Package {HistDAWass}

Type:	Package
Title:	Histogram-Valued Data Analysis
Version:	1.0.9
Date:	2026-10-16
Maintainer:	Antonio Irpino <antonio.irpino@unicampania.it>
Description:	In the framework of Symbolic Data Analysis, a relatively new approach to the statistical analysis of multi-valued data, we consider histogram-valued data, i.e., data described by univariate histograms. The methods and the basic statistics for histogram-valued data are mainly based on the L2 Wasserstein metric between distributions, i.e., the Euclidean metric between quantile functions. The package contains unsupervised classification techniques, least square regression and tools for histogram-valued data and for histogram time series. An introducing paper is Irpino A. Verde R. (2015) <doi:10.1007/s11634-014-0176-4>.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Imports:	graphics, class, FactoMineR, ggplot2 (≥ 3.4.0), ggridges, grid, histogram, grDevices, stats, utils, Rcpp
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
URL:	https://github.com/Airpino/HistDAWass
BugReports:	https://github.com/Airpino/HistDAWass/issues
Depends:	R(≥ 3.1), methods
LazyData:	true
Collate:	'For_Rccp_int.R' 'All_classes.R' 'RcppExports.R' 'Utility.R' 'Met_HTS.R' 'Met_MatH.R' 'Met_distributionH.R' 'Fuzzy_cmeans.R' 'H_time_series.R' 'HistDAWass-package.R' 'Kohonen_maps.R' 'principal_components.R' 'regression.R' 'unsuperv_classification.R' 'Plotting_with_ggplot.R'
Encoding:	UTF-8
NeedsCompilation:	yes
LinkingTo:	Rcpp,RcppArmadillo
Config/roxygen2/version:	8.1.0
Packaged:	2026-10-16 14:51:34 UTC; root
Author:	Antonio Irpino [aut, cre]
Repository:	CRAN
Date/Publication:	2026-10-18 19:20:02 UTC

HistDAWass: Histogram-Valued Data Analysis

Description

Methods for the analysis of histogram-valued data, i.e. data described by univariate histograms, in the framework of Symbolic Data Analysis. The basic statistics and the methods are based on the L2 Wasserstein metric between distributions, i.e. the Euclidean metric between quantile functions.

Data structures

• distributionH: a histogram; data2hist builds one from raw data.

• MatH: a matrix of histograms (individuals by histogram-valued variables).

• HTS-class: a histogram time series.

Basic statistics

Moments of a histogram (meanH, stdH, skewH, kurtH), quantiles and cdf (compQ, compP), the L2 Wasserstein distance (WassSqDistH, WH_MAT_DIST), the mean histogram (WH.vec.mean), variance-covariance and correlation matrices (WH.var.covar).

Methods

• Clustering: WH_kmeans, WH_adaptive.kmeans, WH_fcmeans, WH_adaptive_fcmeans, WH_hclust, Kohonen maps (WH_2d_Kohonen_maps, WH_2d_Adaptive_Kohonen_maps).

• Principal component analysis: WH.1d.PCA, WH.MultiplePCA.

• Regression: WH.regression.two.components.

• Histogram time series: HTS.moving.averages, HTS.exponential.smoothing, HTS.predict.knn.

Author(s)

Maintainer: Antonio Irpino antonio.irpino@unicampania.it (ORCID)

Authors:

• Antonio Irpino antonio.irpino@unicampania.it (ORCID)

References

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

See Also

Useful links:

• https://github.com/Airpino/HistDAWass

• Report bugs at https://github.com/Airpino/HistDAWass/issues

Examples

# a matrix of four histograms (2 individuals x 2 variables)
a <- list(
  distributionH(x = c(80, 100, 120, 135, 150, 165, 180, 200, 240),
                p = c(0, 0.025, 0.1, 0.275, 0.525, 0.725, 0.887, 0.975, 1)),
  distributionH(x = c(80, 100, 120, 135, 150, 165, 180, 195, 210, 240),
                p = c(0, 0.013, 0.101, 0.255, 0.508, 0.718, 0.895, 0.961, 0.987, 1)),
  distributionH(x = c(95, 110, 125, 140, 155, 170, 185, 200, 215, 230, 245),
                p = c(0, 0.012, 0.041, 0.154, 0.36, 0.595, 0.781, 0.929, 0.972, 0.992, 1)),
  distributionH(x = c(105, 120, 135, 150, 165, 180, 195, 210, 225, 240, 260),
                p = c(0, 0.009, 0.035, 0.081, 0.186, 0.385, 0.633, 0.832, 0.932, 0.977, 1))
)
M <- MatH(x = a, nrows = 2, ncols = 2,
          rownames = c("u1", "u2"), varnames = c("Var_1", "Var_2"))
M
WH.vec.mean(M)

Age pyramids of the countries of the World in 2014

Description

Distributions of the population by 5-year age classes (relative frequencies) for 228 countries and the World (first row): whole population, male and female population.

Format

A MatH object with 229 rows (the World and 228 countries) and 3 columns: Both.Sexes.Population, Male.Population, Female.Population.

Source

United States Census Bureau, https://www.census.gov/data.html

Examples

get.MatH.main.info(Age_Pyramids_2014)

Agronomique data

Description

Distributions of the marginal costs of farms in 22 French regions.

Format

A MatH object with 22 rows (regions) and 4 columns: Y_TSC (total costs of a farm), X_Wheat (costs for wheat), X_Pig (costs for pigs), X_Cmilk (costs for cow milk).

Source

Verde, R., Irpino, A., Desbois, D. (2014). How to cope with modelling and privacy concerns? A regression model and a visualization tool for aggregated data, Conference of European Statistics Stakeholders, Rome.

Examples

Agronomique

Arithmetic operators for histograms

Description

Sum, difference and product of distributionH objects and numbers, consistent with the L2 Wasserstein metric: the operations are applied to the quantile functions of the distributions.

• e1 + e2: the sum of the quantile functions. A number is treated as an impulse distribution, so x + a shifts x by a.

• e1 - e2: the difference of the quantile functions. The difference of two distributions is a distribution only if e1 is more variable than e2 (the result must be a non-decreasing quantile function); otherwise it may not be valid. x - a shifts x by -a.

• a * x or x * a: the product of a distribution and a number, i.e. a rescaling of x.

Usage

## S4 method for signature 'distributionH,distributionH'
e1 + e2

## S4 method for signature 'distributionH,distributionH'
e1 - e2

## S4 method for signature 'distributionH,distributionH'
e1 * e2

Arguments

Value

A distributionH object.

Examples

d1 <- distributionH(x = c(1, 2, 3), p = c(0, 0.4, 1))
d2 <- distributionH(x = c(7, 8, 10, 15), p = c(0, 0.2, 0.7, 1))
d1 + d2
d2 + 10
2 * d1
d2 - d1

Blood data

Description

Histograms of cholesterol, hemoglobin and hematocrit for 14 groups of patients (by sex and age).

