MLE.RdComputes the ML estimators for the PFC model
MLE(X,Fy,d)
| X | vector of response variables in the inverse model, \(n x p\) matrix, each row is a response vector |
|---|---|
| Fy | vector of covariates in the inverse problem, vector containing functions of the response variable in the original multiple regression problem. Is a \(n x r\) matrix, each row is the corresponding response vector |
| d | number indicating the reduction subspace dimension |
We consider the Principal Fitted Components (PFC) model given by \(X = \mu + \Gamma\betaf(y) +\Delta\)1/2\(\epsilon\), where the variables are
\(y\) is an observed response variable of the original model. \(f\) is a known vector valued function, that takes values in Rr
\(X\) is the correspondent p x 1 observed covariates vector
\(\epsilon\) is unobserved p dimensional vector, \(\Delta\)1/2\(\epsilon\) is the error vector
and the unknown parameters (to be estimated) are
\(\mu\) a p x 1 vector of intercepts
\(\Gamma\) is a full-rank p x d matrix whose columns span the dimension reduction subspace
\(\beta\) is a full-rank d x r matrix
cov(\(\epsilon\)) = \(\Delta\), is a p x p positive definite matrix
Both coefficient matrices \(\Gamma\) and \(\beta\) are not unique, but their product p x r matrix is unique, with rank d \(\le min(p,r)\). The notation refers to Cook and Forzani (2008).
List with the following components
MLE estimation of the term \(\mu\) in the PFC model
MLE estimation of the parameter \(\beta\) in the PFC model
MLE estimation of the parameter \(\Gamma\) in the PFC model
MLE estimation of the covariance matrix \(\Delta\) in the PFC model
Cook, R. D. and Forzani, L. (2008). Principal Fitted components for dimension reduction in regression. Statistical Science, 23(4):485-501.
p=10 n=200 mutrue=rep(0,p) gamatrue=as.matrix(c(1,rep(0,p-1))) betatrue=t(as.matrix(c(1,1))) data_sim=generate(p,n,mutrue,gamatrue,betatrue,sigmatrue=1) Fy=data_sim$Fy X=data_sim$X MLE(X,Fy,d=1)#> $mu #> [,1] #> [1,] -0.386915109 #> [2,] -0.005455151 #> [3,] 0.080674931 #> [4,] 0.149416645 #> [5,] -0.155483960 #> [6,] 0.065566515 #> [7,] -0.035582895 #> [8,] -0.067898516 #> [9,] 0.070541987 #> [10,] 0.055343925 #> #> $beta #> [,1] [,2] #> [1,] 1.613029 0.9896072 #> #> $gamma #> [,1] #> [1,] 0.9062473086 #> [2,] 0.0043693984 #> [3,] -0.0022197577 #> [4,] -0.0084626687 #> [5,] 0.0006400118 #> [6,] -0.0078993474 #> [7,] 0.0013556167 #> [8,] -0.0044431204 #> [9,] -0.0072377339 #> [10,] -0.0014514545 #> #> $delta #> [,1] [,2] [,3] [,4] [,5] #> [1,] 0.859291480 -0.060957870 0.01926682 -0.03123792 -0.09818991 #> [2,] -0.060957870 0.931105530 -0.00794794 0.05560378 0.03502754 #> [3,] 0.019266818 -0.007947940 0.90187711 0.08464409 0.10149200 #> [4,] -0.031237923 0.055603777 0.08464409 0.82278117 0.09736358 #> [5,] -0.098189913 0.035027541 0.10149200 0.09736358 0.97922967 #> [6,] 0.026857002 0.017771872 -0.09736280 0.05534137 -0.04541982 #> [7,] -0.048017592 -0.039532881 -0.01447534 -0.09163427 -0.04497941 #> [8,] 0.102669746 0.041190755 0.06657439 -0.07255920 -0.03728341 #> [9,] 0.003214433 0.005061948 -0.11356561 0.05590304 0.09922575 #> [10,] -0.086513178 -0.066619509 -0.04242245 -0.10581391 0.08054624 #> [,6] [,7] [,8] [,9] [,10] #> [1,] 0.026857002 -0.04801759 0.102669746 0.003214433 -0.08651318 #> [2,] 0.017771872 -0.03953288 0.041190755 0.005061948 -0.06661951 #> [3,] -0.097362795 -0.01447534 0.066574390 -0.113565609 -0.04242245 #> [4,] 0.055341374 -0.09163427 -0.072559198 0.055903039 -0.10581391 #> [5,] -0.045419820 -0.04497941 -0.037283409 0.099225750 0.08054624 #> [6,] 1.020862224 0.08133097 -0.002449878 0.093072513 -0.02259914 #> [7,] 0.081330967 0.91940901 -0.083409375 0.034813682 0.04635996 #> [8,] -0.002449878 -0.08340938 0.945947593 0.110829472 0.06679265 #> [9,] 0.093072513 0.03481368 0.110829472 1.044779803 0.09807687 #> [10,] -0.022599145 0.04635996 0.066792651 0.098076873 1.06128029 #>