Package 'TULIP'

Adjust tensor for covariates.

Description

Adjusts tensor with respect to covariates to achieve a more accurate performance. Tensor depends on the covariates through a linear regression model. The function returns the coefficients of covariates in regression and adjusted tensor list for further classifier modeling. It estimates coefficients based on training data, and then adjusts training tensor. When testing data is provided, the function will automatically adjust testing data by learned coefficients as well.

Usage

adjten(x, z, y, testx = NULL, testz = NULL, is.centered = FALSE)

Arguments

x	Input tensor or matrix list of length $N$, where $N$ is the number of observations. Each element of the list is a tensor or matrix. The order of tensor can be any integer not less than 2.
z	Input covariate matrix of dimension $N \times q$, where $q<N$. Each row of z is an observation.
y	Class label vector of dimention $N\times 1$. For K class problems, y takes values in $\{1,\cdots,\code{K}\}$.
testx	Input testing tensor or matrix list. Each element of the list is a test case. When testx is not provided, the function will only adjust training data.
testz	Input testing covariate matrix with each row being an observation.
is.centered	Indicates whether the input tensor and covariates have already been centered by their within class mean or not. If is.centered is FALSE, the function adjten will center data by class. If is.centered is TRUE, the function will skip the centering step.

Details

The model CATCH assumes the linear relationship bewteen covariates and tensor as

$\mathbf{X}=\boldsymbol{\mu}_k+\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}+\mathbf{E},$

where $\boldsymbol{\alpha}$ is the matrix of estimated coefficient of covariates. The function removes the effects of covariates on response variable through tensor and obtain $\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}$ as adjusted tensor to fit tensor discriminant analysis model.

In estimating $\boldsymbol{\alpha}$, which is the alpha in the package, adjten first centers both tensor and covariates within their individual classes, then performs tensor response regression which regresses ${\mathbf{X}}$ on ${\mathbf{Z}}$.

Value

gamma	The estimated coefficients of covariates to plug in classifier. gamma is the $\boldsymbol{\gamma}_k$ defined function catch of dimension $q\times (K-1)$, where q is the size of covariates and K is the number of classes.
xres	Adjusted training tensor list $\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}$ after adjusting for covariates. The effect of the covariate is removed.
testxres	Adjusted testing tensor list $\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}$ after adjusting for covariates. The effect of the covariate is removed.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, accepted.

See Also

catch

Examples

n <- 20
p <- 4
k <- 2
nvars <- p*p*p
x <- array(list(),n)
vec_x <- matrix(rnorm(n*nvars),nrow=n,ncol=nvars)
vec_x[1:10,] <- vec_x[1:10,]+2
z <- matrix(rnorm(n*2),nrow=n,ncol=2)
z[1:10,] <- z[1:10,]+0.5
y <- c(rep(1,10),rep(2,10))
for (i in 1:n){
  x[[i]] <- array(vec_x[i,],dim=c(p,p,p))
}
obj <- adjten(x, z, y)

---

Adjust vector for covariates.

Description

Adjusts vector with respect to covariates. Vector depends on the covariates through a linear regression model. The function returns the coefficients of covariates in regression and adjusted predictor matrix for further classifier modeling. It estimates coefficients based on training data, and then adjusts training tensor. When testing data is provided, the function will automatically adjust testing data by learned coefficients as well.

Usage

adjvec(x, z, y, testx = NULL, testz = NULL, is.centered = FALSE)

Arguments

x	Input matrix of dimension $N\times p$, where $N$ is the number of observations and $p$ is the number of variables. Each row is an observation
z	Input covariate matrix of dimension $N \times q$, where $q<N$. Each row of z is an observation.
y	Class label vector of dimention $N\times 1$. For K class problems, y takes values in $\{1,\cdots,\code{K}\}$.
testx	Input testing matrix. Each row is a test case. When testx is not provided, the function will only adjust training data.
testz	Input testing covariate matrix with each row being an observation.
is.centered	Indicates whether the input vector and covariates have already been centered by their within class mean or not. If is.centered is FALSE, the function adjvec will center data by class. If is.centered is TRUE, the function will skip the centering step.

Details

Similar as CATCH model, assume the linear relationship between vector predictors and covariates as

$\mathbf{X}=\boldsymbol{\mu}_k+\boldsymbol{\alpha}\times\mathbf{Z}+\mathbf{E},$

where $\mathbf{X}$ is a $N\times p$ matrix and $\boldsymbol{\alpha}$ is the matrix of estimated coefficient of covariates. The function removes the effects of covariates on response variable through vector and obtain $\mathbf{X}-\boldsymbol{\alpha}\times\mathbf{Z}$ as adjusted predictors to fit MSDA and DSDA model.

Value

gamma	The estimated coefficients of covariates to plug in classifier. gamma is similar as the $\boldsymbol{\gamma}_k$ defined function catch of dimension $q\times (K-1)$, where q is the size of covariates and K is the number of classes.
xres	Adjusted training predictor matrix $\mathbf{X}-\boldsymbol{\alpha}\times\mathbf{Z}$ after adjusting for covariates. The effect of the covariate is removed.
testxres	Adjusted testing predictor matrix $\mathbf{X}-\boldsymbol{\alpha}\times\mathbf{Z}$ after adjusting for covariates. The effect of the covariate is removed.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, accepted.

See Also

adjten

Examples

n <- 50
p <- 200
k <- 2
q <- 2
x <- matrix(rnorm(n*p), n, p)
z <- matrix(rnorm(n*q), n, q)
x[1:20, ] <- x[1:20, ] + 2
z[1:20, ] <- z[1:20, ] + 0.5
y <- c(rep(1, 20), rep(2, 30))
obj <- adjvec(x, z, y)

---

Fit a CATCH model and predict categorical response.

Description

The catch function solves classification problems and selects variables by fitting a covariate-adjusted tensor classification in high-dimensions (CATCH) model. The input training predictors include two parts: tensor data and low dimensional covariates. The tensor data could be matrix as a special case of tensor. In catch, tensor data should be stored in a list form. If the dataset contains no covariate, catch can also fit a classifier only based on the tensor predictors. If covariates are provided, the method will adjust tensor for covariates and then fit a classifier based on the adjusted tensor along with the covariates. If users specify testing data at the same time, predicted response will be obtained as well.

