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Aformula_hal9001 object as outputted byh.

Aformula_hal9001 object as outputted byh.

Anumeric vector of observations of the outcome variable.

An inputmatrix with dimensions number of observations -by-number of covariates that will be used to derive the design matrix of basisfunctions.

A matrix of new observations on which to obtain predictions. Thedefault ofNULL computes predictions on training inputsX.

Afamily object (one that is supportedbyglmnet) specifying the error/link family for ageneralized linear model.

Anumeric vector of observational-level weights.

Anumeric vector of IDs.

The highest order of interaction terms for which basisfunctions ought to be generated.

Aninteger vector of length 1 or greater,specifying the smoothness of the basis functions. See the argumentsmoothness_orders offit_hal for more information.

Aninteger vector of length 1 ormax_degree,specifying the maximum number of knot points (i.e., bins) for eachcovariate for generating basis functions. Seenum_knots argument infit_hal for more information.

Additional arguments tofit_hal.

Sparse matrix containing columns of indicator functions.

the copy map

An object of classMatrix that has a sparse structuresuitable for coercion to a sparse matrix format ofdgCMatrix.

Index or indices (asnumeric) of covariates (columns) ofinterest in the data matrixx for which basis functions ought to begenerated. Note that basis functions for interactions of these columns arecomputed automatically.

Amatrix containing observations in the rows and covariatesin the columns. Basis functions are computed for these covariates.

An integer vector of lengthncol(x)specifying the desired smoothness of the function in each covariate. k = 0is no smoothness (indicator basis), k = 1 is first order smoothness, and soon. For an additive model, the component function for each covariate willhave the degree of smoothness as specified by smoothness_orders. Fornon-additive components (tensor products of univariate basis functions),the univariate basis functions in each tensor product have smoothnessdegree as specified by smoothness_orders.

Alogical, indicating whether the zerothorder basis functions are included for each covariate (ifTRUE), inaddition to the smooth basis functions given bysmoothness_orders.This allows the algorithm to data-adaptively choose the appropriate degreeof smoothness.

Alogical, likeinclude_zero_order,except including all basis functions of lower smoothness degrees thanspecified viasmoothness_orders.

An inputmatrix containing observations and covariatesfollowing standard conventions in problems of statistical learning.

The highest order of interaction terms for which the basisfunctions ought to be generated. The default (NULL) corresponds togenerating basis functions for the full dimensionality of the input matrix.

An integer vector of lengthncol(x)specifying the desired smoothness of the function in each covariate. k = 0is no smoothness (indicator basis), k = 1 is first order smoothness, and soon. For an additive model, the component function for each covariate willhave the degree of smoothness as specified by smoothness_orders. Fornon-additive components (tensor products of univariate basis functions),the univariate basis functions in each tensor product have smoothnessdegree as specified by smoothness_orders.

Alogical, indicating whether the zerothorder basis functions are included for each covariate (ifTRUE), inaddition to the smooth basis functions given bysmoothness_orders.This allows the algorithm to data-adaptively choose the appropriate degreeof smoothness.

Alogical, likeinclude_zero_order,except including all basis functions of lower smoothness degrees thanspecified viasmoothness_orders.

A sparse matrix, to be centered.

A vector of column means to be used for centering X.

An inputmatrix containing observations and covariatesfollowing standard conventions in problems of statistical learning.

The highest order of interaction terms for which the basisfunctions ought to be generated. The default (NULL) corresponds togenerating basis functions for the full dimensionality of the input matrix.

An integer vector of lengthncol(x)specifying the desired smoothness of the function in each covariate. k = 0is no smoothness (indicator basis), k = 1 is first order smoothness, and soon. For an additive model, the component function for each covariate willhave the degree of smoothness as specified by smoothness_orders. Fornon-additive components (tensor products of univariate basis functions),the univariate basis functions in each tensor product have smoothnessdegree as specified by smoothness_orders.

Alogical, indicating whether the zerothorder basis functions are included for each covariate (ifTRUE), inaddition to the smooth basis functions given bysmoothness_orders.This allows the algorithm to data-adaptively choose the appropriate degreeof smoothness.

Alogical, likeinclude_zero_order,except including all basis functions of lower smoothness degrees thanspecified viasmoothness_orders.

