pocre | R Documentation |
Apply POCRE with a pre-specified tuning parameter to build a linear regression model with orthogonal components X\vartheta_1, X\vartheta_2, …,
Y=μ+∑_j (X\varpi_j)\vartheta_j+ε=μ+Xβ+ε,
where var[ε]=σ^2 and β=∑_j \varpi_j\vartheta_j. These orthogonal components are sequentially constructed according to supervised dimension reduction under penalty set by the pre-specified tuning parameter.
While the orthogonal components are constructed using the centralized covariates, the intercept μ and regression coefficients in β are estimated for original covariates. The sequential construction stops when no new component can be constructed (returning bSparse=1), or the new component is constructed with more than maxvar covariates (returning bSparse=0).
pocre(y, x, lambda=1, x.nop=NA, maxvar=dim(x)[1]/2, maxcmp=10, ptype=c('ebtz','ebt','l1','scad','mcp'), maxit=100, tol=1e-6, gamma=3.7, pval=FALSE)
y |
n*q matrix, values of q response variables (allow for multiple response variables). |
x |
n*p matrix, values of p predicting variables (excluding the intercept). |
lambda |
the tuning parameter (=1 by default). |
x.nop |
a vector indicating indices of covariates which are excluded only when evaluating the significance of components. |
maxvar |
maximum number of selected variables. |
maxcmp |
maximum number of components to be constructed. |
ptype |
a character to indicate the type of penalty: |
maxit |
maximum number of iterations to be allowed. |
tol |
tolerance of precision in iterations. |
gamma |
a parameter used by SCAD and MCP (=3.7 by default). |
pval |
a logical value indicating whether to calculate the p-values of components. |
mu |
estimated intercept of the linear regression. |
beta |
estimated coefficients of the linear regression. |
varpi |
loadings of the constructed components. |
vartheta |
the regression coefficients of the constructed components. |
bSparse |
a logical value indicating whether estimated beta has less than maxvar nonzero values. |
lambda |
value of the tuning paramete. |
nCmp |
number of constructed components. |
n |
sample size. |
p |
number of covariates. |
xShift |
the column means of x. |
yShift |
the column means of y. |
sigmae2 |
estimated error variance σ^2. |
rsq |
R^2 value of the fitted regression model. |
nzBeta |
number of non-zero regression coefficients in β. |
omega |
internal matrix. |
theta |
internal matrix. |
pvalue |
p-values of constructed components, available when |
seqpv |
Type I p-values of components when sequentially including them into the model, available when |
indpv |
p-values of components when marginally testing each component, available when pval=TRUE. |
loglik |
the loglikelihood function, available when |
effp |
the effective number of predictors, excluding redundant ones, available when pval=TRUE. |
Dabao Zhang, Zhongli Jiang, Zeyu Zhang, Department of Statistics, Purdue University
Fan J and Li R (2001). Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association, 96:1348-1360
Johnstone IM and Silverman BW (2004). Needles and straw in haystacks: empirical Bayes estimates of possibly sparse sequences. Annals of Statistics, 32: 1594-1649.
Zhang C-H (2010). Nearly unbiased variable selection under minimax concave penalty. The Annals of Statistics, 38: 894-942.
Zhang D, Lin Y, and Zhang M (2009). Penalized orthogonal-components regression for large p small n data. Electronic Journal of Statistics, 3: 781-796.
plot.pocre
, pocrescreen
, pocrepath
, cvpocre
.
data(simdata) xx <- simdata[,-1] yy <- simdata[,1] #pres <- pocre(yy,xx,lambda=0.9) pres <- pocre(yy,xx) # lambda=1 by default
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