This function implements kfold cross validation on complete or incomplete datasets for partial least squares beta regression models (formula specification of the model).

PLS_beta_kfoldcv_formula(formula,data=NULL,nt=2,limQ2set=.0975,
modele="pls", family=NULL, K=nrow(dataX), NK=1, grouplist=NULL,
random=FALSE, scaleX=TRUE, scaleY=NULL, keepcoeffs=FALSE,
keepfolds=FALSE, keepdataY=TRUE, keepMclassed=FALSE, tol_Xi=10^(-12),
weights,subset,start=NULL,etastart,mustart,offset,method,control= list(),
link.phi=NULL,type="ML",verbose=TRUE)

## Arguments

formula an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted. The details of model specification are given under 'Details'. an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which plsRglm is called. number of components to be extracted limit value for the Q2 name of the PLS glm or PLS beta model to be fitted ("pls", "pls-glm-Gamma", "pls-glm-gaussian", "pls-glm-inverse.gaussian", "pls-glm-logistic", "pls-glm-poisson", "pls-glm-polr", "pls-beta"). Use "modele=pls-glm-family" to enable the family option. a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. (See family for details of family functions.) To use the family option, please set modele="pls-glm-family". User defined families can also be defined. See details. number of groups number of times the group division is made to specify the members of the K groups should the K groups be made randomly scale the predictor(s) : must be set to TRUE for modele="pls" and should be for glms pls. scale the response : Yes/No. Ignored since non always possible for glm responses. shall the coefficients for each model be returned shall the groups' composition be returned shall the observed value of the response for each one of the predicted value be returned shall the number of miss classed be returned (unavailable) minimal value for Norm2(Xi) and $$\mathrm{det}(pp' \times pp)$$ if there is any missing value in the dataX. It defaults to $$10^{-12}$$ an optional vector of 'prior weights' to be used in the fitting process. Should be NULL or a numeric vector. an optional vector specifying a subset of observations to be used in the fitting process. starting values for the parameters in the linear predictor. starting values for the linear predictor. starting values for the vector of means. this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases. One or more offset terms can be included in the formula instead or as well, and if more than one is specified their sum is used. See model.offset. for fitting glms with glm ("pls-glm-Gamma", "pls-glm-gaussian", "pls-glm-inverse.gaussian", "pls-glm-logistic", "pls-glm-poisson", "modele=pls-glm-family")the method to be used in fitting the model. The default method "glm.fit" uses iteratively reweighted least squares (IWLS). User-supplied fitting functions can be supplied either as a function or a character string naming a function, with a function which takes the same arguments as glm.fit. If "model.frame", the model frame is returned. pls-glm-polrlogistic, probit, complementary log-log or cauchit (corresponding to a Cauchy latent variable). a list of parameters for controlling the fitting process. For glm.fit this is passed to glm.control. an optional list. See the contrasts.arg of model.matrix.default. should the coefficients of non-significant predictors (<alpha.pvals.expli) be set to 0 should component extraction stop when no significant predictors (<alpha.pvals.expli) are found Use the naive estimates for the Degrees of Freedom in plsR? Default is FALSE. character specification of the link function in the mean model (mu). Currently, "logit", "probit", "cloglog", "cauchit", "log", "loglog" are supported. Alternatively, an object of class "link-glm" can be supplied. character specification of the link function in the precision model (phi). Currently, "identity", "log", "sqrt" are supported. The default is "log" unless formula is of type y~x where the default is "identity" (for backward compatibility). Alternatively, an object of class "link-glm" can be supplied. character specification of the type of estimator. Currently, maximum likelihood ("ML"), ML with bias correction ("BC"), and ML with bias reduction ("BR") are supported. should info messages be displayed ?

## Details

Predicts 1 group with the K-1 other groups. Leave one out cross validation is thus obtained for K==nrow(dataX).

There are seven different predefined models with predefined link functions available :

"pls"

ordinary pls models

"pls-glm-Gamma"

glm gaussian with inverse link pls models

"pls-glm-gaussian"

glm gaussian with identity link pls models

"pls-glm-inverse-gamma"

glm binomial with square inverse link pls models

"pls-glm-logistic"

glm binomial with logit link pls models

"pls-glm-poisson"

glm poisson with log link pls models

"pls-glm-polr"

glm polr with logit link pls models

Using the "family=" option and setting "modele=pls-glm-family" allows changing the family and link function the same way as for the glm function. As a consequence user-specified families can also be used.

The gaussian family

accepts the links (as names) identity, log and inverse.

The binomial family

accepts the links logit, probit, cauchit, (corresponding to logistic, normal and Cauchy CDFs respectively) log and cloglog (complementary log-log).

The Gamma family

accepts the links inverse, identity and log.

The poisson family

accepts the links log, identity, and sqrt.

The inverse.gaussian family

accepts the links 1/mu^2, inverse, identity and log.

The quasi family

accepts the links logit, probit, cloglog, identity, inverse, log, 1/mu^2 and sqrt.

The function power

can be used to create a power link function.

A typical predictor has the form response ~ terms where response is the (numeric) response vector and terms is a series of terms which specifies a linear predictor for response. A terms specification of the form first + second indicates all the terms in first together with all the terms in second with any duplicates removed.

A specification of the form first:second indicates the the set of terms obtained by taking the interactions of all terms in first with all terms in second. The specification first*second indicates the cross of first and second. This is the same as first + second + first:second.

The terms in the formula will be re-ordered so that main effects come first, followed by the interactions, all second-order, all third-order and so on: to avoid this pass a terms object as the formula.

Non-NULL weights can be used to indicate that different observations have different dispersions (with the values in weights being inversely proportional to the dispersions); or equivalently, when the elements of weights are positive integers w_i, that each response y_i is the mean of w_i unit-weight observations.

## Value

results_kfolds

list of NK. Each element of the list sums up the results for a group division:

list of K matrices of size about nrow(dataX)/K * nt with the predicted values for a growing number of components

list of K matrices of size about nrow(dataX)/K * nt with the predicted values for a growing number of components

folds

list of NK. Each element of the list sums up the informations for a group division:

list of K vectors of length about nrow(dataX) with the numbers of the rows of dataX that were used as a training set

list of K vectors of length about nrow(dataX) with the numbers of the rows of dataX that were used as a training set

dataY_kfolds

list of NK. Each element of the list sums up the results for a group division:

list of K matrices of size about nrow(dataX)/K * 1 with the observed values of the response

list of K matrices of size about nrow(dataX)/K * 1 with the observed values of the response

call

the call of the function

## References

Frédéric Bertrand, Nicolas Meyer, Michèle Beau-Faller, Karim El Bayed, Izzie-Jacques Namer, Myriam Maumy-Bertrand (2013). Régression Bêta PLS. Journal de la Société Française de Statistique, 154(3):143-159. http://publications-sfds.math.cnrs.fr/index.php/J-SFdS/article/view/215

## Note

Work for complete and incomplete datasets.

kfolds2coeff, kfolds2Pressind, kfolds2Press, kfolds2Mclassedind, kfolds2Mclassed and kfolds2CVinfos_beta to extract and transform results from kfold cross validation.
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