Apply POCRE with a pre-specified tuning parameter to build a linear regression model with orthogonal components \(X\vartheta_1, X\vartheta_2, \dots\), $$Y=\mu+\sum_j (X\varpi_j)\vartheta_j+\epsilon=\mu+X\beta+\epsilon,$$ where \(var[\epsilon]=\sigma^2\) and \(\beta=\sum_j \varpi_j\vartheta_j\). 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 \(\mu\) and regression coefficients in \(\beta\) 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)n*q matrix, values of q response variables (allow for multiple response variables).
n*p matrix, values of p predicting variables (excluding the intercept).
the tuning parameter (=1 by default).
a vector indicating indices of covariates which are excluded only when evaluating the significance of components.
maximum number of selected variables.
maximum number of components to be constructed.
a character to indicate the type of penalty: 'ebtz' (emprical Bayes thresholding after Fisher's z-transformation, by default), 'ebt' (emprical Bayes thresholding by Johnstone & Silverman (2004)), 'l1' (L_1 penalty), 'scad' (SCAD by Fan & Li (2001)), 'mcp' (MCP by Zhang (2010)).
maximum number of iterations to be allowed.
tolerance of precision in iterations.
a parameter used by SCAD and MCP (=3.7 by default).
a logical value indicating whether to calculate the p-values of components.
estimated intercept of the linear regression.
estimated coefficients of the linear regression.
loadings of the constructed components.
the regression coefficients of the constructed components.
a logical value indicating whether estimated beta has less than maxvar nonzero values.
value of the tuning paramete.
number of constructed components.
sample size.
number of covariates.
the column means of x.
the column means of y.
estimated error variance \(\sigma^2\).
\(R^2\) value of the fitted regression model.
number of non-zero regression coefficients in \(\beta\).
internal matrix.
internal matrix.
p-values of constructed components, available when pval=TRUE.
Type I p-values of components when sequentially including them into the model, available when pval=TRUE.
p-values of components when marginally testing each component, available when pval=TRUE.
the loglikelihood function, available when pval=TRUE.
the effective number of predictors, excluding redundant ones, available when pval=TRUE.
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.
# NOT RUN {
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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