4. Principal component analysis of histogram data

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.

pca1 <- WH.1d.PCA(BLOOD, var = 1, quantiles = 10, listaxes = c(1, 2))

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

round(head(pca1$PCAout$eig, 3), 2)
#>        eigenvalue percentage of variance cumulative percentage of variance
#> comp 1     365.43                  90.38                             90.38
#> comp 2      29.92                   7.40                             97.78
#> comp 3       6.59                   1.63                             99.41
round(head(pca1$PCAout$ind$coord[, 1:2]), 3)
#>            Dim.1 Dim.2
#> u1: F-20 -29.335 4.203
#> u2: F-30 -29.001 4.072
#> u3: F-40 -15.411 2.853
#> u4: F-50   5.048 4.858
#> u5: F-60  12.434 3.239
#> u6: F-70  12.078 2.363

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.

round(pca1$PCAout$quanti.sup$cor[, 1:2], 3)
#>        Dim.1 Dim.2
#> vmeans 0.999 0.011
#> vstds  0.140 0.896

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.

mpca <- WH.MultiplePCA(BLOOD, list.of.vars = 1:3, quantiles = 10, graph = FALSE)
round(head(mpca$eig, 3), 2)
#>        eigenvalue percentage of variance cumulative percentage of variance
#> comp 1       2.27                  68.97                             68.97
#> comp 2       0.71                  21.44                             90.40
#> comp 3       0.16                   4.79                             95.19

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

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):

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:

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