Online robust reducedrank regression with two major estimation methods:
Stochastic MajorisationMinimisation
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 )
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 reducedrank matrix \(AB'\). See 
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 
SAAmethod  Character. The sub solver used in each iteration when the 
...  Additional arguments to function

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 
ProgressBar  Logical. Indicating if a progress bar is shown during the estimation process.
The progress bar requires package 
return_data  Logical. Indicating if the data used is return in the output.
If set to 
A list of the estimated parameters of class ORRRR
.
The estimation method being used
If SAA is the major estimation method, what is the sub solver in each iteration.
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 reducedrank 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 wellchosen majorising surrogate function which can be much easier to optimise.
SMM method is robust in the sense that it assumes a heavytailed Cauchy distribution for the innovations.
update.RRRR
, RRRR
, RRR
#>res#> Online Robust ReducedRank Regression #>  #> Stochastic MajorisationMinimisation #>  #> 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# }