Fits a regularization path for Multi-Class Sparse Discriminant Analysis

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Description

Fits a regularization path for Multi-Class Sparse Discriminant Analysis at a sequence of regularization parameters lambda.

Usage

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msda(x, y, nlambda = 100, 
lambda.factor = ifelse((nobs - nclass) <= nvars, 0.2, 0.001), 
lambda = NULL, 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

matrix of predictors, of dimension N*p; each row is an observation vector.

y

response variable. This argument should be a factor for classification.

nlambda

the number of lambda values - default is 100.

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 the relationship between N (the number of rows in the matrix of predictors) and p (the number of predictors). If N > p, the default is 0.0001, close to zero. If N<p, the default is 0.2. A very small value of lambda.factor will lead to a saturated fit. It takes no effect if there is user-defined lambda sequence.

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.

dfmax

limit the maximum number of variables in the model. Useful for very large p, if a partial path is desired. Default is n.

pmax

limit the maximum number of variables ever to be nonzero. For example once β enters the model, no matter how many times it exits or re-enters model through the path, it will be counted only once. Default is min(dfmax*1.2,p).

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).

eps

convergence threshold for coordinate descent. Each inner coordinate descent loop continues until the relative change in any coefficient. Defaults value is 1e-8.

maxit

maximum number of outer-loop iterations allowed at fixed lambda value. Default is 1e6. If models do not converge, consider increasing maxit.

sml
verbose

whether to print out computation progress. The default is FALSE.

perturb

a scalar number. If it is specified, the number will be added to each diagonal element of the sigma matrix as perturbation. The default is NULL.

Details

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.

Value

An object with S3 class msda.

theta

a list of length(lambda) for fitted coefficients theta, each one corresponding to one lambda value, each stored as a sparse matrix (dgCMatrix class, the standard class for sparse numeric matrices in the Matrix package.). To convert it into normal type matrix use as.matrix().

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 at each lambda.

lambda

the actual sequence of lambda values used.

x

matrix of predictors.

y

response variable.

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.

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

Author(s)

Qing Mai <mai@stat.fsu.edu>, Yi Yang <yiyang@umn.edu>, Hui Zou <hzou@stat.umn.edu>
Maintainer: Yi Yang <yiyang@umn.edu>

References

Mai, Q.*, Yang, Y.*, and Zou, H. (2014), "Multiclass Sparse Discriminant Analysis." Submitted to Journal of the American Statistical Association. (* co-first author)

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

See Also

cv.msda, predict.msda

Examples

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