,
Wenjie Wang [aut] ,
Jun Yan [ctb] | Title: | Reduced-Rank Regression |
|---|---|
| Description: | Multivariate regression methodologies including classical reduced-rank regression (RRR) studied by Anderson (1951) <doi:10.1214/aoms/1177729580> and Reinsel and Velu (1998) <doi:10.1007/978-1-4757-2853-8>, reduced-rank regression via adaptive nuclear norm penalization proposed by Chen et al. (2013) <doi:10.1093/biomet/ast036> and Mukherjee et al. (2015) <doi:10.1093/biomet/asx080>, robust reduced-rank regression (R4) proposed by She and Chen (2017) <doi:10.1093/biomet/asx032>, generalized/mixed-response reduced-rank regression (mRRR) proposed by Luo et al. (2018) <doi:10.1016/j.jmva.2018.04.011>, row-sparse reduced-rank regression (SRRR) proposed by Chen and Huang (2012) <doi:10.1080/01621459.2012.734178>, reduced-rank regression with a sparse singular value decomposition (RSSVD) proposed by Chen et al. (2012) <doi:10.1111/j.1467-9868.2011.01002.x> and sparse and orthogonal factor regression (SOFAR) proposed by Uematsu et al. (2019) <doi:10.1109/TIT.2019.2909889>. |
| Authors: | Kun Chen [aut, cre] ,
Wenjie Wang [aut] ,
Jun Yan [ctb] |
| Maintainer: | Kun Chen <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 0.1-13 |
| Built: | 2024-11-23 04:32:58 UTC |
| Source: | https://github.com/cran/rrpack |
Mixed-response reduced-rank regression with rank selected by cross validation
cv.mrrr( Y, X, is.pca = NULL, offset = NULL, ctrl.id = c(), family = list(gaussian(), binomial(), poisson()), familygroup = NULL, maxrank = min(ncol(Y), ncol(X)), penstr = list(), init = list(), control = list(), nfold = 5, foldid = NULL, nlam = 20, warm = FALSE )cv.mrrr( Y, X, is.pca = NULL, offset = NULL, ctrl.id = c(), family = list(gaussian(), binomial(), poisson()), familygroup = NULL, maxrank = min(ncol(Y), ncol(X)), penstr = list(), init = list(), control = list(), nfold = 5, foldid = NULL, nlam = 20, warm = FALSE )
Y |
response matrix |
X |
covariate matrix |
is.pca |
If TRUE, mixed principal component analysis with X=I |
offset |
matrix of the same dimension as Y for offset |
ctrl.id |
indices of unpenalized predictors |
family |
a list of family functions as used in |
familygroup |
a list of family indices of the responses |
maxrank |
integer giving the maximum rank allowed. |
penstr |
a list of penalty structure of SVD. |
init |
a list of initial values of kappaC0, kappaS0, C0, and S0 |
control |
a list of controling parameters for the fitting |
nfold |
number of folds in cross validation |
foldid |
to specify the folds if desired |
nlam |
number of tuning parameters; not effective when using rank constrained estimation |
warm |
if TRUE, use warm start in fitting the solution paths |
S3 mrrr object, a list containing
fit |
the output from the selected model |
dev |
deviance measures |
## Not run: library(rrpack) simdata <- rrr.sim3(n = 100, p = 30, q.mix = c(5, 20, 5), nrank = 2, mis.prop = 0.2) Y <- simdata$Y Y_mis <- simdata$Y.mis X <- simdata$X X0 <- cbind(1,X) C <- simdata$C family <- simdata$family familygroup <- simdata$familygroup svdX0d1 <- svd(X0)$d[1] init1 = list(kappaC0 = svdX0d1 * 5) offset = NULL control = list(epsilon = 1e-4, sv.tol = 1e-2, maxit = 2000, trace = FALSE, gammaC0 = 1.1, plot.cv = TRUE, conv.obj = TRUE) fit.cv.mrrr <- cv.mrrr(Y_mis, X, family = family, familygroup = familygroup, maxrank = 20, penstr = list(penaltySVD = "rankCon", lambdaSVD = c(1 : 6)), control = control, init = init1, nfold = 10, nlam = 50) summary(fit.cv.mrrr) coef(fit.cv.mrrr) fit.mrrr <- fit.cv.mrrr$fit ## plot(svd(fit.mrrr$coef[- 1,])$d) plot(C ~ fit.mrrr$coef[- 1, ]) abline(a = 0, b = 1) ## End(Not run)## Not run: library(rrpack) simdata <- rrr.sim3(n = 100, p = 30, q.mix = c(5, 20, 5), nrank = 2, mis.prop = 0.2) Y <- simdata$Y Y_mis <- simdata$Y.mis X <- simdata$X X0 <- cbind(1,X) C <- simdata$C family <- simdata$family familygroup <- simdata$familygroup svdX0d1 <- svd(X0)$d[1] init1 = list(kappaC0 = svdX0d1 * 5) offset = NULL control = list(epsilon = 1e-4, sv.tol = 1e-2, maxit = 2000, trace = FALSE, gammaC0 = 1.1, plot.cv = TRUE, conv.obj = TRUE) fit.cv.mrrr <- cv.mrrr(Y_mis, X, family = family, familygroup = familygroup, maxrank = 20, penstr = list(penaltySVD = "rankCon", lambdaSVD = c(1 : 6)), control = control, init = init1, nfold = 10, nlam = 50) summary(fit.cv.mrrr) coef(fit.cv.mrrr) fit.mrrr <- fit.cv.mrrr$fit ## plot(svd(fit.mrrr$coef[- 1,])$d) plot(C ~ fit.mrrr$coef[- 1, ]) abline(a = 0, b = 1) ## End(Not run)
Reduced-rank regression with rank selected by cross validation
cv.rrr( Y, X, nfold = 10, maxrank = min(dim(Y), dim(X)), norder = NULL, coefSVD = FALSE )cv.rrr( Y, X, nfold = 10, maxrank = min(dim(Y), dim(X)), norder = NULL, coefSVD = FALSE )
Y |
response matrix |
X |
covariate matrix |
nfold |
number of folds |
maxrank |
maximum rank allowed |
norder |
for constructing the folds |
coefSVD |
If TRUE, svd of the coefficient is retuned |
a list containing rr estimates from cross validation
Chen, K., Dong, H. and Chan, K.-S. (2013) Reduced rank regression via adaptive nuclear norm penalization. Biometrika, 100, 901–920.
