RRRR | R Documentation |
Majorisation-Minimisation based Estimation for Reduced-Rank Regression with a Cauchy Distribution Assumption.
This method is robust in the sense that it assumes a heavy-tailed Cauchy distribution
for the innovations. This method is an iterative optimization algorithm. See References
for a similar setting.
RRRR( y, x, z = NULL, mu = TRUE, r = 1, itr = 100, earlystop = 1e-04, initial_A = matrix(rnorm(P * r), ncol = r), initial_B = matrix(rnorm(Q * r), ncol = r), initial_D = matrix(rnorm(P * R), ncol = R), initial_mu = matrix(rnorm(P)), initial_Sigma = diag(P), return_data = TRUE )
y |
Matrix of dimension N*P. The matrix for the response variables. See |
x |
Matrix of dimension N*Q. The matrix for the explanatory variables to be projected. See |
z |
Matrix of dimension N*R. The matrix for the explanatory variables not to be projected. See |
mu |
Logical. Indicating if a constant term is included. |
r |
Integer. The rank for the reduced-rank matrix AB'. See |
itr |
Integer. The maximum number of iteration. |
earlystop |
Scalar. The criteria to stop the algorithm early. The algorithm will stop if the improvement on objective function is small than earlystop * objective_from_last_iteration. |
initial_A |
Matrix of dimension P*r. The initial value for matrix A. See |
initial_B |
Matrix of dimension Q*r. The initial value for matrix B. See |
initial_D |
Matrix of dimension P*R. The initial value for matrix D. See |
initial_mu |
Matrix of dimension P*1. The initial value for the constant mu. See |
initial_Sigma |
Matrix of dimension P*P. The initial value for matrix Sigma. See |
return_data |
Logical. Indicating if the data used is return in the output.
If set to |
The formulation of the reduced-rank regression is as follow:
y = μ +AB' x + D z+innov,
where for each realization y is a vector of dimension P for the P response variables, x is a vector of dimension Q for the Q explanatory variables that will be projected to reduce the rank, z is a vector of dimension R for the R explanatory variables that will not be projected, μ is the constant vector of dimension P, innov is the innovation vector of dimension P, D is a coefficient matrix for z with dimension P*R, A is the so called exposure matrix with dimension P*r, and B is the so called factor matrix with dimension Q*r. The matrix resulted from AB' will be a reduced rank coefficient matrix with rank of r. The function estimates parameters μ, A, B, D, and Sigma, the covariance matrix of the innovation's distribution, assuming the innovation has a Cauchy distribution.
A list of the estimated parameters of class RRRR
.
The input specifications. N is the sample size.
The path of all the parameters during optimization and the path of the objective value.
The estimated constant vector. Can be NULL
.
The estimated exposure matrix.
The estimated factor matrix.
The estimated coefficient matrix of z
.
The estimated covariance matrix of the innovation distribution.
The final objective value.
The data used in estimation if return_data
is set to TRUE
. NULL
otherwise.
Yangzhuoran Yang
Z. Zhao and D. P. Palomar, "Robust maximum likelihood estimation of sparse vector error correction model," in2017 IEEE Global Conference on Signal and Information Processing (GlobalSIP), pp. 913–917,IEEE, 2017.
set.seed(2222) data <- RRR_sim() res <- RRRR(y=data$y, x=data$x, z = data$z) res
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