README

#HistDAWass ##(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

\[d_W(f,g)=\\sqrt{\\int\\limits_0^1{(Q_f(p) - Q_g(p))^2 dp}}\]

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

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

#From raw data to histograms

data2hist functions

#Basic statistics for a distributionH (A histogram)

mean
- the mean of a histogram
standard deviation
- the standard deviation of a histogram
skewness
- the third standardized moment of a histogram
kurthosis
- the fourth standardized momemt of a histogram

#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.

#Visualization > plot of a distributionH

plot of a MatH

plot of a HTS

#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)

#Methods for Histogram time series

Smoothing

Moving averages
Exponential smoothing

Predicting

KNN prediction of histogram time series

#Linear regression

A two component model for a linear regression using Least Square method