library(rrpack) p <- 50; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) rfit_cv <- with(mydata, cv.rrr(Y, X, nfold = 10, maxrank = 10)) summary(rfit_cv) coef(rfit_cv)library(rrpack) p <- 50; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) rfit_cv <- with(mydata, cv.rrr(Y, X, nfold = 10, maxrank = 10)) summary(rfit_cv) coef(rfit_cv)
Sparse orthognal factor regression tuned by cross validation
cv.sofar( Y, X, nrank = 1, su = NULL, sv = NULL, nfold = 5, norder = NULL, modstr = list(), control = list(), screening = FALSE )cv.sofar( Y, X, nrank = 1, su = NULL, sv = NULL, nfold = 5, norder = NULL, modstr = list(), control = list(), screening = FALSE )
Y |
response matrix |
X |
covariate matrix |
nrank |
an integer specifying the desired rank/number of factors |
su |
a scaling vector for U such that |
sv |
a scaling vector for V such that |
nfold |
number of fold; used for cv.sofar |
norder |
observation orders to constrct data folds; used for cv.sofar |
modstr |
a list of internal model parameters controlling the model fitting |
control |
a list of internal computation parameters controlling optimization |
screening |
If TRUE, marginal screening via lasso is performed before sofar fitting. |
The model parameters can be specified through argument modstr.
The available elements include
mu: parameter in the augmented Lagrangian function.
mugamma: increament of mu along iterations to speed up computation.
WA: weight matrix for A.
WB: weight matrix for B.
Wd: weight matrix for d.
wgamma: power parameter in constructing adaptive weights.
The model fitting can be controled through argument control.
The avilable elements include
nlam: number of lambda triplets to be used.
lam.min.factor: set the smallest lambda triplets as a fraction of the estimation lambda.max triplets.
lam.max.factor: set the largest lambda triplets as a multiple of the estimation lambda.max triplets.
lam.AB.factor: set the relative penalty level between A/B and D.
penA,penB,penD: if TRUE, penalty is applied.
lamA: sequence of tuning parameters for A.
lamB: sequence of tuning parameters for B.
lamD: sequence of tuning parameters for d.
methodA: penalty for penalizing A.
methodB: penalty for penalizing B.
epsilon: convergence tolerance.
maxit: maximum number of iterations.
innerEpsilon: convergence tolerance for inner subroutines.
innerMaxit: maximum number of iterations for inner subroutines.
sv.tol: tolerance for singular values.
Row-sparse reduced-rank regression tuned by cross validation
cv.srrr( Y, X, nrank = 1, method = c("glasso", "adglasso"), nfold = 5, norder = NULL, A0 = NULL, V0 = NULL, modstr = list(), control = list() )cv.srrr( Y, X, nrank = 1, method = c("glasso", "adglasso"), nfold = 5, norder = NULL, A0 = NULL, V0 = NULL, modstr = list(), control = list() )
Y |
response matrix |
X |
covariate matrix |
nrank |
prespecified rank |
method |
group lasso or adaptive group lasso |
nfold |
fold number |
norder |
for constructing the folds |
A0 |
initial value |
V0 |
initial value |
modstr |
a list of model parameters controlling the model fitting |
control |
a list of parameters for controlling the fitting process |
Model parameters controlling the model fitting can be specified through
argument modstr. The available elements include
lamA: tuning parameter sequence.
nlam: number of tuning parameters; no effect if lamA is
specified.
minLambda: minimum lambda value, no effect if lamA is
specified.
maxLambda: maxmum lambda value, no effect if lamA is specified.
WA: adaptive weights. If NULL, the weights are constructed from RRR.
wgamma: power parameter for constructing adaptive weights.
Similarly, the computational parameters controlling optimization can be
specified through argument control. The available elements include
epsilon: epsilonergence tolerance.
maxit: maximum number of iterations.
inner.eps: used in inner loop.
inner.maxit: used in inner loop.
A list of fitting results
Chen, L. and Huang, J.Z. (2012) Sparse reduced-rank regression for simultaneous dimension reduction and variable selection. Journal of the American Statistical Association. 107:500, 1533–1545.
