---
title: "Introduction to ForeCA"
author: "Georg M. Goerg"
date: "May 21, 2020"
output: html_document
---
<!--
%\VignetteEngine{knitr::knitr}
%\VignetteIndexEntry{An Introduction to the ForeCA R package}
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```{r setup, include=FALSE}
library(knitr)
opts_chunk$set(cache = TRUE)
```

The **ForeCA** R package implements forecastable compoenent analysis (ForeCA).

```{r load-packages, cache=FALSE, message = FALSE}
library(ForeCA)
```


If you don't know about ForeCA yet, the next section gives a quick
overview.  If you know ForeCA already you can skip it and go straight to 
the examples.

## What is ForeCA?

Forecastable component analysis (ForeCA) is a novel dimension reduction (DR)
technique for multivariate time series.  Similar to other DR methods **ForeCA**
tries to find an "__interesting__" linear combination $y_t = \mathbf{w}' X_t$ of
a multivariate time series $\mathbf{X}_t$.  For principal component analysis
(PCA) *high variance* data is interesting; for slow feature analysis (SFA)
*slow* signals are interesting.  **ForeCA** tries to find linear combinations
that are easy to forecast, i.e., __forecastable__ -- hence the name.


The measure of forecastability, denoted as $\Omega(y_t)$, is based on the
spectral entropy of $y_t$, i.e., the entropy of the spectral density
$f_y(\lambda)$: high entropy means low forecastability; low entropy signals are
easy to forecast.

For more details see the original ForeCA paper:
```{r cite-ForeCA}
citation("ForeCA")
```

# Specifying estimation settings

ForeCA uses two main estimation techniques:

  * spectral density estimation of multivariate (or univariate) time series 
    $\mathbf{X}_t$ (or $y_t$);
  * Shannon entropy estimation of a univariate probability mass function.

The **ForeCA** package achieves this via

  * **astsa** package for good non-parametric estimates of the spectrum, and 
  * simple plugin estimates of Shannon entropy
   (`discrete_entropy` / `continuous_entropy`) with optional prior smoothing.

In this package details of those two estimator are specified via
`spectrum.control` and `entropy.control` arguments (lists).  For sake of
simplicity let's fix them here and use those settings all throughout the
examples.

```{r set-controls}
# spectrum control
sc <- list(method = "mvspec")
# entropy control
ec <- list(prior.weight = 1e-2)
```

For more options and detailed explanations see `?complete-controls`.

# Example: European stock market data

```{r load-eu-stock-markets}
data("EuStockMarkets")
```

The `EuStockMarkets` data contains daily closing prices for 
`r ncol(EuStockMarkets)` major European stock markets.  As usual we convert this to
log-returns to make the time series stationary.


```{r eu-stock-markets}
# log-returns in %
ret <- diff(log(EuStockMarkets)) * 100
```

## Time series EDA

`ret` contains time series of `r ncol(ret)` major European stock indices (see
`?EuStockMarkets` for details). 

```{r plot-returns, cache = FALSE, echo = FALSE}
plot(ret)
```

```{r cor-matrix}
cor.ret <- cor(ret)
kable(format(cor.ret, digits = 2), 
      caption = "Correlation matrix")
kable(format(solve(cor.ret), digits = 2),
      caption = "Conditional covariance given other variables")
```

The correlation matrix shows that they are highly correlated with each other and
the partial autocorrelation function (PACF) and spectra show that they are
slightly autocorrelated.

```{r acf-spectra}
ret.spec <- mvspectrum(ret, method = sc$method)
plot(ret.spec)
layout(matrix(seq_len(ncol(ret)), ncol = 2))
for (nn in colnames(ret)) {
  pacf(ret[, nn], main = nn)
}
```

Not surprisingly the PACF shows only very small partial correlations, since
these are stock market return and we should not expect see too large
correlations over time (cf. ["no arbitrage"
hypothesis](https://en.wikipedia.org/wiki/Arbitrage)).

More specifically, we can estimate ForeCA measure of forecastability, $\Omega$,
for each series:

```{r omega-eu-stocks}
ret.omega <- Omega(ret, spectrum.control = sc, entropy.control = ec)
ret.omega
```

According to the estimates `r names(which.min(ret.omega))` is the least
forecastable, `r names(which.max(ret.omega))` is the most forecastable stock
market.

However, we can ask if there are linear combinations of stock markets, i.e., a
*portfolio*, that are even easier to forecast.  That's exactly what ForeCA is
doing.

## foreca(): find forecastable components

The main function in the package is `foreca()`, which should be straightforward
to use as it resembles `princomp` for PCA or `fastICA` for independent component
anlaysis (ICA). In the basic setting users only have to pass the multivariate
time series and the number of components (`n.comp` -- by default it uses `2`).
We also specify `spectrum.control` and `entropy.control` but this is optional.

```{r foreca-returns}
mod.foreca.ret <- foreca(ret, n.comp = ncol(ret), 
                         spectrum.control = sc,
                         entropy.control = ec)
```
```{r show-results}
mod.foreca.ret  # this uses the print.foreca method
```

For ease of use and better integration with existing methods in R, the returned
`foreca` objects are very similar to `princomp` objects, i.e., they have
`$scores` and `$loadings` (and many other useful metrics).

```{r str-foreca-ret, eval = FALSE}
str(mod.foreca.ret) # too much to display; try it out!
```

The console print out showed that ForeCA indeed worked and it could find linear
combinations that were more forecastable than any of the original series. The
$\Omega$ score of the forecastable components (ForeCs) are

```{r foreca-ret-omega}
mod.foreca.ret$Omega
```

Recall that the most forecastable original time series had $\hat{\Omega} = 
`r round(max(ret.omega), 2)`$ (use `summary(mod.foreca.ret)$Omega.orig)` to get
them).  The loadings of the ForeCs tell us what this forecastable portfolio looks like:

```{r foreca-ret-loadings}
mod.foreca.ret$loadings
```

# S3 methods

The **ForeCA** package implements several `S3` methods for objects of class
`foreca` so that results can be quickly visualized and easily interpreted.

```{r foreca-returns-summary}
plot(mod.foreca.ret)
plot(mod.foreca.ret$scores)
```

By design of ForeCA, the returned series are uncorrelated and have zero mean and
unit variance.

```{r foreca-unit-variance-check}
round(colMeans(mod.foreca.ret$scores), 3)
kable(round(cov(mod.foreca.ret$scores), digits = 3))
```

# Session info

```{r session-info}
sessionInfo()
```
