#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 |

#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.

- It is an average
- 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