Type: Package
Title: Sparse Distance Weighted Discrimination
Version: 1.0.5
Date: 2020-10-24
Author: Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang <boxiang-wang@uiowa.edu>
Description: Formulates a sparse distance weighted discrimination (SDWD) for high-dimensional classification and implements a very fast algorithm for computing its solution path with the L1, the elastic-net, and the adaptive elastic-net penalties. More details about the methodology SDWD is seen on Wang and Zou (2016) (<doi:10.1080/10618600.2015.1049700>).
Depends: Matrix
Imports: grDevices, graphics, stats, methods
License: GPL-2
Repository: CRAN
NeedsCompilation: yes
Packaged: 2020-10-24 07:00:42 UTC; boxiangw
Date/Publication: 2020-10-27 14:00:06 UTC

Sparse Distance Weighted Discrimination

Description

This package implements the generalized coordinate descent (GCD) algorithm to efficiently compute the solution path of the sparse distance weighted discrimination (DWD) at a given fine grid of regularization parameters. Sparse distance weighted discrimination is a high-dimensional margin-based classifier.

Details

Package: sdwd
Type: Package
Version: 1.0.3
Date: 2020-02-16
License: GPL-2

Suppose x is the predictors and y is the binary response. With a fixed value lambda2, the package produces the solution path of the sparse DWD over a grid of lambda values. The value of lambda2 can be further tuned by cross-validation.

The package sdwd contains five main functions:
sdwd
cv.sdwd
coef.sdwd
plot.sdwd
plot.cv.sdwd

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

Marron, J.S., Todd, M.J., Ahn, J. (2007) “Distance-Weighted Discrimination"", Journal of the American Statistical Association, 102(408), 1267–1271.
https://www.tandfonline.com/doi/abs/10.1198/016214507000001120

Tibshirani, Robert., Bien, J., Friedman, J.,Hastie, T.,Simon, N.,Taylor, J., and Tibshirani, Ryan. (2012) Strong Rules for Discarding Predictors in Lasso-type Problems, Journal of the Royal Statistical Society, Series B, 74(2), 245–266.
https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2011.01004.x

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

compute coefficients from a "cv.sdwd" object

Description

Computes coefficients at chosen values of lambda from the cv.sdwd object.

Usage

## S3 method for class 'cv.sdwd'
coef(object, s=c("lambda.1se", "lambda.min"),...)

Arguments

object

A fitted cv.sdwd object, obtained by conducting the cross-validation to the sparse DWD model.

s

Value(s) of the L1 tuning parameter lambda for computing coefficients. Default value is "lambda.1se", which represents the largest lambda value achieving the cross-validation error within one standard error of the minimum. An alternative value is "lambda.min", which is the lambda incurring the least cross-validation error. s can also be numeric, being taken as the value(s) to be used.

...

Other arguments that can be passed to sdwd.

Details

This function computes the coefficients at the values of lambda suggested by the cross-validation. This function is modified based on the coef.cv function from the glmnet and the gcdnet packages.

Value

The returned object depends on the choice of s and the ... argument passed on to the sdwd method.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

cv.sdwd and predict.cv.sdwd methods.

Examples

data(colon)
colon$x = colon$x[ , 1:100] # this example only uses the first 100 columns 
set.seed(1)
cv = cv.sdwd(colon$x, colon$y, lambda2=1, nfolds=5)
c1 = coef(cv, s="lambda.1se")

compute coefficients for the sparse DWD

Description

Computes the coefficients or returns the indices of nonzero coefficients at chosen values of lambda from a fitted sdwd object.

Usage

## S3 method for class 'sdwd'
coef(object, s=NULL, type=c("coefficients","nonzero"), ...)

Arguments

object

A fitted sdwd object.

s

Value(s) of the L1 tuning parameter lambda for computing coefficients. Default is the entire lambda sequence obtained by sdwd.

type

"coefficients" or "nonzero"? "coefficients" computes the coefficients at given values for s; "nonzero" returns a list of the indices of the nonzero coefficients for each value of s. Default is "coefficients".

...

Not used. Other arguments to predict.

Details

s is the new vector at which predictions are requested. If s is not in the lambda sequence used for fitting the model, the coef function will use linear interpolation to make predictions. The new values are interpolated using a fraction of coefficients from both left and right lambda indices. This function is modified based on the coef function from the gcdnet and the glmnet packages.