Peforms either rank constrained maximum likelihood estimation or singular value penalized estimation.
mrrr( Y, X, is.pca = NULL, offset = NULL, ctrl.id = c(), family = list(gaussian(), binomial()), familygroup = NULL, maxrank = min(ncol(Y), ncol(X)), penstr = list(), init = list(), control = list() )mrrr( Y, X, is.pca = NULL, offset = NULL, ctrl.id = c(), family = list(gaussian(), binomial()), familygroup = NULL, maxrank = min(ncol(Y), ncol(X)), penstr = list(), init = list(), control = list() )
Y |
response matrix |
X |
covariate matrix |
is.pca |
If TRUE, mixed principal component analysis with X=I |
offset |
matrix of the same dimension as Y for offset |
ctrl.id |
indices of unpenalized predictors |
family |
a list of family functions as used in |
familygroup |
a list of family indices of the responses |
maxrank |
integer giving the maximum rank allowed. Usually this can be set to min(n,p,q) |
penstr |
a list of penalty structure of SVD, contains penstr$penaltySVD is the penalty of SVD, penstr$lambdaSVD is the regularization parameter |
init |
a list of initial values of kappaC0, kappaS0, C0, and S0 |
control |
a list of controling parameters for the fitting |
The model fitting process can be fine tuned through argument control.
The available elements for control include
epsilon: positive convergence tolerance epsilon; the iterations converge when |new - old | / (old + 0.1) < epsilon. treated as zero.
sv.tol: tolerance for singular values.
maxit: integer giving the maximal number of iterations.
trace:logical indicating if tracing the objective is needed.
conv.obj:if TRUE, track objective function.
equal.phi:if TRUE, use a single dispersion parameter for Gaussian responses.
plot.obj:if TRUE, plot obj values along iterations; for checking only
plot.cv:if TRUE, plot cross validation error.
gammaC0:adaptive scaling to speed up computation.
Similarly, the available elements for arguments penstr specifying
penalty structure of SVD include
penaltySVD: penalty for reducing rank
lambdaSVD: tuning parameter. For penaltySVD = rankCon, this is the specified rank.
S3 mrrr object, a list containing
obj |
the objective function tracking |
converged |
TRUE/FALSE for convergence |
coef |
the estimated coefficient matrix |
outlier |
the estimated outlier matrix |
nrank |
the rank of the fitted model |
library(rrpack) simdata <- rrr.sim3(n = 100, p = 30, q.mix = c(5, 20, 5), nrank = 2, mis.prop = 0.2) Y <- simdata$Y Y_mis <- simdata$Y.mis X <- simdata$X X0 <- cbind(1, X) C <- simdata$C family <- simdata$family familygroup <- simdata$familygroup svdX0d1 <- svd(X0)$d[1] init1 = list(kappaC0 = svdX0d1 * 5) offset = NULL control = list(epsilon = 1e-4, sv.tol = 1e-2, maxit = 2000, trace = FALSE, gammaC0 = 1.1, plot.cv = TRUE, conv.obj = TRUE) fit.mrrr <- mrrr(Y_mis, X, family = family, familygroup = familygroup, penstr = list(penaltySVD = "rankCon", lambdaSVD = 2), control = control, init = init1) summary(fit.mrrr) coef(fit.mrrr) par(mfrow = c(1, 2)) plot(fit.mrrr$obj) plot(C ~ fit.mrrr$coef[- 1 ,]) abline(a = 0, b = 1)library(rrpack) simdata <- rrr.sim3(n = 100, p = 30, q.mix = c(5, 20, 5), nrank = 2, mis.prop = 0.2) Y <- simdata$Y Y_mis <- simdata$Y.mis X <- simdata$X X0 <- cbind(1, X) C <- simdata$C family <- simdata$family familygroup <- simdata$familygroup svdX0d1 <- svd(X0)$d[1] init1 = list(kappaC0 = svdX0d1 * 5) offset = NULL control = list(epsilon = 1e-4, sv.tol = 1e-2, maxit = 2000, trace = FALSE, gammaC0 = 1.1, plot.cv = TRUE, conv.obj = TRUE) fit.mrrr <- mrrr(Y_mis, X, family = family, familygroup = familygroup, penstr = list(penaltySVD = "rankCon", lambdaSVD = 2), control = control, init = init1) summary(fit.mrrr) coef(fit.mrrr) par(mfrow = c(1, 2)) plot(fit.mrrr$obj) plot(C ~ fit.mrrr$coef[- 1 ,]) abline(a = 0, b = 1)
S3 methods generating scatter plot for some objects generated by
rrpack using ggplot2. An ggplot2 object is returned so
that users are allowed to easily further customize the plot.
## S3 method for class 'rrr' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'sofar' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'cv.sofar' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'srrr' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'cv.srrr' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'rssvd' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... )## S3 method for class 'rrr' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'sofar' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'cv.sofar' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'srrr' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'cv.srrr' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... ) ## S3 method for class 'rssvd' plot( x, y = NULL, layer = 1L, xlab = paste("latent predictor ", layer, sep = ""), ylab = paste("latent response ", layer, sep = ""), ... )
x |
Some object generated by |
y |
NULL. Do not need to specify. |
layer |
The unit-rank layer to plot; cannot be larger than the estimated rank |
xlab |
Label of X axis. |
ylab |
Label of Y axis. |
... |
Other argumnts for future usage. |
ggplot2 object.
Perform robust reduced-rank regression.
r4( Y, X, maxrank = min(dim(Y), dim(X)), method = c("rowl0", "rowl1", "entrywise"), Gamma = NULL, ic.type = c("AIC", "BIC", "PIC"), modstr = list(), control = list() )r4( Y, X, maxrank = min(dim(Y), dim(X)), method = c("rowl0", "rowl1", "entrywise"), Gamma = NULL, ic.type = c("AIC", "BIC", "PIC"), modstr = list(), control = list() )
Y |
a matrix of response (n by q) |
X |
a matrix of covariate (n by p) |
maxrank |
maximum rank for fitting |
method |
outlier detection method, either entrywise or rowwise |
Gamma |
weighting matrix in the loss function |
ic.type |
information criterion, AIC, BIC or PIC |
modstr |
a list of model parameters controlling the model fitting |
control |
a list of parameters for controlling the fitting process |
The model parameters can be controlled through argument modstr.
