48692121 0.46313131 0.44435556#> [49] 0.22941616 0.49802424 0.12830505 0.31013333 0.23446263 0.20488081#> [55] 0.48862424 0.28376768 0.37487475 0.44553131 0.38107071 0.42778182#> [61] 0.46742828 0.43395758 0.17490101 0.46710303 0.43027879 0.38315152#> [67] 0.49235758 0.48834949 0.05709293 0.44494949 0.40256566 0.19737778#> [73] 0.46515152 0.30163030 0.46011111 0.43029495 0.46189091 0.33044242#> [79] 0.40226263 0.35060202 0.48137374 0.04209091 0.25347273 0.27454747#> [85] 0.25617778 0.28562424 0.39055152 0.34064242 0.38934141 0.32964040#> [91] 0.48462828 0.37410303 0.46637778 0.23939394 0.46542626 0.28972727#> [97] 0.29029697 0.46380606 0.41843434 0.39590101#>#> $Fvalue#> [1] 0.8693445#>#> $ID_outliers#> [1] 8 11 41 42 51 69 82#># }# MULTIVARIATE FUNCTIONAL BOXPLOT - NO ADJUSTMENTset.seed(1618033)P=1e2N=1e2L=2grid=seq(0,1, length.out=1e2)C1=exp_cov_function(grid, alpha=0.3, beta=0.4)C2=exp_cov_function(grid, alpha=0.3, beta=0.4)# Generating a bivariate functional dataset of gaussian data with partially# correlated componentsData=generate_gauss_mfdata(N,L, centerline=matrix(sin(2*pi*grid), nrow=2, ncol=P, byrow=TRUE), correlations=rep(0.5,1), listCov=list(C1,C2))mfD=mfData(grid,Data)dev.new()fbplot(mfD, Fvalue=2.5, xlab='time', ylab=list('Values 1','Values 2'), main=list('First component','Second component'))#> $Depth#> [1] 0.43254949 0.39116364 0.35335556 0.39053131 0.08351313 0.16953131#> [7] 0.43867677 0.20252121 0.21967273 0.45222828 0.41967879 0.25533333#> [13] 0.42095758 0.35050101 0.38179192 0.35996364 0.29859798 0.20294747#> [19] 0.40174545 0.46333535 0.38374949 0.39712121 0.34678384 0.31019192#> [25] 0.48306263 0.31276768 0.35823030 0.28142222 0.31049495 0.44906869#> [31] 0.42941010 0.38867677 0.38472323 0.44426667 0.34880000 0.38969091#> [37] 0.45598182 0.33775758 0.45459192 0.48445051 0.31913737 0.37989091#> [43] 0.15915354 0.34753131 0.37781414 0.27654545 0.48208081 0.15098990#> [49] 0.36521818 0.26237778 0.44220000 0.44574949 0.24915556 0.37333131#> [55] 0.35525455 0.28274343 0.38808283 0.11473333 0.36927071 0.16206869#> [61] 0.33650909 0.39217778 0.39524040 0.44069091 0.04190707 0.47903636#> [67] 0.41923030 0.18813333 0.39609697 0.43845657 0.35709697 0.49089697#> [73] 0.39865657 0.46600606 0.39957576 0.35944444 0.19638990 0.30273333#> [79] 0.41812323 0.27323434 0.27932929 0.29413333 0.40971717 0.20922020#> [85] 0.29652727 0.49594141 0.38739798 0.26582020 0.11922424 0.31874747#> [91] 0.49965859 0.33760808 0.32843232 0.38696970 0.42256364 0.18884444#> [97] 0.46271919 0.44007475 0.36836970 0.22256364#>#> $Fvalue#> [1] 2.5#>#> $ID_outliers#> integer(0)#>