Usage

catch(x, z = NULL, y, testx = NULL, testz = NULL, nlambda = 100, 
lambda.factor = ifelse((nobs - nclass) <= nvars, 0.2, 1E-03), 
lambda = NULL,dfmax = nobs, pmax = min(dfmax * 2 + 20, nvars), 
pf = rep(1, nvars), eps = 1e-04, maxit = 1e+05, sml = 1e-06, 
verbose = FALSE, perturb = NULL)

Arguments

x	Input tensor (or matrix) list of length $N$, where $N$ is the number of observations. Each element of the list is a tensor or matrix. The order of tensor can be any positive integer not less than 2.
z	Input covariate matrix of dimension $N \times q$, where $q<N$. z can be omitted if covariate is absent.
y	Class label. For K class problems, y takes values in $\{1,\cdots,\code{K}\}$.
testx	Input testing tensor or matrix list. Each element of the list is a test case. When testx is not provided, the function will only fit the model and return the classifier. When testx is provided, the function will predict response on testx as well.
testz	Input testing covariate matrix. Can be omitted if covariate is absent. However, training covariates z and testing covariates testz must be provided or not at the same time.
nlambda	The number of tuning values in sequence lambda. If users do not specify lambda values, the package will generate a solution path containing nlambda many tuning values of lambda. Default is 100.
lambda.factor	When lambda is not supplied, catch first finds the largest value in lambda which yields $\boldsymbol{\beta}=0$. Then the minimum value in lambda is obtained by (largest value*lambda.factor). The sequence of lambda is generated by evenly sampling nlambda numbers within the range. Default value of lambda.factor is 0.2 if $N<p$ and 0.0001 if $N>p$.
lambda	A sequence of user-specified lambda values. lambda is the weight of L1 penalty and a smaller lambda allows more variables to be nonzero. If NULL, then the algorithm will generate a sequence of nlambda many potential lambdas according to lambda.factor.
dfmax	The maximum number of selected variables in the model. Default is the number of observations N.
pmax	The maximum number of potential selected variables during iteration. In middle step, the algorithm can select at most pmax variables and then shrink part of them such that the nubmer of final selected variables is less than dfmax. Default is $\min(dfmax\times 2+20, N)$.
pf	Weight of lasso penalty. Default is a vector of value 1 and length p, representing L1 penalty of length $p$. Can be mofidied to use adaptive lasso penalty.
eps	Convergence threshold for coordinate descent difference between iterations. Default value is 1e-04.
maxit	Maximum iteration times for all lambda. Default value is 1e+05.
sml	Threshold for ratio of loss function change after each iteration to old loss function value. Default value is 1e-06.
verbose	Indicates whether print out lambda during iteration or not. Default value is FALSE.
perturb	Perturbation scaler. If it is specified, the value will be added to diagonal of estimated covariance matrix. A small value can be used to accelerate iteration. Default value is NULL.

Details

The catch function fits a linear discriminant analysis model as follows:

$\mathbf{Z}|(Y=k)\sim N(\boldsymbol{\phi_k},\boldsymbol{\psi}),$

$\mathbf{X}|(\mathbf{Z}=\mathbf{z}, Y=k)\sim TN(\boldsymbol{\mu}_k+\boldsymbol{\alpha}\bar{\times}_{M+1}\mathbf{z},\boldsymbol{\Sigma}_1,\cdots,\boldsymbol{\Sigma}_M).$

The categorical response is predicted from the estimated Bayes rule:

$\widehat{Y}=\arg\max_{k=1,\cdots,K}{a_k+\boldsymbol{\gamma}_k^T\mathbf{Z}+<\boldsymbol{\beta}_k,\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}>},$

where $\mathbf{X}$ is the tensor, $\mathbf{Z}$ is the covariates, $a_k$, $\boldsymbol{\gamma}_k$ and $\boldsymbol{\alpha}$ are parameters estimated by CATCH. A detailed explanation can be found in reference. When Z is not NULL, the function will first adjust tensor on covariates by modeling

$\mathbf{X}=\boldsymbol{\mu}_k+\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}+\mathbf{E},$

where $\mathbf{E}$ is an unobservable tensor normal error independent of $\mathbf{Z}$. Then catch fits model on the adjusted training tensor $\mathbf{X}-\boldsymbol{\alpha}\overline{\times}_{M+1}\mathbf{Z}$ and makes predictions on testing data by using the adjusted tensor list. If Z is NULL, it reduces to a simple tensor discriminant analysis model.

The coefficient of tensor $\boldsymbol{\beta}$, represented by beta in package, is estimated by

$\min_{\boldsymbol{\beta}_2,\ldots,\boldsymbol{\beta}_K}\left[\sum_{k=2}^K\left(\langle\boldsymbol{\beta}_k,[\![\boldsymbol{\beta}_k;\widehat{\boldsymbol{\Sigma}}_{1},\dots,\widehat{\boldsymbol{\Sigma}}_{M}]\!]\rangle-2\langle\boldsymbol{\beta}_k,\widehat{\boldsymbol{\mu}}_{k}-\widehat{\boldsymbol{\mu}}_{1}\rangle\right)+\lambda\sum_{j_{1}\dots j_{M}}\sqrt{\sum_{k=2}^{K}\beta_{k,j_{1}\cdots j_{M}}^2}\right].$

When response is multi-class, the group lasso penalty over categories is added to objective function through parameter lambda, and it reduces to a lasso penalty in binary problems.

The function catch will predict categorical response when testing data is provided. If testing data is not provided or if one wishes to perform prediction separately, catch can be used to only fit model with a catch object outcome. The object outcome can be combined with the adjusted tensor list from adjten to perform prediction by predict.catch.

Value

beta	Output variable coefficients for each lambda, which is the estimation of $\boldsymbol{\beta}$ in the Bayes rule. beta is a list of length being the number of lambdas. Each element of beta is a matrix of dimension $nvars\times (nclass-1)$.
df	The number of nonzero variables for each value in sequence lambda.
dim	Dimension of coefficient array.
lambda	The actual lambda sequence used. The user specified sequence or automatically generated sequence could be truncated by constraints on dfmax and pmax.
obj	Objective function value for each value in sequence lambda.
x	The tensor list after adjustment in training data. If covariate is absent, this is the original input tensor list.
y	Class label in training data.
npasses	Total number of iterations.
jerr	Error flag.
sigma	Estimated covariance matrix on each mode. sigma is a list with the ith element being covariance matrix on ith mode.
delta	Estimated delta matrix $(vec(\widehat{\boldsymbol{\mu}}_2-\widehat{\boldsymbol{\mu}}_1),\cdots,vec(\widehat{\boldsymbol{\mu}}_K-\widehat{\boldsymbol{\mu}}_1))$.
mu	Estimated mean array of $\mathbf{X}$.
prior	Proportion of samples in each class.
call	The call that produces this object.
pred	Predicted categorical response for each value in sequence lambda when testx is provided.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, accepted.