The available elements include
nlam: parameter in the augmented Lagrangian function.
adaptive: if TRUE, use leverage values for adaptive
penalization. The default value is FALSE.
weights: user supplied weights for adaptive penalization.
minlam: maximum proportion of outliers.
maxlam: maximum proportion of good observations.
delid: discarded observation indices for initial estimation.
The model fitting can be controlled through argument control.
The available elements include
epsilon: convergence tolerance.
maxit: maximum number of iterations.
qr.tol: tolerance for qr decomposition.
tol: tolerance.
a list consisting of
coef.path |
solutuon path of regression coefficients |
s.path |
solutuon path of sparse mean shifts |
s.norm.path |
solutuon path of the norms of sparse mean shifts |
ic.path |
paths of information criteria |
ic.smooth.path |
smoothed paths of information criteria |
lambda.path |
paths of the tuning parameter |
id.solution |
ids of the selected solutions on the path |
ic.best |
lowest values of the information criteria |
rank.best |
rank values of selected solutions |
coef |
estimated regression coefficients |
s |
estimated sparse mean shifts |
rank |
rank estimate |
She, Y. and Chen, K. (2017) Robust reduced-rank regression. Biometrika, 104 (3), 633–647.
## Not run: library(rrpack) n <- 100; p <- 500; q <- 50 xrank <- 10; nrank <- 3; rmax <- min(n, p, q, xrank) nlam <- 100; gamma <- 2 rho_E <- 0.3 rho_X <- 0.5 nlev <- 0 vlev <- 0 vout <- NULL vlevsd <- NULL nout <- 0.1 * n s2n <- 1 voutsd <- 2 simdata <- rrr.sim5(n, p, q, nrank, rx = xrank, s2n = s2n, rho_X = rho_X, rho_E = rho_E, nout = nout, vout = vout, voutsd = voutsd,nlev = nlev,vlev=vlev,vlevsd=vlevsd) Y <- simdata$Y X <- simdata$X fit <- r4(Y, X, maxrank = rmax, method = "rowl0", ic.type= "PIC") summary(fit) coef(fit) which(apply(fit$s,1,function(a)sum(a^2))!=0) ## End(Not run)## Not run: library(rrpack) n <- 100; p <- 500; q <- 50 xrank <- 10; nrank <- 3; rmax <- min(n, p, q, xrank) nlam <- 100; gamma <- 2 rho_E <- 0.3 rho_X <- 0.5 nlev <- 0 vlev <- 0 vout <- NULL vlevsd <- NULL nout <- 0.1 * n s2n <- 1 voutsd <- 2 simdata <- rrr.sim5(n, p, q, nrank, rx = xrank, s2n = s2n, rho_X = rho_X, rho_E = rho_E, nout = nout, vout = vout, voutsd = voutsd,nlev = nlev,vlev=vlev,vlevsd=vlevsd) Y <- simdata$Y X <- simdata$X fit <- r4(Y, X, maxrank = rmax, method = "rowl0", ic.type= "PIC") summary(fit) coef(fit) which(apply(fit$s,1,function(a)sum(a^2))!=0) ## End(Not run)
S3 methods extracting estimated coefficients for objects generated by
rrpack.
## S3 method for class 'mrrr' coef(object, ...) ## S3 method for class 'cv.mrrr' coef(object, ...) ## S3 method for class 'r4' coef(object, ...) ## S3 method for class 'rrr' coef(object, ...) ## S3 method for class 'rrr.fit' coef(object, ...) ## S3 method for class 'cv.rrr' coef(object, ...) ## S3 method for class 'srrr' coef(object, ...) ## S3 method for class 'sofar' coef(object, ...) ## S3 method for class 'rssvd' coef(object, ...)## S3 method for class 'mrrr' coef(object, ...) ## S3 method for class 'cv.mrrr' coef(object, ...) ## S3 method for class 'r4' coef(object, ...) ## S3 method for class 'rrr' coef(object, ...) ## S3 method for class 'rrr.fit' coef(object, ...) ## S3 method for class 'cv.rrr' coef(object, ...) ## S3 method for class 'srrr' coef(object, ...) ## S3 method for class 'sofar' coef(object, ...) ## S3 method for class 'rssvd' coef(object, ...)
object |
Object generated by |
... |
Other argumnts for future usage. |
A numeric matrix.
Produce solution paths of reduced-rank estimators and adaptive nuclear norm penalized estimators; compute the degrees of freeom of the RRR estimators and select a solution via certain information criterion.
