Compositional Data Analysis

Description

A Collection of Functions for Compositional Data Analysis.

Details

Maintainers

Michail Tsagris <mtsagris@uoc.gr>

Note

Acknowledgments:

Michail Tsagris would like to express his acknowledgments to Professor Andy Wood and Professor Simon Preston from the university of Nottingham for being his supervisors during his PhD in compositional data analysis.

We would also like to express our acknowledgments to Profesor Kurt Hornik (and also the rest of the R core team) for his help with this package.

Manos Papadakis, undergraduate student in the department of computer science, university of Crete, is also acknowledged for his programming tips.

Ermanno Affuso from the university of South Alabama suggested that I have a default value in the function mkde.

Van Thang Hoang from Hasselt university spotted a bug in the function js.compreg.

Claudia Wehrhahn Cortes spotted a bug in the function diri.reg.

Philipp Kynast from Bruker Daltonik GmbH found a mistake in the function mkde which is now fixed.

Jasmine Heyse from the university of Ghent spotted a bug in the function kl.compreg which is now fixed.

Magne Neby suggested to add names in the covariance matrix of the divergence based regression models.

John Barry from the Centre for Environment, Fisheries, and Aquaculture Science (UK) suggested that I should add more explanation in the function diri.est. I hope it is clearer now.

Charlotte Fabri and Laura Byrne spotted a possible problem in the function zadr.

Levi Bankston found a bug in the bootstrap version of the function kl.compreg.

Sucharitha Dodamgodage suggested to add an extra case in the function dirimean.test.

Loic Mangnier found a bug in the function lc.glm which is now fixed and also became faster.

Ravi Varadhan found a bug in diri.reg and he is acknowledged for that.

Author(s)

Michail Tsagris mtsagris@uoc.gr, Giorgos Athineou <gioathineou@gmail.com>, Abdulaziz Alenazi <a.alenazi@nbu.edu.sa> and Christos Adam pada4m4@gmail.com.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

ANOVA for the log-contrast GLM versus the uncostrained GLM

Description

ANOVA for the log-contrast GLM versus the uncostrained GLM.

Usage

lcglm.aov(mod0, mod1)

Arguments

Details

A chi-square test is performed to test the zero-to-sum constraints of the regression coefficients.

Value

A vector with two values, the chi-square test statistic and its associated p-value.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

See Also

lc.glm, ulc.glm

Examples

y <- rbinom(150, 1, 0.5)
x <- as.matrix(iris[, 2:4])
x <- x / rowSums(x)
mod0 <- lc.glm(y, x)
mod1 <- ulc.glm(y, x)
lcglm.aov(mod0, mod1)

ANOVA for the log-contrast regression versus the uncostrained linear regression

Description

ANOVA for the log-contrast regression versus the uncostrained linear regression.

Usage

lcreg.aov(mod0, mod1)

Arguments

Details

An F-test is performed to test the zero-to-sum constraints of the regression coefficients.

Value

A vector with two values, the F test statistic and its associated p-value.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

See Also

lc.reg, ulc.reg, alfa.pcr, alfa.knn.reg

Examples

y <- iris[, 1]
x <- as.matrix(iris[, 2:4])
x <- x / rowSums(x)
mod0 <- lc.reg(y, x)
mod1 <- ulc.reg(y, x)
lcreg.aov(mod0, mod1)

Aitchison's test for two mean vectors and/or covariance matrices

Description

Aitchison's test for two mean vectors and/or covariance matrices.

Usage

ait.test(x1, x2, type = 1, alpha = 0.05)

Arguments

Details

The test is described in Aitchison (2003). See the references for more information.

Value

A vector with the test statistic, the p-value, the critical value and the degrees of freedom of the chi-square distribution.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

John Aitchison (2003). The Statistical Analysis of Compositional Data, p. 153-157. Blackburn Press.

See Also

comp.test

Examples

x1 <- as.matrix(iris[1:50, 1:4])
x1 <- x1 / rowSums(x1)
x2 <- as.matrix(iris[51:100, 1:4])
x2 <- x2 / rowSums(x2)
ait.test(x1, x2, type = 1)
ait.test(x1, x2, type = 2)
ait.test(x1, x2, type = 3)

All pairwise additive log-ratio transformations

Description

All pairwise additive log-ratio transformations.

Usage

alr.all(x)

Arguments

Details

The additive log-ratio transformation with the first component being the commn divisor is applied. Then all the other pairwise log-ratios are computed and added next to each column. For example, divide by the first component, then divide by the second component and so on. This means that no zeros are allowed.

Value

A matrix with all pairwise alr transformed data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

alr, alfa

Examples

x <- as.matrix(iris[, 2:4])
x <- x / rowSums(x)
y <- alr.all(x)

\alpha-generalised correlations between two compositional datasets

Description

\alpha-generalised correlations between two compositional datasets.

Usage

acor(y, x, a, type = "dcor")

Arguments

Details

The \alpha-transformation is applied to each composition and then the distance correlation or the canonical correlation is computed. If one value of \alpha is supplied the type="cancor" will return all eigenvalues. If more than one values of \alpha are provided then the first eigenvalue only will be returned.

Value

A vector or a matrix depending on the length of the values of \alpha and the type of the correlation to be computed.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

G.J. Szekely, M.L. Rizzo and N. K. Bakirov (2007). Measuring and Testing Independence by Correlation of Distances. Annals of Statistics, 35(6): 2769-2794.

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

Tsagris M. and Papadakis M. (2025). Fast and light-weight energy statistics using the R package Rfast. https://arxiv.org/abs/2501.02849v2

See Also

acor.tune, aeqdist.etest, alfa, alfa.profile

Examples

y <- rdiri(30, runif(3) )
x <- rdiri(30, runif(4) )
acor(y, x, a = 0.4)

Beta regression

Description

Beta regression.

Usage

beta.reg(y, x, xnew = NULL)

Arguments

Details

Beta regression is fitted.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ferrari S.L.P. and Cribari-Neto F. (2004). Beta Regression for Modelling Rates and Proportions. Journal of Applied Statistics, 31(7): 799-815.

See Also

beta.est, propreg, diri.reg

Examples

y <- rbeta(300, 3, 5)
x <- matrix( rnorm(300 * 2), ncol = 2)
beta.reg(y, x)

Column-wise MLE of some univariate distributions

Description

Column-wise MLE of some univariate distributions.

Usage

colbeta.est(x, tol = 1e-07, maxiters = 100, parallel = FALSE)
collogitnorm.est(x)
colunitweibull.est(x, tol = 1e-07, maxiters = 100, parallel = FALSE)
colzilogitnorm.est(x)

Arguments

Details

For each column, the same distribution is fitted and its parameters and log-likelihood are computed.

Value

A matrix with two, three or four columns. The first one, two or three columns contain the parameter(s) of the distribution, while the last column contains the relevant log-likelihood.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

N.L. Johnson, S. Kotz & N. Balakrishnan (1994). Continuous Univariate Distributions, Volume 1 (2nd Edition).

N.L. Johnson, S. Kotz & N. Balakrishnan (1970). Distributions in statistics: continuous univariate distributions, Volume 2.

J. Mazucheli, A. F. B. Menezes, L. B. Fernandes, R. P. de Oliveira & M. E. Ghitany (2020). The unit-Weibull distribution as an alternative to the Kumaraswamy distribution for the modeling of quantiles conditional on covariates. Journal of Applied Statistics, DOI:10.1080/02664763.2019.1657813.

See Also

beta.est

Examples

x <- matrix( rbeta(200, 3, 4), ncol = 4 )
a <- colbeta.est(x)

Contour plot of mixtures of Dirichlet distributions in S^2

Description

Contour plot of mixtures of Dirichlet distributions in S^2.

Usage

mixdiri.contour(a, prob, n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The user can plot only the contour lines of a Dirichlet with a given vector of parameters, or can also add the relevant data should he/she wish to.

Value

A ternary diagram with the points and the Dirichlet contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.contour, gendiri.contour, compnorm.contour, comp.kerncontour, mix.compnorm.contour, diri.nr, dda

Examples

a <- matrix( c(12, 30, 45, 32, 50, 16), byrow = TRUE,ncol = 3)
prob <- c(0.5, 0.5)
mixdiri.contour(a, prob)

Contour plot of the Dirichlet distribution in S^2

Description

Contour plot of the Dirichlet distribution in S^2.

Usage

diri.contour(a, n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The user can plot only the contour lines of a Dirichlet with a given vector of parameters, or can also add the relevant data should he/she wish to.

Value

A ternary diagram with the points and the Dirichlet contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

mixdiri.contour, gendiri.contour, compnorm.contour, comp.kerncontour, mix.compnorm.contour

Examples

x <- as.matrix( iris[, 1:3] )
x <- x / rowSums(x)
diri.contour( a = c(3, 4, 2) )

Contour plot of the Flexible Dirichlet distribution in S^2

Description

Contour plot of the Flexible Dirichlet distribution in S^2.

Usage

fd.contour(alpha, prob, tau, n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The user can plot only the contour lines of a Dirichlet with a given vector of parameters, or can also add the relevant data should they wish to.

Value

A ternary diagram with the points and the Flexible Dirichlet contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Ongaro A. and Migliorati S. (2013). A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati S., Ongaro A. and Monti G. S. (2017). A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, 27, 963–983.

See Also

compnorm.contour, folded.contour, bivt.contour, comp.kerncontour, mix.compnorm.contour

Examples

fd.contour(alpha = c(10, 11, 12), prob = c(0.25, 0.25, 0.5), tau = 4)

Contour plot of the Gaussian mixture model in S^2

Description

Contour plot of the Gaussian mixture model in S^2.

Usage

mix.compnorm.contour(mod, type = "alr", n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The contour plot of a Gaussian mixture model is plotted. For this you need the (fitted) model.

Value

A ternary plot with the data and the contour lines of the fitted Gaussian mixture model.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Ryan P. Browne, Aisha ElSherbiny and Paul D. McNicholas (2015). R package mixture: Mixture Models for Clustering and Classification

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

mix.compnorm, bic.mixcompnorm, diri.contour

Examples

x <- as.matrix(iris[, 1:3])
x <- x / rowSums(x)
mod <- mix.compnorm(x, 3, model = "EII")
mix.compnorm.contour(mod, "alr")

Contour plot of the \alpha multivariate normal in S^2

Description

Contour plot of the \alpha multivariate normal in S^2.

Usage

alfa.contour(m, s, a, n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The \alpha-transformation is applied to the compositional data and then for a grid of points within the 2-dimensional simplex, the density of the \alpha multivariate normal is calculated and the contours are plotted.

Value

The contour plot of the \alpha multivariate normal appears.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Tsagris M. and Stewart C. (2022). A Review of Flexible Transformations for Modeling Compositional Data. In Advances and Innovations in Statistics and Data Science, pp. 225–234. https://link.springer.com/chapter/10.1007/978-3-031-08329-7_10

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

folded.contour, compnorm.contour, diri.contour, mix.compnorm.contour, bivt.contour, skewnorm.contour

Examples

x <- as.matrix(iris[, 1:3])
x <- x / rowSums(x)
a <- a.est(x)$best
m <- colMeans(alfa(x, a)$aff)
s <- cov(alfa(x, a)$aff)
alfa.contour(m, s, a)

Contour plot of the \alpha-folded model in S^2

Description

Contour plot of the \alpha-folded model in S^2.

Usage

folded.contour(mu, su, p, a, n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The \alpha-transformation is applied to the compositional data and then for a grid of points within the 2-dimensional simplex the folded model's density is calculated and the contours are plotted.

Value

The contour plot of the folded model appears.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Tsagris M. and Stewart C. (2022). A Review of Flexible Transformations for Modeling Compositional Data. In Advances and Innovations in Statistics and Data Science, pp. 225–234. https://link.springer.com/chapter/10.1007/978-3-031-08329-7_10

Tsagris M. and Stewart C. (2020). A folded model for compositional data analysis. Australian and New Zealand Journal of Statistics, 62(2): 249-277. https://arxiv.org/pdf/1802.07330.pdf

See Also

alfa.contour, compnorm.contour, diri.contour, mix.compnorm.contour, bivt.contour, skewnorm.contour

Examples

x <- as.matrix(iris[, 1:3])
x <- x / rowSums(x)
a <- a.est(x)$best
mod <- alpha.mle(x, a)
folded.contour(mod$mu, mod$su, mod$p, a)

Contour plot of the generalised Dirichlet distribution in S^2

Description

Contour plot of the generalised Dirichlet distribution in S^2.

Usage

gendiri.contour(a, b, n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The user can plot only the contour lines of a Dirichlet with a given vector of parameters, or can also add the relevant data should he/she wish to.

Value

A ternary diagram with the points and the Dirichlet contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.contour, mixdiri.contour, compnorm.contour, comp.kerncontour, mix.compnorm.contour

Examples

x <- as.matrix( iris[, 1:3] )
x <- x / rowSums(x)
gendiri.contour( a = c(3, 4, 2), b = c(1, 2, 3) )

Contour plot of the kernel density estimate in S^2

Description

Contour plot of the kernel density estimate in S^2.

Usage

comp.kerncontour(x, type = "alr", n = 50, cont.line = FALSE)

Arguments

Details

The alr or the ilr transformation are applied to the compositional data. Then, the optimal bandwidth using maximum likelihood cross-validation is chosen. The multivariate normal kernel density is calculated for a grid of points. Those points are the points on the 2-dimensional simplex. Finally the contours are plotted.

Value

A ternary diagram with the points and the kernel contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

M.P. Wand and M.C. Jones (1995). Kernel smoothing, CrC Press.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.contour, mix.compnorm.contour, bivt.contour, compnorm.contour

Examples

x <- as.matrix(iris[, 1:3])
x <- x / rowSums(x)
comp.kerncontour(x, type = "alr", n = 20)
comp.kerncontour(x, type = "ilr", n = 20)

Contour plot of the normal distribution in S^2

Description

Contour plot of the normal distribution in S^2.