See Also

cv.catch, predict.catch, adjten

Examples

#without prediction
n <- 20
p <- 4
k <- 2
nvars <- p*p*p
x <- array(list(),n)
vec_x <- matrix(rnorm(n*nvars), nrow=n, ncol=nvars)
vec_x[1:10,] <- vec_x[1:10,]+2
z <- matrix(rnorm(n*2), nrow=n, ncol=2)
z[1:10,] <- z[1:10,]+0.5
y <- c(rep(1,10),rep(2,10))
for (i in 1:n){
  x[[i]] <- array(vec_x[i,],dim=c(p,p,p))
}
obj <- catch(x,z,y=y)

---

Fit a CATCH model for matrix and predict categorical response.

Description

Fits a classifier for matrix data. catch_matrix is a special case of catch when each observation $\mathbf{X}_i$ is a matrix. Different from catch takes list as input, data need to be formed in an array to call the function (see arguments). The function will perform prediction as well.

Usage

catch_matrix(x, z = NULL, y, testx = NULL, testz = NULL, ...)

Arguments

x	Input matrix array. The array should be organized with dimension $p_1 \times p_2 \times N$.
z	Input covariate matrix of dimension $N \times q$, where $q<N$. z can be omitted if covariate is absent.
y	Class label. For K class problems, y takes values in $\{1,\cdots,\code{K}\}$.
testx	Input testing matrix array. When testx is not provided, the function will only fit model. When testx is provided, the function will predict response on testx as well.
testz	Input testing covariate matrix. Can be omitted if there is no covariate.
...	Other arguments that can be passed to catch.

Details

The function fits a matrix classifier as a special case of catch. The fitted model and predictions should be identical to catch when matrix data is provided. Input matrix should be organized as three-way array where sample size is the last dimension. If the matrix is organized in a list, users can either reorganize it or use catch directly to fit model, which takes a matrix or tensor list as input and has the same output as catch_matrix.

Value

beta	Output variable coefficients for each lambda. beta is a list of length being the number of lambdas. Each element of beta is a matrix of dimension $(p_1\times p_2)\times (nclass-1)$.
df	The number of nonzero variables for each value in sequence lambda.
dim	Dimension of coefficient array.
lambda	The actual lambda sequence used. The user specified sequence or automatically generated sequence could be truncated by constraints on dfmax and pmax.
obj	Objective function value for each value in sequence lambda.
x	The matrix list after adjustment in training data. If covariate is absent, this is the original input matrix.
y	Class label in training data.
npasses	Total number of iterations.
jerr	Error flag.
sigma	Estimated covariance matrix on each mode. sigma is a list with the ith element being covariance matrix on ith mode.
delta	Estimated delta matrix $(vec(\widehat{\boldsymbol{\mu}}_2-\widehat{\boldsymbol{\mu}}_1),\cdots,vec(\widehat{\boldsymbol{\mu}}_K-\widehat{\boldsymbol{\mu}}_1))$.
mu	Estimated mean array.
prior	Prior proportion of observations in each class.
call	The call that produces this object.
pred	Predicted categorical response for each value in sequence lambda when testx is provided.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, accepted.

See Also

catch

Examples

#without prediction
n <- 20
p <- 4
k <- 2
nvars <- p*p
x=array(rnorm(n*nvars),dim=c(p,p,n))
x[,,11:20]=x[,,11:20]+0.3
z <- matrix(rnorm(n*2), nrow=n, ncol=2)
z[1:10,] <- z[1:10,]+0.5
y <- c(rep(1,10),rep(2,10))
obj <- catch_matrix(x,z,y=y)

---

Colorimetric sensor array (CSA) data

Description

A dataset collected from a series of CSA experiments to identify volatile chemical toxicants (VCT). Chemical dyes were exposed to VCT under different concentration conditions and colors of dyes were recorded to identify the class of VCT. There are two concentration conditions PEL (permissible exposure level) and IDLH (immediately dangerous to life of health).

Usage

data(csa)

Format

Two lists, PEL and IDLH, and a numeric vector y. Each list contains 147 matrices of dimension $36 \times 3$.

PEL

A list of matrices containing the observations after exposure at PEL.

IDLH

A list of matrices containing the observations after exposure at IDLH level.

y

Class label ranging from 1 to 21.

Details

This dataset is provided in the Supplementary matrial of Zhong (2015). In each concentration case, there are 147 observations and 21 classes. We reorganize the data into a list to be directly called by catch. For matrices in the list, each row represents a dye and the three columns correspond to red, green and blue.

Source

Wenxuan Zhong and Kenneth S. Suslick (2015). "Matrix discriminant analysis with application to colorimetric sensor array data" Technometrics 57(4), 524–534.

---

Cross-validation for CATCH

Description

Performs k-fold cross validation for CATCH and returns the best tuning parameter $\lambda$ in the user-specified or automatically generated choices.

Usage

cv.catch(x, z = NULL, y, nfolds = 5, lambda = NULL, 
lambda.opt = "min",...)

Arguments

x	Input tensor or matrix list of length $N$, where $N$ is the number of observations. Each element of the list is a tensor or matrix. The order of tensor can be any number and not limited to three.
z	Input covariate matrix of dimension $N \times q$, where $q<N$. z can be omitted if covariate is absent.
y	Class label. For K class problems, y takes values in $\{1,\cdots,\code{K}\}$.
nfolds	Number of folds. Default value is 5.
lambda	User-specified lambda sequence for cross validation. If not specified, the algorithm will generate a sequence of lambdas based on all data and cross validate on the sequence.
lambda.opt	The optimal criteria when multiple elements in lambda return the same minimum classification error. "min" will return the smallest lambda with minimum cross validation error. "max" will return the largest lambda with the minimum cross validation error.
...	Other arguments that can be passed to catch.

Details

The function cv.catch runs function catch nfolds+1 times. The first one fits model on all data. If lambda is specified, it will check if all lambda satisfies the constraints of dfmax and pmax in catch. If not, a lambda sequence will be generated according to lambda.factor in catch. Then the rest nfolds many replicates will fit model on nfolds-1 many folds data and predict on the omitted fold, repectively. Return the lambda with minimum average cross validation error and the largest lambda within one standard error of the minimum.

Value

lambda	The actual lambda sequence used. The user specified sequence or automatically generated sequence could be truncated by constraints on dfmax and pmax.
cvm	The mean of cross validation errors for each lambda.
cvsd	The standard error of cross validaiton errors for each lambda.
lambda.min	The lambda with minimum cross validation error. If lambda.opt is min, then returns the smallest lambda with minimum cross validation error. If lambda.opt is max, then returns the largest lambda with minimum cross validation error.
lambda.1se	The largest lambda with cross validation error within one standard error of the minimum.
catch.fit	The fitted catchobj object.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, accepted.