A vector of lengthmax_degree, which determines howgranular the knot points to generate basis functions should be for eachdegree of basis function. The first entry ofnum_knots determinesthe number of knot points to be used for each univariate basis function.More generally, The kth entry ofnum_knots determines the number ofknot points to be used for the kth degree basis functions. Specifically,for a kth degree basis function, which is the tensor product of kunivariate basis functions, this determines the number of knot points to beused for each univariate basis function in the tensor product.

An inputmatrix containing observations and covariatesfollowing standard conventions in problems of statistical learning.

The highest order of interaction terms for which the basisfunctions ought to be generated. The default (NULL) corresponds togenerating basis functions for the full dimensionality of the input matrix.

An integer vector of lengthncol(x)specifying the desired smoothness of the function in each covariate. k = 0is no smoothness (indicator basis), k = 1 is first order smoothness, and soon. For an additive model, the component function for each covariate willhave the degree of smoothness as specified by smoothness_orders. Fornon-additive components (tensor products of univariate basis functions),the univariate basis functions in each tensor product have smoothnessdegree as specified by smoothness_orders.

Alogical, indicating whether the zerothorder basis functions are included for each covariate (ifTRUE), inaddition to the smooth basis functions given bysmoothness_orders.This allows the algorithm to data-adaptively choose the appropriate degreeof smoothness.

Alogical, likeinclude_zero_order,except including all basis functions of lower smoothness degrees thanspecified viasmoothness_orders.

The basis function.

The design matrix, containing the original data.

The HAL design matrix, containing indicator functions.

Numeric indicating which column to populate.

An inputmatrix with dimensions number of observations -by-number of covariates that will be used to derive the design matrix of basisfunctions.

Anumeric vector of observations of the outcome variable. Forfamily="mgaussian",Y is a matrix of observations of theoutcome variables.

A character string formula to be used informula_hal. See its documentation for details.

An inputmatrix with the same number of rows asX, for which no L1 penalization will be performed. Note thatX_unpenalized is directly appended to the design matrix; no basisexpansion is performed onX_unpenalized.

The highest order of interaction terms for which basisfunctions ought to be generated.

Aninteger, specifying the smoothness of thebasis functions. See details forsmoothness_orders for moreinformation.

Aninteger vector of length 1 ormax_degree,specifying the maximum number of knot points (i.e., bins) for any covariatefor generating basis functions. Ifnum_knots is a unit-lengthvector, then the samenum_knots are used for each degree (this isnot recommended). The default settings fornum_knots arerecommended, and these defaults decreasenum_knots with increasingmax_degree andsmoothness_orders, which prevents (expensive)combinatorial explosions in the number of higher-degree and higher-orderbasis functions generated. This allows the complexity of the optimizationproblem to grow scalably. See details ofnum_knots more information.

Am optionalnumeric value bounded in the openunit interval indicating the minimum proportion of 1's in a basis functioncolumn needed for the basis function to be included in the procedure to fitthe lasso. Any basis functions with a lower proportion of 1's than thecutoff will be removed. Defaults to 1 over the square root of the number ofobservations. Only applicable for models fit with zero-order splines, i.e.smoothness_orders = 0.

Acharacter or afamily object(supported byglmnet) specifying the error/linkfamily for a generalized linear model.character options are limitedto "gaussian" for fitting a standard penalized linear model, "binomial" forpenalized logistic regression, "poisson" for penalized Poisson regression,"cox" for a penalized proportional hazards model, and "mgaussian" formultivariate penalized linear model. Note that passing infamily objects leads to slower performance relative to passing in acharacter family (if supported). For example, one should setfamily = "binomial" instead offamily = binomial() whencallingfit_hal.

User-specified sequence of values of the regularizationparameter for the lasso L1 regression. IfNULL, the default sequenceincv.glmnet will be used. The cross-validatedoptimal value of this regularization parameter will be selected withcv.glmnet. Iffit_control'scv_selectargument is set toFALSE, then the lasso model will be fit viaglmnet, and regularized coefficient values for eachlambda in the input array will be returned.

A vector of ID values that is used to generate cross-validationfolds forcv.glmnet. This argument is ignored whenfit_control'scv_select argument isFALSE.

observation weights; defaults to 1 per observation.

a vector of offset values, used in fitting.

