# RRRR: Robust Reduced-Rank Regression using... In RRRR: Online Robust Reduced-Rank Regression Estimation

## Description

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.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```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 ) ```

## Arguments

 `y` Matrix of dimension N*P. The matrix for the response variables. See `Detail`. `x` Matrix of dimension N*Q. The matrix for the explanatory variables to be projected. See `Detail`. `z` Matrix of dimension N*R. The matrix for the explanatory variables not to be projected. See `Detail`. `mu` Logical. Indicating if a constant term is included. `r` Integer. The rank for the reduced-rank matrix AB'. See `Detail`. `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 `Detail`. `initial_B` Matrix of dimension Q*r. The initial value for matrix B. See `Detail`. `initial_D` Matrix of dimension P*R. The initial value for matrix D. See `Detail`. `initial_mu` Matrix of dimension P*1. The initial value for the constant mu. See `Detail`. `initial_Sigma` Matrix of dimension P*P. The initial value for matrix Sigma. See `Detail`. `return_data` 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.

## Details

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.

## Value

A list of the estimated parameters of class `RRRR`.

spec

The input specifications. N is the sample size.

history

The path of all the parameters during optimization and the path of the objective value.

mu

The estimated constant vector. Can be `NULL`.

A

The estimated exposure matrix.

B

The estimated factor matrix.

D

The estimated coefficient matrix of `z`.

Sigma

The estimated covariance matrix of the innovation distribution.

obj

The final objective value.

data

The data used in estimation if `return_data` is set to `TRUE`. `NULL` otherwise.

Yangzhuoran Yang

## References

Z. Zhao and D. P. Palomar, "Robust maximum likelihood estimation of sparse vector error correction model," in2017 IEEE Global Conferenceon Signal and Information Processing (GlobalSIP), pp. 913–917,IEEE, 2017.

## Examples

 ```1 2 3 4``` ```set.seed(2222) data <- RRR_sim() res <- RRRR(y=data\$y, x=data\$x, z = data\$z) res ```

RRRR documentation built on July 8, 2020, 5:51 p.m.