rrr( Y, X, penaltySVD = c("rank", "ann"), ic.type = c("GIC", "AIC", "BIC", "BICP", "GCV"), df.type = c("exact", "naive"), maxrank = min(dim(Y), dim(X)), modstr = list(), control = list() )rrr( Y, X, penaltySVD = c("rank", "ann"), ic.type = c("GIC", "AIC", "BIC", "BICP", "GCV"), df.type = c("exact", "naive"), maxrank = min(dim(Y), dim(X)), modstr = list(), control = list() )
Y |
a matrix of response (n by q) |
X |
a matrix of covariate (n by p) |
penaltySVD |
‘rank’: rank-constrainted estimation; ‘ann’: adaptive nuclear norm estimation. |
ic.type |
the information criterion to be used; currently supporting ‘AIC’, ‘BIC’, ‘BICP’, ‘GCV’, and ‘GIC’. |
df.type |
‘exact’: the exact degrees of freedoms based on SURE theory; ‘naive’: the naive degress of freedoms based on counting number of free parameters |
maxrank |
an integer of maximum desired rank. |
modstr |
a list of model parameters controlling the model fitting |
control |
a list of parameters for controlling the fitting process: ‘sv.tol’ controls the tolerence of singular values; ‘qr.tol’ controls the tolerence of QR decomposition for the LS fit |
Model parameters can be specified through argument modstr. The
available include
gamma: A scalar power parameter of the adaptive weights in
penalty == "ann".
nlambda: The number of lambda values; no effect if
penalty == "count".
lambda: A vector of user-specified rank values if
penalty == "count" or a vector of penalty values if penalty ==
"ann".
The available elements for argument control include
sv.tol: singular value tolerence.
qr.tol: QR decomposition tolerence.
S3 rrr object, a list consisting of
call |
original function call |
Y |
input matrix of response |
X |
input matrix of covariate |
A |
right singular matri x of the least square fitted matrix |
Ad |
a vector of squared singular values of the least square fitted matrix |
coef.ls |
coefficient estimate from LS |
Spath |
a matrix, each column containing shrinkage factors of the singular values of a solution; the first four objects can be used to recover all reduced-rank solutions |
df.exact |
the exact degrees of freedom |
df.naive |
the naive degrees of freedom |
penaltySVD |
the method of low-rank estimation |
sse |
a vecotr of sum of squard errors |
ic |
a vector of information criterion |
coef |
estimated coefficient matrix |
U |
estimated left singular matrix such that XU/sqrtn is orthogonal |
V |
estimated right singular matrix that is orthogonal |
D |
estimated singular value matrix such that C = UDVt |
rank |
estimated rank |
Chen, K., Dong, H. and Chan, K.-S. (2013) Reduced rank regression via adaptive nuclear norm penalization. Biometrika, 100, 901–920.
library(rrpack) p <- 50; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) rfit <- with(mydata, rrr(Y, X, maxrank = 10)) summary(rfit) coef(rfit) plot(rfit)library(rrpack) p <- 50; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) rfit <- with(mydata, rrr(Y, X, maxrank = 10)) summary(rfit) coef(rfit) plot(rfit)
Compute Cook's distance for model diagnostics in rrr estimation.
rrr.cookD(Y, X = NULL, nrank = 1, qr.tol = 1e-07)rrr.cookD(Y, X = NULL, nrank = 1, qr.tol = 1e-07)
Y |
response matrix |
X |
covariate matrix |
nrank |
model rank |
qr.tol |
tolerance |
a list containing diagnostics measures
Chen, K. Model diagnostics in reduced-rank estimation. Statistics and Its interface, 9, 469–484.
Given a response matrix and a covariate matrix, this function fits reduced rank regression for a specified rank. It reduces to singular value decomposition if the covariate matrix is the identity matrix.
rrr.fit(Y, X, nrank = 1, weight = NULL, coefSVD = FALSE)rrr.fit(Y, X, nrank = 1, weight = NULL, coefSVD = FALSE)
Y |
a matrix of response (n by q) |
X |
a matrix of covariate (n by p) |
nrank |
an integer specifying the desired rank |
weight |
a square matrix of weight (q by q); The default is the identity matrix |
coefSVD |
logical indicating the need for SVD for the coeffient matrix in the output; used in ssvd estimation |
S3 rrr object, a list consisting of
coef |
coefficient of rrr |
coef.ls |
coefficient of least square |
fitted |
fitted value of rrr |
fitted.ls |
fitted value of least square |
A |
right singular matrix |
Ad |
a vector of sigular values |
rank |
rank of the fitted rrr |
Y <- matrix(rnorm(400), 100, 4) X <- matrix(rnorm(800), 100, 8) rfit <- rrr.fit(Y, X, nrank = 2) coef(rfit)Y <- matrix(rnorm(400), 100, 4) X <- matrix(rnorm(800), 100, 8) rfit <- rrr.fit(Y, X, nrank = 2) coef(rfit)
Compute leverage scores and Cook's distance for model diagnostics
in rrr estimation.
rrr.leverage(Y, X = NULL, nrank = 1, qr.tol = 1e-07)rrr.leverage(Y, X = NULL, nrank = 1, qr.tol = 1e-07)
Y |
a matrix of response (n by q) |
X |
a matrix of covariate (n by p) |
nrank |
an integer specifying the desired rank |
qr.tol |
tolerence to be passed to ‘qr’ |
‘rrr.leverage’ returns a list containing a vector of leverages and a scalar of the degrees of freedom (sum of leverages). ‘rrr.cooks’ returns a list containing
residuals |
resisuals matrix |
mse |
mean squared error |
leverage |
leverage |
cookD |
Cook's distance |
df |
degrees of freedom |
Chen, K. Model diagnostics in reduced-rank estimation. Statistics and Its interface, 9, 469–484.
Similar to the the RSSVD simulation model in Chen, Chan, Stenseth (2012), JRSSB.
rrr.sim1( n = 50, p = 25, q = 25, nrank = 3, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0 )rrr.sim1( n = 50, p = 25, q = 25, nrank = 3, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0 )
n, p, q
|
model dimensions |
nrank |
model rank |
s2n |
signal to noise ratio |
sigma |
error variance. If specfied, then s2n has no effect |
rho_X |
correlation parameter in the generation of predictors |
rho_E |
correlation parameter in the generation of random errors |
similated model and data
Chen, K., Chan, K.-S. and Stenseth, N. C. (2012) Reduced rank stochastic regression with a sparse singular value decomposition. Journal of the Royal Statistical Society: Series B, 74, 203–221.
