Package 'abundant'

Abundant regression and high-dimensional principal fitted components

Description

Fit and predict with the high-dimensional principal fitted components model.

Details

The main functions are fit.pfc, pred.response.

Author(s)

Adam J. Rothman

Maintainer: Adam J. Rothman <[email protected]>

References

Cook, R. D., Forzani, L., and Rothman, A. J. (2012). Estimating sufficient reductions of the predictors in abundant high-dimensional regressions. Annals of Statistics 40(1), 353-384.

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Fit a high-dimensional principal fitted components model using the method of Cook, Forzani, and Rothman (2012).

Description

Let $(x_1, y_1), \ldots, (x_n, y_n)$ denote the $n$ measurements of the predictor and response, where $x_i\in R^p$ and $y_i \in R$. The model assumes that these measurements are a realization of $n$ independent copies of the random vector $(X,Y)'$, where

$X = \mu_X + \Gamma \beta\{f(Y) - \mu_f\}+ \epsilon,$

$\mu_X\in R^p$; $\Gamma\in R^{p\times d}$ with rank $d$; $\beta \in R^{d\times r}$ with rank $d$; $f: R \rightarrow R^r$ is a known vector valued function; $\mu_f = E\{f(Y)\}$; $\epsilon \sim N_p(0, \Delta)$; and $Y$ is independent of $\epsilon$. The central subspace is $\Delta^{-1} {\rm span}(\Gamma)$.

This function computes estimates of these model parameters by imposing constraints for identifiability. The mean parameters $\mu_X$ and $\mu_f$ are estimated with $\bar x = n^{-1}\sum_{i=1}^n x_i$ and $\bar f = n^{-1} \sum_{i=1}^n f(y_i)$. Let $\widehat\Phi = n^{-1}\sum_{i=1}^{n} \{f(y_i) - \bar f\}\{f(y_i) - \bar f\}'$, which we require to be positive definite. Given a user-specified weight matrix $\widehat W$, let

$(\widehat\Gamma, \widehat\beta) = \arg\min_{G\in R^{p\times d}, B \in R^{d\times r}} \sum_{i=1}^n [x_i - \bar x - GB\{f(y_i) - \bar f\}]'\widehat W [x_i - \bar x - GB\{f(y_i) - \bar f\}],$

subject to the constraints that $G'\widehat W G$ is diagonal and $B \widehat\Phi B' = I$. The sufficient reduction estimate $\widehat R: R^p \rightarrow R^d$ is defined by

$\widehat R(x) = (\widehat\Gamma'\widehat W \widehat\Gamma)^{-1} \widehat\Gamma' \widehat W(x - \bar x).$

Usage

fit.pfc(X, y, r=4, d=NULL, F.user=NULL, weight.type=c("sample", "diag", "L1"), 
        lam.vec=NULL, kfold=5, silent=TRUE, qrtol=1e-10, cov.tol=1e-4, 
        cov.maxit=1e3, NPERM=1e3, level=0.01)

Arguments

X	The predictor matrix with $n$ rows and $p$ columns. The $i$th row is $x_i$ defined above.
y	The vector of measured responses with $n$ entries. The $i$th entry is $y_i$ defined above.
r	When polynomial basis functions are used (which is the case when F.user=NULL), r is the polynomial order, i.e, $f(y) = (y, y^2, \ldots, y^r)'$. The default is r=4. This argument is not used when F.user is specified.
d	The dimension of the central subspace defined above. This must be specified by the user when weight.type="L1". If unspecified by the user this function will use the sequential permutation testing procedure, described in Section 8.2 of Cook, Forzani, and Rothman (2012), to select d.
F.user	A matrix with $n$ rows and $r$ columns, where the $i$th row is $f(y_i)$ defined above. This argument is optional, and will typically be used when polynomial basis functions are not desired.
weight.type	The type of weight matrix estimate $\widehat W$ to use. Let $\widehat\Delta$ be the observed residual sample covariance matrix for the multivariate regression of X on $f$(Y) with $n-r-1$ scaling. There are three options for $\widehat W$: weight.type="sample" uses a Moore-Penrose generalized inverse of $\widehat\Delta$ for $\widehat W$, when $p \leq n-r-1$ this becomes the inverse of $\widehat\Delta$; weight.type="diag" uses the inverse of the diagonal matrix with the same diagonal as $\widehat\Delta$ for $\widehat W$; weight.type="L1" uses the L1-penalized inverse of $\widehat\Delta$ described in equation (5.4) of Cook, Forzani, and Rothman (2012). In this case, lam.vec and d must be specified by the user. The glasso algorithm of Friedman et al. (2008) is used through the R package glasso.
lam.vec	A vector of candidate tuning parameter values to use when weight.type="L1". If this vector has more than one entry, then kfold cross validation will be performed to select the optimal tuning parameter value.
kfold	The number of folds to use in cross-validation to select the optimal tuning parameter when weight.type="L1". Only used if lam.vec has more than one entry.
silent	Logical. When silent=FALSE, progress updates are printed.
qrtol	The tolerance for calls to qr.solve().
cov.tol	The convergence tolerance for the QUIC algorithm used when weight.type="L1".
cov.maxit	The maximum number of iterations allowed for the QUIC algorithm used when weight.type="L1".
NPERM	The number of permutations to used in the sequential permutation testing procedure to select $d$. Only used when d is unspecified.
level	The significance level to use to terminate the sequential permutation testing procedure to select $d$.