Format

A MatH object with 14 rows (groups of patients) and 3 columns (variables).

Source

Billard L. and Diday E. (2006). Symbolic Data Analysis: Conceptual Statistics and Data Mining, Wiley.

Examples

BLOOD
plot(BLOOD)

Blood data from Brito

Description

Histograms of hematocrit and hemoglobin for 10 patients.

Format

A MatH object with 10 rows (patients) and 2 columns (variables).

Source

Dias, S. and Brito P. (2013). Distribution and Symmetric Distribution Regression Model for Histogram-Valued Variables, arXiv:1303.6199.

Examples

BloodBRITO

Centre the histograms of a matrix

Description

Shifts each histogram of a MatH object so that its mean is zero.

Usage

Center.cell.MatH(object)

Arguments

Value

A MatH object whose histograms have zero mean.

Examples

CEN_BLOOD <- Center.cell.MatH(BLOOD)
get.MatH.stats(CEN_BLOOD, stat = "mean")

Monthly climatic data of China

Description

Distributions of 14 climatic variables observed in 60 stations of China in each month (168 histogram-valued variables).

Format

A MatH object with 60 rows (stations) and 168 columns (14 variables x 12 months).

Details

The 14 variables are: mean station pressure (mb), mean temperature, mean maximum temperature, mean minimum temperature, total precipitation (mm), sunshine duration (h), mean cloud amount (percentage of sky cover), mean relative humidity (%), snow days (days with snow cover), dominant wind direction (degrees), mean wind speed (m/s), dominant wind frequency (%), extreme maximum temperature (C), extreme minimum temperature (C). The coordinates of the stations are in stations_coordinates.

Source

Climate Data Bases of the People's Republic of China 1841-1988 (TR055), doi:10.3334/CDIAC/cli.tr055

Examples

get.MatH.main.info(China_Month)$ncols

Seasonal climatic data of China

Description

Distributions of 14 climatic variables observed in 60 stations of China in each season (56 histogram-valued variables).

Format

A MatH object with 60 rows (stations) and 56 columns (14 variables x 4 seasons).

Details

The 14 variables are: mean station pressure (mb), mean temperature, mean maximum temperature, mean minimum temperature, total precipitation (mm), sunshine duration (h), mean cloud amount (percentage of sky cover), mean relative humidity (%), snow days (days with snow cover), dominant wind direction (degrees), mean wind speed (m/s), dominant wind frequency (%), extreme maximum temperature (C), extreme minimum temperature (C). The coordinates of the stations are in stations_coordinates.

Source

Climate Data Bases of the People's Republic of China 1841-1988 (TR055), doi:10.3334/CDIAC/cli.tr055

Examples

get.MatH.main.info(China_Seas)$ncols

Ramer-Douglas-Peucker algorithm

Description

Approximates a curve, given by a sequence of 2d points, with a polyline (Ramer-Douglas-Peucker algorithm). It is used by data2hist with algo = "PolyLine".

Usage

DouglasPeucker(points, epsilon)

Arguments

Value

A two-column matrix with the vertices of the polyline.

See Also

data2hist, ShortestDistance

Examples

x <- seq(0, 1, length.out = 50)
DouglasPeucker(cbind(x, x^2), epsilon = 0.01)

Histogram time series: the HTS class

Description

The HTS class describes a histogram time series, i.e. a sequence of histograms (TdistributionH objects) observed along time.

Slots

data

A list of TdistributionH objects.

Creating objects

Objects are created with new("HTS", epocs, ListOfTimedElements), where epocs is the number of time points and ListOfTimedElements a list of TdistributionH objects.

See Also

RetHTS for an example dataset; subsetHTS, plot,HTS-method, HTS.moving.averages, HTS.exponential.smoothing, HTS.predict.knn.

Examples

RetHTS
subsetHTS(RetHTS, from = 1, to = 5)

Exponential smoothing of a histogram time series

Description

(Beta version) Exponential smoothing of a histogram time series, based on the L2 Wasserstein distance.

Usage

HTS.exponential.smoothing(HTS, alpha = 0.9)

Arguments

Value

A list with smoothing.alpha, the smoothing parameter, and SmoothedHTS, the smoothed HTS object.

See Also

HTS.moving.averages

Examples

mov.expo.smooth <- HTS.exponential.smoothing(HTS = RetHTS, alpha = 0.8)
mov.expo.smooth$SmoothedHTS

Moving averages of a histogram time series

Description

(Beta version) Smoothing of a histogram time series by (weighted) moving averages, based on the L2 Wasserstein distance.

Usage

HTS.moving.averages(HTS, k = 3, weights = rep(1, k))

Arguments

Value

A list with k, the (normalised) weights and AveragedHTS, the smoothed HTS object.

See Also

HTS.exponential.smoothing

Examples

mov.av.smoothed <- HTS.moving.averages(HTS = RetHTS, k = 5)
mov.av.smoothed$AveragedHTS

K-NN forecast of a histogram time series

Description

(Beta version) Forecasts a histogram of a time series with the k-nearest neighbours method, based on the L2 Wasserstein distance: the forecast is the mean of the histograms following the k histograms most similar to the one at position.

Usage

HTS.predict.knn(HTS, position = length(HTS@data), k = 3)

Arguments

Value

A distributionH object, the forecast.

References

Arroyo, J., Mate, C. (2009). Forecasting histogram time series with k-nearest neighbours methods. International Journal of Forecasting, 25(1), 192-207. doi:10.1016/j.ijforecast.2008.07.003

Examples

prediction <- HTS.predict.knn(HTS = RetHTS, position = 108, k = 3)

Matrices of histograms: the MatH class

Description

The MatH class describes a matrix of histograms: each cell contains a distributionH object, rows are the individuals and columns the (histogram-valued) variables.

Usage

MatH(
  x = NULL,
  nrows = 1,
  ncols = 1,
  rownames = NULL,
  varnames = NULL,
  by.row = FALSE
)

Arguments

Value

MatH() returns a MatH object.

Slots

M

A matrix of mode list, each cell containing a distributionH object.

Creating objects

Objects are created with MatH() or with new("MatH", nrows, ncols, ListOfDist, names.rows, names.cols, by.row), where ListOfDist is a list of distributionH objects and names.rows, names.cols are the names of rows and columns.

References

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

See Also

get.MatH.main.info, plot,MatH-method, WH.vec.mean, WH.var.covar.