Similar to the the SRRR simulation model in Chen and Huang (2012), JASA
rrr.sim2( n = 100, p = 50, p0 = 10, q = 50, q0 = 10, nrank = 3, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0 )rrr.sim2( n = 100, p = 50, p0 = 10, q = 50, q0 = 10, nrank = 3, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0 )
n |
sample size |
p |
number of predictors |
p0 |
number of relevant predictors |
q |
number of responses |
q0 |
number of relevant responses |
nrank |
model rank |
s2n |
signal to noise ratio |
sigma |
error variance. If specfied, then s2n has no effect |
rho_X |
correlation parameter in the generation of predictors |
rho_E |
correlation parameter in the generation of random errors |
similated model and data
Chen, L. and Huang, J.Z. (2012) Sparse reduced-rank regression for simultaneous dimension reduction and variable selection. Journal of the American Statistical Association, 107:500, 1533–1545.
Generate data from a mixed-response reduced-rank regression model
rrr.sim3( n = 100, p = 30, q.mix = c(5, 20, 5), nrank = 2, intercept = rep(0.5, 30), mis.prop = 0.2 )rrr.sim3( n = 100, p = 30, q.mix = c(5, 20, 5), nrank = 2, intercept = rep(0.5, 30), mis.prop = 0.2 )
n |
sample size |
p |
number of predictors |
q.mix |
numbers of Gaussian, Bernolli and Poisson responses |
nrank |
model rank |
intercept |
a vector of intercept |
mis.prop |
missing proportion |
similated model and data
Chen, K., Luo, C., and Liang, J. (2017) Leveraging mixed and incomplete outcomes through a mixed-response reduced-rank regression. Technical report.
Generate data from a mean-shifted reduced-rank regression model
rrr.sim4( n = 100, p = 12, q = 8, nrank = 3, s2n = 1, rho_X = 0, rho_E = 0, nout = 10, vout = NULL, voutsd = 2, nlev = 10, vlev = 10, vlevsd = NULL, SigmaX = "CorrCS", SigmaE = "CorrCS" )rrr.sim4( n = 100, p = 12, q = 8, nrank = 3, s2n = 1, rho_X = 0, rho_E = 0, nout = 10, vout = NULL, voutsd = 2, nlev = 10, vlev = 10, vlevsd = NULL, SigmaX = "CorrCS", SigmaE = "CorrCS" )
n |
sample size |
p |
number of predictors |
q |
numbers of responses |
nrank |
model rank |
s2n |
signal to noise ratio |
rho_X |
correlation parameter for predictors |
rho_E |
correlation parameter for errors |
nout |
number of outliers; should be smaller than n |
vout |
control mean-shifted value of outliers |
voutsd |
control mean-shifted magnitude of outliers |
nlev |
number of high-leverage outliers |
vlev |
control value of leverage |
vlevsd |
control magnitude of leverage |
SigmaX |
correlation structure of predictors |
SigmaE |
correlation structure of errors |
similated model and data
She, Y. and Chen, K. (2017) Robust reduced-rank regression. Biometrika, 104 (3), 633–647.
Generate data from a mean-shifted reduced-rank regression model
rrr.sim5( n = 40, p = 100, q = 50, nrank = 5, rx = 10, s2n = 1, rho_X = 0, rho_E = 0, nout = 10, vout = NULL, voutsd = 2, nlev = 10, vlev = 10, vlevsd = NULL, SigmaX = "CorrCS", SigmaE = "CorrCS" )rrr.sim5( n = 40, p = 100, q = 50, nrank = 5, rx = 10, s2n = 1, rho_X = 0, rho_E = 0, nout = 10, vout = NULL, voutsd = 2, nlev = 10, vlev = 10, vlevsd = NULL, SigmaX = "CorrCS", SigmaE = "CorrCS" )
n |
sample size |
p |
number of predictors |
q |
numbers of responses |
nrank |
model rank |
rx |
rank of the design matrix |
s2n |
signal to noise ratio |
rho_X |
correlation parameter for predictors |
rho_E |
correlation parameter for errors |
nout |
number of outliers; should be smaller than n |
vout |
control mean-shifted value of outliers |
voutsd |
control mean-shifted magnitude of outliers |
nlev |
number of high-leverage outliers |
vlev |
control value of leverage |
vlevsd |
control magnitude of leverage |
SigmaX |
correlation structure of predictors |
SigmaE |
correlation structure of errors |
similated model and data
She, Y. and Chen, K. (2017) Robust reduced-rank regression. Biometrika, 104 (3), 633–647.
Fitting reduced-rank ridge regression with given rank and shrinkage penalty
rrs.fit(Y, X, nrank = min(ncol(Y), ncol(X)), lambda = 1, coefSVD = FALSE)rrs.fit(Y, X, nrank = min(ncol(Y), ncol(X)), lambda = 1, coefSVD = FALSE)
Y |
a matrix of response (n by q) |
X |
a matrix of covariate (n by p) |
nrank |
an integer specifying the desired rank |
lambda |
tunging parameter for the ridge penalty |
coefSVD |
logical indicating the need for SVD for the coeffient matrix int the output |
S3 rrr object, a list consisting of
coef |
coefficient of rrs |
coef.ls |
coefficient of least square |
fitted |
fitted value of rrs |
fitted.ls |
fitted value of least square |
A |
right singular matrix |
Ad |
sigular value vector |
nrank |
rank of the fitted rrr |
Mukherjee, A. and Zhu, J. (2011) Reduced rank ridge regression and its kernal extensions.
