tauestimate.RdComputes the tau estimators for the PFC model
tauestimate(X, Fy, d, aux, inic)
| 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 |
| aux | list containing the constants for the tau-scale |
| inic | list containing initial estimators for the coefficient matrix and the covariance matrix |
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
tau-estimator for the term \(\mu\) in the PFC model
tau-estimator for the parameter \(\beta\) in the PFC model
tau-estimator for the parameter \(\Gamma\) in the PFC model
tau-estimator for the covariance matrix \(\Delta\) in the PFC model
Bergesio, A., Szretter Noste, M. E. and Yohai, V. J. (2020). A robust proposal of estimation for the sufficient dimension reduction problem
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))) # to generate a dataset of n observations following the # above PFC model, r=2 data_sim=generate(p,n,mutrue,gamatrue,betatrue,sigmatrue=1) Fy=data_sim$Fy X=data_sim$X aux=kappa_and_c(p,efficiency=.9) inic=initial(X,Fy,aux,efficiency=0.85) tauestimate(X,Fy,d=1,aux,inic)#> $mu #> [,1] #> [1,] -0.243511154 #> [2,] 0.007365935 #> [3,] -0.200036468 #> [4,] -0.093890646 #> [5,] -0.134315877 #> [6,] 0.030943733 #> [7,] 0.074739821 #> [8,] -0.135978958 #> [9,] -0.027015493 #> [10,] -0.017969105 #> #> $beta #> [,1] [,2] #> [1,] 1.547598 1.041942 #> #> $gamma #> [1] 0.8843475901 -0.0065011529 0.0154540745 -0.0008476076 0.0143403897 #> [6] -0.0089488234 -0.0051934122 0.0131187660 -0.0070593616 0.0081187554 #> #> $delta #> [,1] [,2] [,3] [,4] [,5] [,6] #> [1,] 0.85722126 0.08886269 -0.08602559 0.20562160 -0.07067790 -0.06341411 #> [2,] 0.08886269 1.00504058 -0.15587777 0.07747573 0.11216468 -0.06147560 #> [3,] -0.08602559 -0.15587777 1.13532471 -0.02428829 -0.08094360 -0.02444700 #> [4,] 0.20562160 0.07747573 -0.02428829 0.98825636 -0.01437046 -0.02647680 #> [5,] -0.07067790 0.11216468 -0.08094360 -0.01437046 0.94346170 0.03741923 #> [6,] -0.06341411 -0.06147560 -0.02444700 -0.02647680 0.03741923 0.95824002 #> [7,] -0.04152928 -0.09162647 0.02903779 0.18310443 -0.14848678 -0.12948328 #> [8,] 0.05703462 0.05696580 0.01562002 0.16128339 0.04806727 -0.02803199 #> [9,] 0.01748799 -0.10981577 0.22372711 0.01997940 -0.13777782 0.10564786 #> [10,] 0.04809085 0.07390956 -0.11027696 0.09451212 -0.13027690 0.04562906 #> [,7] [,8] [,9] [,10] #> [1,] -0.04152928 0.05703462 0.01748799 0.04809085 #> [2,] -0.09162647 0.05696580 -0.10981577 0.07390956 #> [3,] 0.02903779 0.01562002 0.22372711 -0.11027696 #> [4,] 0.18310443 0.16128339 0.01997940 0.09451212 #> [5,] -0.14848678 0.04806727 -0.13777782 -0.13027690 #> [6,] -0.12948328 -0.02803199 0.10564786 0.04562906 #> [7,] 1.20949379 -0.05146638 0.03213792 0.02045262 #> [8,] -0.05146638 0.88917606 0.09836186 -0.05668995 #> [9,] 0.03213792 0.09836186 0.90890435 0.11477070 #> [10,] 0.02045262 -0.05668995 0.11477070 0.88562579 #>