Examples

## a list of six distributionH objects
ListOfDist <- vector("list", 6)
ListOfDist[[1]] <- distributionH(c(1, 2, 3), c(0, 0.4, 1))
ListOfDist[[2]] <- distributionH(c(7, 8, 10, 15), c(0, 0.2, 0.7, 1))
ListOfDist[[3]] <- distributionH(c(9, 11, 20), c(0, 0.5, 1))
ListOfDist[[4]] <- distributionH(c(2, 5, 8), c(0, 0.3, 1))
ListOfDist[[5]] <- distributionH(c(8, 10, 15), c(0, 0.75, 1))
ListOfDist[[6]] <- distributionH(c(20, 22, 24), c(0, 0.12, 1))
## a 3 x 2 MatH filled by columns
MyMAT <- MatH(
  x = ListOfDist, nrows = 3, ncols = 2,
  rownames = c("I1", "I2", "I3"), varnames = c("Var1", "Var2")
)
MyMAT
## an empty 10 x 4 matrix of histograms
MAT <- MatH(nrows = 10, ncols = 4)

Ozone data (stations without missing data)

Description

Histograms of four variables recorded by 78 stations in the USA, without missing data: ozone concentration (ppb), temperature (C), solar radiation (W/m2) and wind speed (m/s).

Format

A MatH object with 78 rows (stations) and 4 columns (variables).

Source

US EPA, CASTNET (Clean Air Status and Trends Network), meteorological data.

Examples

get.MatH.main.info(OzoneFull)

Ozone data (all the stations)

Description

Histograms of four variables recorded by 84 stations in the USA: ozone concentration (ppb), temperature (C), solar radiation (W/m2) and wind speed (m/s). Some stations have missing data.

Format

A MatH object with 84 rows (stations) and 4 columns (variables).

Source

US EPA, CASTNET (Clean Air Status and Trends Network), meteorological data.

Examples

get.MatH.main.info(OzoneH)

Histogram time series of returns

Description

Daily histograms of the returns of the US dollar / Japanese yen exchange rate.

Format

A HTS-class object with 108 histograms.

Examples

summaryHTS(subsetHTS(RetHTS, from = 1, to = 5))
plot(subsetHTS(RetHTS, from = 1, to = 20))

Distance between a point and a line

Description

Computes the Euclidean distance between a point and the line through two points (used by DouglasPeucker).

Usage

ShortestDistance(p, line)

Arguments

Value

A number, the distance of p from line.

See Also

DouglasPeucker

Examples

ShortestDistance(c(0, 1), rbind(c(0, 0), c(1, 0)))

Matrices of histograms with a time reference: the TMatH class

Description

The TMatH class extends MatH with a time reference (timestamps or time periods).

Slots

tstamp

A numeric vector of timestamps.

period

A list of time periods (alternative to tstamp).

M

The matrix of histograms, as in MatH-class.

Creating objects

Objects are created with new("TMatH", mat, tstamp, period), where mat is a MatH object.

See Also

MatH-class, HTS-class

Histograms with a time reference: the TdistributionH class

Description

The TdistributionH class extends distributionH with a time reference: a timestamp or a time period. It is the element of a histogram time series (HTS-class).

Slots

tstamp

A number, the timestamp.

period

A list with two elements, start and end, the bounds of the time period (alternative to tstamp).

x

The bounds of the bins (see distributionH-class).

p

The cdf at x.

m

The mean.

s

The standard deviation.

Creating objects

Objects are created with new("TdistributionH", tstamp, period, x, p, m, s), where x, p, m and s are as in distributionH.

See Also

HTS-class, plot,TdistributionH-method

Examples

td <- new("TdistributionH", tstamp = 1, x = c(1, 2, 3), p = c(0, 0.4, 1))
td@tstamp

Principal component analysis of a histogram variable

Description

Principal component analysis (PCA) of one histogram variable based on the L2 Wasserstein distance: a centred (not standardised) PCA of the quantiles of the histograms. It provides a dimensional reduction and a visualization of the distributions.

Usage

WH.1d.PCA(
  data,
  var,
  quantiles = 10,
  plots = TRUE,
  listaxes = c(1:4),
  axisequal = FALSE,
  qcut = 1,
  outl = 0
)

Arguments

Value

A list with PCAout, the results of PCA, WASSVARIANCE (not computed, always 0) and INERTIA, the total inertia.

References

Verde, R., Irpino, A., Balzanella, A. (2016). Dimension reduction techniques for distributional symbolic data. IEEE Transactions on Cybernetics, 46(2), 344-355. doi:10.1109/TCYB.2015.2389653

See Also

WH.MultiplePCA, WH.plot_multiple_Spanish.funs

Examples

results <- WH.1d.PCA(data = BLOOD, var = 1, listaxes = c(1:2))

Principal component analysis of a set of histogram variables

Description

(Beta version) Principal component analysis of several histogram variables based on the L2 Wasserstein distance: a multiple factor analysis (MFA) of the quantiles of the histograms, where each variable is a group of quantiles. It extends WH.1d.PCA.

Usage

WH.MultiplePCA(data, list.of.vars, quantiles = 10, outl = 0, graph = TRUE)

Arguments

Value

The results of MFA.

See Also

WH.plot_multiple_Spanish.funs, WH.plot_multiple_indivs

Examples

res <- WH.MultiplePCA(BLOOD, list.of.vars = c(1:3), graph = FALSE)

Bind matrices of histograms

Description

WH.bind.row binds by rows two MatH objects with the same number of columns, WH.bind.col binds by columns two MatH objects with the same number of rows. WH.bind does one of the two according to byrow.

Usage

WH.bind.row(object1, object2)

WH.bind.col(object1, object2)

WH.bind(object1, object2, byrow)

Arguments

Value

A MatH object.

Examples

WH.bind.row(BLOOD[1:3, ], BLOOD[5:8, ])
WH.bind.col(BLOOD[1:10, 1], BLOOD[1:10, 3])
WH.bind(BLOOD[1:10, 1], BLOOD[1:10, 3], byrow = FALSE)

Product of two matrices of histograms

Description

Matrix product of two MatH objects, where the product of two cells is the dot product of the histograms inducing the L2 Wasserstein metric (see dotpW).

Usage

## S4 method for signature 'MatH,MatH'
WH.mat.prod(object1, object2, traspose1 = FALSE, traspose2 = FALSE)

Arguments

Value

A numeric matrix.

Examples

MAT <- WH.mat.prod(BLOOD[1:5, ], BLOOD[6:10, ], traspose1 = TRUE)

Sum of two matrices of histograms

Description

Sums two MatH objects of the same dimensions, cell by cell, by summing the quantile functions of the histograms (a sum consistent with the L2 Wasserstein metric).

Usage

WH.mat.sum(object1, object2)

Arguments

Value

A MatH object.

Examples

MAT <- WH.mat.sum(BLOOD[1:5, ], BLOOD[6:10, ])

Plot the variables of a PCA of histogram variables (Spanish-fan plot)

Description

Plots the quantiles of the histogram variables on a factorial plane after WH.MultiplePCA (or WH.1d.PCA): the quantiles of each variable form a fan.

Usage

WH.plot_multiple_Spanish.funs(
  res,
  axes = c(1, 2),
  var = 1,
  LABS = TRUE,
  multi = TRUE,
  corplot = TRUE
)

Arguments

Value

The plot is drawn on the current device and the ggplot object is returned invisibly.