See Also

catch

Examples

n <- 20
p <- 4
k <- 2
nvars <- p*p*p
x <- array(list(),n)
vec_x <- matrix(rnorm(n*nvars), nrow=n, ncol=nvars)
vec_x[1:10,] <- vec_x[1:10,]+2
z <- matrix(rnorm(n*2),nrow=n,ncol=2)
z[1:10,] <- z[1:10,]+0.5
y <- c(rep(1,10),rep(2,10))
for (i in 1:n){
  x[[i]] <- array(vec_x[i,], dim=c(p,p,p))
}
objcv <- cv.catch(x, z, y=y)

---

Cross validation for direct sparse discriminant analysis

Description

Choose the optimal lambda for direct sparse discriminant analysis by cross validation.

Usage

cv.dsda(x, y, nfolds = 5, lambda=lambda, lambda.opt="min", 
 standardize=FALSE, alpha=1, eps=1e-7)

Arguments

x	An n by p matrix containing the predictors.
y	An n-dimensional vector containing the class labels.
nfolds	The number of folds to be used in cross validation. Default is 5.
lambda	A sequence of lambda's.
lambda.opt	Should be either "min" or "max", specifying whether the smallest or the largest lambda with the smallest cross validation error should be used for the final classification rule.
standardize	A logic object indicating whether x.matrix should be standardized before performing DSDA. Default is FALSE.
alpha	The elasticnet mixing parameter, the same as in glmnet. Default is alpha=1 so that the lasso penalty is used.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Value

lambda	The sequence of lambda's used in cross validation.
cvm	Cross validation errors.
cvsd	The standard error of the cross validation errors.
lambda.min	The optimal lambda chosen by cross validation.
model.fit	The fitted model.

References

Mai, Q., Zou, H. and Yuan, M. (2013). A direct approach to sparse discriminant analysis in ultra-high dimensions. Biometrika, 99, 29-42.

See Also

cv.dsda predict.dsda dsda

---

Cross-validation for DSDA/MSDA through function msda

Description

Performs K-fold cross validation for msda and returns the best tuning parameter $\lambda$ in the user-specified or automatically generated choices.

Usage

cv.msda(x, y, model = NULL, nfolds = 5, lambda = NULL,
 lambda.opt = "min", ...)

Arguments

x	Input matrix of predictors. x is of dimension $N \times p$; each row is an observation vector.
y	Class label. For K class problems, y takes values in $\{1,\cdots,\code{K}\}$.
model	Method type. The model argument can be one of 'binary', 'multi.original', 'multi.modified' and the default is NULL. The function supports fitting DSDA and MSDA models by specifying method type. Without specification, the function will automatically choose one of the methods. If the response variable is binary, the function will fit a DSDA model. If the response variable is multi-class, the function will fit an original MSDA model for dimension $p<=2000$ and a modified MSDA model for dimension $p>2000$.
nfolds	Number of folds. Default value is 5. Although nfolds can be as large as the sample size (leave-one-out CV), it is not recommended for large datasets. Smallest value allowable is nfolds=3 for multi.original and multi.modified.
lambda	User-specified lambda sequence for cross validation. If not specified, the algorithm will generate a sequence of lambdas based on all data and cross validate on the sequence.
lambda.opt	The optimal criteria when multiple elements in lambda return the same minimum classification error. "min" will return the smallest lambda with minimum cross validation error. "max" will return the largest lambda with the minimum cross validation error.
...	other arguments that can be passed to msda.

Details

The function cv.msda runs function msda nfolds+1 times. The first one fits model on all data. If lambda is specified, it will check if all lambda satisfies the constraints of dfmax and pmax in msda. If not, a lambda sequence will be generated according to lambda.factor in msda. Then the rest nfolds many replicates will fit model on nfolds-1 many folds data and predict on the omitted fold, repectively. Return the lambda with minimum average cross validation error and the largest lambda within one standard error of the minimum.

Similar as msda, user can specify which method to use by inputing argument model. Without specification, the function can automatically decide the method by number of classes and variables.

Value

An object of class cv.dsda or cv.msda.original or cv.msda.modified is returned, which is a list with the ingredients of the cross-validation fit.

lambda	The actual lambda sequence used. The user specified sequence or automatically generated sequence could be truncated by constraints on dfmax and pmax.
cvm	The mean of cross validation errors for each lambda.
cvsd	The standard error of cross validaiton errors for each lambda.
lambda.min	The lambda with minimum cross validation error. If lambda.opt is min, then returns the smallest lambda with minimum cross validation error. If lambda.opt is max, then returns the largest lambda with minimum cross validation error.
lambda.1se	The largest value of lambda such that error is within one standard error of the minimum. This arguement is only available for object cv.msda.original and cv.msda.modified.
model.fit	A fitted cv.dsda or cv.msda.original or cv.msda.modified object for the full data.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q., Zou, H. and Yuan, M. (2012), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrica, 99, 29-42.

Mai, Q., Yang, Y., and Zou, H. (2017), "Multiclass sparse discriminant analysis." Statistica Sinica, in press.

URL: https://github.com/emeryyi/msda

See Also

msda

Examples

data(GDS1615)
x <- GDS1615$x
y <- GDS1615$y
obj.cv <- cv.msda(x=x, y=y, nfolds=5, lambda.opt="max")
lambda.min <- obj.cv$lambda.min
obj <- msda(x=x, y=y, lambda=lambda.min)
pred <- predict(obj,x)

---

Cross validation for semiparametric sparse discriminant analysis

Description

Choose the optimal lambda for semiparametric sparse discriminant analysis by cross validation.

Usage

cv.SeSDA(x, y, nfolds = 5, lambda=NULL, lambda.opt="min",
  standardize=FALSE, alpha=1, eps=1e-7)

Arguments

x	An n by p matrix containing the predictors.
y	An n-dimensional vector containing the class labels.
nfolds	The number of folds to be used in cross validation. Default is 5.
lambda	A sequence of lambda's.
lambda.opt	Should be either "min" or "max", specifying whether the smallest or the largest lambda with the smallest cross validation error should be used for the final classification rule.
standardize	A logic object indicating whether x.matrix should be standardized before performing DSDA. Default is FALSE.
alpha	The elasticnet mixing parameter, the same as in glmnet. Default is alpha=1 so that the lasso penalty is used.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Value

transform	The transformation functions.
objdsda	The output of cross validation from cv.dsda on transformed data.

