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.
Matrix of dimension N*P. The matrix for the response variables. See Detail
.
Matrix of dimension N*Q. The matrix for the explanatory variables to be projected. See Detail
.
Matrix of dimension N*R. The matrix for the explanatory variables not to be projected. See Detail
.
Logical. Indicating if a constant term is included.
Integer. The rank for the reduced-rank matrix \(AB'\). See Detail
.
Integer. The maximum number of iteration.
Scalar. The criteria to stop the algorithm early. The algorithm will stop if the improvement on objective function is small than \(earlystop * objective_from_last_iteration\).
Matrix of dimension P*r. The initial value for matrix \(A\). See Detail
.
Matrix of dimension Q*r. The initial value for matrix \(B\). See Detail
.
Matrix of dimension P*R. The initial value for matrix \(D\). See Detail
.
Matrix of dimension P*1. The initial value for the constant \(mu\). See Detail
.
Matrix of dimension P*P. The initial value for matrix Sigma. See Detail
.
Logical. Indicating if the data used is return in the output.
If set to TRUE
, update.RRRR
can update the model by simply provide new data.
Set to FALSE
to save output size.
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.
The formulation of the reduced-rank regression is as follow: $$y = \mu +AB' x + D z+innov,$$ 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, \(\mu\) 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 \(\mu\), \(A\), \(B\), \(D\), and \(Sigma\), the covariance matrix of the innovation's distribution, assuming the innovation has a Cauchy distribution.
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
#> Robust Reduced-Rank Regression
#> ------
#> Majorisation-Minimisation
#> ------------
#> Specifications:
#> N P Q R r
#> 1000 3 3 1 1
#>
#> Coefficients:
#> mu A B D Sigma1 Sigma2 Sigma3
#> 1 0.077140 -0.167090 1.557873 0.205806 0.652482 -0.044401 0.048752
#> 2 0.140989 0.442582 0.922494 1.138489 -0.044401 0.652799 -0.064597
#> 3 0.103221 0.799325 -0.694877 1.954476 0.048752 -0.064597 0.693794