Usage

compnorm.contour(m, s, type = "alr", n = 100, x = NULL, cont.line = FALSE)

Arguments

Details

The alr or the ilr transformation is applied to the compositional data at first. Then for a grid of points within the 2-dimensional simplex the bivariate normal density is calculated and the contours are plotted along with the points.

Value

A ternary diagram with the points (if appear = TRUE) and the bivariate normal contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

See Also

diri.contour, mix.compnorm.contour, bivt.contour, skewnorm.contour

Examples

x <- as.matrix(iris[, 1:3])
x <- x / rowSums(x)
y <- Compositional::alr(x)
m <- colMeans(y)
s <- cov(y)
compnorm.contour(m, s)

Contour plot of the skew skew-normal distribution in S^2

Description

Contour plot of the skew skew-normal distribution in S^2.

Usage

skewnorm.contour(x, type = "alr", n = 100, appear = TRUE, cont.line = FALSE)

Arguments

Details

The alr or the ilr transformation is applied to the compositional data at first. Then for a grid of points within the 2-dimensional simplex the bivariate skew skew-normal density is calculated and the contours are plotted along with the points.

Value

A ternary diagram with the points (if appear = TRUE) and the bivariate skew skew-normal contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Azzalini A. and Valle A. D. (1996). The multivariate skew-skewnormal distribution. Biometrika 83(4): 715–726.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.contour, mix.compnorm.contour, bivt.contour, compnorm.contour

Examples

x <- as.matrix(iris[51:100, 1:3])
x <- x / rowSums(x)
skewnorm.contour(x)

Contour plot of the t distribution in S^2

Description

Contour plot of the t distribution in S^2.

Usage

bivt.contour(x, type = "alr", n = 100, appear = TRUE, cont.line = FALSE)

Arguments

Details

The alr or the ilr transformation is applied to the compositional data at first and the location, scatter and degrees of freedom of the bivariate t distribution are computed. Then for a grid of points within the 2-dimensional simplex the bivariate t density is calculated and the contours are plotted along with the points.

Value

A ternary diagram with the points (if appear = TRUE) and the bivariate t contour lines.

Author(s)

Michail Tsagris and Christos Adam.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Christos Adam pada4m4@gmail.com.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.contour, mix.compnorm.contour, compnorm.contour, skewnorm.contour

Examples

x <- as.matrix( iris[, 1:3] )
x <- x / rowSums(x)
bivt.contour(x)
bivt.contour(x, type = "ilr")

Cross validation for some compositional regression models

Description

Cross validation for some compositional regression models.

Usage

cv.comp.reg(y, x, type = "comp.reg", nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

A k-fold cross validation for a compositional regression model is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

See Also

comp.reg, kl.compreg, compppr.tune, aknnreg.tune

Examples

y <- as.matrix( iris[, 1:3] )
y <- y / rowSums(y)
x <- iris[, 4]
mod <- cv.comp.reg(y, x)

Cross validation for the TFLR model

Description

Cross validation for the TFLR model.

Usage

cv.tflr(y, x, nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

A k-fold cross validation for the transformation-free linear regression for compositional responses and predictors is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Fiksel J., Zeger S. and Datta A. (2022). A transformation-free linear regression for compositional outcomes and predictors. Biometrics, 78(3): 974–987.

Tsagris. M. (2025). Constrained least squares simplicial-simplicial regression. Statistics and Computing, 35(27).

See Also

tflr, cv.scls, klalfapcr.tune

Examples

library(MASS)
y <- rdiri(100, runif(3, 1, 3))
x <- as.matrix(fgl[1:100, 2:9])
x <- x / rowSums(x)
mod <- cv.tflr(y, x)
mod

Cross validation for the \alpha-k-NN regression with compositional predictor variables

Description

Cross validation for the \alpha-k-NN regression with compositional predictor variables.

Usage

alfaknnreg.tune(y, x, a = seq(-1, 1, by = 0.1), k = 2:10, nfolds = 10,
apostasi = "euclidean", method = "average", folds = NULL, seed = NULL, graph = FALSE)

Arguments

Details

A k-fold cross validation for the \alpha-k-NN regression for compositional response data is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M., Alenazi A. and Stewart C. (2023). Flexible non-parametric regression models for compositional response data with zeros. Statistics and Computing, 33(106).

https://link.springer.com/article/10.1007/s11222-023-10277-5

See Also

alfa.rda, alfa.fda

Examples

library(MASS)
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
y <- fgl[, 1]
mod <- alfaknnreg.tune(y, x, a = seq(0.2, 0.4, by = 0.1), k = 2:4, nfolds = 5)

Cross validation for the \alpha-k-NN regression with compositional response data

Description

Cross validation for the \alpha-k-NN regression with compositional response data.

Usage

aknnreg.tune(y, x, a = seq(0.1, 1, by = 0.1), k = 2:10, apostasi = "euclidean",
nfolds = 10, folds = NULL, seed = NULL, rann = FALSE)

Arguments

Details

A k-fold cross validation for the \alpha-k-NN regression for compositional response data is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M., Alenazi A. and Stewart C. (2023). Flexible non-parametric regression models for compositional response data with zeros. Statistics and Computing, 33(106).

https://link.springer.com/article/10.1007/s11222-023-10277-5

See Also

aknn.reg, akernreg.tune, akern.reg, alfa.rda, alfa.fda

Examples

y <- as.matrix( iris[, 1:3] )
y <- y / rowSums(y)
x <- iris[, 4]
mod <- aknnreg.tune(y, x, a = c(0.4, 0.6), k = 2:4, nfolds = 5)

Cross validation for the \alpha-kernel regression with compositional response data

Description

Cross validation for the \alpha-kernel regression with compositional response data.

Usage

akernreg.tune(y, x, a = seq(0.1, 1, by = 0.1), h = seq(0.1, 1, length = 10),
type = "gauss", nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

A k-fold cross validation for the \alpha-kernel regression for compositional response data is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M., Alenazi A. and Stewart C. (2023). Flexible non-parametric regression models for compositional response data with zeros. Statistics and Computing, 33(106).

https://link.springer.com/article/10.1007/s11222-023-10277-5

See Also

akern.reg, aknnreg.tune, aknn.reg, alfa.rda, alfa.fda

Examples

y <- as.matrix( iris[, 1:3] )
y <- y / rowSums(y)
x <- iris[, 4]
mod <- akernreg.tune(y, x, a = c(0.4, 0.6), h = c(0.1, 0.2), nfolds = 5)

Cross validation for the kernel regression with Euclidean response data

Description

Cross validation for the kernel regression with Euclidean response data.

Usage

kernreg.tune(y, x, h = seq(0.1, 1, length = 10), type = "gauss",
nfolds = 10, folds = NULL, seed = NULL, graph = FALSE, ncores = 1)

Arguments

Details

A k-fold cross validation for the kernel regression with a euclidean response is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Wand M. P. and Jones M. C. (1994). Kernel smoothing. CRC press.

See Also

kern.reg, aknnreg.tune, aknn.reg

Examples

y <- iris[, 1]
x <- iris[, 2:4]
mod <- kernreg.tune(y, x, h = c(0.1, 0.2, 0.3) )

Cross validation for the regularised and flexible discriminant analysis with compositional data using the \alpha-transformation

Description

Cross validation for the regularised and flexible discriminant analysis with compositional data using the \alpha-transformation.

Usage

alfarda.tune(x, ina, a = seq(-1, 1, by = 0.1), nfolds = 10,
gam = seq(0, 1, by = 0.1), del = seq(0, 1, by = 0.1),
ncores = 1, folds = NULL, stratified = TRUE, seed = NULL)

alfafda.tune(x, ina, a = seq(-1, 1, by = 0.1), nfolds = 10,
folds = NULL, stratified = TRUE, seed = NULL, graph = FALSE)

Arguments

Details

A k-fold cross validation is performed.

Value

For the alfa.rda a list including:

For the alfa.fda a graph (if requested) with the estimated performance for each value of \alpha and a list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Friedman Jerome, Trevor Hastie and Robert Tibshirani (2009). The elements of statistical learning, 2nd edition. Springer, Berlin

Tsagris M.T., Preston S. and Wood A.T.A. (2016). Improved classification for compositional data using the \alpha-transformation. Jounal of Classification, 33(2):243-261.

Hastie, Tibshirani and Buja (1994). Flexible Disriminant Analysis by Optimal Scoring. Journal of the American Statistical Association, 89(428):1255-1270.

See Also

alfa.rda, alfanb.tune, cv.dda, compknn.tune cv.compnb

Examples

library(MASS)
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
ina <- fgl[, 10]
moda <- alfarda.tune(x, ina, a = seq(0.7, 1, by = 0.1), nfolds = 10,
gam = seq(0.1, 0.3, by = 0.1), del = seq(0.1, 0.3, by = 0.1) )

Cross validation for the ridge regression

Description

Cross validation for the ridge regression is performed. There is an option for the GCV criterion which is automatic.

Usage

ridge.tune(y, x, nfolds = 10, lambda = seq(0, 2, by = 0.1), folds = NULL,
ncores = 1, seed = NULL, graph = FALSE)

Arguments

Details

A k-fold cross validation is performed. This function is used by alfaridge.tune.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Giorgos Athineou <gioathineou@gmail.com> and Michail Tsagris mtsagris@uoc.gr.

References

Hoerl A.E. and R.W. Kennard (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1):55-67.

Brown P. J. (1994). Measurement, Regression and Calibration. Oxford Science Publications.

See Also

ridge.reg, alfaridge.tune

Examples

y <- as.vector(iris[, 1])
x <- as.matrix(iris[, 2:4])
ridge.tune( y, x, nfolds = 10, lambda = seq(0, 2, by = 0.1), graph = TRUE )

Cross validation for the ridge regression with compositional data as predictor using the \alpha-transformation

Description

Cross validation for the ridge regression is performed. There is an option for the GCV criterion which is automatic. The predictor variables are compositional data and the \alpha-transformation is applied first.

Usage

alfaridge.tune(y, x, nfolds = 10, a = seq(-1, 1, by = 0.1),
lambda = seq(0, 2, by = 0.1), folds = NULL, ncores = 1,
graph = TRUE, col.nu = 15, seed = NULL)

Arguments

Details

A k-fold cross validation is performed.

Value

If graph is TRUE a fileld contour a filled contour will appear. A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Giorgos Athineou <gioathineou@gmail.com> and Michail Tsagris mtsagris@uoc.gr.

References

Hoerl A.E. and R.W. Kennard (1970). Ridge regression: Biased estimation for nonorthogonal problems. Technometrics, 12(1):55-67.

Brown P. J. (1994). Measurement, Regression and Calibration. Oxford Science Publications.

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa.ridge, ridge.tune

Examples

library(MASS)
y <- as.vector(fgl[, 1])
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
alfaridge.tune( y, x, nfolds = 10, a = seq(0.1, 1, by = 0.1),
lambda = seq(0, 1, by = 0.1) )

Cross-validation for LASSO with compositional predictors using the alpha-transformation

Description

Cross-validation for LASSO with compositional predictors using the alpha-transformation.

Usage

alfalasso.tune(y, x, a = seq(-1, 1, by = 0.1), model = "gaussian", lambda = NULL,
type.measure = "mse", nfolds = 10, folds = NULL, stratified = FALSE)

Arguments

Details

The function uses the glmnet package to perform LASSO penalised regression. For more details see the function in that package.

Value

A matrix with two columns and number of rows equal to the number of \alpha values used. Each row contains, the optimal value of the \lambda penalty parameter for the LASSO and the optimal value of the loss function, for each value of \alpha.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Friedman, J., Hastie, T. and Tibshirani, R. (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), 1–22.

See Also

alfa.lasso, cv.lasso.klcompreg, lasso.compreg, alfa.knn.reg

Examples

y <- iris[, 1]
x <- rdiri(150, runif(20, 2, 5) )
mod <- alfalasso.tune( y, x, a = c(0.2, 0.5, 1) )

Cross-validation for the Dirichlet discriminant analysis

Description

Cross-validation for the Dirichlet discriminant analysis.

Usage

cv.dda(x, ina, nfolds = 10, folds = NULL, stratified = TRUE, seed = NULL)

Arguments

Details

This function estimates the performance of the Dirichlet discriminant analysis via k-fold cross-validation.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Friedman J., Hastie T. and Tibshirani R. (2017). The elements of statistical learning. New York: Springer.

Thomas P. Minka (2003). Estimating a Dirichlet distribution. http://research.microsoft.com/en-us/um/people/minka/papers/dirichlet/minka-dirichlet.pdf

See Also

dda, alfanb.tune, alfarda.tune, compknn.tune, cv.compnb

Examples

x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
mod <- cv.dda(x, ina = iris[, 5] )

Cross-validation for the LASSO Kullback-Leibler divergence based regression

Description

Cross-validation for the LASSO Kullback-Leibler divergence based regression.

Usage

cv.lasso.klcompreg(y, x, alpha = 1, type = "grouped", nfolds = 10,
folds = NULL, seed = NULL, graph = FALSE)

Arguments

Details

The K-fold cross validation is performed in order to select the optimal value for \lambda, the penalty parameter in LASSO.

Value

The outcome is the same as in the R package glmnet. The extra addition is that if "graph = TRUE", then the plot of the cross-validated object is returned. The contains the logarithm of \lambda and the deviance. The numbers on top of the figure show the number of set of coefficients for each component, that are not zero.

Author(s)

Michail Tsagris and Abdulaziz Alenazi.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Abdulaziz Alenazi a.alenazi@nbu.edu.sa.

References

Alenazi, A. A. (2022). f-divergence regression models for compositional data. Pakistan Journal of Statistics and Operation Research, 18(4): 867–882.

Friedman, J., Hastie, T. and Tibshirani, R. (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), 1-22.

