kerndwd: solve Linear DWD and Kernel DWD

Description Usage Arguments Details Value Author(s) References See Also Examples

View source: R/kerndwd.R

Description

Fit the linear generalized distance weighted discrimination (DWD) model and the generalized DWD on Reproducing kernel Hilbert space. The solution path is computed at a grid of values of tuning parameter lambda.

Usage

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kerndwd(x, y, kern, lambda, qval=1, wt, eps=1e-05, maxit=1e+05)

Arguments

x

A numerical matrix with N rows and p columns for predictors.

y

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

kern

A kernel function; see dots.

lambda

A user supplied lambda sequence.

qval

The exponent index of the generalized DWD. Default value is 1.

wt

A vector of length n for weight factors. When wt is missing or wt=NULL, an unweighted DWD is fitted.

eps

The algorithm stops when (i.e. sum(j)(beta_new[j]-beta_old[j])^2 is less than eps, where j=0,…, p. Default value is 1e-5.

maxit

The maximum of iterations allowed. Default is 1e5.

Details

Suppose that the generalized DWD loss is V_q(u) = 1 - u if u <= q/(q+1) and (1/u)^q * q^q/(q+1)^{(q+1)} if u > q/(q+1). The value of λ, i.e., lambda, is user-specified.

In the linear case (kern is the inner product and N > p), the kerndwd fits a linear DWD by minimizing the L2 penalized DWD loss function,

(1/N) * sum_i [V_q(y_i(β_0 + X_i'β))] + λ β' β.

If a linear DWD is fitted when N < p, a kernel DWD with the linear kernel is actually solved. In such case, the coefficient β can be obtained from β = X'α.

In the kernel case, the kerndwd fits a kernel DWD by minimizing

(1/N) * sum_i [V_q(y_i(β_0 + K_i' α))] + λ α' K α,

where K is the kernel matrix and K_i is the ith row.

The weighted linear DWD and the weighted kernel DWD are formulated as follows,

(1/N) * sum_i [w_i * V_q(y_i(β_0 + X_i'β))] + λ β' β,

(1/N) * sum_i [w_i * V_q(y_i(β_0 + K_i' α))] + λ α' K α,

where w_i is the ith element of wt. The choice of weight factors can be seen in the reference below.

Value

An object with S3 class kerndwd.

alpha

A matrix of DWD coefficients at each lambda value. The dimension is (p+1)*length(lambda) in the linear case and (N+1)*length(lambda) in the kernel case.

lambda

The lambda sequence.

npass

Total number of MM iterations for all lambda values.

jerr

Warnings and errors; 0 if none.

info

A list including parameters of the loss function, eps, maxit, kern, and wt if a weight vector was used.

call

The call that produced this object.

Author(s)

Boxiang Wang and Hui Zou
Maintainer: Boxiang Wang [email protected]

References

Wang, B. and Zou, H. (2018) “Another Look at Distance Weighted Discrimination," Journal of Royal Statistical Society, Series B, 80(1), 177–198.
https://rss.onlinelibrary.wiley.com/doi/10.1111/rssb.12244
Karatzoglou, A., Smola, A., Hornik, K., and Zeileis, A. (2004) “kernlab – An S4 Package for Kernel Methods in R", Journal of Statistical Software, 11(9), 1–20.
http://www.jstatsoft.org/v11/i09/paper
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.
http://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://faculty.franklin.uga.edu/jyahn/sites/faculty.franklin.uga.edu.jyahn/files/DWD3.pdf
Qiao, X., Zhang, H., Liu, Y., Todd, M., Marron, J.S. (2010) “Weighted distance weighted discrimination and its asymptotic properties", Journal of the American Statistical Association, 105(489), 401–414.
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2996856/

See Also

predict.kerndwd, plot.kerndwd, and cv.kerndwd.

Examples

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data(BUPA)
# standardize the predictors
BUPA$X = scale(BUPA$X, center=TRUE, scale=TRUE)

# a grid of tuning parameters
lambda = 10^(seq(3, -3, length.out=10))

# fit a linear DWD
kern = vanilladot()
DWD_linear = kerndwd(BUPA$X, BUPA$y, kern,
  qval=1, lambda=lambda, eps=1e-5, maxit=1e5)

# fit a DWD using Gaussian kernel
kern = rbfdot(sigma=1)
DWD_Gaussian = kerndwd(BUPA$X, BUPA$y, kern,
  qval=1, lambda=lambda, eps=1e-5, maxit=1e5)

# fit a weighted kernel DWD
kern = rbfdot(sigma=1)
weights = c(1, 2)[factor(BUPA$y)]
DWD_wtGaussian = kerndwd(BUPA$X, BUPA$y, kern,
  qval=1, lambda=lambda, wt = weights, eps=1e-5, maxit=1e5)

kerndwd documentation built on April 22, 2018, 1:04 a.m.