Description Usage Arguments Details Value Author(s) References See Also Examples
Let (x_1, y_1), …, (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 = μ_X + Γ β\{f(Y)  μ_f\}+ ε,
μ_X\in R^p; Γ\in R^{p\times d} with rank d; β \in R^{d\times r} with rank d; f: R \rightarrow R^r is a known vector valued function; μ_f = E\{f(Y)\}; ε \sim N_p(0, Δ); and Y is independent of ε. The central subspace is Δ^{1} {\rm span}(Γ).
This function computes estimates of these model parameters by imposing constraints for identifiability. The mean parameters μ_X and μ_f are estimated with \bar x = n^{1}∑_{i=1}^n x_i and \bar f = n^{1} ∑_{i=1}^n f(y_i). Let \widehatΦ = n^{1}∑_{i=1}^{n} \{f(y_i)  \bar f\}\{f(y_i)  \bar f\}', which we require to be positive definite. Given a userspecified weight matrix \widehat W, let
(\widehatΓ, \widehatβ) = \arg\min_{G\in R^{p\times d}, B \in R^{d\times r}} ∑_{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 \widehatΦ B' = I. The sufficient reduction estimate \widehat R: R^p \rightarrow R^d is defined by
\widehat R(x) = (\widehatΓ'\widehat W \widehatΓ)^{1} \widehatΓ' \widehat W(x  \bar x).
1 2 3 
X 
The predictor matrix with n rows and p columns. The ith row is x_i defined above. 
y 
The vector of measured responses with n entries. The ith entry is y_i defined above. 
r 
When polynomial basis functions are used (which is the case when 
d 
The dimension of the central subspace defined above. This must be specified by the user
when 
F.user 
A matrix with n rows and r columns, where the ith 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 \widehatΔ be the observed residual sample covariance matrix for the multivariate
regression of

lam.vec 
A vector of candidate tuning parameter values to use when 
kfold 
The number of folds to use in crossvalidation to select the optimal tuning parameter when 
silent 
Logical. When 
qrtol 
The tolerance for calls to 
cov.tol 
The convergence tolerance for the QUIC algorithm used when 
cov.maxit 
The maximum number of iterations allowed for the QUIC algorithm used when 
NPERM 
The number of permutations to used in the sequential permutation testing procedure to select d.
Only used when 
level 
The significance level to use to terminate the sequential permutation testing procedure to select d. 
See Cook, Forzani, and Rothman (2012) more information.
A list with
Gamhat 
this is \widehatΓ described above. 
bhat 
this is \widehatβ described above. 
Rmat 
this is \widehat W\widehatΓ(\widehatΓ'\widehat W \widehatΓ)^{1}. 
What 
this is \widehat W described above. 
d 
this is d described above. 
r 
this is r described above. 
GWG 
this is \widehatΓ'\widehat W \widehatΓ 
fc 
a matrix with n rows and r columns where the ith row is f(y_i)  \bar f. 
Xc 
a matrix with n rows and p columns where the ith 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 
lam.vec 
this is the vector of candidate tuning parameter values used when

err.vec 
this is the vector of validation errors from cross validation, one error for each entry in 
test.info 
a dataframe that summarizes the results from the sequential testing procedure. Will be 
Adam J. Rothman
Cook, R. D., Forzani, L., and Rothman, A. J. (2012). Estimating sufficient reductions of the predictors in abundant highdimensional regressions. Annals of Statistics 40(1), 353384.
ChoJui Hsieh, Matyas A. Sustik, Inderjit S. Dhillon, and Pradeep Ravikumar (2011). Sparse inverse covariance matrix estimation using quadratic approximation. Advances in Neural Information Processing Systems 24, 2011, p. 2330–2338.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  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$vec)
Fyc=cbind(ysqrt(3),y^24)%*%V.msqrt
X=0+Fyc%*%t(beta)%*%t(Gam) + E
fit=fit.pfc(X=X, y=y, r=3, weight.type="sample")
## display hypothesis testing information for selecting d
fit$test.info
## make a response versus fitted values plot
plot(pred.response(fit), y)

Loading required package: QUIC
d0 test.statistic pvalue
1 0 0.109842773 0.000
2 1 0.001617754 0.003
3 2 0.001565799 0.406
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