See Also

lasso.klcompreg, lassocoef.plot, lasso.compreg, cv.lasso.compreg, kl.compreg

Examples

library(MASS)
y <- rdiri( 214, runif(4, 1, 3) )
x <- as.matrix( fgl[, 2:9] )
mod <- cv.lasso.klcompreg(y, x)

Cross-validation for the LASSO log-ratio regression with compositional response

Description

Cross-validation for the LASSO log-ratio regression with compositional response.

Usage

cv.lasso.compreg(y, x, alpha = 1, nfolds = 10,
folds = NULL, seed = NULL, graph = FALSE)

Arguments

Details

The K-fold cross validation is performed in order to select the optimal value for \lambda, the penalty parameter in LASSO.

Value

The outcome is the same as in the R package glmnet. The extra addition is that if "graph = TRUE", then the plot of the cross-validated object is returned. The contains the logarithm of \lambda and the mean squared error. The numbers on top of the figure show the number of set of coefficients for each component, that are not zero.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Friedman, J., Hastie, T. and Tibshirani, R. (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), 1-22.

See Also

lasso.compreg, lasso.klcompreg, lassocoef.plot, cv.lasso.klcompreg, comp.reg

Examples

library(MASS)
y <- rdiri( 214, runif(4, 1, 3) )
x <- as.matrix( fgl[, 2:9] )
mod <- cv.lasso.compreg(y, x)

Cross-validation for the SCLS model

Description

Cross-validation for the SCLS model.

Usage

cv.scls(y, x, nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

The function performs k-fold cross-validation for the least squares regression where the beta coefficients are constained to be positive and sum to 1.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris. M. (2025). Constrained least squares simplicial-simplicial regression. Statistics and Computing, 35(27).

See Also

scls, cv.tflr, klalfapcr.tune

Examples

library(MASS)
set.seed(1234)
y <- rdiri(214, runif(3, 1, 3))
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
mod <- cv.scls(y, x, nfolds = 5, seed = 12345)
mod

Cross-validation for the SCRQ model

Description

Cross-validation for the SCRQ model.

Usage

cv.scrq(y, x, nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

The function performs k-fold cross-validation for the absolute regression where the beta coefficients are constained to be positive and sum to 1.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris. M. (2025). Constrained least squares simplicial-simplicial regression. Statistics and Computing, 35(27).

See Also

scrq, cv.scls, cv.tflr

Examples

y <- rdiri(500, runif(3, 1, 3))
x <- rdiri(500, runif(3, 1, 3))
mod <- scrq(y, x)

Cross-validation for the alpha-SCLS model

Description

Cross-validation for the alpha-SCLS model.

Usage

cv.ascls(y, x, a = seq(0.1, 1, by = 0.1), nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

The K-fold cross validation is performed in order to select the optimal value for \alpha of the \alpha-SCLS model.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris. M. (2025). Constrained least squares simplicial-simplicial regression. Statistics and Computing, 35(27).

See Also

ascls, cv.atflr

Examples

library(MASS)
y <- rdiri( 214, runif(4, 1, 3) )
x <- as.matrix( fgl[, 2:9] )
mod <- cv.ascls(y, x, nfolds = 5)

Cross-validation for the alpha-TFLR model

Description

Cross-validation for the alpha-TFLR model.

Usage

cv.atflr(y, x, a = seq(0.1, 1, by = 0.1), nfolds = 10, folds = NULL, seed = NULL)

Arguments

Details

The K-fold cross validation is performed in order to select the optimal value for \alpha of the \alpha-TFLR model.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Fiksel J., Zeger S. and Datta A. (2022). A transformation-free linear regression for compositional outcomes and predictors. Biometrics, 78(3): 974–987.

Tsagris. M. (2025). Constrained least squares simplicial-simplicial regression. Statistics and Computing, 35(27).

See Also

atflr, cv.ascls

Examples

library(MASS)
y <- rdiri( 214, runif(4, 1, 3) )
x <- as.matrix( fgl[, 2:9] )
mod <- cv.ascls(y, x, nfolds = 2, a = c(0.5, 1))

Cross-validation for the naive Bayes classifiers for compositional data

Description

Cross-validation for the naive Bayes classifiers for compositional data.

Usage

cv.compnb(x, ina, type = "beta", folds = NULL, nfolds = 10,
      stratified = TRUE, seed = NULL, pred.ret = FALSE)

Arguments

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Friedman J., Hastie T. and Tibshirani R. (2017). The elements of statistical learning. New York: Springer.

See Also

comp.nb

Examples

x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
mod <- cv.compnb(x, ina = iris[, 5] )

Cross-validation for the naive Bayes classifiers for compositional data using the \alpha-transformation

Description

Cross-validation for the naive Bayes classifiers for compositional data using the \alpha-transformation.

Usage

alfanb.tune(x, ina, a = seq(-1, 1, by = 0.1), type = "gaussian",
folds = NULL, nfolds = 10, stratified = TRUE, seed = NULL)

Arguments

Details

This function estimates the performance of the naive Bayes classifier for each value of \alpha of the \alpha-transformation.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Friedman J., Hastie T. and Tibshirani R. (2017). The elements of statistical learning. New York: Springer.

See Also

alfa.nb, alfarda.tune, compknn.tune, cv.dda, cv.compnb

Examples

x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
mod <- alfanb.tune(x, ina = iris[, 5], a = c(0, 0.1, 0.2) )

Simulation of compositional data from Gaussian mixture models

Description

Simulation of compositional data from Gaussian mixture models.

Usage

dmix.compnorm(x, mu, sigma, prob, type = "alr", logged = TRUE)

Arguments

Details

A sample from a multivariate Gaussian mixture model is generated.

Value

A vector with the density values.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ryan P. Browne, Aisha ElSherbiny and Paul D. McNicholas (2015). R package mixture: Mixture Models for Clustering and Classification.

See Also

mix.compnorm, bic.mixcompnorm

Examples

p <- c(1/3, 1/3, 1/3)
mu <- matrix(nrow = 3, ncol = 4)
s <- array( dim = c(4, 4, 3) )
x <- as.matrix(iris[, 1:4])
ina <- as.numeric(iris[, 5])
mu <- rowsum(x, ina) / 50
s[, , 1] <- cov(x[ina == 1, ])
s[, , 2] <- cov(x[ina == 2, ])
s[, , 3] <- cov(x[ina == 3, ])
y <- rmixcomp(100, p, mu, s, type = "alr")$x
mod <- dmix.compnorm(y, mu, s, p)

Density of the Flexible Dirichlet distribution

Description

Density of the Flexible Dirichlet distribution

Usage

dfd(x, alpha, prob, tau)

Arguments

Details

For more information see the references and the package FlxeDir.

Value

The density value(s).

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ongaro A. and Migliorati S. (2013). A generalization of the Dirichlet distribution. Journal of Multivariate Analysis, 114, 412–426.

Migliorati S., Ongaro A. and Monti G. S. (2017). A structured Dirichlet mixture model for compositional data: inferential and applicative issues. Statistics and Computing, 27, 963–983.

See Also

rfd

Examples

alpha <- c(12, 11, 10)
prob <- c(0.25, 0.25, 0.5)
tau <- 8
x <- rfd(20, alpha, prob, tau)
dfd(x, alpha, prob, tau)

Density of the folded model normal distribution

Description

Density of the folded model normal distribution.

Usage

dfolded(x, a, p, mu, su, logged = TRUE)

Arguments

Details

Density values of the folded model.

Value

The density value(s).

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M. and Stewart C. (2020). A folded model for compositional data analysis. Australian and New Zealand Journal of Statistics, 62(2): 249-277. https://arxiv.org/pdf/1802.07330.pdf

See Also

rfolded, a.est, folded.contour

Examples

s <- c(0.1490676523, -0.4580818209,  0.0020395316, -0.0047446076, -0.4580818209,
1.5227259250,  0.0002596411,  0.0074836251,  0.0020395316,  0.0002596411,
0.0365384838, -0.0471448849, -0.0047446076,  0.0074836251, -0.0471448849,
0.0611442781)
s <- matrix(s, ncol = 4)
m <- c(1.715, 0.914, 0.115, 0.167)
x <- rfolded(100, m, s, 0.5)
mod <- a.est(x)
den <- dfolded(x, mod$best, mod$p, mod$mu, mod$su)

Density values of a Dirichlet distribution

Description

Density values of a Dirichlet distribution.

Usage

ddiri(x, a, logged = TRUE)

Arguments

Details

The density of the Dirichlet distribution for a vector or a matrix of compositional data is returned.

Value

A vector with the density values.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

See Also

dgendiri, diri.nr, diri.est, diri.contour, rdiri, dda

Examples

x <- rdiri( 100, c(5, 7, 4, 8, 10, 6, 4) )
a <- diri.est(x)
f <- ddiri(x, a$param)
sum(f)
a

Density values of a generalised Dirichlet distribution

Description

Density values of a generalised Dirichlet distribution.

Usage

dgendiri(x, a, b, logged = TRUE)

Arguments

Details

The density of the Dirichlet distribution for a vector or a matrix of compositional data is returned.

Value

A vector with the density values.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

ddiri, rgendiri, diri.est, diri.contour, rdiri, dda

Examples

a <- c(1, 2, 3)
b <- c(2, 3, 4)
x <- rgendiri(100, a, b)
y <- dgendiri(x, a, b)

Density values of a mixture of Dirichlet distributions

Description

Density values of a mixture of Dirichlet distributions.

Usage

dmixdiri(x, a, prob, logged = TRUE)

Arguments

Details

The density of the mixture of Dirichlet distribution for a vector or a matrix of compositional data is returned.

Value

A vector with the density values.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ye X., Yu Y. K. and Altschul S. F. (2011). On the inference of Dirichlet mixture priors for protein sequence comparison. Journal of Computational Biology, 18(8), 941-954.

See Also

rmixdiri, mixdiri.contour

Examples

a <- matrix( c(12, 30, 45, 32, 50, 16), byrow = TRUE,ncol = 3)
prob <- c(0.5, 0.5)
x <- rmixdiri(100, a, prob)$x
f <- dmixdiri(x, a, prob)

Dirichlet discriminant analysis

Description

Dirichlet discriminant analysis.

Usage

dda(xnew, x, ina)

Arguments

Details

The funcitons performs maximum likelihood discriminant analysis using the Dirichlet distribution.

Value

A vector with the estimated group.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Friedman J., Hastie T. and Tibshirani R. (2017). The elements of statistical learning. New York: Springer.

Thomas P. Minka (2003). Estimating a Dirichlet distribution. http://research.microsoft.com/en-us/um/people/minka/papers/dirichlet/minka-dirichlet.pdf

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

cv.dda, comp.nb, alfa.rda, alfa.knn, comp.knn, mix.compnorm, diri.reg, zadr

Examples

x <- Compositional::rdiri(100, runif(5) )
ina <- rbinom(100, 1, 0.5) + 1
mod <- dda(x, x, ina )

Dirichlet random values simulation

Description

Dirichlet random values simulation.

Usage

rdiri(n, a)

Arguments

Details

The algorithm is straightforward, for each vector, independent gamma values are generated and then divided by their total sum.

Value

A matrix with the simulated data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.est, diri.nr, diri.contour, rgendiri

Examples

x <- rdiri( 100, c(5, 7, 1, 3, 10, 2, 4) )
diri.est(x)

Dirichlet regression

Description

Dirichlet regression.

Usage

diri.reg(y, x, plot = FALSE, xnew = NULL)

diri.reg2(y, x, xnew = NULL)

diri.reg3(y, x, xnew = NULL)

Arguments

Details

A Dirichlet distribution is assumed for the regression. This involves numerical optimization. The function "diri.reg2()" allows for the covariates to be linked with the precision parameter \phi via the exponential link function \phi = e^{x*b}. The function "diri.reg3()" links the covariates to the alpha parameters of the Dirichlet distribution, i.e. it uses the classical parametrization of the distribution. This means, that there is a set of regression parameters for each component.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Maier, Marco J. (2014) DirichletReg: Dirichlet Regression for Compositional Data in R. Research Report Series/Department of Statistics and Mathematics, 125. WU Vienna University of Economics and Business, Vienna. http://epub.wu.ac.at/4077/1/Report125.pdf

Gueorguieva, Ralitza, Robert Rosenheck, and Daniel Zelterman (2008). Dirichlet component regression and its applications to psychiatric data. Computational statistics & data analysis 52(12): 5344-5355.

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

js.compreg, kl.compreg, ols.compreg, comp.reg, alfa.reg, diri.nr, dda

Examples

x <- as.vector(iris[, 4])
y <- as.matrix(iris[, 1:3])
y <- y / rowSums(y)
mod1 <- diri.reg(y, x)
mod2 <- diri.reg2(y, x)
mod3 <- comp.reg(y, x)

Distance based regression models for proportions

Description

Distance based regression models for proportions.

Usage

ols.prop.reg(y, x, cov = FALSE, tol = 1e-07, maxiters = 100)
helling.prop.reg(y, x, tol = 1e-07, maxiters = 100)

Arguments

Details

We are using the Newton-Raphson, but unlike R's built-in function "glm" we do no checks and no extra calculations, or whatever. Simply the model. The functions accept binary responses as well (0 or 1).

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Papke L. E. & Wooldridge J. (1996). Econometric methods for fractional response variables with an application to 401(K) plan participation rates. Journal of Applied Econometrics, 11(6): 619–632.

McCullagh, Peter, and John A. Nelder. Generalized linear models. CRC press, USA, 2nd edition, 1989.

See Also

propreg, beta.reg

Examples

y <- rbeta(100, 1, 4)
x <- matrix(rnorm(100 * 2), ncol = 2)
a1 <- ols.prop.reg(y, x)
a2 <- helling.prop.reg(y, x)

Divergence based regression for compositional data

Description

Regression for compositional data based on the Kullback-Leibler the Jensen-Shannon divergence and the symmetric Kullback-Leibler divergence.