Examples

res <- WH.MultiplePCA(BLOOD, list.of.vars = c(1:3), graph = FALSE)
WH.plot_multiple_Spanish.funs(res, axes = c(1, 2), var = c(1:3))

Plot the histograms of the individuals on a factorial plane

Description

(Beta version) Plots the histograms (or densities) of one variable for each individual, placed at the coordinates of the individuals on a factorial plane of WH.MultiplePCA.

Usage

WH.plot_multiple_indivs(
  data,
  res,
  axes = c(1, 2),
  indiv = 0,
  var = 1,
  strx = 0.1,
  stry = 0.1,
  HISTO = TRUE,
  coor = 0,
  stat = "mean"
)

Arguments

Value

The two plots are drawn on the current device and a list with the two ggplot objects is returned invisibly.

Examples

results <- WH.MultiplePCA(BLOOD, list.of.vars = c(1:3), graph = FALSE)
WH.plot_multiple_indivs(BLOOD, results, axes = c(1, 2), var = 1, HISTO = TRUE)
WH.plot_multiple_indivs(BLOOD, results, axes = c(1, 2), var = 1, HISTO = FALSE)

Goodness of fit of a two-component regression of histogram variables

Description

Computes goodness of fit indices of the predictions of a WH.regression.two.components model.

Usage

WH.regression.GOF(observed, predicted)

Arguments

Value

A list with the root mean squared error based on the L2 Wasserstein distance (RMSE_W), the OMEGA index and the pseudo-R2 index (PSEUDOR2, with the details of the decomposition of the sum of squares).

References

Irpino, A., Verde, R. (2015). Linear regression for numeric symbolic variables: a least squares approach based on Wasserstein Distance. Advances in Data Analysis and Classification, 9, 81-106. doi:10.1007/s11634-015-0197-7

Examples

model.parameters <- WH.regression.two.components(data = BLOOD, Yvar = 1, Xvars = c(2:3))
Predicted.BLOOD <- WH.regression.two.components.predict(
  data = BLOOD[, 2:3],
  parameters = model.parameters
)
WH.regression.GOF(observed = BLOOD[, 1], predicted = Predicted.BLOOD)

Two-component regression of histogram variables

Description

Linear regression of a histogram variable on a set of histogram variables, based on the decomposition of the L2 Wasserstein distance into a component for the means and a component for the centred distributions. The parameters are estimated by least squares (non-negative least squares for the centred component).

Usage

WH.regression.two.components(data, Yvar, Xvars, simplify = FALSE, qua = 20)

Arguments

Value

A named vector with the estimated parameters: the intercept and the coefficients of the means (prefix AV_) and the coefficients of the centred distributions (prefix CEN_).

References

Irpino, A., Verde, R. (2015). Linear regression for numeric symbolic variables: a least squares approach based on Wasserstein Distance. Advances in Data Analysis and Classification, 9, 81-106. doi:10.1007/s11634-015-0197-7

See Also

WH.regression.two.components.predict, WH.regression.GOF

Examples

model.parameters <- WH.regression.two.components(data = BLOOD, Yvar = 1, Xvars = c(2:3))
model.parameters

Predictions of a two-component regression of histogram variables

Description

Predicts the histograms of the dependent variable using the parameters estimated by WH.regression.two.components.

Usage

WH.regression.two.components.predict(data, parameters)

Arguments

Value

A MatH object with one column, the predicted histograms.

See Also

WH.regression.GOF

Examples

model.parameters <- WH.regression.two.components(data = BLOOD, Yvar = 1, Xvars = c(2:3))
Predicted.BLOOD <- WH.regression.two.components.predict(
  data = BLOOD[, 2:3],
  parameters = model.parameters
)

Sum of squares, covariance and correlation of histogram variables

Description

Compute the matrix of the sums of squared deviations from the mean (WH.SSQ), the variance-covariance matrix (WH.var.covar) and the correlation matrix (WH.correlation) of the variables of a MatH object, according to the L2 Wasserstein metric. The functions with suffix 2 compute the same (rectangular) matrices between the variables of two MatH objects with the same number of rows.

Usage

## S4 method for signature 'MatH'
WH.SSQ(object, w = numeric(0))

## S4 method for signature 'MatH'
WH.var.covar(object, w = numeric(0))

## S4 method for signature 'MatH'
WH.correlation(object, w = numeric(0))

## S4 method for signature 'MatH,MatH'
WH.SSQ2(object1, object2, w = numeric(0))

## S4 method for signature 'MatH,MatH'
WH.var.covar2(object1, object2, w = numeric(0))

## S4 method for signature 'MatH,MatH'
WH.correlation2(object1, object2, w = numeric(0))

Arguments

Value

A numeric matrix.

References

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

Examples

WH.SSQ(BLOOD)
WH.var.covar(BLOOD)
WH.correlation(BLOOD)
# weighted, with random weights
RN <- runif(get.MatH.nrows(BLOOD))
WH.var.covar(BLOOD, w = RN)
# between two sets of variables
WH.correlation2(BLOOD[, 1], BLOOD[, 2:3])

Weighted sum and mean of the histograms of a matrix

Description

WH.vec.sum and WH.vec.mean compute the (weighted) sum and the (weighted) mean of all the histograms of a MatH object, consistently with the L2 Wasserstein metric: the result is the histogram whose quantile function is the (weighted) sum or mean of the quantile functions. The mean is the Wasserstein barycenter of the histograms.

Usage

## S4 method for signature 'MatH'
WH.vec.sum(object, w = numeric(0))

## S4 method for signature 'MatH'
WH.vec.mean(object, w = numeric(0))

Arguments

Value

A distributionH object.

Examples

WH.vec.mean(BLOOD)
WH.vec.sum(BLOOD)
# weighted mean with random weights
RN <- runif(get.MatH.nrows(BLOOD) * get.MatH.ncols(BLOOD))
WH.vec.mean(BLOOD, w = RN)

Batch Kohonen self-organising maps of histogram data with adaptive distances

Description

Batch Kohonen self-organising 2d map of the individuals of a MatH object, based on adaptive L2 Wasserstein distances.

Usage

WH_2d_Adaptive_Kohonen_maps(
  x,
  net = list(xdim = 4, ydim = 3, topo = c("rectangular")),
  kern.param = 2,
  TMAX = -9999,
  Tmin = -9999,
  niter = 30,
  repetitions,
  simplify = FALSE,
  qua = 10,
  standardize = FALSE,
  schema = 6,
  init.weights = "EQUAL",
  weight.sys = "PROD",
  theta = 2,
  Wfix = FALSE,
  verbose = FALSE,
  atleast = 2
)

Arguments

Details

Batch Self-Organising Maps (BSOM) for histogram data, with a cost function based on the L2 Wasserstein distance, i.e. the Euclidean distance between quantile functions. The adaptive version introduces an automatic system of weights in the cost function, to take into account the different effect of the variables on the map.

