---
title: "4. Principal component analysis of histogram data"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{4. Principal component analysis of histogram data}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(collapse = TRUE, comment = "#>",
                      fig.width = 7, fig.height = 5, dpi = 72, out.width = "100%")
```

```{r setup}
library(HistDAWass)
```

The principal component analysis (PCA) of histogram data is based on the
quantile functions: each histogram is represented by a set of its quantiles,
and a PCA of the quantiles is consistent with the L2 Wasserstein distance
(Verde, Irpino and Balzanella, 2016). The results are shown by
"Spanish-fan" plots of the quantiles and by plotting the histograms on the
factorial planes.

We use the `BLOOD` data (14 groups of patients, three histogram variables).

## PCA of one histogram variable

`WH.1d.PCA()` analyses a single variable. By default it draws, for each
factorial plane, four plots: the quantiles (Spanish-fan plot), the
individuals, and the individuals coloured by their mean and by their standard
deviation.

```{r, results = "hide", fig.height = 7}
pca1 <- WH.1d.PCA(BLOOD, var = 1, quantiles = 10, listaxes = c(1, 2))
```

The results of `FactoMineR::PCA()` are in `PCAout`:

```{r}
round(head(pca1$PCAout$eig, 3), 2)
round(head(pca1$PCAout$ind$coord[, 1:2]), 3)
```

The first axis is mainly related to the position (the mean) of the histograms,
the second one to their variability: the supplementary variables `vmeans` and
`vstds` (means and standard deviations of the histograms) show it.

```{r}
round(pca1$PCAout$quanti.sup$cor[, 1:2], 3)
```

`outl` removes a share of the probability from the tails of each histogram
(e.g. `outl = 0.05` removes 5% from each tail), to reduce the effect of
outlying values.

## PCA of several histogram variables

`WH.MultiplePCA()` analyses several variables together: it is a multiple
factor analysis (`FactoMineR::MFA()`) where each variable is a group of
quantiles.

```{r}
mpca <- WH.MultiplePCA(BLOOD, list.of.vars = 1:3, quantiles = 10, graph = FALSE)
round(head(mpca$eig, 3), 2)
```

The Spanish-fan plot of the quantiles of all the variables on the first
plane (correlations with the axes):

```{r}
WH.plot_multiple_Spanish.funs(mpca, axes = c(1, 2), var = 1:3)
```

The histograms of a variable, for each individual, plotted at the coordinates
of the individuals on the first factorial plane (each histogram is centred on
its mean). A second plot colours the histograms by a statistic (here the mean):

```{r, fig.height = 5}
WH.plot_multiple_indivs(BLOOD, mpca, axes = c(1, 2), var = 1,
                        HISTO = TRUE, stat = "mean")
```

The histograms are measured on the scale of the variable, while the plane has
the scale of the principal components, so their size is set automatically: by
default the average width of the histograms is one tenth of the range of the
coordinates on the x axis (`strx = 0.1`) and the highest density is one eighth
of the range on the y axis (`stry = 1/8`). All the histograms share the same
scales, so their spreads and their densities remain comparable. Larger or
smaller values zoom the histograms:

```{r, fig.height = 5}
WH.plot_multiple_indivs(BLOOD, mpca, axes = c(1, 2), var = 2,
                        strx = 0.15, stry = 0.2, HISTO = FALSE)
```

With `HISTO = FALSE` smoothed densities are drawn instead of histograms.

## Reference

Verde, R., Irpino, A., Balzanella, A. (2016) Dimension reduction techniques
for distributional symbolic data. *IEEE Transactions on Cybernetics*, 46(2),
344-355. <https://doi.org/10.1109/TCYB.2015.2389653>