List of arguments, including the following, and anyothers to be passed tocv.glmnet orglmnet.

• cv_select: Alogical specifying if the sequence ofspecifiedlambda values should be passed tocv.glmnet in order for a single, optimal value oflambda to be selected according to cross-validation. Whencv_select = FALSE, aglmnet model will beused to fit the sequence of (or single)lambda.

• use_min: Specify the choice of lambda to be selected bycv.glmnet. WhenTRUE,"lambda.min" isused; otherwise,"lambda.1se". Only used whencv_select = TRUE.

• lambda.min.ratio: Aglmnet argumentspecifying the smallest value forlambda, as a fraction oflambda.max, the (data derived) entry value (i.e. the smallest valuefor which all coefficients are zero). We've seen that not settinglambda.min.ratio can lead to nolambda values that fit thedata sufficiently well.

• prediction_bounds: An optional vector of size two that providesthe lower and upper bounds predictions; not used whenfamily = "cox". Whenprediction_bounds = "default", thepredictions are bounded betweenmin(Y) - sd(Y) andmax(Y) + sd(Y) for each outcome (whenfamily = "mgaussian",each outcome can have different bounds). Bounding ensures that there isno extrapolation.

The full set of basis functions generated fromX.

Alogical indicating whether or not to returntheglmnet fit object of the lasso model.

Alogical indicating whether or not to returnthe matrix of (possibly reduced) basis functions used infit_hal.

Alogical indicating whether to print one of a curatedselection of quotes from the HAL9000 computer, from the criticallyacclaimed epic science-fiction film "2001: A Space Odyssey" (1968).

Aformula_hal9001 object as outputted byh.

A default value fors if not providedexplicitly to the functionh.

A default value fork if not provided explicitly tothe functionh.

Controls inheritance of the variableX from parent environment.WhenNULL (the default), such a variable is inherited.

Variables for which to generate multivariate interaction basisfunction where the variables can be found in a matrixX in a parentenvironment/frame. Note, just like standardformula objects, thevariables should not be characters (e.g. do h(W1,W2) not h("W1", "W2"))h(W1,W2,W3) will generate three-way HAL basis functions between W1, W2, andW3. It willnot generate the lower dimensional basis functions.

The number of knots for each univariate basis function used togenerate the tensor product basis functions. If a single value then thisvalue is used for the univariate basis functions for each variable.Otherwise, this should be a variable named list that specifies for eachvariable how many knots points should be used.h(W1,W2,W3, k = list(W1 = 3, W2 = 2, W3=1)) is equivalent to firstbinning the variablesW1,W2 andW3 into3,2 and1 uniquevalues and then callingh(W1,W2,W3). This coarsening of the data ensuresthat fewer basis functions are generated, which can lead to substantialcomputational speed-ups. If not provided and the variablenum_knotsis in the parent environment, thens will be set tonum_knots'.

Thesmoothness_orders for the basis functions. The possiblevalues are0 for piece-wise constant zero-order splines or1 forpiece-wise linear first-order splines. If not provided and the variablesmoothness_orders is in the parent environment, thens willbe set tosmoothness_orders.

Apenalty.factor value the generated basis functions that isused byglmnet in the LASSO penalization procedure.pf = 1(default) is the standard penalization factor used byglmnet andpf = 0 means the generated basis functions are unpenalized.

Whether the basis functions should enforce monotonicity ofthe interaction term. If⁠\code{s} = 0⁠, this is monotonicity of thefunction, and, if⁠\code{s} = 1⁠, this is monotonicity of its derivative(e.g., enforcing a convex fit). Set"none" for no constraints,"i" fora monotone increasing constraint, and"d" for a monotone decreasingconstraint. Using"i" constrains the basis functions to have positivecoefficients in the fit, and"d" constrains the basis functions to havenegative coefficients.

Just like withformula,. as inh(.) orh(.,.) istreated as a wildcard variable that generates terms using all variables inthe data. The argument. should be a character vector of variablenames that. iterates over. Specifically,h(., k=1, . = c("W1", "W2", "W3")) is equivalent toh(W1, k=1) + h(W2, k=1) + h(W3, k=1), andh(., ., k=1, . = c("W1", "W2", "W3")) is equivalent toh(W1,W2, k=1) + h(W2,W3, k=1) + h(W1, W3, k=1)

Whether the arguments... are characters orcharacter vectors and should thus be evaluated directly. WhenTRUE, theexpression h("W1", "W2") can be used.