Value

A list with:

and further details on the parameters of the algorithm. The final map is plotted on the current graphic device.

References

Irpino A., Verde R., De Carvalho F.A.T. (2012). Batch self organizing maps for interval and histogram data. In: Proceedings of COMPSTAT 2012, ISI/IASC, 143-154.

See Also

WH_2d_Kohonen_maps

Examples

results <- WH_2d_Adaptive_Kohonen_maps(
  x = BLOOD,
  net = list(xdim = 2, ydim = 3, topo = c("rectangular")),
  repetitions = 2, simplify = TRUE,
  qua = 10, standardize = TRUE
)

Batch Kohonen self-organising maps of histogram data

Description

Batch Kohonen self-organising 2d map of the individuals of a MatH object, based on the L2 Wasserstein distance.

Usage

WH_2d_Kohonen_maps(
  x,
  net = list(xdim = 4, ydim = 3, topo = c("rectangular")),
  kern.param = 2,
  TMAX = 2,
  Tmin = 0.2,
  niter = 30,
  repetitions = 5,
  simplify = FALSE,
  qua = 10,
  standardize = FALSE,
  verbose = FALSE
)

Arguments

Details

Batch Self-Organising Maps (BSOM) for histogram data, with a cost function based on the L2 Wasserstein distance, i.e. the Euclidean distance between quantile functions. The adaptive version introduces an automatic system of weights in the cost function, to take into account the different effect of the variables on the map.

Value

A list with:

and further details on the parameters of the algorithm. The final map is plotted on the current graphic device.

References

Irpino A., Verde R., De Carvalho F.A.T. (2012). Batch self organizing maps for interval and histogram data. In: Proceedings of COMPSTAT 2012, ISI/IASC, 143-154.

See Also

WH_2d_Adaptive_Kohonen_maps

Examples

results <- WH_2d_Kohonen_maps(
  x = BLOOD,
  net = list(xdim = 2, ydim = 3, topo = c("rectangular")),
  repetitions = 2, simplify = TRUE,
  qua = 10, standardize = TRUE
)

L2 Wasserstein distance matrix

Description

Computes the matrix of the L2 Wasserstein distances between the individuals (rows) of a MatH object, summing the squared distances over the variables.

Usage

WH_MAT_DIST(x, simplify = FALSE, qua = 10, standardize = FALSE)

Arguments

Value

A symmetric matrix of L2 Wasserstein distances.

References

Irpino A., Verde R. (2006). A new Wasserstein based distance for the hierarchical clustering of histogram symbolic data. In: Batanjeli et al. (eds.) Data Science and Classification, IFCS 2006, Springer, 185-192.

See Also

WassSqDistH, WH_hclust

Examples

DMAT <- WH_MAT_DIST(x = BLOOD, simplify = TRUE)

K-means clustering of histogram data with adaptive distances

Description

K-means of the individuals of a MatH object based on adaptive L2 Wasserstein distances: the algorithm also estimates a weight for each variable (or for each component of the variables, and/or for each cluster).

Usage

WH_adaptive.kmeans(
  x,
  k,
  schema = 1,
  init,
  rep,
  simplify = FALSE,
  qua = 10,
  standardize = FALSE,
  weight.sys = "PROD",
  theta = 2,
  init.weights = "EQUAL",
  verbose = FALSE
)

Arguments

Value

A list with IDX (the cluster of each individual), cardinality (the size of each cluster), proto (a MatH object with the prototypes), weights (the weights of the variables or components), Crit (the criterion), TOTSSQ, BSQ, WSQ (total, between and within sums of squares) and quality (the proportion of the total sum of squares explained by the partition).

References

Irpino A., Verde R., De Carvalho F.A.T. (2014). Dynamic clustering of histogram data based on adaptive squared Wasserstein distances. Expert Systems with Applications, 41, 3351-3366. doi:10.1016/j.eswa.2013.12.001

See Also

WH_kmeans, WH_adaptive_fcmeans

Examples

results <- WH_adaptive.kmeans(x = BLOOD, k = 2, rep = 10,
            simplify = TRUE, qua = 10, standardize = TRUE)
results$weights

Fuzzy c-means clustering of histogram data with adaptive distances

Description

Fuzzy c-means of the individuals of a MatH object based on adaptive L2 Wasserstein distances: the algorithm also estimates a weight for each variable (or for each component of the variables, and/or for each cluster).

Usage

WH_adaptive_fcmeans(
  x,
  k = 5,
  schema,
  m = 1.6,
  rep,
  simplify = FALSE,
  qua = 10,
  standardize = FALSE,
  init.weights = "EQUAL",
  weight.sys = "PROD",
  theta = 2,
  verbose = FALSE
)

Arguments

Value

A list with:

and further details on the decomposition of the sum of squares.

See Also

WH_fcmeans, WH_adaptive.kmeans

Examples

results <- WH_adaptive_fcmeans(
  x = BLOOD, k = 2, schema = 4, m = 1.5, rep = 3, simplify = TRUE,
  qua = 10, standardize = TRUE, init.weights = "EQUAL", weight.sys = "PROD"
)
results$solution$var.weights

Fuzzy c-means clustering of histogram data

Description

Fuzzy c-means of the individuals of a MatH object, based on the L2 Wasserstein distance between histograms.

Usage

WH_fcmeans(x, k, m = 1.6, rep, simplify = FALSE, qua = 10, standardize = FALSE)

Arguments

Value

A list with:

and further details on the decomposition of the sum of squares.

See Also

WH_adaptive_fcmeans, WH_kmeans

Examples

results <- WH_fcmeans(x = BLOOD, k = 2, m = 1.5, rep = 10,
                      simplify = TRUE, qua = 10, standardize = TRUE)
results$solution$membership

Hierarchical clustering of histogram data

Description

Hierarchical clustering of the individuals of a MatH object, based on the L2 Wasserstein distance. It is a wrapper of hclust.

Usage

WH_hclust(
  x,
  simplify = FALSE,
  qua = 10,
  standardize = FALSE,
  distance = "WDIST",
  method = "complete"
)

Arguments

Value

An object of class hclust.

References

Irpino A., Verde R. (2006). A new Wasserstein based distance for the hierarchical clustering of histogram symbolic data. In: Batanjeli et al. (eds.) Data Science and Classification, IFCS 2006, Springer, 185-192.

See Also

WH_MAT_DIST

Examples

results <- WH_hclust(x = BLOOD, simplify = TRUE, method = "complete")
plot(results)
cutree(results, k = 5)

K-means clustering of histogram data

Description

K-means (dynamic clustering) of the individuals of a MatH object, based on the L2 Wasserstein distance between histograms.