Usage

kl.compreg(y, x, con = TRUE, B = 1, ncores = 1, xnew = NULL, tol = 1e-07, maxiters = 50)
js.compreg(y, x, con = TRUE, B = 1, ncores = 1, xnew = NULL)
tv.compreg(y, x, con = TRUE, B = 1, ncores = 1, xnew = NULL)
symkl.compreg(y, x, con = TRUE, B = 1, ncores = 1, xnew = NULL)
hellinger.compreg(y, x, con = TRUE, B = 1, ncores = 1, xnew = NULL)

Arguments

Details

In the kl.compreg() the Kullback-Leibler divergence is adopted as the objective function. In case of problematic convergence the "multinom" function by the "nnet" package is employed. This will obviously be slower. The js.compreg() uses the Jensen-Shannon divergence and the symkl.compreg() uses the symmetric Kullback-Leibler divergence. The tv.compreg() uses the Total Variation divergence. There is no actual log-likelihood for the last three regression models. The hellinger.compreg() minimizes the Hellinger distance.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Murteira Jose MR, and Joaquim JS Ramalho (2016). Regression analysis of multivariate fractional data. Econometric Reviews 35(4): 515-552.

Tsagris Michail (2015). A novel, divergence based, regression for compositional data. Proceedings of the 28th Panhellenic Statistics Conference, 15-18/4/2015, Athens, Greece. https://arxiv.org/pdf/1511.07600.pdf

Endres D. M. and Schindelin J. E. (2003). A new metric for probability distributions. Information Theory, IEEE Transactions on 49, 1858-1860.

Osterreicher F. and Vajda I. (2003). A new class of metric divergences on probability spaces and its applicability in statistics. Annals of the Institute of Statistical Mathematics 55, 639-653.

Alenazi A. A. (2022). f-divergence regression models for compositional data. Pakistan Journal of Statistics and Operation Research, 18(4): 867–882.

See Also

diri.reg, ols.compreg, comp.reg

Examples

library(MASS)
x <- as.vector(fgl[, 1])
y <- as.matrix(fgl[, 2:9])
y <- y / rowSums(y)
mod1<- kl.compreg(y, x, B = 1, ncores = 1)
mod2 <- js.compreg(y, x, B = 1, ncores = 1)

Divergence based regression for compositional data with compositional data in the covariates side using the \alpha-transformation

Description

Divergence based regression for compositional data with compositional data in the covariates side using the \alpha-transformation.

Usage

kl.alfapcr(y, x, covar = NULL, a, k, xnew = NULL, B = 1, ncores = 1, tol = 1e-07,
maxiters = 50)

Arguments

Details

The \alpha-transformation is applied to the compositional data first, the first k principal component scores are calcualted and used as predictor variables for the Kullback-Leibler divergence based regression model.

Value

A list including:

Author(s)

Initial code by Abdulaziz Alenazi. Modifications by Michail Tsagris.

R implementation and documentation: Abdulaziz Alenazi a.alenazi@nbu.edu.sa and Michail Tsagris mtsagris@uoc.gr.

References

Alenazi A. (2019). Regression for compositional data with compositional data as predictor variables with or without zero values. Journal of Data Science, 17(1): 219-238. https://jds-online.org/journal/JDS/article/136/file/pdf

Tsagris M. (2015). Regression analysis with compositional data containing zero values. Chilean Journal of Statistics, 6(2): 47-57. http://arxiv.org/pdf/1508.01913v1.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. http://arxiv.org/pdf/1106.1451.pdf

See Also

klalfapcr.tune, tflr, glm.pcr, alfapcr.tune

Examples

library(MASS)
y <- rdiri(214, runif(4, 1, 3))
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
mod <- alfa.pcr(y = y, x = x, a = 0.7, k = 1)
mod

Divergence matrix of compositional data

Description

Divergence matrix of compositional data.

Usage

divergence(x, type = "kullback_leibler", vector = FALSE)

Arguments

Details

The function produces the distance matrix either using the Kullback-Leibler (distance) or the Jensen-Shannon (metric) divergence. The Kullback-Leibler refers to the symmetric Kullback-Leibler divergence.

Value

if the vector argument is FALSE a symmetric matrix with the divergences, otherwise a vector with the divergences.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Endres, D. M. and Schindelin, J. E. (2003). A new metric for probability distributions. Information Theory, IEEE Transactions on 49, 1858-1860.

Osterreicher, F. and Vajda, I. (2003). A new class of metric divergences on probability spaces and its applicability in statistics. Annals of the Institute of Statistical Mathematics 55, 639-653.

See Also

comp.knn, js.compreg

Examples

x <- as.matrix(iris[1:20, 1:4])
x <- x / rowSums(x)
divergence(x)

Empirical likelihood hypothesis testing for two mean vectors

Description

Empirical likelihood hypothesis testing for two mean vectors.

Usage

el.test2(y1, y2, R = 0, ncores = 1, graph = FALSE)

Arguments

Details

The H_0 is that \pmb{\mu}_1 = \pmb{\mu}_2 and the two constraints imposed by EL are

\frac{1}{n_j}\sum_{i=1}^{n_j}\left\lbrace\left[1+\pmb{\lambda}_j^T\left({\bf x}_{ji}-\pmb{\mu} \right)\right]^{-1}\left({\bf x}_{ij}-\pmb{\mu}\right)\right\rbrace={\bf 0},

where j=1,2 and the \pmb{\lambda}_js are Lagrangian parameters introduced to maximize the above expression. Note that the maximization of is with respect to the \pmb{\lambda}_js. The probabilities of the j-th sample have the following form

p_{ji}=\frac{1}{n_j} \left[1+\pmb{\lambda}_j^T \left({\bf x}_{ji}-\pmb{\mu} \right)\right]^{-1}

. The log-likelihood ratio test statistic can be written as

\Lambda=\sum_{j=1}^2\sum_{i=1}^{n_j}\log{n_jp_{ij}}.

The test is implemented by searching for the mean vector that minimizes the sum of the two one sample EL test statistics.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Amaral G.J.A., Dryden I.L. and Wood A.T.A. (2007). Pivotal bootstrap methods for k-sample problems in directional statistics and shape analysis. Journal of the American Statistical Association, 102(478): 695–707.

Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.

Owen A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75(2): 237–249.

Preston S.P. and Wood A.T.A. (2010). Two-Sample Bootstrap Hypothesis Tests for Three-Dimensional Labelled Landmark Data. Scandinavian Journal of Statistics, 37(4): 568–587.

See Also

eel.test2, maovjames, hotel2T2, james

Examples

el.test2( y1 = as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 0 )
el.test2( y1 = as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 1 )
el.test2( y1 =as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 2 )

Energy test of equality of distributions using the \alpha-transformation

Description

Energy test of equality of distributions using the \alpha-transformation.

Usage

aeqdist.etest(x, sizes, a = 1, R = 999, ms = FALSE)

Arguments

Details

The \alpha-transformation is applied to each composition and then the energy distance of equality of distributions is applied for each value of \alpha or for the single value of \alpha.

Value

A numerical value or a numerical vector, depending on the length of the values of \alpha, with the permutation based p-value(s) of the energy test.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal Distributions in High Dimension. InterStat, November (5).

Szekely, G. J. (2000) Technical Report 03-05: E-statistics: Energy of Statistical Samples. Department of Mathematics and Statistics, Bowling Green State University.

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451

Sevinc V. and Tsagris. M. (2024). Energy Based Equality of Distributions Testing for Compositional Data. https://arxiv.org/pdf/2412.05199

See Also

acor, acor.tune, alfa, alfa.profile

Examples

y <- rdiri(50, c(3, 4, 5) )
x <- rdiri(60, c(3, 4, 5) )
aeqdist.etest( rbind(x, y), c(dim(x)[1], dim(y)[1]), a = c(-1, 0, 1) )

Energy test of equality of two distributions

Description

Energy test of equality of two distributions.

Usage

eqdist.etest(x, y, R = 999)

Arguments

Details

The energy distance of equality of two distributions is applied. The main advantage of this implementation is that it is light-weight, memory saving, however it works for two distributions only.

Value

The permutation based p-value of the energy test.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Szekely, G. J. and Rizzo, M. L. (2004) Testing for Equal Distributions in High Dimension. InterStat, November (5).

Szekely, G. J. (2000) Technical Report 03-05: E-statistics: Energy of Statistical Samples. Department of Mathematics and Statistics, Bowling Green State University.

See Also

aeqdist.etest, acor, acor.tune, alfa

Examples

x <- as.matrix(iris[1:50, 1:4])
y <- as.matrix(iris[51:100, 1:4])
eqdist.etest(x, y)

Estimating location and scatter parameters for compositional data

Description

Estimating location and scatter parameters for compositional data in a robust and non robust way.

Usage

comp.den(x, type = "alr", dist = "normal", tol = 1e-07)

Arguments

Details

This function calculates robust and non robust estimates of location and scatter.

Value

A list including: The mean vector and covariance matrix mainly. Other parameters are also returned depending on the value of the argument "dist".

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

P. J. Rousseeuw and K. van Driessen (1999) A fast algorithm for the minimum covariance determinant estimator. Technometrics 41, 212–223.

Mardia K.V., Kent J.T., and Bibby J.M. (1979). Multivariate analysis. Academic press.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

T. Karkkaminen and S. Ayramo (2005). On computation of spatial median for robust data mining. Evolutionary and Deterministic Methods for Design, Optimization and Control with Applications to Industrial and Societal Problems EUROGEN 2005.

A Durre, D Vogel, DE Tyler (2014). The spatial sign covariance matrix with unknown location. Journal of Multivariate Analysis, 130: 107–117.

J. T. Kent, D. E. Tyler and Y. Vardi (1994) A curious likelihood identity for the multivariate t-distribution. Communications in Statistics-Simulation and Computation 23, 441–453.

Azzalini A. and Dalla Valle A. (1996). The multivariate skew-normal distribution. Biometrika 83(4): 715–726.

See Also

spatmed.reg, multivt

Examples

library(MASS)
x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
comp.den(x)
comp.den(x, type = "alr", dist = "t")
comp.den(x, type = "alr", dist = "spatial")

Estimation of the probability left outside the simplex when using the alpha-transformation

Description

Estimation of the probability left outside the simplex when using the alpha-transformationn.

Usage

probout(mu, su, a) 

Arguments

Details

When applying the \alpha-transformation based on a multivariate normal there might be probability left outside the simplex as the space of this transformation is a subspace of the Euclidean space. The function estimates the missing probability via Monte Carlo simulation using 40 million generated vectors.

Value

The estimated probability left outside the simplex.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M. and Stewart C. (2020). A folded model for compositional data analysis. Australian and New Zealand Journal of Statistics, 62(2): 249-277. https://arxiv.org/pdf/1802.07330.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa, alpha.mle, a.est, rfolded

Examples

s <-  c(0.1490676523, -0.4580818209,  0.0020395316, -0.0047446076, -0.4580818209,
1.5227259250,  0.0002596411,  0.0074836251,  0.0020395316,  0.0002596411,
0.0365384838, -0.0471448849, -0.0047446076,  0.0074836251, -0.0471448849,
0.0611442781)
s <- matrix(s, ncol = 4)
m <- c(1.715, 0.914, 0.115, 0.167)
probout(m, s, 0.5)

Estimation of the value of \alpha in the folded model

Description

Estimation of the value of \alpha in the folded model.

Usage

a.est(x)

Arguments

Details

This is a function for choosing or estimating the value of \alpha in the folded model (Tsagris and Stewart, 2020).

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M. and Stewart C. (2022). A Review of Flexible Transformations for Modeling Compositional Data. In Advances and Innovations in Statistics and Data Science, pp. 225–234. https://link.springer.com/chapter/10.1007/978-3-031-08329-7_10

Tsagris M. and Stewart C. (2020). A folded model for compositional data analysis. Australian and New Zealand Journal of Statistics, 62(2): 249-277. https://arxiv.org/pdf/1802.07330.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa.profile, alfa, alfainv, alpha.mle

Examples

x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
alfa.tune(x)
a.est(x)

Estimation of the value of \alpha via the alfa profile log-likelihood

Description

Estimation of the value of \alpha via the alfa profile log-likelihood.

Usage

alfa.profile(x, a = seq(-1, 1, by = 0.01))

Arguments

Details

For every value of \alpha the normal likelihood (see the refernece) is computed. At the end, the plot of the values is constructed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa.tune, alfa, alfainv

Examples

x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
alfa.tune(x)
alfa.profile(x)

Exponential empirical likelihood hypothesis testing for two mean vectors

Description

Exponential empirical likelihood hypothesis testing for two mean vectors.

Usage

eel.test2(y1, y2, tol = 1e-07, R = 0, graph = FALSE)

Arguments

Details

Exponential empirical likelihood or exponential tilting was first introduced by Efron (1981) as a way to perform a "tilted" version of the bootstrap for the one sample mean hypothesis testing. Similarly to the empirical likelihood, positive weights p_i, which sum to one, are allocated to the observations, such that the weighted sample mean {\bf \bar{x}} is equal to some population mean \pmb{\mu}, under the H_0. Under H_1 the weights are equal to \frac{1}{n}, where n is the sample size. Following Efron (1981), the choice of p_is will minimize the Kullback-Leibler distance from H_0 to H_1

D\left(L_0,L_1\right)=\sum_{i=1}^np_i\log\left(np_i\right),

subject to the constraint \sum_{i=1}^np_i{\bf x}_i=\pmb{\mu}. The probabilities take the form

p_i=\frac{e^{\pmb{\lambda}^T{\bf x}_i}}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}

and the constraint becomes

\frac{\sum_{i=1}^ne^{\pmb{\lambda}^T{\bf x}_i}\left({\bf x}_i-\pmb{\mu}\right)}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}=0 \Rightarrow \frac{\sum_{i=1}^n{\bf x}_ie^{\pmb{\lambda}^T{\bf x}_i}}{\sum_{j=1}^ne^{\pmb{\lambda}^T{\bf x}_j}}-\pmb{\mu}=0.

