minQuad: minQuad

In aucm: AUC Maximization

Description

minimizes the objective function 0.5a'Qa + b'a with respect to "a" subject to 0 <= a <= C, or to the constraints lower <= a <= upper, v <= Ax <= v + r.

Usage

       
minQuad(H,b,C = 1.0,n1=0,n2=0, 
    mem.efficient = FALSE,alpha = NULL,
    lower = NULL,upper = NULL,mat.constr = NULL, lhs.constr = NULL,rhs.constr = NULL,
    control = list(DUP = TRUE,maxit = 1e4, tol = 1e-04, 
        verbose = FALSE, ret.ws = FALSE,ret.data = FALSE,
        rank = 0,
        method = c("default","tron","loqo","exhaustive","x"),
        optim.control = list(),
        q = 2, 
        ws =  c("v","v2","greedy","rv2wg","rvwg","rv","rv2")
    )
) 

Arguments

H	A symmetric matrix whose typeof returns "double" of dimension (n x n). If mem.efficient = FALSE n = n1*n2 matches the length of the vector 'b' else n = n1 + n2, see details, defaults to NULL.
b	a numeric vector of length 'n' whose typeof returns "double".
C	a numeric variable whose typeof returns "double", defaults to 1.0 . It is the upper bound on alpha's, where the lower bound is 0.0 .
n1,n2	integer variables giving the specific values for n1 = #{diseased}, n2 = #{non-diseased}

subjects if mem.efficient = TRUE.

mem.efficient	logical, if FALSE then 'H' is represented by the (n1n2 x n1n2) matrix 'Q' else by the (n1+n2 x n1+n2) matrix 'K', defaults to FALSE.
alpha	a length-n1n2 vector vector of initial values for "alpha", whose typeof returns "double", defaults to NULL, in which case it is set to 0.5C.
control	a list with control parameters. See control.minQuad
mat.constr	m x n constraint matrix for loqo optimizer
lhs.constr	numeric of length 'm', the left hand side constraints for loqo optimizer
rhs.constr	numeric of length 'm', theleft hand side for constraints for loqo optimizer
lower	numeric of length 'n', the lower bounds on primal valriables for loqo optimizer
upper	numeric of length 'n', the upper bounds on primal valriables for loqo optimizer

Details

The function minQuad passes its arguments by "reference" via the .C function call to C if DUP = FALSE to avoid copying the large matrix "Q". When 'H' = 'Q', 'Q' is a symmetric matrix and should have numeric type "double", be of type "matrix" not of "data.frame": is.matrix(.) should return "TRUE". We do not make an extra copy by tranposing Q but access the 'flattened' vector in C directly since 'Q' is symmetric. When 'mem.efficient' = TRUE 'H' = K_{n1+n2 x n1+n2} and may be obtained by the function getK. 'K' is relevant to AUC estimation, see rauc for more details. The "ws" argument sets the type of strategy to select the working set. Denote the two sets of violators as V0 = {1 if (a[p] > 0.0),(df[p] > 0.0), 0 ow.}, VC = {1 if (a[p] < C),(df[p] < 0.0), 0 ow.} where "df[a]" stands for the gradient of the objective function at 'a'. "greedy" selects the extremes pairs (max,min) from the two sets (M,m) where M = {-df[i] , a[i] < C} and m = {-df[j] | a[j] > 0}. "v" selects from violators V = V0 U VC ranked by |df|. "v2" selects separately from V0 and VC separately, ranked by |df|. "rv" selects without replacement (WOR) from all violators. "rvwg" selects WOR from all violators V with probability ~ |df[V]|. "rv2wg" selects WOR from the two sets of violators V0 and VC with probability ~ |df[V]|. Three methods are available, "tron","hideo" and "exhaustive". Optimizer 'x' is a slightly faster implementation of the exhaustive method, whereas 'default' is a fast implementation for q = 2 only. The "exhaustive" method should probably not be used beyond working set size q = 8,10 on most computers. The 'loqo' optimizer accepts constraints of the form v <= A*x <= v + r, lower <= x <= upper. v = lhs.constr and r = rhs.constr - lhs.constr . The entries in 'v','r' and 'A' must be finite. When verbose is TRUE, each DCA iteration prints one line. Delta means, for the linear kernel, max(abs((beta.new-beta.init)/beta.init)), and for the nonlinear kernel the difference in penalized RAUC loss. epsilon is the KKT criterion of minQuad.

Value

A list with the following elements:

convergence	0 if converged, 1 if maximum iteration is reached.
alpha	estimated vector of coefficients.
value	value of the objective function.
iterations	number of iterations until convergence or "maxit" reached.
epsilon	stopping rule value to be compared to "tol".
n
nSV	#{0 < a}, no. of support vectors.
nBSV	#{a==C}, no. of bounded support vectors.
nFSV	#{0 < alpha < C}, no. of unbounded support vectors.
control	the control argument.
ws	if requested, the working set selected at current iteration.

Author(s)

Krisztian Sebestyen [e-mail]
Youyi Fong [e-mail]
Shuxin Yin

References

Combining Biomarkers Nonlinearly for Classification Using the Area Under the ROC Curve Y. FONG, S. YIN, Y. HUANG Biometrika (2012), pp 1-28

Newton's method for large bound-constrained optimization problems Lin, C.J. and More, J.J. SIAM Journal on Optimization (1999), volume 9, pp 1100-1127.

kernlab - An S4 Package for Kernel Methods in R. Alexandros Karatzoglou, Alex Smola, Kurt Hornik, Achim Zeileis Journal of Statistical Software (2004) 11(9), 1-20. URL http://www.jstatsoft.org/v11/i09/

See Also

control.minQuad,rauc

aucm documentation built on Jan. 11, 2020, 9:43 a.m.