Value

Either the coefficients at the requested values of lambda, or a list of the indices of the nonzero coefficients for each lambda.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22
https://www.jstatsoft.org/v33/i01/paper

See Also

predict.sdwd

Examples

data(colon)
fit = sdwd(colon$x, colon$y, lambda2=1)
c1 = coef(fit, type="coef",s=c(0.1, 0.005))
c2 = coef(fit, type="nonzero")

simplified gene expression data from Alon et al. (1999)

Description

Gene expression data (2000 genes for 62 samples) from a DNA microarray experiments of colon tissue samples (Alon et al., 1999).

Usage

data(colon)

Details

This data set contains 62 colon tissue samples with 2000 gene expression levels. Among 62 samples, 40 are tumor tissues (coded 1) and 22 are normal tissues (coded -1).

Value

A list with the following elements:

x

A matrix of 2000 columns and 62 rows standing for 2000 gene expression levels and 62 colon tissue samples. Each row corresponds to a patient.

y

A numeric vector of length 62 representing the tissue type (1 for tumor; -1 for normal).

Source

The data were introduced in Alon et al. (1999).

References

Alon, U., Barkai, N., Notterman, D.A., Gish, K., Ybarra, S., Mack, D., and Levine, A.J. (1999). “Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissues probed by oligonucleotide arrays”, Proceedings of the National Academy of Sciences, 96(12), 6745–6750.

Examples

# load sdwd library
library(sdwd)

# load data set
data(colon)

# how many samples and how many predictors?
dim(colon$x)

# how many samples of class -1 and 1 respectively?
sum(colon$y == -1)
sum(colon$y == 1)

cross-validation for the sparse DWD

Description

Conducts a k-fold cross-validation for sdwd and returns the suggested values of the L1 parameter lambda.

Usage

cv.sdwd(x, y, lambda = NULL, pred.loss = c("misclass", "loss"), nfolds = 5, foldid, ...)

Arguments

x

A matrix of predictors, i.e., the x matrix used in sdwd.

y

A vector of binary class labels, i.e., the y used in sdwd.

lambda

Default is NULL, and the sequence generated by sdwd is used. User can also provide a new lambda sequence to use in cross-validation.

pred.loss

misclass for classification error, loss for DWD loss.

nfolds

The number of folds. Default value is 5. The allowable range is from 3 to the sample size. Larger nfolds needs more timing.

foldid

An optional vector with values between 1 and nfold, representing the folder indices for each observation. If supplied, nfold can be missing.

...

Other arguments that can be passed to sdwd.

Details

This function runs sdwd to the sparse DWD by excluding every fold alternatively, and then computes the mean cross-validation error and the standard deviation. This function is modified based on the cv function from the gcdnet and the glmnet packages.

Value

A cv.sdwd object is returned, which includes the cross-validation fit.

lambda

The lambda sequence used in sdwd.

cvm

A vector of length length(lambda) for the mean cross-validated error.

cvsd

A vector of length length(lambda) for estimates of standard error of cvm.

cvupper

The upper curve: cvm + cvsd.

cvlower

The lower curve: cvm - cvsd.

nzero

Numbers of non-zero coefficients at each lambda.

name

“Mis-classification error", for plotting purposes.

sdwd.fit

A fitted sdwd object using the full data.

lambda.min

The lambda incurring the minimum cross validation error cvm.

lambda.1se

The largest value of lambda such that error is within one standard error of the minimum.

cv.min

The minimum cross-validation error.

cv.1se

The cross-validation error associated with lambda.1se.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

sdwd, plot.cv.sdwd, predict.cv.sdwd, and coef.cv.sdwd methods.

Examples

data(colon)
colon$x = colon$x[ , 1:100] # this example only uses the first 100 columns 
n = nrow(colon$x)
set.seed(1)
id = sample(n, trunc(n/3))
cvfit = cv.sdwd(colon$x[-id, ], colon$y[-id], lambda2=1, nfolds=5)
plot(cvfit)
predict(cvfit, newx=colon$x[id, ], s="lambda.min")

plot the cross-validation curve of the sparse DWD

Description

Plots the cross-validation curve against a function of lambda values. The function also provides the upper and lower standard deviation curves.

Usage

## S3 method for class 'cv.sdwd'
plot(x, sign.lambda, ...)

Arguments

x

A fitted cv.sdwd object.

sign.lambda

Whether to plot against log(lambda) (default) or its negative if sign.lambda=-1.