References

Mai, Q., Zou, H. and Yuan, M. (2013). A direct approach to sparse discriminant analysis in ultra-high dimensions. Biometrika, 99, 29-42.

See Also

cv.dsda SeSDA

---

Solution path for direct sparse discriminant analysis

Description

Compute the solution path for direct sparse discriminant analysis (DSDA).

Usage

dsda(x, z=NULL, y, testx=NULL, testz=NULL, standardize=FALSE, 
 lambda=lambda, alpha=1, eps=1e-7)

Arguments

x	Input matrix of predictors. x is of dimension $N \times p$; each row is an observation vector.
z	Input covariate matrix of dimension $N \times q$, where $q<N$. z can be omitted if covariate is absent.
y	An n-dimensional vector containing the class labels. The classes have to be labeled as 1 and 2.
testx	Input testing matrix. Each row is a test case. When testx is not provided, the function will only fit the model and return the classifier. When testx is provided, the function will predict response on testx as well.
testz	Input testing covariate matrix. Can be omitted if covariate is absent. However, training covariates z and testing covariates testz must be provided or not at the same time.
standardize	A logic object indicating whether x should be standardized before performing DSDA. Default is FALSE.
lambda	A sequence of lambda's. If lambda is missed, the function will automatically generates a sequence of lambda's to fit model.
alpha	The elasticnet mixing parameter, the same as in glmnet. Default is alpha=1 so that the lasso penalty is used.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Value

beta	Output variable coefficients for each lambda. The first element of each solution is the intercept.
lambda	The sequence of lambda's used in computing the solution path.
x	The predictor matrix in training data.
y	The class label in training data.
pred	Predicted categorical response for each value in sequence lambda when testx is provided.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q., Zou, H. and Yuan, M. (2013). A direct approach to sparse discriminant analysis in ultra-high dimensions. Biometrika, 99, 29-42.

Examples

data(GDS1615)   ##load the prostate data
  x<-GDS1615$x
  y<-GDS1615$y
  x=x[which(y<3),]
  y=y[which(y<3)]
  obj.path <- dsda(x, y=y)

---

Direct sparse discriminant analysis

Description

Performs direct sparse discriminant analysis, with the optimal lambda chosen by cross validation. The function can perform prediction on test data as well.

Usage

dsda.all(x, y, x.test.matrix=NULL, y.test=NULL, standardize=FALSE, 
lambda.opt="min", nfolds=10, lambda=lambda, alpha=1, eps=1e-7)

Arguments

x	An n by p matrix containing the predictors.
y	An n-dimensional vector containing the class labels 1 and 2.
x.test.matrix	The predictors of a testing set. (Optional.)
y.test	The class labels of the testing set. (Required if x.test.matrix is supplied, but otherwise optional.)
standardize	A logic object indicating whether x.matrix should be standardized before performing DSDA. Default is FALSE.
lambda.opt	Should be either "min" or "max", specifying whether the smallest or the largest lambda with the smallest cross validation error should be used for the final classification rule.
nfolds	The number of folds to be used in cross validation. Default is 10.
lambda	A sequence of lambda's.
alpha	The elasticnet mixing parameter, the same as in glmnet. Default is alpha=1 so that the lasso penalty is used.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Value

error	Testing error if x.test.matrix is supplied.
beta	The coefficients of the classification rule corresponding to the optimal lambda chosen by cross validation.
s	The optimal lambda chosen by cross validation.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q., Zou, H. and Yuan, M., (2012), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrika, 99, 29-42.

See Also

dsda

Examples

data(GDS1615)   ##load the prostate data
x<-GDS1615$x
y<-GDS1615$y

x=x[which(y<3),]
y=y[which(y<3)]

n<-length(y)    ##split the original dataset to a training set and a testing set
n.test<-round(n/3)
set.seed(20120822)
id<-sample(n,n.test,replace=FALSE)
x.train<-x[-id,]
x.test<-x[id,]
y.train<-y[-id]
y.test<-y[id]

set.seed(123)
##perform direct sparse discriminant analysis
obj<-dsda.all(x.train,y.train,x.test,y.test)  
obj$error

---

GDS1615 data introduced in Burczynski et al. (2012).

Description

The dataset is a subset of the dataset available on Gene Expression Omnibus with the accession number GDS1615. The original dataset contains 22283 gene expression levels and the disease states of the observed subjects. In Mai, Yang and Zou, the dimension of the original dataset was first reduced to 127 by F-test screening.

Usage

data(GDS1615)

Value

This data frame contains the following:

x	Gene expression levels.
y	Disease state that is coded as 1,2,3. 1: normal; 2: ulcerative colitis; 3: Crohn's disease.

References

M. E. Burczynski, R. L Peterson, N. C. Twine, K. A. Zuberek, B. J. Brodeur, L. Casciotti, V. Maganti, P. S. Reddy, A. Strahs, F. Immermann, W. Spinelli, U. Schwertschlag, A. M. Slager, M. M. Cotreau, and A. J. Dorner. (2012), "Molecular classification of crohn's disease and ulcerative colitis patients using transcriptional profiles in peripheral blood mononuclear cells". Journal of Molecular Diagnostics, 8:51–61.

Mai, Q., Zou, H. and Yuan, M. (2012), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrica, 99, 29-42.

Examples

data(GDS1615)

---

Direct sparse discriminant analysis

Description

Transform the predictors to achieve normality.

Usage

getnorm(x, y, type="pooled")

Arguments

x	an n dimensional vector containing n observations for one predictor.
y	an n-dimensional vector containing the class labels.
type	The type of estimator. Two estimators were proposed in Mai & Zou (2015), the naive estimator and the pooled estimator. The function getnorm() uses the naive estimator if type="naive", and it uses the pooled estimator if type="pooled". The default is "pooled". When the naive estimator is used, it is recommended to label the class with more samples as Class 0.

Value

x.norm	Transformed x.
f0	The transformation computed based on observations from Class 0. Not applicable if type="naive".
f1	The transformation computed based on observations from Class 1. Not applicable if type="naive".
mu.hat	The sample mean for transformed x from Class 1.
transform	The transformation that was actually used to transform x.

References

Mai, Q., Zou, H. and Yuan, M. (2013). A direct approach to sparse discriminant analysis in ultra-high dimensions. Biometrika, 99, 29-42.

Mai, Q. and Zou, H. (2015). Sparse semiparametric discriminant analysis. Journal of Multivariate Analysis, 135, 175-188.