Mukherjee, A., Chen, K., Wang, N. and Zhu, J. (2015) On the degrees of freedom of reduced-rank estimators in multivariate regression. Biometrika, 102, 457–477.
library(rrpack) Y <- matrix(rnorm(400), 100, 4) X <- matrix(rnorm(800), 100, 8) rfit <- rrs.fit(Y, X)library(rrpack) Y <- matrix(rnorm(400), 100, 4) X <- matrix(rnorm(800), 100, 8) rfit <- rrs.fit(Y, X)
Reduced-rank regression with a sparse singular value decomposition using the iterative exclusive extraction algorithm.
rssvd( Y, X, nrank, ic.type = c("BIC", "BICP", "AIC"), orthX = FALSE, control = list(), screening = FALSE )rssvd( Y, X, nrank, ic.type = c("BIC", "BICP", "AIC"), orthX = FALSE, control = list(), screening = FALSE )
Y |
response matrix |
X |
covariate matrix |
nrank |
integer specification of the desired rank |
ic.type |
character specifying which information criterion to use to select the best: ‘BIC’, ‘BICP’, and ‘AIC’ |
orthX |
logical indicating if X is orthogonal, in which case a faster algorithm is used |
control |
a list of parameters controlling the fitting process |
screening |
If TRUE, marginal screening via glm is performed before srrr fitting. |
The model fitting can be controled through argument control.
The available elements include
maxit: maximum number of iterations.
epsilon: convergence tolerance.
innerMaxit: maximum number of iterations for inner steps.
innerEpsilon: convergence tolerance for inner steps.
nlambda: number of tuning parameters.
adaptive: if Ture, use adaptive penalization.
gamma0: power parameter for constructing adaptive weights.
minLambda: multiplicate factor to determine the minimum lambda.
niter.eea: the number of iterations in the iterative exclusive extraction algorithm.
df.tol: tolerance.
S3 rssvd.path object, a list consisting of
Upath |
solution path of U |
Vpath |
solution path of V |
Dpath |
solution path of D |
U |
estimated left singular matrix that is orthogonal |
V |
estimated right singular matrix that is orthogonal |
D |
estimated singular values such that C=UDVt |
rank |
estimated rank |
Chen, K., Chan, K.-S. and Stenseth, N. C. (2012) Reduced rank stochastic regression with a sparse singular value decomposition. Journal of the Royal Statistical Society: Series B, 74, 203–221.
library(rrpack) ## Simulate data from a sparse factor regression model p <- 50; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) fit1 <- with(mydata, rssvd(Y, X, nrank = nrank + 1)) summary(fit1) plot(fit1)library(rrpack) ## Simulate data from a sparse factor regression model p <- 50; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) fit1 <- with(mydata, rssvd(Y, X, nrank = nrank + 1)) summary(fit1) plot(fit1)
Compute solution paths of sparse orthogonal factor regression
sofar( Y, X, nrank = 1, su = NULL, sv = NULL, ic.type = c("GIC", "BIC", "AIC", "GCV"), modstr = list(), control = list(), screening = FALSE )sofar( Y, X, nrank = 1, su = NULL, sv = NULL, ic.type = c("GIC", "BIC", "AIC", "GCV"), modstr = list(), control = list(), screening = FALSE )
Y |
response matrix |
X |
covariate matrix |
nrank |
an integer specifying the desired rank/number of factors |
su |
a scaling vector for U such that |
sv |
a scaling vector for V such that |
ic.type |
select tuning method; the default is GIC |
modstr |
a list of internal model parameters controlling the model fitting |
control |
a list of internal computation parameters controlling optimization |
screening |
If TRUE, marginal screening via lasso is performed before sofar fitting. |
The model parameters can be specified through argument modstr.
The available elements include
mu: parameter in the augmented Lagrangian function.
mugamma: increament of mu along iterations to speed up computation.
WA: weight matrix for A.
WB: weight matrix for B.
Wd: weight matrix for d.
wgamma: power parameter in constructing adaptive weights.
The model fitting can be controled through argument control.
The avilable elements include
nlam: number of lambda triplets to be used.
lam.min.factor: set the smallest lambda triplets as a fraction of the estimation lambda.max triplets.
lam.max.factor: set the largest lambda triplets as a multiple of the estimation lambda.max triplets.
lam.AB.factor: set the relative penalty level between A/B and D.
penA,penB,penD: if TRUE, penalty is applied.
lamA: sequence of tuning parameters for A.
lamB: sequence of tuning parameters for B.
lamD: sequence of tuning parameters for d.
methodA: penalty for penalizing A.
methodB: penalty for penalizing B.
epsilon: convergence tolerance.
maxit: maximum number of iterations.
innerEpsilon: convergence tolerance for inner subroutines.
innerMaxit: maximum number of iterations for inner subroutines.
sv.tol: tolerance for singular values.