...

Other graphical parameters to plot.

Details

This function depicts the cross-validation curves. This function is modified based on the plot.cv function from the glmnet and the gcdnet packages.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

cv.sdwd.

Examples

data(colon)
colon$x = colon$x[ , 1:100] # this example only uses the first 100 columns 
set.seed(1)
cv = cv.sdwd(colon$x, colon$y, lambda2=1, nfolds=5)
plot(cv)

plot coefficients for the sparse DWD

Description

Plots the solution paths for a fitted sdwd object.

Usage

## S3 method for class 'sdwd'
plot(x, xvar=c("norm", "lambda"), color=FALSE, label=FALSE, ...)

Arguments

x

A fitted sdwd model.

xvar

Specifies the X-axis. If xvar == "norm", plots against the L1-norm of the coefficients; if xvar == "lambda" against the log-lambda sequence.

color

If TRUE, plots the curves with rainbow colors; otherwise, with gray colors (default).

label

If TRUE, labels the curves with variable sequence numbers. Default is FALSE.

...

Other graphical parameters to plot.

Details

Plots the solution paths as a coefficient profile plot. This function is modified based on the plot function from the gcdnet and the glmnet packages.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

print.sdwd, predict.sdwd, coef.sdwd, plot.sdwd, and cv.sdwd.

Examples

data(colon)
fit = sdwd(colon$x, colon$y)
par(mfrow=c(1,3))
# plots against the L1-norm of the coefficients
plot(fit) 
# plots against the log-lambda sequence
plot(fit, xvar="lambda", label=TRUE)
# plots with colors
plot(fit, color=TRUE)

make predictions from a "cv.sdwd" object

Description

This function predicts the class labels of new observations by the sparse DWD at the lambda values suggested by cv.sdwd.

Usage

## S3 method for class 'cv.sdwd'
predict(object, newx, s=c("lambda.1se","lambda.min"),...)

Arguments

object

A fitted cv.sdwd object.

newx

A matrix of new values for x at which predictions are to be made. Must be a matrix. See documentation for predict.sdwd.

s

Value(s) of the L1 tuning parameter lambda for making predictions. Default is the s="lambda.1se" saved on the cv.sdwd object. An alternative choice is s="lambda.min". s can also be numeric, being taken as the value(s) to be used.

...

Not used. Other arguments to predict.

Details

This function uses the cross-validation results to making predictions. This function is modified based on the predict.cv function from the glmnet and the gcdnet packages.

Value

Predicted class labels or fitted values, depending on the choice of s and the ... argument passed on to the sdwd method.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

cv.sdwd, and coef.cv.sdwd methods.

Examples

data(colon)
colon$x = colon$x[ , 1:100] # this example only uses the first 100 columns 
set.seed(1)
cv = cv.sdwd(colon$x, colon$y, lambda2=1, nfolds=5)
predict(cv$sdwd.fit, newx=colon$x[2:5, ], 
  s=cv$lambda.1se, type="class")

make predictions for the sparse DWD

Description

This function predicts the binary class labels or the fitted values of an sdwd object.

Usage

## S3 method for class 'sdwd'
predict(object, newx, s=NULL, type=c("class", "link"), ...)

Arguments

object

A fitted sdwd object.

newx

A matrix of new values for x at which predictions are to be made. We note that newx must be a matrix, predict function does not accept a vector or other formats of newx.

s

Value(s) of the L1 tuning parameter lambda for computing coefficients. Default is the entire lambda sequence obtained by sdwd.

type

"class" or "link"? "class" produces the predicted binary class labels."link" returns the fitted values. Default is "class".

...

Not used. Other arguments to predict.

Details

s stands for the new lambda values for making predictions. If s is not in the original lambda sequence generated by sdwd, the predict.sdwd function will use linear interpolation by using a fraction of predicted values from the lambda values in the original sequence adjacent to the s to make predictions. The predict.sdwd function is modified based on the predict function from the glmnet and the gcdnet packages.

Value

Returns either the predicted class labels or the fitted values, depending on the choice of type.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

coef.sdwd

Examples

data(colon)
fit = sdwd(colon$x, colon$y, lambda2=1)
print(predict(fit ,type="class",newx=colon$x[2:5,]))

print an sdwd object

Description

Print a summary of the sdwd solution paths.

