README

(Histogram-valued Data analysis using Wasserstein

metric)

In this document we describe the main features of the HistDAWass package. The name is the acronym for Histogram-valued Data analysis using Wasserstein metric. The implemented classes and functions are related to the analysis of data tables containing histograms in each cell instead of the classical numeric values.

In this document we describe the main features of the HistDAWass package. The name is the acronym for Histogram-valued Data analysis using Wasserstein metric. The implemented classes and functions are related to the anlysis of data tables containing histograms in each cell instead of the classical numeric values.

What is the L2 Wasserstein metric?

given two probability density functions f and g, each one has a cumulative distribution function F and G and thei respectively quantile functions (the inverse of a cumulative distribution function) Q_f and Q_g. The L2 Wasserstein distance is

${d_W(f,g)=\sqrt{\int_0^1{(Q_f(p) - Q_g(p))^2 dp}}}$

The implemented classes are those described in the following table

Class	wrapper function for initializing	Description
`distributionH`	`distributionH(x,p)`	A class describing a histogram distibution
`MatH`	`MatH(x, nrows, ncols,rownames,varnames, by.row )`	A class describing a matrix of distributions
`TdistributionH`	`TdistributionH()`	A class derived from distributionH equipped with a timestamp or a time window
`HTS`	`HTS()`	A class describing a Histgram-valued time series

distributionH

distributionH(x,p)

A class describing a histogram distibution

MatH

MatH(x, nrows, ncols,rownames,varnames, by.row )

A class describing a matrix of distributions

TdistributionH

TdistributionH()

A class derived from distributionH equipped with a timestamp or a time window

HTS

HTS()

A class describing a Histgram-valued time series

library(HistDAWass)
mydist=distributionH(x=c(0,1,2),p=c(0,0.3,1))

Basic statistics for a MatH (A matrix of histogrm-valued data)

The average hisogram of a column

It is an average histogram that minimizes the sum of squared Wasserstein distances.

The standard deviation of a variable

It is a number that measures the dispersion of a set of histograms.

The covarince matrix of a MatH

It is a matrix that measures the covariances into a set of hitogram variables.

The correlation matrix of a MatH

It is a matrix that measures the correlation into a set of hitogram variables.

Data Analysis methods

Clustering

Kmeans

Adaptive distance based Kmeans

Fuzzy cmeans

Fuzzy cmeans based on adaptive Wasserstein distances

Kohonen batch self organizing maps

Kohonen batch self organizing maps with Wasserstein adaptive distances

Hierarchical clustering

Dimension reduction techniques

Principal components analysis of a single histogram variable

Principal components analysis of a set of histogram variables (using Multiple Factor Analysis)

HistDAWass

(Histogram-valued Data analysis using Wasserstein

From raw data to histograms

Basic statistics for a distributionH (A histogram)

Basic statistics for a MatH (A matrix of histogrm-valued data)

Visualization > plot of a distributionH

Data Analysis methods

Methods for Histogram time series

Linear regression