Usage

WH_kmeans(
  x,
  k,
  rep = 5,
  simplify = FALSE,
  qua = 10,
  standardize = FALSE,
  verbose = FALSE
)

Arguments

Value

A list with:

References

Irpino A., Verde R., Lechevallier Y. (2006). Dynamic clustering of histograms using Wasserstein metric. In: Rizzi A., Vichi M. (eds.) COMPSTAT 2006 - Advances in computational statistics, Physica-Verlag, 869-876.

See Also

WH_adaptive.kmeans, WH_fcmeans, WH_hclust

Examples

results <- WH_kmeans(
  x = BLOOD, k = 2, rep = 10, simplify = TRUE,
  qua = 10, standardize = TRUE
)
results$solution$IDX
results$quality

Squared L2 Wasserstein distance between two histograms

Description

Computes the squared L2 Wasserstein distance between two distributionH objects, i.e. the integral of the squared difference of their quantile functions. Optionally it returns its decomposition into a position, a size and a shape component.

Usage

## S4 method for signature 'distributionH,distributionH'
WassSqDistH(object1 = object1, object2 = object2, details = FALSE)

Arguments

Value

If details = FALSE, a number, the squared distance. If details = TRUE, a named vector with the squared distance (SQ_W_dist), its position (POSITION), size (SIZE) and shape (SHAPE) components, and the correlation between the quantile functions (rQQ).

References

Irpino, A. and Romano, E. (2007): Optimal histogram representation of large data sets: Fisher vs piecewise linear approximations. RNTI E-9, 99-110.

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

See Also

WH_MAT_DIST for the distance matrix of a MatH.

Examples

mydist1 <- distributionH(x = c(1, 2, 3), p = c(0, 0.4, 1))
mydist2 <- distributionH(x = c(7, 8, 10, 15), p = c(0, 0.2, 0.7, 1))
WassSqDistH(mydist1, mydist2)
WassSqDistH(mydist1, mydist2, details = TRUE)

Fix the empty bins of histograms

Description

Checks for empty bins, i.e. bins with zero weight (two consecutive equal values of the cdf). Each empty bin gets a weight of 1e-8 and the cdf is renormalised. Since version 1.0.9 empty bins are fixed when a distributionH object is created (see distributionH), so this method is useful for objects created with previous versions of the package or modified by assigning their slots directly. The MatH method applies the check to all the cells of the matrix.

Usage

checkEmptyBins(object)

Arguments

Value

An object of the same class of object, without empty bins.

Examples

## an empty bin in an object modified by slot assignment
mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.5, 0.6, 1))
mydist@p <- c(0, 0.5, 0.5, 1)
diff(checkEmptyBins(mydist)@p)
## all the distributions of a MatH
newMat <- checkEmptyBins(China_Seas[1:5, 1:3])

Cumulative distribution function of a histogram at a value

Description

Computes the value F(q) of the cumulative distribution function of a distributionH object.

Usage

compP(object, q)

Arguments

Value

A number between 0 and 1.

See Also

compQ

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
compP(mydist, 5)

Quantile of a histogram

Description

Computes the quantile of a distributionH object at a given level, i.e. the value Q(p) = F^{-1}(p) of its quantile function.

Usage

compQ(object, p)

Arguments

Value

A number, the quantile of order p.

See Also

compP

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
compQ(mydist, 0.5) # the median
compQ(mydist, 0) # the minimum
compQ(mydist, 1) # the maximum
compQ(mydist, 0.25) # the first quartile

Centres, radii and weights of the bins of a histogram

Description

Computes the centre (midpoint), the radius (half-width) and the weight (probability) of each bin of a distributionH object. This representation allows a fast computation of the statistics based on the L2 Wasserstein distance.

Usage

crwtransform(object)

Arguments

Value

A list with three numeric vectors: Centers, the midpoints of the bins; Radii, the half-widths of the bins; Weights, the probabilities of the bins (summing to one).

References

Irpino, A., Verde, R., Lechevallier, Y. (2006) Dynamic clustering of histograms using Wasserstein metric, In: Proceedings of COMPSTAT 2006, Physica-Verlag, 869-876.

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
crwtransform(mydist)

Build a histogram from raw data

Description

Builds a distributionH object from a vector of numeric values, using one of several algorithms for choosing the bins.

Usage

data2hist(
  data,
  algo = "histogram",
  type = "combined",
  qua = 10,
  breaks = numeric(0),
  epsilon = 0.01
)

Arguments

Value

A distributionH object. Empty bins (without observations) get a weight of 1e-8.

See Also

distributionH, histogram

Examples

data <- rnorm(n = 1000, mean = 2, sd = 3)
mydist <- data2hist(data)
plot(mydist)
data2hist(data, algo = "FixedQuantiles", qua = 20)

Histogram objects: the distributionH class

Description

The distributionH class describes a histogram (a distribution) by means of its cumulative distribution function (cdf): the bounds of the bins and the values of the cdf at the bounds. All the methods of the package are based on the L2 Wasserstein distance between distributions.

Usage

distributionH(x = numeric(0), p = numeric(0))

Arguments

Value

distributionH() returns a distributionH object.

Slots

x

A non-decreasing numeric vector: the bounds of the bins (the domain of the distribution).

p

A non-decreasing numeric vector of the same length of x, from 0 to 1: the cdf at the values in x.

m

The mean of the distribution.

s

The standard deviation of the distribution.

Creating objects

Objects are created with distributionH(x, p) or, equivalently, with new("distributionH", x, p, m, s), where the optional m and s (mean and standard deviation) are computed when not given.

Empty bins, i.e. bins with zero weight (two consecutive equal values in p), may produce numerical problems. Since version 1.0.9 each empty bin gets a weight of 1e-8 and the cdf is renormalised when the object is created (see also checkEmptyBins).

References

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

See Also

data2hist to build a histogram from raw data; meanH, stdH, plot,distributionH-method.

Examples

# the histogram
# | Bins  | Prob | cdf |
# | [1,2) | 0.4  | 0.4 |
# | [2,3] | 0.6  | 1.0 |
mydist <- distributionH(x = c(1, 2, 3), p = c(0, 0.4, 1))
mydist
str(mydist)

Dot product of two histograms

Description

Computes the dot product, inducing the L2 Wasserstein metric, between two distributionH objects, or between a distributionH and a number (considered as an impulse distribution).

Usage

dotpW(e1, e2)

Arguments

Value

A number.

References

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

Examples

mydist1 <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
mydist2 <- distributionH(x = c(5, 7, 15), p = c(0, 0.7, 1))
dotpW(mydist1, mydist2) # 39.51429
dotpW(mydist1, 3) # 13.2
dotpW(3, mydist1) # 13.2

Extract a sub-matrix of histograms

Description

Extracts rows and/or columns of a MatH object, as for an ordinary matrix.

Usage

## S4 method for signature 'MatH,ANY,ANY,ANY'
x[i, j, ..., drop = TRUE]

Arguments

Value

A MatH object.

