Principal axes analysis of symbolic histogram variables

Abstract

We present a new method to perform a principal axes analysis of symbolic histogram variables. In the symbolic data analysis framework, several Histogram Principal component Analysis (Histogram PCA) have been proposed. Some approaches focus on the relationships between some specific features of histograms such as the means or the quantiles. Others use the association for distributional variables based on the squared Wasserstein distance. In this paper, we propose two new approaches. The first one uses new correlation measures based on Fisher's z scores between corresponding bins of histogram variables. We also suggest the use of the estimator proposed by Olkin and Pratt. In the first approach, histogram variables must have the same number of bins. The second proposed approach, by contrast, extends the previous proposed correlations by considering the corresponding quantiles. This second approach can be used when histograms do not have the same number of bins.

Principal component analysis: a review and recent developments

“…A recent review [50] covers several proposed definitions of PCA-type analyses for histogram data. Most of them require the definition of concepts such as distances between histograms (the Wasserstein distance being a common choice) or the sum and mean of histograms.…”

“…Another common type of symbolic data is given by histograms, which can be considered a generalization of interval-valued data where for each observation there are several intervals (the histogram bins) and associated frequencies. A recent review [50] covers several proposed definitions of PCA-type analyses for histogram data. Most of them require the definition of concepts such as distances between histograms (the Wasserstein distance being a common choice) or the sum and mean of histograms.…”

Loss-Aware Histogram Binning and Principal Component Analysis for Customer Fleet Analytics

“…According to the findings in section A, when p ≫ mk, m and n are sufficiently large, we can outline the dependence between the parameters and the information losses empirically in (19).…”

“…Billard et al [18] put out the PCA method for interval data. Makosso et al [19] extended the scope from interval data to symbolic histogram variables.…”