---
title: "TRMF (Temporally Regularized Matrix Factorization)"
author: "Chad Hammerquist"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{TRMF (Temporally Regularized Matrix Factorization)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```

## 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:

$$\left (X,F\right) = \arg \min \limits_{X_m,F_m \in \Theta} \left( ||W\circ \left(X_m F_m-A \right)||_F^2+\lambda_f^2 ||F_m||_F^2 + \sum\limits_{s} R_s(X_m)\right)$$ where $W$ is a weighting matrix the same size as $A$ and has 0's where $A$ has missing values and $||\cdot||_F$ is the Frobenius norm. $\Theta$ is a constraint set for $Fm$, 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.

#### TRMF plus Regression

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

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

#### 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

```{r,eval=FALSE}
obj = create_TRMF(A)
```

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

```{=html}
<!-- -->
```
2.  Add a constraint and regularization for $F_m$ to TRMF object

```{r,eval=FALSE}
obj = TRMF_columns(obj,reg_type = "nnls",lambda=1)
```

3.  Add temporal regularization model for $X_m$ to TRMF object

```{r,eval=FALSE}
obj = TRMF_trend(obj,numTS = 2,order = 2,lambdaD=1)
```

4.  Maybe add another temporal regularization model for $X_m$ to TRMF object

```{r,eval=FALSE}
obj = TRMF_trend(obj,numTS = 3,order = 0.5,lambdaD=10)
```

5.  Maybe add an external regressor

```{r,eval=FALSE}
obj = TRMF_regression(obj, Xreg, type = "global")
```

6.  Train object

```{r,eval=FALSE}
out = train(obj)
```

7.  Evaluate solution

```{r,eval=FALSE}
summary(out)
plot(out)
resid(out)
fitted(out)

```

8.  Get solution

```{r,eval=FALSE}
impute_TRMF(out)
coef(out)
Fm = out$Factors$Fm
Xm =out$Factors$Xm 
predict(out)

```