A sofar object containing
call |
original function call |
Y |
input response matrix |
X |
input predictor matrix |
Upath |
solution path of U |
Dpath |
solution path of D |
Vpath |
solution path of D |
Rpath |
path of estimated rank |
icpath |
path of information criteria |
lam.id |
ids of selected lambda for GIC, BIC, AIC and GCV |
p.index |
ids of predictors which passed screening |
q.index |
ids of responses which passed screening |
lamA |
tuning sequence for A |
lamB |
tuning sequence for B |
lamD |
tuning sequence for D |
U |
estimated left singular matrix that is orthogonal (factor weights) |
V |
estimated right singular matrix that is orthogonal (factor loadings) |
D |
estimated singular values |
rank |
estimated rank |
Uematsu, Y., Fan, Y., Chen, K., Lv, J., & Lin, W. (2019). SOFAR: large-scale association network learning. IEEE Transactions on Information Theory, 65(8), 4924–4939.
## Not run: library(rrpack) ## Simulate data from a sparse factor regression model p <- 100; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) Y <- mydata$Y X <- mydata$X fit1 <- sofar(Y, X, ic.type = "GIC", nrank = nrank + 2, control = list(methodA = "adlasso", methodB = "adlasso")) summary(fit1) plot(fit1) fit1$U crossprod(fit1$U) #check orthogonality fit1$V crossprod(fit1$V) #check orthogonality ## End(Not run)## Not run: library(rrpack) ## Simulate data from a sparse factor regression model p <- 100; q <- 50; n <- 100; nrank <- 3 mydata <- rrr.sim1(n, p, q, nrank, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0.3) Y <- mydata$Y X <- mydata$X fit1 <- sofar(Y, X, ic.type = "GIC", nrank = nrank + 2, control = list(methodA = "adlasso", methodB = "adlasso")) summary(fit1) plot(fit1) fit1$U crossprod(fit1$U) #check orthogonality fit1$V crossprod(fit1$V) #check orthogonality ## End(Not run)
Row-sparse reduced-rank regresssion for a prespecified rank; produce a solution path for selecting predictors
srrr( Y, X, nrank = 2, method = c("glasso", "adglasso"), ic.type = c("BIC", "BICP", "AIC", "GCV", "GIC"), A0 = NULL, V0 = NULL, modstr = list(), control = list(), screening = FALSE )srrr( Y, X, nrank = 2, method = c("glasso", "adglasso"), ic.type = c("BIC", "BICP", "AIC", "GCV", "GIC"), A0 = NULL, V0 = NULL, modstr = list(), control = list(), screening = FALSE )
Y |
response matrix |
X |
covariate matrix |
nrank |
prespecified rank |
method |
group lasso or adaptive group lasso |
ic.type |
information criterion |
A0 |
initial value |
V0 |
initial value |
modstr |
a list of model parameters controlling the model fitting |
control |
a list of parameters for controlling the fitting process |
screening |
If TRUE, marginal screening via glm is performed before srrr fitting. |
Model parameters controlling the model fitting can be specified through
argument modstr. The available elements include
lamA: tuning parameter sequence.
nlam: number of tuning parameters; no effect if lamA is
specified.
minLambda: minimum lambda value, no effect if lamA is
specified.
maxLambda: maxmum lambda value, no effect if lamA is specified.
WA: adaptive weights. If NULL, the weights are constructed from RRR.
wgamma: power parameter for constructing adaptive weights.
Similarly, the computational parameters controlling optimization can be
specified through argument control. The available elements include
epsilon: epsilonergence tolerance.
maxit: maximum number of iterations.
inner.eps: used in inner loop.
inner.maxit: used in inner loop.
A list of fitting results
Chen, L. and Huang, J. Z. (2012) Sparse reduced-rank regression for simultaneous dimension reduction and variable selection. Journal of the American Statistical Association. 107:500, 1533–1545.
library(rrpack) p <- 100; n <- 100; nrank <- 3 mydata <- rrr.sim2(n, p, p0 = 10,q = 50, q0 = 10, nrank = 3, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0) fit1 <- with(mydata, srrr(Y, X, nrank = 3)) summary(fit1) coef(fit1) plot(fit1)library(rrpack) p <- 100; n <- 100; nrank <- 3 mydata <- rrr.sim2(n, p, p0 = 10,q = 50, q0 = 10, nrank = 3, s2n = 1, sigma = NULL, rho_X = 0.5, rho_E = 0) fit1 <- with(mydata, srrr(Y, X, nrank = 3)) summary(fit1) coef(fit1) plot(fit1)
S3 methods summarizing objects generated by rrpack.
## S3 method for class 'mrrr' summary(object, ...) ## S3 method for class 'cv.mrrr' summary(object, ...) ## S3 method for class 'r4' summary(object, ...) ## S3 method for class 'rrr' summary(object, ...) ## S3 method for class 'cv.rrr' summary(object, ...) ## S3 method for class 'sofar' summary(object, ...) ## S3 method for class 'cv.sofar' summary(object, ...) ## S3 method for class 'srrr' summary(object, ...) ## S3 method for class 'cv.srrr' summary(object, ...) ## S3 method for class 'rssvd' summary(object, ...)## S3 method for class 'mrrr' summary(object, ...) ## S3 method for class 'cv.mrrr' summary(object, ...) ## S3 method for class 'r4' summary(object, ...) ## S3 method for class 'rrr' summary(object, ...) ## S3 method for class 'cv.rrr' summary(object, ...) ## S3 method for class 'sofar' summary(object, ...) ## S3 method for class 'cv.sofar' summary(object, ...) ## S3 method for class 'srrr' summary(object, ...) ## S3 method for class 'cv.srrr' summary(object, ...) ## S3 method for class 'rssvd' summary(object, ...)
object |
Object generated from |
... |
Other argumnts for future usage. |