Similarly to empirical likelihood a numerical search over \pmb{\lambda} is required.

We can derive the asymptotic form of the test statistic in the two sample means case but in a simpler form, generalizing the approach of Jing and Robinson (1997) to the multivariate case as follows. The three constraints are

{\begin{array}{ccc} \left(\sum_{j=1}^{n_1}e^{\pmb {\lambda}_1^T{\bf x}_j}\right)^{-1}\left(\sum_{i=1}^{n_1}{\bf x}_ie^{\pmb{\lambda}_1^T {\bf x}_i}\right) -\pmb{\mu} & = & {\bf 0} \\ \left(\sum_{j=1}^{n_2}e^{\pmb {\lambda}_2^T{\bf y}_j}\right)^{-1}\left(\sum_{i=1}^{n_2}{\bf y}_ie^{\pmb{\lambda}_2^T {\bf y}_i}\right) -\pmb{\mu} & = & {\bf 0} \\ n_1\pmb{\lambda}_1+n_2\pmb{\lambda}_2 & = & {\bf 0}. \end{array}}

Similarly to EL the sum of a linear combination of the \pmb{\lambda}s is set to zero. We can equate the first two constraints of

\left(\sum_{j=1}^{n_1}e^{\pmb {\lambda}_1^T{\bf x}_j}\right)^{-1}\left(\sum_{i=1}^{n_1}{\bf x}_ie^{\pmb{\lambda}_1^T {\bf x}_i}\right)= \left(\sum_{j=1}^{n_2}e^{\pmb {\lambda}_2^T{\bf y}_j}\right)^{-1}\left(\sum_{i=1}^{n_2}{\bf y}_ie^{\pmb{\lambda}_2^T {\bf y}_i}\right).

Also, we can write the third constraint of as \pmb{\lambda}_2=-\frac{n_1}{n_2}\pmb{\lambda}_1 and thus rewrite the first two constraints as

\left(\sum_{j=1}^{n_1}e^{\pmb{\lambda}^T{\bf x}_j}\right)^{-1}\left(\sum_{i=1}^{n_1}{\bf x}_ie^{\pmb{\lambda}^T {\bf x}_i}\right) = \left(\sum_{j=1}^{n_2}e^{-\frac{n_1}{n_2}\pmb{\lambda}^T{\bf y}_j}\right)^{-1}\left(\sum_{i=1}^{n_2}{\bf y}_ie^{-\frac{n_1}{n_2}\pmb{\lambda}^T {\bf y}_i}\right).

This trick allows us to avoid the estimation of the common mean. It is not possible though to do this in the empirical likelihood method. Instead of minimisation of the sum of the one-sample test statistics from the common mean, we can define the probabilities by searching for the \pmb{\lambda} which makes the last equation hold true. The third constraint of is a convenient constraint, but Jing and Robinson (1997) mention that even though as a constraint is simple it does not lead to second-order accurate confidence intervals unless the two sample sizes are equal. Asymptotically, the test statistic follows a \chi^2_d under the null hypothesis.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Efron B. (1981) Nonparametric standard errors and confidence intervals. Canadian Journal of Statistics, 9(2): 139–158.

Jing B.Y. and Wood A.T.A. (1996). Exponential empirical likelihood is not Bartlett correctable. Annals of Statistics, 24(1): 365–369.

Jing B.Y. and Robinson J. (1997). Two-sample nonparametric tilting method. Australian Journal of Statistics, 39(1): 25–34.

Owen A.B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.

Preston S.P. and Wood A.T.A. (2010). Two-Sample Bootstrap Hypothesis Tests for Three-Dimensional Labelled Landmark Data. Scandinavian Journal of Statistics 37(4): 568–587.

Tsagris M., Preston S. and Wood A.T.A. (2017). Nonparametric hypothesis testing for equality of means on the simplex. Journal of Statistical Computation and Simulation, 87(2): 406–422.

See Also

el.test2, maovjames, hotel2T2, james

Examples

y1 = as.matrix(iris[1:25, 1:4])
y2 = as.matrix(iris[26:50, 1:4])
eel.test2(y1, y2)
eel.test2(y1, y2 )
eel.test2( y1, y2 )

Fast estimation of the value of \alpha

Description

Fast estimation of the value of \alpha.

Usage

alfa.tune(x, B = 1, ncores = 1)

Arguments

Details

This is a faster function than alfa.profile for choosing the value of \alpha.

Value

A vector with the best alpha, the maximised log-likelihood and the log-likelihood at \alpha=0, when B = 1 (no bootstrap). If B>1 a list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa.profile, alfa, alfainv

Examples

library(MASS)
x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
alfa.tune(x)
alfa.profile(x)

Gaussian mixture models for compositional data

Description

Gaussian mixture models for compositional data.

Usage

mix.compnorm(x, g, model, type = "alr", veo = FALSE)

Arguments

Details

A log-ratio transformation is applied and then a Gaussian mixture model is constructed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ryan P. Browne, Aisha ElSherbiny and Paul D. McNicholas (2015). R package mixture: Mixture Models for Clustering and Classification.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

bic.mixcompnorm, rmixcomp, mix.compnorm.contour, alfa.mix.norm, alfa.knn, alfa.rda, comp.nb

Examples

x <- as.matrix(iris[, 1:4])
x <- x/ rowSums(x)
mod1 <- mix.compnorm(x, 3, model = "EII" )
mod2 <- mix.compnorm(x, 4, model = "VII")

Gaussian mixture models for compositional data using the \alpha-transformation

Description

Gaussian mixture models for compositional data using the \alpha-transformation.

Usage

alfa.mix.norm(x, g, a, model, veo = FALSE)

Arguments

Details

A log-ratio transformation is applied and then a Gaussian mixture model is constructed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ryan P. Browne, Aisha ElSherbiny and Paul D. McNicholas (2015). R package mixture: Mixture Models for Clustering and Classification.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

bic.alfamixnorm, bic.mixcompnorm, rmixcomp, mix.compnorm.contour, mix.compnorm, alfa, alfa.knn, alfa.rda, comp.nb

Examples

x <- as.matrix(iris[, 1:4])
x <- x/ rowSums(x)
mod1 <- alfa.mix.norm(x, 3, 0.4, model = "EII" )
mod2 <- alfa.mix.norm(x, 4, 0.7, model = "VII")

Generalised Dirichlet random values simulation

Description

Generalised Dirichlet random values simulation.

Usage

rgendiri(n, a, b)

Arguments

Details

The algorithm is straightforward, for each vector, independent gamma values are generated and then divided by their total sum. The difference with rdiri is that here the Gamma distributed variables are not equally scaled.

Value

A matrix with the simulated data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

rdiri, diri.est, diri.nr, diri.contour

Examples

a <- c(1, 2, 3)
b <- c(2, 3, 4)
x <- rgendiri(100, a, b)

Generate random folds for cross-validation

Description

Random folds for use in a cross validation are generated. There is the option for stratified splitting as well.

Usage

makefolds(ina, nfolds = 10, stratified = TRUE, seed = NULL)

Arguments

Details

I was inspired by the command in the package TunePareto in order to do the stratified version.

Value

A list with nfolds elements where each elements is a fold containing the indices of the data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

See Also

compknn.tune

Examples

a <- makefolds(iris[, 5], nfolds = 5, stratified = TRUE)
table(iris[a[[1]], 5])  ## 10 values from each group

Greenacre's power transformation

Description

Greenacre's power transformation.

Usage

green(x, theta)

Arguments

Details

Greenacre's transformation is applied to the compositional data.

Value

A matrix with the power transformed data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Greenacre, M. (2009). Power transformations in correspondence analysis. Computational Statistics & Data Analysis, 53(8): 3107-3116. http://www.econ.upf.edu/~michael/work/PowerCA.pdf

See Also

alfa

Examples

library(MASS)
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
y1 <- green(x, 0.1)
y2 <- green(x, 0.2)
rbind( colMeans(y1), colMeans(y2) )

Helper Frechet mean for compositional data

Description

Helper Frechet mean for compositional data.

Usage

frechet2(x, di, a, k)

Arguments

Details

The power transformation is applied to the compositional data and the mean vector is calculated. Then the inverse of it is calculated and the inverse of the power transformation applied to the last vector is the Frechet mean.

What this helper function do is to speed up the Frechet mean when used in the \alpha-k-NN regression. The \alpha-k-NN regression computes the Frechet mean of the k nearest neighbours for a value of \alpha and this function does exactly that. Suppose you want to predict the compositional value of some new predictors. For each predictor value you must use the Frechet mean computed at various nearest neighbours. This function performs these computations in a fast way. It is not the fastest way, yet it is a pretty fast way. This function is being called inside the function aknn.reg.

Value

A list where eqch element contains a matrix. Each matrix contains the Frechet means computed at various nearest neighbours.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa, alfainv, profile

Examples

library(MASS)
library(Rfast)
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
xnew <- x[1:10, ]
x <- x[-c(1:10), ]
k <- 2:5
di <- Rfast::dista( xnew, x, k = max(k), index = TRUE, square = TRUE )
est <- frechet2(x, di, 0.2, k)

Helper functions for the Kullback-Leibler regression

Description

Helper functions for the Kullback-Leibler regression.

Usage

kl.compreg2(y, x, con = TRUE, xnew = NULL, tol = 1e-07, maxiters = 50)
klcompreg.boot(y, x, der, der2, id, b1, n, p, d, tol = 1e-07, maxiters = 50)

Arguments

Details

These are help functions for the kl.compreg function. They are not to be called directly by the user.

Value

For kl.compreg2 a list including:

For klcompreg.boot a list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Murteira, Jose MR, and Joaquim JS Ramalho 2016. Regression analysis of multivariate fractional data. Econometric Reviews 35(4): 515-552.

See Also

diri.reg, js.compreg, ols.compreg, comp.reg

Examples

  library(MASS)
  x <- as.vector(fgl[, 1])
  y <- as.matrix(fgl[, 2:9])
  y <- y / rowSums(y)
  mod1<- kl.compreg(y, x, B = 1, ncores = 1)
  mod2 <- js.compreg(y, x, B = 1, ncores = 1)

Hotelling's multivariate version of the 2 sample t-test for Euclidean data

Description

Hotelling's test for testing the equality of two Euclidean population mean vectors.

Usage

hotel2T2(x1, x2, a = 0.05, R = 999, graph = FALSE)

Arguments

Details

The fist case scenario is when we assume equality of the two covariance matrices. This is called the two-sample Hotelling's T^2 test (Mardia, Kent and Bibby, 1979, pg. 131-140) and Everitt (2005, pg. 139). The test statistic is defined as

T^2=\frac{n_1n_2}{n_1+n_2}\left(\bar{{\bf X}}_1- \bar{{\bf X}}_2\right)^T{\bf S}^{-1}\left(\bar{{\bf X}}_1- \bar{{\bf X}}_2\right),

where \bf S is the pooled covariance matrix calculated under the assumption of equal covariance matrices {\bf S}=\frac{\left(n_1-1\right){\bf S}_1+\left(n_2-1\right){\bf S}_2}{n_1+n_2-2}. Under H_0 the statistic F given by

F=\frac{\left( n_1+n_2-p-1 \right)T^2}{\left(n_1+n_2-2 \right)p}

follows the F distribution with p and n_1+n_2-p-1 degrees of freedom. Similar to the one-sample test, an extra argument (R) indicates whether bootstrap calibration should be used or not. If R=1, then the asymptotic theory applies, if R>1, then the bootstrap p-value will be applied and the number of re-samples is equal to R. The estimate of the common mean used in the bootstrap to transform the data under the null hypothesis the mean vector of the combined sample, of all the observations.

The built-in command manova does the same thing exactly. Try it, the asymptotic F test is what you have to see. In addition, this command allows for more mean vector hypothesis testing for more than two groups. I noticed this command after I had written my function and nevertheless as I mention in the introduction this document has an educational character as well.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Everitt B. (2005). An R and S-Plus Companion to Multivariate Analysis. Springer.

Mardia K.V., Kent J.T. and Bibby J.M. (1979). Multivariate Analysis. London: Academic Press.

Tsagris M., Preston S. and Wood A.T.A. (2017). Nonparametric hypothesis testing for equality of means on the simplex. Journal of Statistical Computation and Simulation, 87(2): 406–422.

See Also

james, el.test2, eel.test2

Examples

hotel2T2( as.matrix(iris[1:25, 1:4]), as.matrix(iris[26:50, 1:4]) )
hotel2T2( as.matrix(iris[1:25, 1:4]), as.matrix(iris[26:50, 1:4]), R = 1 )

Hypothesis testing for two or more compositional mean vectors

Description

Hypothesis testing for two or more compositional mean vectors.

Usage

comp.test(x, ina, test = "james", R = 0, ncores = 1, graph = FALSE)

Arguments

Details

The idea is to apply the \alpha-transformation, with \alpha=1, to the compositional data and then use a test to compare their mean vectors. See the help page of each test for more information. The function is visible so you can see exactly what is going on.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Tsagris M., Preston S. and Wood A.T.A. (2017). Nonparametric hypothesis testing for equality of means on the simplex. Journal of Statistical Computation and Simulation, 87(2): 406–422.

G.S. James (1954). Tests of Linear Hypothese in Univariate and Multivariate Analysis when the Ratios of the Population Variances are Unknown. Biometrika, 41(1/2): 19–43

Krishnamoorthy K. and Yanping Xia (2006). On Selecting Tests for Equality of Two Normal Mean Vectors. Multivariate Behavioral Research 41(4): 533–548.

Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.

Owen A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75(2): 237–249.

Amaral G.J.A., Dryden I.L. and Wood A.T.A. (2007). Pivotal bootstrap methods for k-sample problems in directional statistics and shape analysis. Journal of the American Statistical Association 102(478): 695–707.