Usage

## S3 method for class 'sdwd'
print(x, digits=max(3, getOption("digits") - 3), ...)

Arguments

x

A fitted sdwd object.

digits

Specify the significant digits.

...

Additional print arguments.

Details

This function prints a two-column matrix with columns Df and Lambda, where the Df column exhibits the number of nonzero coefficients and the Lambda column displays the corresponding lambda value. This function is modified based on the print function from the gcdnet and the glmnet packages.

Value

A two-column matrix with one column of the number of nonzero coefficients and a second column of lambda values.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

See Also

print.sdwd, predict.sdwd, coef.sdwd, plot.sdwd, and cv.sdwd.

Examples

data(colon)
fit = sdwd(colon$x, colon$y)
print(fit)

fit the sparse DWD

Description

Fits the sparse distance weighted discrimination (SDWD) model with imposing L1, elastic-net, or adaptive elastic-net penalties. The solution path is computed at a grid of values of tuning parameter lambda. This function is modified based on the glmnet and the gcdnet packages.

Usage

sdwd(x, y, nlambda=100, 
     lambda.factor=ifelse(nobs < nvars, 0.01, 1e-04), 
     lambda=NULL, lambda2=0, pf=rep(1, nvars), 
     pf2=rep(1, nvars), exclude, dfmax=nvars + 1, 
     pmax=min(dfmax * 1.2, nvars), standardize=TRUE, 
     eps=1e-8, maxit=1e6, strong=TRUE)

Arguments

x

A matrix with N rows and p columns for predictors.

y

A vector of length p for binary responses. The element of y is either -1 or 1.

nlambda

The number of lambda values, i.e., length of the lambda sequence. Default is 100.

lambda.factor

The ratio of the smallest to the largest lambda in the sequence: lambda.factor = min(lambda) / max(lambda). max(lambda) is the least lambda to make all coefficients to be zero. The default value of lambda.factor is 0.0001 if N >= p or 0.01 if N < p. Takes no effect when user specifies a lambda sequence.

lambda

An optional user-supplied lambda sequence. If lambda = NULL (default), the program computes its own lambda sequence based on nlambda and lambda.factor; otherwise, the program uses the user-specified one. Since the program will automatically sort user-defined lambda sequence in decreasing order, it is better to supply a decreasing sequence.

lambda2

The L2 tuning parameter \lambda_2.

pf

A vector of length p representing the L1 penalty weights to each coefficient of \beta for adaptive L1 or adaptive elastic net. pf can be 0 for some predictor(s), leading to including the predictor(s) all the time. One suggested choice of pf is {(\beta + 1/n)}^{-1}, where n is the sample size and \beta is the coefficents obtained by L1 DWD or enet DWD. Default is 1 for all predictors (and infinity if some predictors are listed in exclude).

pf2

A vector of length p for L2 penalty factor for adaptive L1 or adaptive elastic net. To allow different L2 shrinkage, user can set pf2 to be different L2 penalty weights for each coefficient of \beta. pf2 can be 0 for some variables, indicating no L2 shrinkage. Default is 1 for all predictors.

exclude

Whether to exclude some predictors from the model. This is equivalent to adopting an infinite penalty factor when excluding some predictor. Default is none.

dfmax

Restricts at most how many predictors can be incorporated in the model. Default is p+1. This restriction is helpful when p is large, provided that a partial path is acceptable.

pmax

Restricts the maximum number of variables ever to be nonzero; e.g, once some \beta enters the model, it counts once. The count will not change when the \beta exits or re-enters the model. Default is min(dfmax*1.2,p).

standardize

Whether to standardize the data. If TRUE, sdwd normalizes the predictors such that each column has sum squares\sum^N_{i=1}x_{ij}^2/N=1 of one. Note that x is always centered (i.e. \sum^N_{i=1}x_{ij}=0) no matter standardize is TRUE or FALSE. sdwd always returns coefficient beta on the original scale. Default value is TRUE.

eps

The algorithm stops when (i.e. 4\max_j(\beta_j^{new}-\beta_j^{old})^2 is less than eps, where j=0,\ldots, p. Defaults value is 1e-8.

maxit

Restricts how many outer-loop iterations are allowed. Default is 1e6. Consider increasing maxit when the algorithm does not converge.

strong

If TRUE, adopts the strong rule to accelerate the algorithm.