An optional design matrix where the variables given in...can be found. Otherwise,X is taken from the parent environment.

Sparse matrix containing columns of indicator functions.

A subset of the columns of X, the original design matrix.

An index of the columns that were reduced to by sub-setting.

A vector with length the original unsubsetted matrix X which specifies the smoothness of the function in each covariate.

A design matrix consisting of basis (indicator) functions forcovariates (X) and terms for interactions thereof.

Matrix of covariates containing observed data in the columns.

List of basis functions with which to build HAL design matrix.

Sparse matrix pre-allocation proportion. Default value is 0.5.If one expects a dense HAL design matrix, it is useful to set p_reserve to a higher value.

A matrix of basis functions with all redundant basisfunctions already removed.

A scalarnumeric value bounded in the openinterval (0,1) indicating the minimum proportion of 1's in a basis functioncolumn needed for the basis function to be included in the procedure to fitthe Lasso. Any basis functions with a lower proportion of 1's than thespecified cutoff will be removed. This argument defaults toNULL, inwhich case all basis functions are used in the lasso-fitting stage of theHAL algorithm.

The design matrix, containing the original data.

Numeri for a row index over which to evaluate.

Numeric for the column indices of the basis function.

Numeric providing thresholds.

Numeric providing smoothness orders

interaction degree.

seefit_hal.

The base number of knots for zeroth-order smoothnessbasis functions. The number of knots by degree interaction decays asbase_num_knots_0/2^(d-1) whered is the interaction degree of the basisfunction.

The base number of knots for 1 or greater ordersmoothness basis functions. The number of knots by degree interactiondecays asbase_num_knots_1/2^(d-1) whered is the interaction degree ofthe basis function.

A fitted object of classhal9001.

A matrix of new observations on which to obtain predictions.

Not used.

An object of classhal9001, containing the results offitting the Highly Adaptive Lasso, as produced byfit_hal.

Amatrix ordata.frame containing new data(i.e., observations not used for fitting thehal9001 object that'spassed in via theobject argument) for which thehal9001object will compute predicted values.

If the user suppliedX_unpenalized duringtraining, then user should also supply this matrix with the same number ofobservations asnew_data.

A vector of offsets. Must be provided if provided at training.

Either "response" for predictions of the response, or "link" forun-transformed predictions (on the scale of the link function).

Additional arguments passed topredict as necessary.

A formula_hal9001 object.

Other arguments (ignored).

An object of classsummary.hal9001.

The number of ranked coefficients to be summarized.

Other arguments (ignored).

Anumeric vector to be discretized.

Anumeric scalar indicating the number of bins into whichX should be discretized..

An object of classhal9001, containing the results offitting the Highly Adaptive LASSO, as produced by a call tofit_hal.

An object of classhal9001, containing the results offitting the Highly Adaptive Lasso, as produced byfit_hal.

Optionalnumeric value of the lambda tuningparameter, for which corresponding coefficient values will be summarized.Defaults tofit_hal's optimal value,lambda_star, orthe minimum value oflambda_star.

Alogical specifying whether the summaryshould include only terms with non-zero coefficients.

Alogical specifying whether thesummary should remove so-called "redundant terms". We define a redundantterm (say x1) as a term (1) with basis function corresponding to anexisting basis function, a duplicate; and (2) the duplicate contains thex1 term as part of its term, so that x1 terms inclusion would be redundant.For example, say the same coefficient corresponds to these three terms:(1) "I(age >= 50)*I(bmi >= 18)", (2) "I(age >= 50)", and (3)"I(education >= 16)". Wheninclude_redundant_terms isFALSE (default), the second basis function is omitted.

Aninteger indicating the number of decimalplaces to be used for rounding cutoff values in the term. For example, if"bmi" was numeric that was rounded to the third decimal, in the exampleabove we would have needed to specifyround_cutoffs = 0 in order toyield a term like "I(bmi >= 18)" opposed to something like"I(bmi >= 18.111)". This rounding is intended to simplify the term-wisepart of the output and only rounds the basis cutoffs, thehal9001model's coefficients are not rounded.

Additional arguments passed tosummary, not supported.