Preston S.P. and Wood A.T.A. (2010). Two-Sample Bootstrap Hypothesis Tests for Three-Dimensional Labelled Landmark Data. Scandinavian Journal of Statistics 37(4): 568–587.

Jing Bing-Yi and Andrew TA Wood (1996). Exponential empirical likelihood is not Bartlett correctable. Annals of Statistics 24(1): 365–369.

See Also

hd.meantest2, dptest

Examples

ina <- rep(1:2, each = 50)
x <- as.matrix(iris[1:100, 1:4])
x <- x/ rowSums(x)
comp.test( x, ina, test = "james" )
comp.test( x, ina, test = "hotel" )
comp.test( x, ina, test = "el" )
comp.test( x, ina, test = "eel" )

ICE plot for projection pursuit regression with compositional predictor variables

Description

ICE plot for projection pursuit regression with compositional predictor variables.

Usage

ice.pprcomp(model, x, k = 1, frac = 0.1, type = "log")

Arguments

Details

This function implements the Individual Conditional Expecation plots of Goldstein et al. (2015). See the references for more details.

Value

A graph with several curves. The horizontal axis contains the selected variable, whereas the vertical axis contains the centered predicted values. The black curves are the effects for each observation and the blue line is their average effect.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

https://christophm.github.io/interpretable-ml-book/ice.html

Goldstein, A., Kapelner, A., Bleich, J. and Pitkin, E. (2015). Peeking inside the black box: Visualizing statistical learning with plots of individual conditional expectation. Journal of Computational and Graphical Statistics 24(1): 44-65.

Friedman, J. H. and Stuetzle, W. (1981). Projection pursuit regression. Journal of the American Statistical Association, 76, 817-823. doi: 10.2307/2287576.

See Also

pprcomp, pprcomp.tune, ice.kernreg, alfa.pcr, lc.reg, comp.ppr

Examples

x <- as.matrix( iris[, 2:4] )
x <- x/ rowSums(x)
y <- iris[, 1]
model <- pprcomp(y, x)
ice <- ice.pprcomp(model, x, k = 1)

ICE plot for the \alpha-k-NN regression

Description

ICE plot for the \alpha-k-NN regression.

Usage

ice.aknnreg(y, x, a, k, apostasi = "euclidean", rann = FALSE,
ind = 1, frac = 0.2, qpos = 0.9)

Arguments

Details

This function implements the Individual Conditional Expecation plots of Goldstein et al. (2015). See the references for more details.

Value

A graph with several curves, one for each component. The horizontal axis contains the selected variable, whereas the vertical axis contains the locally smoothed predicted compositional lines.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

https://christophm.github.io/interpretable-ml-book/ice.html

Goldstein, A., Kapelner, A., Bleich, J. and Pitkin, E. (2015). Peeking inside the black box: Visualizing statistical learning with plots of individual conditional expectation. Journal of Computational and Graphical Statistics 24(1): 44-65.

See Also

ice.akernreg, ice.pprcomp

Examples

y <- as.matrix( iris[, 2:4] )
x <- iris[, 1]
ice <- ice.aknnreg(y, x, a = 0.6, k = 5, ind = 1)

ICE plot for the \alpha-kernel regression

Description

ICE plot for the \alpha-kernel regression.

Usage

ice.akernreg(y, x, a, h, type = "gauss", ind = 1, frac = 0.1, qpos = 0.9)

Arguments

Details

This function implements the Individual Conditional Expecation plots of Goldstein et al. (2015). See the references for more details.

Value

A graph with several curves, one for each component. The horizontal axis contains the selected variable, whereas the vertical axis contains the locally smoothed predicted compositional lines.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

https://christophm.github.io/interpretable-ml-book/ice.html

Goldstein, A., Kapelner, A., Bleich, J. and Pitkin, E. (2015). Peeking inside the black box: Visualizing statistical learning with plots of individual conditional expectation. Journal of Computational and Graphical Statistics 24(1): 44-65.

See Also

ice.aknnreg, ice.pprcomp

Examples

y <- as.matrix( iris[, 2:4] )
x <- iris[, 1]
ice <- ice.akernreg(y, x, a = 0.6, h = 0.1, ind = 1)

ICE plot for univariate kernel regression

Description

ICE plot for univariate kernel regression.

Usage

ice.kernreg(y, x, h, type = "gauss", k = 1, frac = 0.1)

Arguments

Details

This function implements the Individual Conditional Expecation plots of Goldstein et al. (2015). See the references for more details.

Value

A graph with several curves. The horizontal axis contains the selected variable, whereas the vertical axis contains the centered predicted values. The black curves are the effects for each observation and the blue line is their average effect.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

https://christophm.github.io/interpretable-ml-book/ice.html

Goldstein, A., Kapelner, A., Bleich, J. and Pitkin, E. (2015). Peeking inside the black box: Visualizing statistical learning with plots of individual conditional expectation. Journal of Computational and Graphical Statistics 24(1): 44-65.

See Also

ice.pprcomp, kernreg.tune, alfa.pcr, lc.reg

Examples

x <- as.matrix( iris[, 2:4] )
y <- iris[, 1]
ice <- ice.kernreg(y, x, h = 0.1, k = 1)

Inverse of the \alpha-transformation

Description

The inverse of the \alpha-transformation.

Usage

alfainv(x, a, h = TRUE)

Arguments

Details

The inverse of the \alpha-transformation is applied to the data. If the data lie outside the \alpha-space, NAs will be returned for some values.

Value

A matrix with the pairwise distances.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Tsagris M. and Stewart C. (2022). A Review of Flexible Transformations for Modeling Compositional Data. In Advances and Innovations in Statistics and Data Science, pp. 225–234. https://link.springer.com/chapter/10.1007/978-3-031-08329-7_10

Tsagris M.T., Preston S. and Wood A.T.A. (2016). Improved classification for compositional data using the \alpha-transformation. Journal of Classification 33(2): 243–261. https://arxiv.org/pdf/1506.04976v2.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa, alfadist

Examples

library(MASS)
x <- as.matrix(fgl[1:10, 2:9])
x <- x / rowSums(x)
y <- alfa(x, 0.5)$aff
alfainv(y, 0.5)

James multivariate version of the t-test

Description

James test for testing the equality of two population mean vectors without assuming equality of the covariance matrices.

Usage

james(y1, y2, a = 0.05, R = 999, graph = FALSE)

Arguments

Details

Here we show the modified version of the two-sample T^2 test (function hotel2T2) in the case where the two covariances matrices cannot be assumed to be equal.

James (1954) proposed a test for linear hypotheses of the population means when the variances (or the covariance matrices) are not known. Its form for two p-dimensional samples is:

T^2_u=\left(\bar{{\bf X}}_1-\bar{{\bf X}}_2\right)^T\tilde{{\bf S}}^{-1}\left(\bar{{\bf X}}_1-\bar{{\bf X}}_2\right),

where \tilde{{\bf S}}=\tilde{{\bf S}_1}+\tilde{{\bf S}_2}=\frac{{\bf S}_1}{n_1}+\frac{{\bf S}_2}{n_2} .

James (1954) suggested that the test statistic is compared with 2h\left(\alpha\right), a corrected \chi^2 distribution whose form is

2h\left(\alpha\right)=\chi^2\left(A+B\chi^2\right),

where A=1+\frac{1}{2p}\sum_{i=1}^2\frac{\left(tr \tilde{{\bf S}}^{-1}\tilde{{\bf S}_i}\right)^2}{n_i-1} and B=\frac{1}{p\left(p+2\right)}\left[\sum_{i=1}^2\frac{tr\left(\tilde{{\bf S}}^{-1}\tilde{{\bf S}_i}\right)^2}{n_i-1}+\frac{1}{2}\sum_{i=1}^2\frac{\left(\text{tr} \tilde{{\bf S}}^{-1}\tilde{{\bf S}_i}\right)^2}{n_i-1} \right].

If you want to do bootstrap to get the p-value, then you must transform the data under the null hypothesis. The estimate of the common mean is given by Aitchison (1986)

\hat{\pmb{\mu}}_c = \left(n_1{\bf S}_1^{-1}+n_2{\bf S}_2^{-1}\right)^{-1}\left(n_1{\bf S}_1^{-1}\bar{{\bf X}}_1+n_2{\bf S}_2^{-1}\bar{{\bf X}}_2\right)= \left(\tilde{{\bf S}}_1^{-1}+\tilde{{\bf S}}_2^{-1}\right)^{-1}\left(\tilde{{\bf S}}_1^{-1}\bar{{\bf X}}_1+\tilde{{\bf S}}_2^{-1}\bar{{\bf X}}_2\right).

The modified Nel and van der Merwe (1986) test is based on the same quadratic form as that of James (1954) but the distribution used to compare the value of the test statistic is different. It is shown in Krishnamoorthy and Yanping (2006) that T^2_u \sim \frac{\nu p}{\nu-p+1}F_{p,\nu-p+1} approximately, where \nu=\frac{p+p^2}{\frac{1}{n_1}\left\lbrace \text{tr}\left[ \left( {\bf S}_1\tilde{{\bf S}} \right)^2\right]+ \text{tr}\left[ \left( {\bf S}_1\tilde{{\bf S}} \right)\right]^2 \right\rbrace + \frac{1}{n_2}\left\lbrace \text{tr}\left[ \left( {\bf S}_2\tilde{{\bf S}}\right)^2\right]+ \text{tr}\left[ \left( {\bf S}_2\tilde{{\bf S}} \right)\right]^2 \right\rbrace }.

The algorithm is taken by Krishnamoorthy and Yu (2004).

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

James G.S. (1954). Tests of Linear Hypothese in Univariate and Multivariate Analysis when the Ratios of the Population Variances are Unknown. Biometrika, 41(1/2): 19–43.

Krishnamoorthy K. and Yu J. (2004). Modified Nel and Van der Merwe test for the multivariate Behrens-Fisher problem. Statistics & Probability Letters, 66(2): 161–169.

Krishnamoorthy K. and Yanping Xia (2006). On Selecting Tests for Equality of Two Normal Mean Vectors. Multivariate Behavioral Research, 41(4): 533–548.

Tsagris M., Preston S. and Wood A.T.A. (2017). Nonparametric hypothesis testing for equality of means on the simplex. Journal of Statistical Computation and Simulation, 87(2): 406–422.

See Also

hotel2T2, maovjames, el.test2, eel.test2

Examples

james( as.matrix(iris[1:25, 1:4]), as.matrix(iris[26:50, 1:4]), R = 1 )
james( as.matrix(iris[1:25, 1:4]), as.matrix(iris[26:50, 1:4]), R = 2 )
james( as.matrix(iris[1:25, 1:4]), as.matrix(iris[26:50, 1:4]) )

Kernel regression with a numerical response vector or matrix

Description

Kernel regression (Nadaraya-Watson estimator) with a numerical response vector or matrix.

Usage

kern.reg(xnew, y, x, h = seq(0.1, 1, length = 10), type = "gauss" )

Arguments

Details

The Nadaraya-Watson estimator regression is applied.

Value

The fitted values. If a single bandwidth is considered then this is a vector or a matrix, depeding on the nature of the response. If multiple bandwidth values are considered then this is a matrix, if the response is a vector, or a list, if the response is a matrix.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Wand M. P. and Jones M. C. (1994). Kernel smoothing. CRC press.

See Also

kernreg.tune, ice.kernreg, akern.reg, aknn.reg

Examples

y <- iris[, 1]
x <- iris[, 2:4]
est <- kern.reg(x, y, x, h = c(0.1, 0.2) )

Kullback-Leibler divergence and Bhattacharyya distance between two Dirichlet distributions

Description

Kullback-Leibler divergence and Bhattacharyya distance between two Dirichlet distributions.

Usage

kl.diri(a, b, type = "KL")

Arguments

Details

Note that the order is important in the Kullback-Leibler divergence, since this is asymmetric, but not in the Bhattacharyya distance, since it is a metric.

Value

The value of the Kullback-Leibler divergence or the Bhattacharyya distance.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

See Also

diri.est, diri.nr

Examples

library(MASS)
a <- runif(10, 0, 20)
b <- runif(10, 1, 10)
kl.diri(a, b)
kl.diri(b, a)
kl.diri(a, b, type = "bhatt")
kl.diri(b, a, type = "bhatt")

LASSO Kullback-Leibler divergence based regression

Description

LASSO Kullback-Leibler divergence based regression.

Usage

lasso.klcompreg(y, x, alpha = 1, lambda = NULL,
nlambda = 100, type = "grouped", xnew = NULL)

Arguments

Details

The function uses the glmnet package to perform LASSO penalised regression. For more details see the function in that package.

Value

A list including:

Author(s)

Michail Tsagris and Abdulaziz Alenazi.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Abdulaziz Alenazi a.alenazi@nbu.edu.sa.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Alenazi A. A. (2022). f-divergence regression models for compositional data. Pakistan Journal of Statistics and Operation Research, 18(4): 867–882.

Friedman J., Hastie T. and Tibshirani R. (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), 1–22.

See Also

lassocoef.plot, cv.lasso.klcompreg, kl.compreg, lasso.compreg, ols.compreg, alfa.pcr, alfa.knn.reg

Examples

y <- as.matrix(iris[, 1:4])
y <- y / rowSums(y)
x <- matrix( rnorm(150 * 30), ncol = 30 )
a <- lasso.klcompreg(y, x)

LASSO log-ratio regression with compositional response

Description

LASSO log-ratio regression with compositional response.

Usage

lasso.compreg(y, x, alpha = 1, lambda = NULL,
nlambda = 100, xnew = NULL)

Arguments

Details

The function uses the glmnet package to perform LASSO penalised regression. For more details see the function in that package.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Friedman, J., Hastie, T. and Tibshirani, R. (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), 1-22.