Examples

data(GDS1615)   ##load the prostate data
x<-GDS1615$x
y<-GDS1615$y
x<-exp(x[which(y<3),])
y<-y[which(y<3)]

n<-length(y)
n1<-sum(y==1)
n2<-n-n1
n1.test<-round(n1/2)
n2.test<-round(n2/2)
n.test<-n1.test+n2.test
n.train<-n-n.test
id.test<-c(sample(which(y==1),n1.test),sample(which(y==2),n2.test))

p<-ncol(x)
x.train<-x[-id.test,]
y.train<-y[-id.test]
x.test<-x[id.test,]
y.test<-y[id.test]

 x.norm<-matrix(0,n.train,p)
  x.test.norm<-matrix(0,n.test,p)
  for(i in 1:p){
    obj.norm<-getnorm(x.train[,i],y.train)
    x.norm[,i]<-obj.norm$x.norm
    x.test.norm[,i]<-obj.norm$transform(x.test[,i])
  }
    
  obj<-dsda.all(x.norm,y.train,x.test.norm,y.test)

---

Fits a regularization path of Sparse Discriminant Analysis and predicts

Description

Fits a regularization path of Sparse Discriminant Analysis at a sequence of regularization parameters lambda. Performs prediction when testing data is provided. The msda function solves classification problem by fitting a sparse discriminant analysis model. When covariates are provided, the function will first make adjustment on the training data. It provides three models: binary for fitting DSDA model to solve binary classification problems, multi.original and multi.modified for fitting MSDA model to solve multi-class classification problems. multi.original runs faster for small dimension case but the computation ability is limited to a relatively large dimension. multi.modified has no such limitation and works in ultra-high dimensions. User can specify method by argument or use the default settings.

Usage

msda(x, z=NULL, y, testx=NULL,testz=NULL, model = NULL, lambda = NULL, 
 standardize=FALSE, alpha=1, nlambda = 100, 
 lambda.factor = ifelse((nobs - nclass)<= nvars, 0.2, 1e-03), dfmax = nobs, 
 pmax = min(dfmax * 2 + 20, nvars), pf = rep(1, nvars), eps = 1e-04, 
 maxit = 1e+06, sml = 1e-06, verbose = FALSE, perturb = NULL)

Arguments

x	Input matrix of predictors. x is of dimension $N \times p$; each row is an observation vector.
z	Input covariate matrix of dimension $N \times q$, where $q<N$. z can be omitted if covariate is absent.
y	Class labl. This argument should be a factor for classification. For model='binary', y should be a binary variable with values 1 and 2. For model='multi.original' or 'multi.modified', y should be a multi-class variable starting from 1.
testx	Input testing matrix. Each row is a test case. When testx is not provided, the function will only fit the model and return the classifier. When testx is provided, the function will predict response on testx as well.
testz	Input testing covariate matrix. Can be omitted if covariate is absent. However, training covariates z and testing covariates testz must be provided or not at the same time.
model	Method type. The model argument can be one of 'binary', 'multi.original', 'multi.modified' and the default is NULL. The function supports fitting DSDA and MSDA models by specifying method type. Without specification, the function will automatically choose one of the methods. If the response variable is binary, the function will fit a DSDA model. If the response variable is multi-class, the function will fit an original MSDA model for dimension $p<=2000$ and a modified MSDA model for dimension $p>2000$.
lambda	A user supplied lambda sequence. Typically, by leaving this option unspecified users can have the program compute its own lambda sequence based on nlambda and lambda.factor. Supplying a value of lambda overrides this. It is better to supply a decreasing sequence of lambda values than a single (small) value, if not, the program will sort user-defined lambda sequence in decreasing order automatically.
standardize	A logic object indicating whether x should be standardized before performing DSDA. Default is FALSE. This argument is only valid for model = 'binary'.
alpha	The elasticnet mixing parameter, the same as in glmnet. Default is alpha=1 so that the lasso penalty is used in DSDA. This argument is only valid for model = 'binary'.
nlambda	The number of tuning values in sequence lambda. If users do not specify lambda values, the package will generate a solution path containing nlambda many tuning values of lambda. Default is 100 for model = 'multi.original' and 50 for model = 'multi.modified'.
lambda.factor	The factor for getting the minimal lambda in lambda sequence, where min(lambda) = lambda.factor * max(lambda). max(lambda) is the smallest value of lambda for which all coefficients are zero. The default depends on $p$ (the number of predictors) and its relationship with $N$ (the number of rows in the matrix of predictors). For Original MSDA, if $N > p$, the default is 0.0001, close to zero. If $N<p$, the default is 0.2. For Modified MSDA, if $p\le 5000$, the default is 0.2. If $5000<p\le 30000$, the default is 0.4. If $p>30000$, the default is 0.5. A very small value of lambda.factor will lead to a saturated fit. It takes no effect if there is user-defined lambda sequence. This argument is only valid for multi.original and multi.modified.
dfmax	The maximum number of selected variables in the model. Default is the number of observations N. This argument is only valid for multi.original and multi.modified.
pmax	The maximum number of potential selected variables during iteration. In middle step, the algorithm can select at most pmax variables and then shrink part of them such that the nubmer of final selected variables is less than dfmax. Default is $\min(dfmax\times 2+20, N)$.
pf	L1 penalty factor of length $p$. Separate L1 penalty weights can be applied to each coefficient of $\theta$ to allow differential L1 shrinkage. Can be 0 for some variables, which implies no L1 shrinkage, and results in that variable always being included in the model. Default is 1 for all variables (and implicitly infinity for variables listed in exclude). This argument is only valid for multi.original and multi.modified.
eps	Convergence threshold for coordinate descent. Each inner coordinate descent loop continues until the relative change in any coefficient. Defaults value is 1e-4.
maxit	Maximum number of outer-loop iterations allowed at fixed lambda value. Default is 1e6. If models do not converge, consider increasing maxit. This argument is only valid for multi.original and multi.modified.
sml	Threshold for ratio of loss function change after each iteration to old loss function value. Default is 1e-06. This argument is only valid for multi.original and multi.modified.
verbose	Whether to print out computation progress. The default is FALSE. This argument is only valid for multi.original and multi.modified.
perturb	A scalar number. If it is specified, the number will be added to each diagonal element of the covariance matrix as perturbation. The default is NULL. This argument is only valid for multi.original and multi.modified.