Details

See Cook, Forzani, and Rothman (2012) more information.

Value

A list with

Gamhat	this is $\widehat\Gamma$ described above.
bhat	this is $\widehat\beta$ described above.
Rmat	this is $\widehat W\widehat\Gamma(\widehat\Gamma'\widehat W \widehat\Gamma)^{-1}$.
What	this is $\widehat W$ described above.
d	this is $d$ described above.
r	this is $r$ described above.
GWG	this is $\widehat\Gamma'\widehat W \widehat\Gamma$
fc	a matrix with $n$ rows and $r$ columns where the $i$th row is $f(y_i) - \bar f$.
Xc	a matrix with $n$ rows and $p$ columns where the $i$th row is $x_i - \bar x$.
y	the vector of $n$ response measurements.
mx	this is $\bar x$ described above.
mf	this is $\bar f$ described above.
best.lam	this is selected tuning parameter value used when weight.type="L1", will be NULL otherwise.
lam.vec	this is the vector of candidate tuning parameter values used when weight.type="L1", will be NULL otherwise.
err.vec	this is the vector of validation errors from cross validation, one error for each entry in lam.vec. Will be NULL unless weight.type="L1" and lam.vec has more than one entry.
test.info	a dataframe that summarizes the results from the sequential testing procedure. Will be NULL unless d is unspecified.

Author(s)

Adam J. Rothman

References

Cook, R. D., Forzani, L., and Rothman, A. J. (2012). Estimating sufficient reductions of the predictors in abundant high-dimensional regressions. Annals of Statistics 40(1), 353-384.

Friedman, J., Hastie, T., and Tibshirani R. (2008). Sparse inverse covariance estimation with the lasso. Biostatistics 9(3), 432-441.

See Also

pred.response

Examples

set.seed(1)
n=20
p=30
d=2
y=sqrt(12)*runif(n)
Gam=matrix(rnorm(p*d), nrow=p, ncol=d)
beta=diag(2)
E=matrix(0.5*rnorm(n*p), nrow=n, ncol=p)
V=matrix(c(1, sqrt(12), sqrt(12), 12.8), nrow=2, ncol=2)
tmp=eigen(V, symmetric=TRUE)
V.msqrt=tcrossprod(tmp$vec*rep(tmp$val^(-0.5), each=2), tmp$vec)
Fyc=cbind(y-sqrt(3),y^2-4)%*%V.msqrt
X=0+Fyc%*%t(beta)%*%t(Gam) + E

fit=fit.pfc(X=X, y=y, r=3, weight.type="sample")
## display hypothesis testing information for selecting d
fit$test.info
##  make a response versus fitted values plot
plot(pred.response(fit), y)

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Predict the response with the fitted high-dimensional principal fitted components model

Description

Let $x\in R^p$ denote the values of the $p$ predictors. This function computes $\widehat E(Y|X=x)$ using equation (8.1) of Cook, Forzani, and Rothman (2012).

Usage

pred.response(fit, newx=NULL)

Arguments

fit	The object returned by fit.pfc().
newx	A matrix with $N$ rows and $p$ columns where each row is an instance of $x$ described above. If this argument is unspecified, then the fitted values are returned, i.e, newx=X, where X was the predictor matrix used in the call to fit.pfc().

Details

See Cook, Forzani, and Rothman (2012) for more information.

Value

A vector of response prediction with nrow(newx) entries.

Author(s)

Adam J. Rothman

References

Cook, R. D., Forzani, L., and Rothman, A. J. (2012). Estimating sufficient reductions of the predictors in abundant high-dimensional regressions. Annals of Statistics 40(1), 353-384.

See Also

fit.pfc

Examples

set.seed(1)
n=25
p=50
d=1
true.G = matrix(rnorm(p*d), nrow=p, ncol=d)
y=rnorm(n)
fy = y
E=matrix(rnorm(n*p), nrow=n, ncol=p) 
X=fy%*%t(true.G) + E
fit=fit.pfc(X=X, r=4, d=d, y=y, weight.type="diag")
fitted.values=pred.response(fit)
mean((y-fitted.values)^2)
plot(fitted.values, y)

n.new=100
y.new=rnorm(n.new)
fy.new=y.new
E.new=matrix(rnorm(n.new*p), nrow=n.new, ncol=p) 
X.new = fy.new%*%t(true.G) + E.new
mean((y.new - pred.response(fit, newx=X.new))^2)

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