Details

The sdwd minimizes the sparse penalized DWD loss function,

L(y, X, \beta)/N + \lambda_1||\beta||_1 + 0.5\lambda_2||\beta||_2^2,

where L(u)=1-u if u \le 1/2, 1/(4u) if u > 1/2 is the DWD loss. The value of lambda2 is user-specified.

To use the L1 penalty (lasso), set lambda2=0. To use the elastic net, set lambda2 as nonzero. To use the adaptive L1, set lambda2=0 and specify pf and pf2. To use the adaptive elastic net, set lambda2 as nonzero and specify pf and pf2 as well.

When the algorithm do not converge or run slow, consider increasing eps, decreasing nlambda, or increasing lambda.factor before increasing maxit.

Value

An object with S3 class sdwd.

b0

A vector of length length(lambda) representing the intercept at each lambda value.

beta

A matrix of dimension p*length(lambda) representing the coefficients at each lambda value. The matrix is stored as a sparse matrix (Matrix package). To convert it into normal type matrix use as.matrix().

df

The number of nonzero coefficients at each lambda.

dim

The dimension of coefficient matrix, i.e., p*length(lambda).

lambda

The lambda sequence that was actually used.

npasses

Total number of iterations for all lambda values.

jerr

Warnings and errors; 0 if no error.

call

The call that produced this object.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent", Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper

Marron, J.S., Todd, M.J., and Ahn, J. (2007) “Distance-Weighted Discrimination", Journal of the American Statistical Association, 102(408), 1267–1271.
https://www.tandfonline.com/doi/abs/10.1198/016214507000001120

Tibshirani, Robert., Bien, J., Friedman, J.,Hastie, T.,Simon, N.,Taylor, J., and Tibshirani, Ryan. (2012) Strong Rules for Discarding Predictors in Lasso-type Problems, Journal of the Royal Statistical Society, Series B, 74(2), 245–266.
https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9868.2011.01004.x

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

See Also

print.sdwd, predict.sdwd, coef.sdwd, plot.sdwd, and cv.sdwd.

Examples

# load the data
data(colon)
# fit the elastic-net penalized DWD with lambda2=1
fit = sdwd(colon$x, colon$y, lambda2=1)
print(fit)
# coefficients at some lambda value
c1 = coef(fit, s=0.005)
# make predictions
predict(fit, newx=colon$x[1:10, ], s=c(0.01, 0.005))

internal sdwd functions

Description

Internal sdwd functions.

Usage

cv.sdwdNET(outlist, lambda, x, y, foldid, pred.loss)
cvcompute(mat, foldid, nlams)
err(n, maxit, pmax)
error.bars(x, upper, lower, width=0.02, ...)
getmin(lambda, cvm, cvsd)
getoutput(fit, maxit, pmax, nvars, vnames)
lambda.interp(lambda, s)
lamfix(lam)
nonzero(beta, bystep=FALSE)
zeromat(nvars, nalam, vnames, stepnames)

Details

These internal functions are not intended for use by users. coef.sdwdNET computes the coefficient of the sdwd object. cv.sdwdNET does cross-validation for the sdwd object. cvcompute computes the mean and the standard deviation of the cross-validation error. err obtains the error message from fortran code. error.bars helps to plot the cross-validation error curve. getmin addresses the best lambda through the cross-validation either using or not using the one-standard-deviation rule. getoutput organizes the output of the sdwd object. lambda.interp conducts the linear interpolation of the lambdas values to obtain the coefficients at new lambda values. Note the obtained coefficients are not the exact values. lamfix fixes the largest lambda value in the lambda sequence. nonzero and zeromat organize the nonzero coefficients. Most of the aforementioned functions are modified or directly copied from the gcdnet and the glmnet packages.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang boxiang-wang@uiowa.edu

References

Wang, B. and Zou, H. (2016) “Sparse Distance Weighted Discrimination", Journal of Computational and Graphical Statistics, 25(3), 826–838.
https://www.tandfonline.com/doi/full/10.1080/10618600.2015.1049700

Yang, Y. and Zou, H. (2013) “An Efficient Algorithm for Computing the HHSVM and Its Generalizations", Journal of Computational and Graphical Statistics, 22(2), 396–415.
https://www.tandfonline.com/doi/full/10.1080/10618600.2012.680324

Friedman, J., Hastie, T., and Tibshirani, R. (2010), "Regularization paths for generalized linear models via coordinate descent," Journal of Statistical Software, 33(1), 1–22.
https://www.jstatsoft.org/v33/i01/paper