Online robust reduced-rank regression with two major estimation methods:

SMM

Stochastic Majorisation-Minimisation

SAA

Sample Average Approximation

ORRRR(
y,
x,
z = NULL,
mu = TRUE,
r = 1,
initial_size = 100,
method = c("SMM", "SAA"),
SAAmethod = c("optim", "MM"),
...,
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),
ProgressBar = requireNamespace("lazybar"),
return_data = TRUE
)

## Arguments

y 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 number of data points to be used in the first iteration. Integer. The number of data points to be added in the algorithm in each iteration after the first. Character. The estimation method. Either "SMM" or "SAA". See Description and Detail. Character. The sub solver used in each iteration when the method is chosen to be "SAA". See Detail. Additional arguments to function optimwhen the method is "SAA" and the SAAmethod is "optim" RRRRwhen the method is "SAA" and the SAAmethod is "MM" 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 a progress bar is shown during the estimation process. The progress bar requires package dplyr to work. 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.

## Value

A list of the estimated parameters of class ORRRR.

method

The estimation method being used

SAAmethod

If SAA is the major estimation method, what is the sub solver in each iteration.

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.

## Details

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.

The algorithm is online in the sense that the data is continuously incorporated and the algorithm can update the parameters accordingly. See ?update.RRRR for more details.

At each iteration of SAA, a new realisation of the parameters is achieved by solving the minimisation problem of the sample average of the desired objective function using the data currently incorporated. This can be computationally expensive when the objective function is highly nonconvex. The SMM method overcomes this difficulty by replacing the objective function by a well-chosen majorising surrogate function which can be much easier to optimise.

SMM method is robust in the sense that it assumes a heavy-tailed Cauchy distribution for the innovations.

update.RRRR, RRRR, RRR

## Examples

# \donttest{
set.seed(2222)
data <- RRR_sim()
res <- ORRRR(y=data$y, x=data$x, z = data\$z)#> Loading required namespace: lazybarres#> Online Robust Reduced-Rank Regression
#> ------
#> Stochastic Majorisation-Minimisation
#> ------------
#> Specifications:
#>            N            P            R            r initial_size        addon
#>         1000            3            1            1          100           10
#>
#> Coefficients:
#>          mu         A         B         D    Sigma1    Sigma2    Sigma3
#> 1  0.078343 -0.167661  1.553252  0.204748  0.656940 -0.044872  0.050316
#> 2  0.139471  0.442293  0.919832  1.138335 -0.044872  0.657402 -0.063890
#> 3  0.106746  0.801818 -0.693768  1.955019  0.050316 -0.063890  0.698777# }