Details

The msda function fits a linear discriminant analysis model for vector $X$ as follows:

$\mathbf{X}|Y=k\sim N(\boldsymbol{\mu}_k,\boldsymbol{\Sigma}).$

The categorical response is predicted from the Bayes rule:

$\widehat{Y}=\arg\max_{k=1,\cdots,K}{(\mathbf{X}-\frac{\boldsymbol{\mu}_k}{2})^T\boldsymbol{\beta}_k+\log\pi_k}.$

The parameter model specifies which method to use in estimating $\boldsymbol{\beta}$. Users can use binary for binary problems and binary and multi.modified for multi-class problems. In multi.original, the algorithm first computes and stores $\boldsymbol{\Sigma}$, while it doesn't compute or store the entire covariance matrix in multi.modified. Since the algorithm is element-wise based, multi.modified computes each element of covariance matrix when needed. Therefore, multi.original is faster for low dimension but multi.modified can fit model for a much higher dimension case.

Note that for computing speed reason, if models are not converging or running slow, consider increasing eps and sml, or decreasing nlambda, or increasing lambda.factor before increasing maxit. Users can also reduce dfmax to limit the maximum number of variables in the model.

The arguments list out all parameters in the three models, but not all of them are necessary in applying one of the methods. See the specific explaination of each argument for more detail. Meanwhile, the output of DSDA model only includes beta and lambda.

Value

An object with S3 class dsda or msda.original and msda.modified.

beta	Output variable coefficients for each lambda, which is the estimation of $\boldsymbol{\beta}$ in the Bayes rule. beta is a list of length being the number of lambdas. Each element of beta is a matrix of dimension $nvars\times (nclass-1)$. For model = 'dsda', beta is a vector of length $nvars+1$, where the first element is intercept.
df	The number of nonzero coefficients for each value of lambda.
obj	The fitted value of the objective function for each value of lambda.
dim	Dimension of each coefficient matrix.
lambda	The actual lambda sequence used. The user specified sequence or automatically generated sequence could be truncated by constraints on dfmax and pmax.
x	The input matrix of predictors for training.
y	Class label in training data.
npasses	Total number of iterations (the most inner loop) summed over all lambda values
jerr	Error flag, for warnings and errors, 0 if no error.
sigma	Estimated sigma matrix. This argument is only available in object msda.original.
delta	Estimated delta matrix. delta[k] = mu[k]-mu[1].
mu	Estimated mu vector.
prior	Prior probability that y belong to class k, estimated by mean(y that belong to k).
call	The call that produced this object
pred	Predicted categorical response for each value in sequence lambda when testx is provided.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q., Zou, H. and Yuan, M. (2012), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrica, 99, 29-42.

Mai, Q., Yang, Y., and Zou, H. (2017), "Multiclass sparse discriminant analysis." Statistica Sinica, in press.

URL: https://github.com/emeryyi/msda

See Also

cv.msda, predict.msda

Examples

data(GDS1615)
x<-GDS1615$x
y<-GDS1615$y
obj <- msda(x = x, y = y)

---

Predict categorical responses for matrix/tensor data.

Description

Predict categorical responses on new matrix/tensor data given the fitted CATCH model input.

Usage

## S3 method for class 'catch'
predict(object, newx, z = NULL, ztest = NULL, gamma = NULL,...)

Arguments

object	Input catchobj class object as fitted model.
newx	Input adjusted testing tensor or matrix list. Each element of the list is a tensor. The tensor should of the same dimension as training data.
z	Input training covariates matrix. z can be omitted if there is no covariate.
ztest	Input testing covariates matrix. ztest can be omitted if there is no covariate.
gamma	Coefficients of covariates obtained from adjten. gamma is NULL if there is no covariate.
...	Other arguments that can be passed to predict.

Details

The function fits LDA model on selected discriminant vectors. Call predict or predict.catch to perform predictions.

There are two ways to make predictions. One way is to directly predict at the same time as fitting model by catch since predict.catch has already been embedded in catch and it will predicts response when testing data is provided. The other way is to first use adjten to adjuste tensor and catch to fit model. predict.catch will take the input adjusted tensor list newx, covariate coefficient gamma from adjten and the fitted model from catch to perform prediction. The prediction is identical to providing catch testing data.

Value

Predicted response of newx for each lambda in model object.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Pan, Y., Mai, Q., and Zhang, X. (2018) Covariate-Adjusted Tensor Classification in High-Dimensions, arXiv:1805.04421.

See Also

catch, adjten

Examples

#generate training data
n <- 20
p <- 4
k <- 2
nvars <- p*p*p
x <- array(list(),n)
vec_x <- matrix(rnorm(n*nvars),nrow=n,ncol=nvars)
vec_x[1:10,] <- vec_x[1:10,]+2
z <- matrix(rnorm(n*2),nrow=n,ncol=2)
z[1:10,] <- z[1:10,]+0.5
y <- c(rep(1,10),rep(2,10))
for (i in 1:n){
  x[[i]] <- array(vec_x[i,],dim=c(p,p,p))
}

#generate testing data
newx <- array(list(),n)
vec_newx <- matrix(rnorm(n*nvars),nrow=n,ncol=nvars)
vec_newx[1:10,] <- vec_newx[1:10,]+2
newz <- matrix(rnorm(n*2),nrow=n,ncol=2)
newz[1:10,] <- newz[1:10,]+0.5
for (i in 1:n){
  newx[[i]] <- array(vec_newx[i,],dim=c(p,p,p))
}

#Make adjustment and fit model
obj <- adjten(x, z, y, newx, newz)
fit <- catch(x, z, y)
#Predict
pred <- predict(fit, obj$testxres, z, newz, obj$gamma)

#The adjusting, fitting model and predicting step can also be completed
#by one command.
pred <- catch(x, z, y, newx, newz)$pred

---

Prediction for direct sparse discriminant analysis

Description

Predict the class labels by direct sparse discriminant analysis.

Usage

## S3 method for class 'dsda'
predict(object, newx, z=NULL, ztest=NULL, gamma=NULL,...)

Arguments

object	An object returned by dsda or msda with binary setting.
newx	An n by p matrix containing the predictors.
z	Input training covariates matrix. z can be omitted if there is no covariate.
ztest	Input testing covariates matrix. ztest can be omitted if there is no covariate.
gamma	Coefficients of covariates obtained from adjvec. gamma is NULL if there is no covariate.
...	Other arguments that can be passed to predict.

Value

pred	The the predicted class labels.

References

Mai, Q., Zou, H. and Yuan, M. (2013), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrika, 99, 29-42.

See Also

dsda, dsda.all, predict.msda

---

Predict categorical responses for vector data.

Description

Predict categorical responses on new vector data given the fitted DSDA/MSDA model input.