See Also

cv.lasso.compreg, lassocoef.plot, lasso.klcompreg, cv.lasso.klcompreg, comp.reg

Examples

y <- as.matrix(iris[, 1:4])
y <- y / rowSums(y)
x <- matrix( rnorm(150 * 30), ncol = 30 )
a <- lasso.compreg(y, x)

LASSO with compositional predictors using the alpha-transformation

Description

LASSO with compositional predictors using the alpha-transformation.

Usage

alfa.lasso(y, x, a = seq(-1, 1, by = 0.1), model = "gaussian", lambda = NULL,
xnew = NULL)

Arguments

Details

The function uses the glmnet package to perform LASSO penalised regression. For more details see the function in that package.

Value

A list including sublists for each value of \alpha:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Friedman, J., Hastie, T. and Tibshirani, R. (2010) Regularization Paths for Generalized Linear Models via Coordinate Descent. Journal of Statistical Software, Vol. 33(1), 1–22.

See Also

alfalasso.tune, cv.lasso.klcompreg, lasso.compreg, alfa.knn.reg

Examples

y <- as.matrix(iris[, 1])
x <- rdiri(150, runif(20, 2, 5) )
mod <- alfa.lasso(y, x, a = c(0, 0.5, 1))

Log-contrast GLMS with compositional predictor variables

Description

Log-contrast GLMs with compositional predictor variables.

Usage

lc.glm(y, x, z = NULL, model = "logistic", xnew = NULL, znew = NULL)

Arguments

Details

The function performs the log-contrast logistic or Poisson regression model. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients equals 0. If you want the regression without the zum-to-zero contraints see ulc.glm. Extra predictors variables are allowed as well, for instance categorical or continuous.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Lu J., Shi P. and Li H. (2019). Generalized linear models with linear constraints for microbiome compositional data. Biometrics, 75(1): 235–244.

See Also

ulc.glm, lc.glm2, ulc.glm2, lcglm.aov

Examples

y <- rbinom(150, 1, 0.5)
x <- rdiri(150, runif(3, 1, 4) )
mod1 <- lc.glm(y, x)

Log-contrast logistic or Poisson regression with with multiple compositional predictors

Description

Log-contrast logistic or Poisson regression with with multiple compositional predictors.

Usage

lc.glm2(y, x, z = NULL, model = "logistic", xnew = NULL, znew = NULL)

Arguments

Details

The function performs the log-contrast logistic or Poisson regression model. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients equals 0. If you want the regression without the zum-to-zero contraints see ulc.glm2. Extra predictors variables are allowed as well, for instance categorical or continuous.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Lu J., Shi P. and Li H. (2019). Generalized linear models with linear constraints for microbiome compositional data. Biometrics, 75(1): 235–244.

See Also

ulc.glm2, ulc.glm, lc.glm

Examples

y <- rbinom(150, 1, 0.5)
x <- list()
x1 <- as.matrix(iris[, 2:4])
x1 <- x1 / rowSums(x1)
x[[ 1 ]] <- x1
x[[ 2 ]] <- rdiri(150, runif(4) )
x[[ 3 ]] <- rdiri(150, runif(5) )
mod <- lc.glm2(y, x)

Log-contrast quantile regression with compositional predictor variables

Description

Log-contrast quantile regression with compositional predictor variables.

Usage

lc.rq(y, x, z = NULL, tau, xnew = NULL, znew = NULL)

Arguments

Details

The function performs the quantile regression model. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients equals 0. If you want the regression without the zum-to-zero contraints see ulc.rq. Extra predictor variables are allowed as well, for instance categorical or continuous.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Koenker R. W. and Bassett G. W. (1978). Regression Quantiles, Econometrica, 46(1): 33–50.

Koenker R. W. and d'Orey V. (1987). Algorithm AS 229: Computing Regression Quantiles. Applied Statistics, 36(3): 383–393.

See Also

lc.rq2, ulc.rq

Examples

y <- rnorm(150)
x <- rdiri(150, runif(3, 1, 4) )
mod1 <- lc.rq(y, x)

Log-contrast quantile regression with with multiple compositional predictors

Description

Log-contrast quantile regression with with multiple compositional predictors.

Usage

lc.rq2(y, x, z = NULL, tau = 0.5, xnew = NULL, znew = NULL)

Arguments

Details

The function performs the log-contrast quantile regression model. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients equals 0. If you want the regression without the zum-to-zero contraints see ulc.rq2. Extra predictor variables are allowed as well, for instance categorical or continuous.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Koenker R. W. and Bassett G. W. (1978). Regression Quantiles, Econometrica, 46(1): 33–50.

Koenker R. W. and d'Orey V. (1987). Algorithm AS 229: Computing Regression Quantiles. Applied Statistics, 36(3): 383–393.

See Also

lc.rq, ulc.rq

Examples

y <- rnorm(150)
x <- list()
x1 <- as.matrix(iris[, 2:4])
x1 <- x1 / rowSums(x1)
x[[ 1 ]] <- x1
x[[ 2 ]] <- rdiri(150, runif(4) )
x[[ 3 ]] <- rdiri(150, runif(5) )
mod <- lc.rq2(y, x)

Log-contrast regression with compositional predictor variables

Description

Log-contrast regression with compositional predictor variables.

Usage

lc.reg(y, x, z = NULL, xnew = NULL, znew = NULL)

Arguments

Details

The function performs the log-contrast regression model as described in Aitchison (2003), pg. 84-85. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients equals 0. Hence, we apply constrained least squares, which has a closed form solution. The constrained least squares is described in Chapter 8.2 of Hansen (2019). The idea is to minimise the sum of squares of the residuals under the constraint R^T \beta = c, where c=0 in our case. If you want the regression without the zum-to-zero contraints see ulc.reg. Extra predictors variables are allowed as well, for instance categorical or continuous.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Hansen, B. E. (2022). Econometrics. Princeton University Press.

See Also

ulc.reg, lcreg.aov, lc.reg2, alfa.pcr, alfa.knn.reg

Examples

y <- iris[, 1]
x <- as.matrix(iris[, 2:4])
x <- x / rowSums(x)
mod1 <- lc.reg(y, x)
mod2 <- lc.reg(y, x, z = iris[, 5])

Log-contrast regression with multiple compositional predictors

Description

Log-contrast regression with multiple compositional predictors.

Usage

lc.reg2(y, x, z = NULL, xnew = NULL, znew = NULL)

Arguments

Details

The function performs the log-contrast regression model as described in Aitchison (2003), pg. 84-85. The logarithm of the compositional predictor variables is used (hence no zero values are allowed). The response variable is linked to the log-transformed data with the constraint that the sum of the regression coefficients for each composition equals 0. Hence, we apply constrained least squares, which has a closed form solution. The constrained least squares is described in Chapter 8.2 of Hansen (2019). The idea is to minimise the sum of squares of the residuals under the constraint R^T \beta = c, where c=0 in our case. If you want the regression without the zum-to-zero contraints see ulc.reg2. Extra predictors variables are allowed as well, for instance categorical or continuous. The difference with lc.reg is that instead of one, there are multiple compositions treated as predictor variables.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Hansen, B. E. (2022). Econometrics. Princeton University Press.

Xiaokang Liu, Xiaomei Cong, Gen Li, Kendra Maas and Kun Chen (2020). Multivariate Log-Contrast Regression with Sub-Compositional Predictors: Testing the Association Between Preterm Infants' Gut Microbiome and Neurobehavioral Outcome.

See Also

ulc.reg2, lc.reg, ulc.reg, lcreg.aov, alfa.pcr, alfa.knn.reg

Examples

y <- iris[, 1]
x <- list()
x1 <- as.matrix(iris[, 2:4])
x1 <- x1 / rowSums(x1)
x[[ 1 ]] <- x1
x[[ 2 ]] <- rdiri(150, runif(4) )
x[[ 3 ]] <- rdiri(150, runif(5) )
mod <- lc.reg2(y, x)
be <- mod$be
sum(be[2:4])
sum(be[5:8])
sum(be[9:13])

Log-likelihood ratio test for a Dirichlet mean vector

Description

Log-likelihood ratio test for a Dirichlet mean vector.

Usage

dirimean.test(x, a)

Arguments

Details

Log-likelihood ratio test is performed for the hypothesis the given vector of parameters "a" describes the compositional data well.

Value

If there are no zeros in the data, a list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

See Also

sym.test, diri.nr, diri.est, rdiri, ddiri

Examples

x <- rdiri( 100, c(1, 2, 3) )
dirimean.test(x, c(1, 2, 3) )
dirimean.test( x, c(1, 2, 3)/6 )

Log-likelihood ratio test for a symmetric Dirichlet distribution

Description

Log-likelihood ratio test for a symmetric Dirichlet distribution.

Usage

sym.test(x)

Arguments

Details

Log-likelihood ratio test is performed for the hypothesis that all Dirichelt parameters are equal.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

See Also

diri.nr, diri.est, rdiri, dirimean.test

Examples

x <- rdiri( 100, c(5, 7, 1, 3, 10, 2, 4) )
sym.test(x)
x <- rdiri( 100, c(5, 5, 5, 5, 5) )
sym.test(x)

MLE for the multivariate t distribution

Description

MLE of the parameters of a multivariate t distribution.

Usage

multivt(y, plot = FALSE)

Arguments

Details

The parameters of a multivariate t distribution are estimated. This is used by the functions comp.den and bivt.contour.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Nadarajah, S. and Kotz, S. (2008). Estimation methods for the multivariate t distribution. Acta Applicandae Mathematicae, 102(1):99-118.

See Also

bivt.contour, comp.den

Examples

x <- as.matrix(iris[, 1:4])
multivt(x)

MLE of distributions defined in the (0, 1) interval

Description

MLE of distributions defined in the (0, 1) interval.

Usage

beta.est(x, tol = 1e-07)
logitnorm.est(x)
hsecant01.est(x, tol = 1e-07)
kumar.est(x, tol = 1e-07)
unitweibull.est(x, tol = 1e-07, maxiters = 100)
ibeta.est(x, tol = 1e-07)
zilogitnorm.est(x)

Arguments

Details

Maximum likelihood estimation of the parameters of some distributions are performed, some of which use the Newton-Raphson. Some distributions and hence the functions do not accept zeros. "logitnorm.mle" fits the logistic normal, hence no Newton-Raphson is required and the "hypersecant01.mle" use the golden ratio search as is it faster than the Newton-Raphson (less computations). The "zilogitnorm.est" stands for the zero inflated logistic normal distribution. The "ibeta.est" fits the zero or the one inflated beta distribution.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Kumaraswamy, P. (1980). A generalized probability density function for double-bounded random processes. Journal of Hydrology. 46(1-2): 79-88.

Jones, M.C. (2009). Kumaraswamy's distribution: A beta-type distribution with some tractability advantages. Statistical Methodology. 6(1): 70-81.

You can also check the relevant wikipedia pages.

See Also

diri.est

Examples

x <- rbeta(1000, 1, 4)
beta.est(x)
ibeta.est(x)

x <- runif(1000)
hsecant01.est(x)
logitnorm.est(x)
ibeta.est(x)

x <- rbeta(1000, 2, 5)
x[sample(1:1000, 50)] <- 0
ibeta.est(x)

MLE of the a Dirichlet distribution

Description

MLE of the parameters of a Dirichlet distribution.

Usage

diri.est(x, type = "mle")

Arguments

Details

Maximum likelihood estimation of the parameters of a Dirichlet distribution is performed.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Ng Kai Wang, Guo-Liang Tian and Man-Lai Tang (2011). Dirichlet and related distributions: Theory, methods and applications. John Wiley & Sons.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

diri.nr, diri.contour, rdiri, ddiri, dda, diri.reg

Examples

x <- rdiri( 100, c(5, 7, 1, 3, 10, 2, 4) )
diri.est(x)
diri.est(x, type = "prec")

MLE of the Dirichlet distribution via Newton-Rapshon

Description

MLE of the Dirichlet distribution via Newton-Rapshon.

Usage

diri.nr(x, type = 1, tol = 1e-07)

Arguments

Details

Maximum likelihood estimation of the parameters of a Dirichlet distribution is performed via Newton-Raphson. Initial values suggested by Minka (2003) are used. The estimation is super faster than "diri.est" and the difference becomes really apparent when the sample size and or the dimensions increase. In fact this will work with millions of observations. So in general, I trust this one more than "diri.est".

The only problem I have seen with this method is that if the data are concentrated around a point, say the center of the simplex, it will be hard for this and the previous methods to give estimates of the parameters. In this extremely difficult scenario I would suggest the use of the previous function with the precision parametrization "diri.est(x, type = "prec")". It will be extremely fast and accurate.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Thomas P. Minka (2003). Estimating a Dirichlet distribution. http://research.microsoft.com/en-us/um/people/minka/papers/dirichlet/minka-dirichlet.pdf

See Also

diri.est, diri.contour rdiri, ddiri, dda

Examples

x <- rdiri( 100, c(5, 7, 5, 8, 10, 6, 4) )
diri.nr(x)
diri.nr(x, type = 2)
diri.est(x)

MLE of the folded model for a given value of \alpha

Description

MLE of the folded model for a given value of \alpha.

Usage

alpha.mle(x, a)
a.mle(a, x)

Arguments

Details

This is a function for choosing or estimating the value of \alpha in the \alpha-folded model (Tsagris and Stewart, 2020). It is called by a.est.

Value

If "alpha.mle" is called, a list including:

If "a.mle" is called, the log-likelihood is returned only.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M. and Stewart C. (2022). A Review of Flexible Transformations for Modeling Compositional Data. In Advances and Innovations in Statistics and Data Science, pp. 225–234. https://link.springer.com/chapter/10.1007/978-3-031-08329-7_10

Tsagris M. and Stewart C. (2020). A folded model for compositional data analysis. Australian and New Zealand Journal of Statistics, 62(2): 249-277. https://arxiv.org/pdf/1802.07330.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa.profile, alfa, alfainv, a.est

Examples

x <- as.matrix(iris[, 1:4])
x <- x / rowSums(x)
mod <- alfa.tune(x)
mod
alpha.mle(x, mod[1])

MLE of the zero adjusted Dirichlet distribution

Description

MLE of the zero adjusted Dirichlet distribution.

