Online Robust Reduced-Rank Regression

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,
  addon = 10,
  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.

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.

initial_size

Integer. The number of data points to be used in the first iteration.

addon

Integer. The number of data points to be added in the algorithm in each iteration after the first.

method

Character. The estimation method. Either "SMM" or "SAA". See Description and Detail.

SAAmethod

Character. The sub solver used in each iteration when the method is chosen to be "SAA". See Detail.

...

Additional arguments to function

optim: when the method is "SAA" and the SAAmethod is "optim"
RRRR: when the method is "SAA" and the SAAmethod is "MM"

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.

ProgressBar

Logical. Indicating if a progress bar is shown during the estimation process. The progress bar requires package lazybar to work.

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.

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.

Author

Yangzhuoran Yang

Examples