Usage

## S3 method for class 'msda'
predict(object, newx, z = NULL, ztest = NULL, gamma = NULL,...)

Arguments

object	Fitted model object from msda. The model object can be anyone of binary, multi.original and multi.modified.
newx	The matrix of new values for x at which predictions are to be made. If covariates exist, then newx should be adjusted matrix.
z	Input training covariates matrix. z can be omitted if there is no covariate.
ztest	Input testing covariates matrix. ztest can be omitted if there is no covariate.
gamma	Coefficients of covariates obtained from adjvec. gamma is NULL if there is no covariate.
...	Other arguments that can be passed to predict.

Details

The function fits LDA model on selected discriminant vectors. Call predict or predict.msda to perform prediction. When covariates exist, users could first call adjvec to make adjustment and obtain obtain gamma. The fitted model from msda should also takes adjusted vector as input. The newx in predict.msda shoudl be adjusted vector as well.

Value

Predicted class label(s) at the entire sequence of the penalty parameter lambda used to create the model.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q., Zou, H. and Yuan, M. (2012), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrica, 99, 29-42.

Mai, Q., Yang, Y., and Zou, H. (2017), "Multiclass sparse discriminant analysis." Statistica Sinica, in press.

Pan, Y., Mai, Q., and Zhang, X. (2018), "Covariate-Adjusted Tensor Classification in High-Dimensions." Journal of the American Statistical Association, accepted.

See Also

msda

Examples

data(GDS1615)
x<-GDS1615$x
y<-GDS1615$y
obj <- msda(x = x, y = y)
pred<-predict(obj,x)

---

Prediction for semiparametric sparse discriminant analysis

Description

Predict the class labels by semiparametric sparse discriminant analysis.

Usage

## S3 method for class 'SeSDA'
predict(object, x.test,...)

Arguments

object	An object returned by SeSDA.
x.test	An n by p matrix containing the predictors.
...	Other arguments that can be passed to predict.

Value

pred	The the predicted class labels.

References

Mai, Q., Zou, H. and Yuan, M. (2013), "A direct approach to sparse discriminant analysis in ultra-high dimensions." Biometrika, 99, 29-42.

See Also

dsda, SeSDA

---

Solution path for regularized optimal affine discriminant

Description

Compute the solution path for regularized optimal affine discriminant (ROAD).

Usage

ROAD(x,y,standardize=FALSE,lambda=NULL,eps=1e-7)

Arguments

x	Input matrix of predictors. x is of dimension $N \times p$; each row is an observation vector.
y	An n-dimensional vector containing the class labels. The classes have to be labeled as 1 and 2.
standardize	A logic object indicating whether x should be standardized before performing ROAD. Default is FALSE.
lambda	A sequence of lambda's. If lambda is missed, the function will automatically generates a sequence of lambda's to fit model.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Details

The function obtains the solution path of ROAD through dsda.

Value

beta	Output variable coefficients for each lambda.
lambda	The sequence of lambda's used in computing the solution path.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q. and Zou, H. (2013), "A note on the connection and equivalence of three sparse linear discriminant analysis methods." Technometrics, 55, 243-246.

Examples

data(GDS1615)   ##load the prostate data
    x<-GDS1615$x
    y<-GDS1615$y
    x=x[which(y<3),]
    y=y[which(y<3)]
    obj.path <- ROAD(x, y)

---

Solution path for semiparametric sparse discriminant analysis

Description

Compute the solution path for semiparametric sparse discriminant analysis.

Usage

SeSDA(x,y,standardize=FALSE,lambda=NULL,alpha=1,eps=1e-7)

Arguments

x	Input matrix of predictors. x is of dimension $N \times p$; each row is an observation vector.
y	An n-dimensional vector containing the class labels. The classes have to be labeled as 1 and 2.
standardize	A logic object indicating whether x should be standardized after transformation but before fitting classifier. Default is FALSE.
lambda	A sequence of lambda's. If lambda is missed or NULL, the function will automatically generates a sequence of lambda's to fit model.
alpha	The elasticnet mixing parameter, the same as in glmnet. Default is alpha=1 so that the lasso penalty is used.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Value

transform	The tranformation functions.
objdsda	A DSDA object fitted on transformed data.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q., Zou, H. and Yuan, M. (2013). A direct approach to sparse discriminant analysis in ultra-high dimensions. Biometrika, 99, 29-42.

Examples

data(GDS1615)   ##load the prostate data
  x<-GDS1615$x
  y<-GDS1615$y
  x=x[which(y<3),]
  y=y[which(y<3)]
  obj.path <- SeSDA(x,y)

---

Simulate data

Description

Simulate a binary data set with vector predictor.

Usage

sim.bi.vector(tesize = 100)

Arguments

tesize

Number of observations in testing data.

Details

The function simulates a data set with $p=500$. Response are binary.

Value

x	Simulated vector predictor.
testx	Simulated testing vector predictor.
y	Response corresponding to x.
testy	Response corresponding to testx.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

---

Simulate data

Description

Simulate a data set with tensor predictor and covariates.

Usage

sim.tensor.cov(tesize = 100)

Arguments

tesize

Number of observations in testing data.

Details

The function simulates a data set with $10\times 10\times 10$ tensor and covariate being a two-dimensional vector. Response are binary.

Value

x	Simulated tensor predictor.
z	Simulated covariate.
testx	Simulated testing tensor predictor.
testz	Simualted testing covariate.
vec_x	Vectorization of x.
vec_testx	Vectorization of testx.
y	Response corresponding to x and z.
testy	Response corresponding to testx and testz.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

---

Solution path for sparse discriminant analysis

Description

Compute the solution path for sparse optimal scoring (SOS).

Usage

SOS(x,y,standardize=FALSE,lambda=NULL,eps=1e-7)

Arguments

x	Input matrix of predictors. x is of dimension $N \times p$; each row is an observation vector.
y	An n-dimensional vector containing the class labels. The classes have to be labeled as 1 and 2.
standardize	A logic object indicating whether x should be standardized before performing SOS. Default is FALSE.
lambda	A sequence of lambda's. If lambda is missed, the function will automatically generates a sequence of lambda's to fit model.
eps	Convergence threshold for coordinate descent, the same as in glmnet. Default is 1e-7.

Details

The function obtains the solution path of sparse optimal scoring model through dsda.

Value

beta	Output variable coefficients for each lambda.
lambda	The sequence of lambda's used in computing the solution path.

Author(s)

Yuqing Pan, Qing Mai, Xin Zhang

References

Mai, Q. and Zou, H. (2013), "A note on the connection and equivalence of three sparse linear discriminant analysis methods." Technometrics, 55, 243-246.

Examples

data(GDS1615)   ##load the prostate data
    x<-GDS1615$x
    y<-GDS1615$y
    x=x[which(y<3),]
    y=y[which(y<3)]
    obj.path <- SOS(x, y)

---