TRMF (Temporally Regularized Matrix Factorization)

Chad Hammerquist

Temporally Regularized Matrix Factorization

This package contains a set of functions that factor a time series matrix into a set of latent time series. Given a time series matrix \(A\), alternating least squares is used to estimate the solution to the following equation:

\[\left (X,F\right) = \arg \min \limits_{X_m,F_m \in \Theta} \left( ||W\circ \left(X_m F_m-A \right)||_F+\lambda_f^2 ||F_m||_F + \lambda_x^2 ||X_m||_F + \sum\limits_{s} R_s(X_m)\right)\] where \(W\) is a weighting matrix the same size as \(A\) and has 0’s where \(A\) has missing values. \(\Theta\) is a constraint set for \(F\), possible values are non-negative solutions, or in the interval \([0,1]\) or non-negative and sum row-wise to 1 for probability-like solutions.

The last term does the temporal regularization \[R_s(X) = \lambda_D^2||W_s(LX_s)||_2^2\] where \(L\) is a graph-Laplacian matrix, \(X_s\) is a subset of the columns of \(X\) and \(W_s\) is a diagonal weight matrix. An example of \(L\) is a finite difference matrix \(D_{\alpha}\) approximating a derivative of order \(\alpha\). In this case, if \(\alpha = 2\) then the regularization prefers penalized cubic spline solutions. If \(\alpha=1\) then it can be used to fit a random walk.

TRMF plus Regression

If necessary, external regressors can be included in matrix factorization by modifying the first term to include the external regressor:

\[\left (X,F\right) = \arg \min \limits_{X_m,F_m \in \Theta} \left( ||W\circ \left(X_m F^{(1)}_m + E_xF^{(2)}_m -A \right)||_F+\lambda_f^2 ||F_m||_F + \lambda_x^2 ||X_m||_F + \sum\limits_{s} R_s(X_m)\right)\]

References:

Yu, Hsiang-Fu, Nikhil Rao, and Inderjit S. Dhillon. “High-dimensional time series prediction with missing values.” arXiv preprint arXiv:1509.08333 (2015).

How to use

To use the TRMF package to factor a time series matrix:

  1. Create TRMF object for your time series matrix
obj = create_TRMF(A)
  1. It is recommended to scale the matrix using one of the scaling option in create_TRMF
  2. Missing values are imputed as default
  1. Add a constraint and regularization for \(F_m\) to TRMF object
obj = TRMF_coefficients(obj,reg_type = "nnls",lambda=1)
  1. Add temporal regularization model for \(X_m\) to TRMF object
obj = TRMF_trend(obj,numTS = 2,order = 2,lambdaD=1)
  1. Maybe add another temporal regularization model for \(X_m\) to TRMF object
obj = TRMF_trend(obj,numTS = 3,order = 0.5,lambdaD=10)
  1. Maybe add an external regressor
obj = TRMF_regression(obj, Xreg, type = "global")
  1. Train object
out = train(obj)
  1. Evaluate solution
summary(out)
plot(out)
resid(out)
fitted(out)
  1. Get solution
impute_TRMF(out)
coef(out)
Fm = out$Factors$Fm
Xm =out$Factors$Xm 
predict(out)