Examples

SUB_D <- BLOOD[c(1, 2, 5), c(1, 2)]
SUB_D

Dimensions and names of a matrix of histograms

Description

Functions returning the number of rows and of columns, the names of the rows (individuals) and of the columns (variables) of a MatH object.

Usage

get.MatH.nrows(object)

get.MatH.ncols(object)

get.MatH.rownames(object)

get.MatH.varnames(object)

get.MatH.main.info(object)

Arguments

Value

get.MatH.nrows and get.MatH.ncols return an integer; get.MatH.rownames and get.MatH.varnames return a character vector; get.MatH.main.info returns a list with all of them (nrows, ncols, rownames, varnames).

Examples

get.MatH.nrows(BLOOD)
get.MatH.ncols(BLOOD)
get.MatH.rownames(BLOOD)
get.MatH.varnames(BLOOD)
get.MatH.main.info(BLOOD)

Statistics of the histograms of a matrix

Description

Computes a statistic (mean, median, standard deviation, ...) for each histogram of a MatH object.

Usage

## S4 method for signature 'MatH'
get.MatH.stats(object, stat = "mean", prob = 0.5)

Arguments

Value

A list with the name of the statistic (stat), the level prob (only when stat = "quantile") and the matrix of values (mat).

Examples

get.MatH.stats(BLOOD) # means
get.MatH.stats(BLOOD, stat = "median")
get.MatH.stats(BLOOD, stat = "quantile", prob = 0.05) # 5th percentiles
get.MatH.stats(BLOOD, stat = "std")

Get or set a cell of a matrix of histograms

Description

get.cell.MatH returns the histogram in a cell of a MatH object; set.cell.MatH assigns a histogram to a cell.

Usage

get.cell.MatH(object, r, c)

set.cell.MatH(object, mat, r, c)

Arguments

Value

get.cell.MatH returns a distributionH object, set.cell.MatH the modified MatH object.

Examples

get.cell.MatH(BLOOD, r = 1, c = 1)
mydist <- distributionH(x = c(0, 1, 2, 3, 4), p = c(0, 0.1, 0.6, 0.9, 1))
MAT <- set.cell.MatH(mydist, BLOOD, r = 1, c = 1)

Cumulative distribution function of a histogram

Description

Returns the points of the cumulative distribution function (cdf) of a distributionH object.

Usage

get.distr(object)

Arguments

Value

A data.frame with two columns: x, the bounds of the bins, and p, the values of the cdf at x.

See Also

get.histo

Examples

D <- distributionH(x = c(1, 2, 3, 4), p = c(0, 0.2, 0.6, 1))
get.distr(D)

Bins of a histogram

Description

Returns the bins of a distributionH object and their probabilities.

Usage

get.histo(object)

Arguments

Value

A data.frame with one row for each bin and three columns: min.x and max.x, the bounds of the bin, and p, its probability (relative frequency).

See Also

get.distr

Examples

D <- distributionH(x = c(1, 2, 3, 4), p = c(0, 0.2, 0.6, 1))
get.histo(D)

Mean of a histogram (stored value)

Description

Returns the mean stored in the m slot of a distributionH object. To compute it from the distribution use meanH.

Usage

get.m(object)

Arguments

Value

A number, the mean of the distribution.

See Also

meanH, get.s

Examples

D <- distributionH(x = c(1, 2, 3, 4), p = c(0, 0.2, 0.6, 1))
get.m(D)

Standard deviation of a histogram (stored value)

Description

Returns the standard deviation stored in the s slot of a distributionH object. To compute it from the distribution use stdH.

Usage

get.s(object)

Arguments

Value

A non-negative number, the standard deviation of the distribution.

See Also

stdH, get.m

Examples

D <- distributionH(x = c(1, 2, 3, 4), p = c(0, 0.2, 0.6, 1))
get.s(D)

Kurtosis of a histogram

Description

Computes the kurtosis (fourth standardised moment) of a distributionH object.

Usage

kurtH(object)

Arguments

Value

A number, the kurtosis index (3 for a Gaussian distribution).

See Also

meanH, stdH, skewH

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
kurtH(mydist) # 2.133533

Mean of a histogram

Description

Computes the mean (first moment) of a distributionH object.

Usage

meanH(object)

Arguments

Value

A number, the mean of the distribution.

See Also

stdH, skewH, kurtH

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
meanH(mydist) # 4.4

Plot a histogram time series

Description

Plots a HTS object as a sequence of violin plots or boxplots, one for each timestamp.

Usage

## S4 method for signature 'HTS'
plot(x, y = "missing", type = "VIOLIN", border = "black", maxno.perplot = 30)

Arguments

Value

The plot is drawn on the current device and the ggplot object is returned invisibly, so that it can be further modified.

Examples

plot(subsetHTS(RetHTS, from = 1, to = 10))

plot(RetHTS, type = "BOXPLOT", border = "blue", maxno.perplot = 20)
plot(RetHTS, type = "VIOLIN", maxno.perplot = 20)

Plot a matrix of histograms

Description

Plots all the histograms of a MatH object: one panel for each variable, one row for each individual.

Usage

## S4 method for signature 'MatH'
plot(x, y = "missing", type = "HISTO", border = "black", angL = 330)

Arguments

Value

The plot is drawn on the current device and the ggplot object is returned invisibly, so that it can be further modified.

Examples

plot(BLOOD)

plot(BLOOD, type = "DENS", border = "blue")
plot(BLOOD, type = "BOXPLOT")

Plot a histogram with a time reference

Description

Plots a TdistributionH object: see plot,distributionH-method for the types of plot.

Usage

## S4 method for signature 'TdistributionH'
plot(x, type = "HISTO", col = "green", border = "black")

Arguments

Value

The plot is drawn on the current device and the ggplot object is returned invisibly.

Examples

td <- new("TdistributionH", tstamp = 1, x = c(1, 2, 3), p = c(0, 0.4, 1))
plot(td)

Plot a histogram

Description

Plots a distributionH object as a histogram, a cumulative distribution function, a quantile function, a density or a boxplot.

Usage

## S4 method for signature 'distributionH'
plot(x, type = "HISTO", col = "green", border = "black")

Arguments

Value

The plot is drawn on the current device and the ggplot object is returned invisibly, so that it can be further modified, e.g. p <- plot(mydist); p + ggplot2::theme_bw().

Examples

mydist <- distributionH(x = c(7, 8, 10, 15), p = c(0, 0.2, 0.7, 1))
plot(mydist)
plot(mydist, type = "HISTO", col = "red", border = "blue")
plot(mydist, type = "DENS", col = "red", border = "blue")
plot(mydist, type = "HBOXPLOT")
plot(mydist, type = "VBOXPLOT")
plot(mydist, type = "CDF")
plot(mydist, type = "QF")

Plot observed versus predicted histograms

Description

Compares observed and predicted histograms (e.g. from WH.regression.two.components.predict), one panel for each individual.