Usage

zad.est(y)

Arguments

Details

A zero adjusted Dirichlet distribution is being fitted and its parameters are estimated.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M. and Stewart C. (2018). A Dirichlet regression model for compositional data with zeros. Lobachevskii Journal of Mathematics, 39(3): 398–412.

Preprint available from https://arxiv.org/pdf/1410.5011.pdf

See Also

zadr, diri.nr, zilogitnorm.est, zeroreplace

Examples

y <- as.matrix(iris[, 1:3])
y <- y / rowSums(y)
mod1 <- diri.nr(y)
y[sample(1:450, 15) ] <- 0
mod2 <- zad.est(y)

Minimized Kullback-Leibler divergence between Dirichlet and logistic normal

Description

Minimized Kullback-Leibler divergence between Dirichlet and logistic normal distributions.

Usage

kl.diri.normal(a)

Arguments

Details

The function computes the minimized Kullback-Leibler divergence from the Dirichlet distribution to the logistic normal distribution.

Value

The minimized Kullback-Leibler divergence from the Dirichlet distribution to the logistic normal distribution.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data, p. 127. Chapman & Hall.

See Also

diri.nr, diri.contour, rdiri, ddiri, dda, diri.reg

Examples

a <- runif(5, 1, 5)
kl.diri.normal(a)

Mixture model selection via BIC

Description

Mixture model selection via BIC.

Usage

bic.mixcompnorm(x, G, type = "alr", veo = FALSE, graph = TRUE)

Arguments

Details

The alr or the ilr-transformation is applied to the compositional data first and then mixtures of multivariate Gaussian distributions are fitted. BIC is used to decide on the optimal model and number of components.

Value

A plot with the BIC of the best model for each number of components versus the number of components. A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ryan P. Browne, Aisha ElSherbiny and Paul D. McNicholas (2018). mixture: Mixture Models for Clustering and Classification. R package version 1.5.

Ryan P. Browne and Paul D. McNicholas (2014). Estimating Common Principal Components in High Dimensions. Advances in Data Analysis and Classification, 8(2), 217-226.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

mix.compnorm, mix.compnorm.contour, rmixcomp, bic.alfamixnorm

Examples

x <- as.matrix( iris[, 1:4] )
x <- x/ rowSums(x)
bic.mixcompnorm(x, 1:3, type = "alr", graph = FALSE)
bic.mixcompnorm(x, 1:3, type = "ilr", graph = FALSE)

Mixture model selection with the \alpha-transformation using BIC

Description

Mixture model selection with the \alpha-transformation using BIC.

Usage

bic.alfamixnorm(x, G, a = seq(-1, 1, by = 0.1), veo = FALSE, graph = TRUE)

Arguments

Details

The \alpha-transformation is applied to the compositional data first and then mixtures of multivariate Gaussian distributions are fitted. BIC is used to decide on the optimal model and number of components.

Value

A list including:

If graph is set equal to TRUE a plot with the BIC of the best model for each number of components versus the number of components and a list with the results of the Gaussian mixture model for each value of \alpha.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Ryan P. Browne, Aisha ElSherbiny and Paul D. McNicholas (2018). mixture: Mixture Models for Clustering and Classification. R package version 1.5.

Ryan P. Browne and Paul D. McNicholas (2014). Estimating Common Principal Components in High Dimensions. Advances in Data Analysis and Classification, 8(2), 217-226.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

alfa.mix.norm, mix.compnorm, mix.compnorm.contour, rmixcomp, alfa, alfa.knn, alfa.rda, comp.nb

Examples

x <- as.matrix( iris[, 1:4] )
x <- x/ rowSums(x)
bic.alfamixnorm(x, 1:3, a = c(0.4, 0.5, 0.6), graph = FALSE)

Multivariate analysis of variance (James test)

Description

Multivariate analysis of variance without assuming equality of the covariance matrices.

Usage

maovjames(x, ina, a = 0.05)

Arguments

Details

James (1954) also proposed an alternative to MANOVA when the covariance matrices are not assumed equal. The test statistic for k samples is

J=\sum_{i=1}^k\left(\bar{{\bf x}}_i-\bar{{\bf X}}\right)^T{\bf W}_i\left(\bar{{\bf x}}_i-\bar{{\bf X}}\right),

where \bar{{\bf x}}_i and n_i are the sample mean vector and sample size of the i-th sample respectively and {\bf W}_i=\left(\frac{{\bf S}_i}{n_i}\right)^{-1}, where {\bf S}_i is the covariance matrix of the i-sample mean vector and \bar{{\bf X}} is the estimate of the common mean \bar{{\bf X}}=\left(\sum_{i=1}^k{\bf W}_i\right)^{-1}\sum_{i=1}^k{\bf W}_i\bar{{\bf x}}_i.

Normally one would compare the test statistic with a \chi^2_{r,1-\alpha}, where r=p\left(k-1\right) are the degrees of freedom with k denoting the number of groups and p the dimensionality of the data. There are r constraints (how many univariate means must be equal, so that the null hypothesis, that all the mean vectors are equal, holds true), that is where these degrees of freedom come from. James (1954) compared the test statistic with a corrected \chi^2 distribution instead. Let A and B be A= 1+\frac{1}{2r}\sum_{i=1}^k\frac{\left[\text{tr}\left({\bf I}_p-{\bf W}^{-1}{\bf W}_i\right)\right]^2}{n_i-1} and B= \frac{1}{r\left(r+2\right)}\sum_{i=1}^k\left\lbrace\frac{\text{tr}\left[\left({\bf I}_p-{\bf W}^{-1}{\bf W}_i\right)^2\right]}{n_i-1}+\frac{\left[\text{tr}\left({\bf I}_p-{\bf W}^{-1}{\bf W}_i\right)\right]^2}{2\left(n_i-1\right)}\right\rbrace.

The corrected quantile of the \chi^2 distribution is given as before by 2h\left(\alpha\right)=\chi^2\left(A+B\chi^2\right).

Value

A vector with the next 4 elements:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

James G.S. (1954). Tests of Linear Hypotheses in Univariate and Multivariate Analysis when the Ratios of the Population Variances are Unknown. Biometrika, 41(1/2): 19–43.

See Also

hotel2T2, james

Examples

maovjames( as.matrix(iris[,1:4]), iris[,5] )

Multivariate analysis of variance assuming equality of the covariance matrices

Description

Multivariate analysis of variance assuming equality of the covariance matrices.

Usage

maov(x, ina)

Arguments

Details

Multivariate analysis of variance assuming equality of the covariance matrices.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Johnson R.A. and Wichern D.W. (2007, 6th Edition). Applied Multivariate Statistical Analysis, pg. 302–303.

Todorov V. and Filzmoser P. (2010). Robust Statistic for the One-way MANOVA. Computational Statistics & Data Analysis, 54(1): 37–48.

See Also

maovjames, hotel2T2, james

Examples

maov( as.matrix(iris[,1:4]), iris[,5] )
maovjames( as.matrix(iris[,1:4]), iris[,5] )

Multivariate kernel density estimation

Description

Multivariate kernel density estimation.

Usage

mkde(x, h = NULL, thumb = "silverman")

Arguments

Details

The multivariate kernel density estimate is calculated with a (not necssarily given) bandwidth value.

Value

A vector with the density estimates calculated for every vector.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Arsalane Chouaib Guidoum (2015). Kernel Estimator and Bandwidth Selection for Density and its Derivatives. The kedd R package.

M.P. Wand and M.C. Jones (1995). Kernel smoothing, pages 91-92.

B.W. Silverman (1986). Density estimation for statistics and data analysis, pages 76-78.

See Also

mkde.tune, comp.kerncontour

Examples

mkde( as.matrix(iris[, 1:4]), thumb = "scott" )
mkde( as.matrix(iris[, 1:4]), thumb = "silverman" )

Multivariate kernel density estimation for compositional data

Description

Multivariate kernel density estimation for compositional data.

Usage

comp.kern(x, type= "alr", h = NULL, thumb = "silverman")

Arguments

Details

The multivariate kernel density estimate is calculated with a (not necssarily given) bandwidth value.

Value

A vector with the density estimates calculated for every vector.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Arsalane Chouaib Guidoum (2015). Kernel Estimator and Bandwidth Selection for Density and its Derivatives.

The kedd R package.

M.P. Wand and M.C. Jones (1995). Kernel smoothing, pages 91-92.

B.W. Silverman (1986). Density estimation for statistics and data analysis, pages 76-78.

See Also

comp.kerncontour, mkde

Examples

x <- as.matrix(iris[, 1:3])
x <- x / rowSums(x)
f <- comp.kern(x)

Multivariate linear regression

Description

Multivariate linear regression.

Usage

multivreg(y, x, plot = TRUE, xnew = NULL)

Arguments

Details

The classical multivariate linear regression model is obtained.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

K.V. Mardia, J.T. Kent and J.M. Bibby (1979). Multivariate Analysis. Academic Press.

See Also

diri.reg, js.compreg, kl.compreg, ols.compreg, comp.reg

Examples

library(MASS)
x <- as.matrix(iris[, 1:2])
y <- as.matrix(iris[, 3:4])
multivreg(y, x, plot = TRUE)

Multivariate normal random values simulation on the simplex

Description

Multivariate normal random values simulation on the simplex.

Usage

rcompnorm(n, m, s, type = "alr")

Arguments

Details

The algorithm is straightforward, generate random values from a multivariate normal distribution in R^d and brings the values to the simplex S^d using the inverse of a log-ratio transformation.

Value

A matrix with the simulated data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

comp.den, rdiri, rcompt, rcompsn

Examples

x <- as.matrix(iris[, 1:2])
m <- colMeans(x)
s <- var(x)
y <- rcompnorm(100, m, s)
comp.den(y)
ternary(y)

Multivariate or univariate regression with compositional data in the covariates side using the \alpha-transformation

Description

Multivariate or univariate regression with compositional data in the covariates side using the \alpha-transformation.

Usage

alfa.pcr(y, x, a, k, model = "gaussian", xnew = NULL)

Arguments

Details

The \alpha-transformation is applied to the compositional data first ,the first k principal component scores are calcualted and used as predictor variables for a regression model. The family of distributions can be either, "normal" for continuous response and hence normal distribution, "binomial" corresponding to binary response and hence logistic regression or "poisson" for count response and poisson regression.

Value

A list tincluding:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Tsagris M. (2015). Regression analysis with compositional data containing zero values. Chilean Journal of Statistics, 6(2): 47-57. https://arxiv.org/pdf/1508.01913v1.pdf

Tsagris M.T., Preston S. and Wood A.T.A. (2011). A data-based power transformation for compositional data. In Proceedings of the 4th Compositional Data Analysis Workshop, Girona, Spain. https://arxiv.org/pdf/1106.1451.pdf

See Also

glm.pcr, alfapcr.tune

Examples

library(MASS)
y <- as.vector(fgl[, 1])
x <- as.matrix(fgl[, 2:9])
x <- x / rowSums(x)
mod <- alfa.pcr(y = y, x = x, 0.7, 1)
mod

Multivariate regression with compositional data

Description

Multivariate regression with compositional data.

Usage

comp.reg(y, x, type = "classical", xnew = NULL, yb = NULL)

Arguments

Details

The additive log-ratio transformation is applied and then the chosen multivariate regression is implemented. The alr is easier to explain than the ilr and that is why the latter is avoided here.

Value

A list including:

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr and Giorgos Athineou <gioathineou@gmail.com>.

References

Mardia K.V., Kent J.T., and Bibby J.M. (1979). Multivariate analysis. Academic press.

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

multivreg, spatmed.reg, js.compreg, diri.reg

Examples

library(MASS)
y <- as.matrix(iris[, 1:3])
y <- y / rowSums(y)
x <- as.vector(iris[, 4])
mod1 <- comp.reg(y, x)
mod2 <- comp.reg(y, x, type = "spatial")

Multivariate skew normal random values simulation on the simplex

Description

Multivariate skew normal random values simulation on the simplex.

Usage

rcompsn(n, xi, Omega, alpha, dp = NULL, type = "alr")

Arguments

Details

The algorithm is straightforward, generate random values from a multivariate t distribution in R^d and brings the values to the simplex S^d using the inverse of a log-ratio transformation.

Value

A matrix with the simulated data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Azzalini, A. and Dalla Valle, A. (1996). The multivariate skew-normal distribution. Biometrika, 83(4): 715–726.

Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew normal distribution. Journal of the Royal Statistical Society Series B, 61(3):579-602. Full-length version available from http://arXiv.org/abs/0911.2093

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

comp.den, rdiri, rcompnorm

Examples

x <- as.matrix(iris[, 1:2])
par <- sn::msn.mle(y = x)$dp
y <- rcompsn(100, dp = par)
comp.den(y, dist = "skewnorm")
ternary(y)

Multivariate t random values simulation on the simplex

Description

Multivariate t random values simulation on the simplex.

Usage

rcompt(n, m, s, dof, type = "alr")

Arguments

Details

The algorithm is straightforward, generate random values from a multivariate t distribution in R^d and brings the values to the simplex S^d using the inverse of a log-ratio transformation.

Value

A matrix with the simulated data.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris mtsagris@uoc.gr.

References

Aitchison J. (1986). The statistical analysis of compositional data. Chapman & Hall.

See Also

comp.den, rdiri, rcompnorm

Examples

x <- as.matrix(iris[, 1:2])
m <- Rfast::colmeans(x)
s <- var(x)
y <- rcompt(100, m, s, 10)
comp.den(y, dist = "t")
ternary(y)

Naive Bayes classifiers for compositional data