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:

\[\left (X,F\right) = \arg \min \limits_{X_m,F_m \in \Theta} \left( ||W\circ \left(X_m F_m-A \right)||_F^2+\lambda_f^2 ||F_m||_F^2 + \sum\limits_{s} R_s(X_m)\right)\] 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 for NNLS-type 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 \[R_s(X) = \lambda_D^2||W_s(LX_s)||_2^2+\lambda_A^2||X_s||_2^2\] 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.

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

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

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

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

- Create TRMF object for your time series matrix

`= create_TRMF(A) obj `

- It is recommended to scale the matrix using one of the scaling option in create_TRMF
- Missing values are imputed as default

- Add a constraint and regularization for \(F_m\) to TRMF object

`= TRMF_columns(obj,reg_type = "nnls",lambda=1) obj `

- Add temporal regularization model for \(X_m\) to TRMF object

`= TRMF_trend(obj,numTS = 2,order = 2,lambdaD=1) obj `

- Maybe add another temporal regularization model for \(X_m\) to TRMF object

`= TRMF_trend(obj,numTS = 3,order = 0.5,lambdaD=10) obj `

- Maybe add an external regressor

`= TRMF_regression(obj, Xreg, type = "global") obj `

- Train object

`= train(obj) out `

- Evaluate solution

```
summary(out)
plot(out)
resid(out)
fitted(out)
```

- Get solution

```
impute_TRMF(out)
coef(out)
= out$Factors$Fm
Fm =out$Factors$Xm
Xm predict(out)
```