Usage

plotPredVsObs(PRED, OBS, type = "HISTO", ncolu = 2)

Arguments

Value

The plot is drawn on the current device and the ggplot object is returned invisibly.

See Also

plot_errors

Examples

pars <- WH.regression.two.components(BLOOD, Yvar = 1, Xvars = c(2:3))
PRED <- WH.regression.two.components.predict(data = BLOOD[, 2:3], parameters = pars)
OBS <- BLOOD[, 1]
plotPredVsObs(PRED, OBS, "HISTO")

plotPredVsObs(PRED, OBS, "CDF")
plotPredVsObs(PRED, OBS, "DENS")

Plot the errors of predicted histograms

Description

Plots the differences between observed and predicted histograms (e.g. from WH.regression.two.components.predict) and prints the root mean squared error based on the L2 Wasserstein distance.

Usage

plot_errors(PRED, OBS, type = "HISTO_QUA", np = 200)

Arguments

Value

The root mean squared error, invisibly. The plot is drawn on the current device.

See Also

plotPredVsObs, WH.regression.GOF

Examples

pars <- WH.regression.two.components(BLOOD, Yvar = 1, Xvars = c(2:3))
PRED <- WH.regression.two.components.predict(data = BLOOD[, 2:3], parameters = pars)
OBS <- BLOOD[, 1]
plot_errors(PRED, OBS, "HISTO_QUA")
plot_errors(PRED, OBS, "HISTO_DEN")
plot_errors(PRED, OBS, "DENS_KDE")

Correlation between the quantile functions of two histograms

Description

Computes the correlation between the quantile functions of two distributionH objects.

Usage

rQQ(e1, e2)

Arguments

Value

A number, the correlation between the quantile functions.

References

Irpino, A., Verde, R. (2015) Basic statistics for distributional symbolic variables: a new metric-based approach, Advances in Data Analysis and Classification, 9(2), 143-175. doi:10.1007/s11634-014-0176-4

Examples

mydist1 <- distributionH(x = c(1, 2, 3), p = c(0, 0.4, 1))
mydist2 <- distributionH(x = c(7, 8, 10, 15), p = c(0, 0.2, 0.7, 1))
rQQ(mydist1, mydist2) # 0.916894

Register two histograms on a common cdf

Description

Given two distributionH objects, returns two equivalent distributions sharing the same cdf values (the union of the two p vectors). This is the basis for computing the statistics based on the L2 Wasserstein distance.

Usage

register(object1, object2)

Arguments

Value

A list: the two distributionH objects, with the same p slot, and a matrix with their quantiles and the common cdf.

References

Irpino, A., Lechevallier, Y. and Verde, R. (2006): Dynamic clustering of histograms using Wasserstein metric In: Rizzi, A., Vichi, M. (eds.) COMPSTAT 2006. Physica-Verlag, Berlin, 869-876.

Examples

mydist1 <- distributionH(c(1, 2, 3), c(0, 0.4, 1))
mydist2 <- distributionH(c(7, 8, 10, 15), c(0, 0.2, 0.7, 1))
regDist <- register(mydist1, mydist2)
regDist[[1]]@p
regDist[[1]]@x
regDist[[2]]@x

Register the histograms of a matrix on a common cdf

Description

registerMH recomputes all the histograms of a MatH object so that they share the same cdf values (the same p slot), without changing the distributions. This allows a fast computation of all the methods based on the L2 Wasserstein metric. is.registeredMH checks whether the histograms already share the same cdf.

Usage

is.registeredMH(object)

registerMH(object)

Arguments

Value

registerMH returns a MatH object whose histograms share the same cdf; is.registeredMH returns TRUE or FALSE.

References

Irpino, A., Lechevallier, Y. and Verde, R. (2006): Dynamic clustering of histograms using Wasserstein metric In: Rizzi, A., Vichi, M. (eds.) COMPSTAT 2006. Physica-Verlag, Berlin, 869-876.

Examples

mydist1 <- distributionH(c(1, 2, 3), c(0, 0.4, 1))
mydist2 <- distributionH(c(7, 8, 10, 15), c(0, 0.2, 0.7, 1))
mydist3 <- distributionH(c(9, 11, 20), c(0, 0.8, 1))
MyMAT <- MatH(x = list(mydist1, mydist2, mydist3), nrows = 1, ncols = 3)
is.registeredMH(MyMAT) # FALSE
MATregistered <- registerMH(MyMAT)
is.registeredMH(MATregistered) # TRUE
MATregistered@M[[1, 1]]@p

Print a matrix of histograms

Description

Prints a MatH object showing, for each histogram, its mean and its standard deviation.

Usage

## S4 method for signature 'MatH'
show(object)

Arguments

Value

Invisibly NULL; called for its side effect.

Examples

BLOOD

Print a histogram

Description

Prints a distributionH object as a table of bins and probabilities. If the histogram has many bins, only the first and the last five are shown.

Usage

## S4 method for signature 'distributionH'
show(object)

Arguments

Value

Invisibly NULL; called for its side effect.

Examples

mydist <- distributionH(x = c(7, 8, 10, 15), p = c(0, 0.2, 0.7, 1))
mydist

Skewness of a histogram

Description

Computes the skewness (third standardised moment) of a distributionH object.

Usage

skewH(object)

Arguments

Value

A number, the skewness index.

See Also

meanH, stdH, kurtH

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
skewH(mydist) # 0.7148855

Coordinates of the stations of the China datasets

Description

Geographical coordinates of the 60 stations of China_Month and China_Seas.

Format

A data frame with 60 rows and 9 columns: ID, Name_ST (name of the station), Code_ST (code of the station), LAT.deg, LAT.min, LON.deg, LON.min (latitude and longitude, degrees and minutes), Elevation, Beginning (first year of observation).

Source

Climate Data Bases of the People's Republic of China 1841-1988 (TR055), doi:10.3334/CDIAC/cli.tr055

Examples

head(stations_coordinates)

Standard deviation of a histogram

Description

Computes the standard deviation (square root of the second centred moment) of a distributionH object.

Usage

stdH(object)

Arguments

Value

A non-negative number, the standard deviation of the distribution.

See Also

meanH, skewH, kurtH

Examples

mydist <- distributionH(x = c(1, 2, 3, 10), p = c(0, 0.1, 0.5, 1))
stdH(mydist) # 2.563851

Subset of a histogram time series

Description

Extracts the elements of a histogram time series between two positions.

Usage

subsetHTS(object, from, to)

Arguments

Value

A HTS object.

Examples

SUB_RetHTS <- subsetHTS(RetHTS, from = 1, to = 20) # the first 20 elements

Summary statistics of a histogram time series

Description

Computes basic statistics for each histogram of a HTS object.

Usage

summaryHTS(x)

Arguments

Value

A data frame with one row for each histogram: timestamp and period, mean, standard deviation, skewness, kurtosis, minimum, quartiles and maximum.

Examples

summaryHTS(subsetHTS(RetHTS, from = 1, to = 10))