---
title: "2. Multimodal Independent Component Analysis (MICA) and Group Independent Component Analysis (GroupICA)"
author:
- name: Koki Tsuyuzaki
  affiliation: Department of Artificial Intelligence Medicine,
    Graduate School of Medicine, Chiba University
  email: k.t.the-answer@hotmail.co.jp
date: "`r Sys.Date()`"
bibliography: bibliography.bib
package: iTensor
output: rmarkdown::html_vignette
vignette: |
  %\VignetteIndexEntry{2. Multimodal Independent Component Analysis (MICA) and Group Independent Component Analysis (GroupICA)}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

# Introduction

In this vignette we consider approximating multiple data matrices as a product of multiple low-rank matrices (a.k.a., factor matrices).

Test data is available from `toyModel`.

```{r data, echo=TRUE}
library("iTensor")
data1 <- iTensor::toyModel("MICA")
data2 <- iTensor::toyModel("GroupICA")
str(data1, 2)
str(data2, 2)
```

Both `data1` and `data2` contain two time-series data `X` and `Y` as follows.

```{r data2, echo=TRUE, fig.height=4, fig.width=4}
plot.ts(data1$X[7700:8000,], main="data1 (X)")
plot.ts(data1$Y[7700:8000,], main="data1 (Y)")
```

```{r data3, echo=TRUE, fig.height=4, fig.width=8}
plot.ts(data2$X[4700:5000,], main="data2 (X)")
plot.ts(data2$Y[4700:5000,], main="data2 (Y)")
```

# Multimodal Independent Component Analysis (MICA)

As a formulation that extends ICA (independent component analysis) to the multiple matrices case, Multimodal Independent Component Analysis (MICA) was proposed ([@mica]). MICA extracts statistically dependent pairs of features from the sources, where the components of feature vector extracted from each source are independent.

`MICA` can be performed as follows.

```{r mica, echo=TRUE, fig.height=4, fig.width=4}
t_series <- seq(from = 0.00, to = 1.000, by = 1e-4)
out.MICA <- MICA(data1$X, data1$Y, J=3, gamma_ts = 1 - 1 / (1 + exp(-100 * (t_series - 0.3))))
```

`J` is the rank parameter for ICA and `gamma_ts` is the weighting factor for dependence on independence.
You will see that `MICA` could extract some time-series signals.

```{r pairs_mica, echo=TRUE, fig.height=4, fig.width=4}
plot.ts(out.MICA$U[7700:8000, ], main="Source Signal (X)")
plot.ts(out.MICA$V[7700:8000, ], main="Source Signal (Y)")
```

# Group Independent Component Analysis (GroupICA)

Another formulation of the decomposition is Group Independent Component Analysis (GroupICA [@groupica1; @groupica2]). `GroupICA` can be performed as follows.

```{r groupica, echo=TRUE}
out_groupica <- GroupICA(data2, J1=6, algorithm="pooled")
```

The rank for each factor matrix can be set as `J1` and the decomposition algorithm can be easily switched by `ica.algorithm` (`algorithm` of `ICA` and `ICA2` can be specified). To pool the results of ICA against each data matrix, we implemented three algorithms such as `pooled`, `Calhoun2009`, and `Pfister2018`. For the details, see the references [@groupica1; @groupica2].

You will see that `GroupICA` could extract some time-series signals.

```{r plot_groupica, echo=TRUE, fig.height=4, fig.width=8}
plot.ts(out_groupica$Ss[[1]], main="Source Signal (X)")
plot.ts(out_groupica$Ss[[2]], main="Source Signal (Y)")
```

Unlike `MICA`, `GroupICA` can also be applied to more than three matrices.

# Session Information {.unnumbered}

```{r sessionInfo, echo=FALSE}
sessionInfo()
```

# References