Source document: https://docs-lawep.s3.us-east-2.amazonaws.com/1698058828404.pdf (Liberia)
Package ‘ManlyMix’ April 29, 2023 Version 0.1.15 Date 2023-04-27 Title Manly Mixture Modeling and Model-Based Clustering Depends R (>= 3.0.0) LazyLoad yes LazyData no Description The utility of this package includes ﬁnite mixture modeling and model-based cluster- ing through Manly mixture models by Zhu and Mel- nykov (2016) <DOI:10.1016/j.csda.2016.01.015>.

It also provides capabilities for for- ward and backward model selection procedures.
License GPL (>= 2) Author Xuwen Zhu [aut, cre], Volodymyr Melnykov [aut], Michael Hutt [ctb, cph] (NM optimization in c), Stephen Moshier [ctb, cph] (eigen calculations in c), Rouben Rostamian [ctb, cph] (memory allocation in c) Maintainer Xuwen Zhu <xzhu20@cba.ua.edu> NeedsCompilation yes Repository CRAN Date/Publication 2023-04-29 06:40:02 UTC R topics documented: .

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17 19 ManlyMix-package Finite mixture modeling and model-based clustering based on Manly mixture models.

Description The utility of this package includes ﬁnite mixture modeling and model-based clustering based on Manly mixtures as well as forward and backward model selection procedures.

Details Package: Type: Version: Date: License: LazyLoad: ManlyMix Package 0.1.7 2016-12-01 GPL (>= 2) no Function ’Manly.sim’ simulates Manly mixture datasets.
Function ’Manly.overlap’ estimates the pairwise overlaps for a Manly mixture.

Function ’Manly.EM’ runs the EM algorithm for Manly mixture models.

Function ’Manly.select’ runs forward and backward model selection procedures.

Function ’Manly.Kmeans’ runs k-means model with Manly transformation.

Function ’Manly.var’ produces the variance-covariance matrix of the parameter estimates from Manly mixture model.

Function ’Manly.plot’ produces the density plot or contour plot of Manly mixture.
Function ’Manly.model’ incorporates all Manly mixture related functionality.

Author(s) Xuwen Zhu and Volodymyr Melnykov.

Maintainer: Xuwen Zhu <xuwen.zhu@louisville.edu> References Zhu, X. and Melnykov, V. (2016) “Manly Transformation in Finite Mixture Modeling”, Journal of Computational Statistics and Data Analysis, doi:10.1016/j.csda.2016.01.015.
acidity 3 Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of ﬁnite mixture model- ing and clustering algorithms”, Journal of Computational and Graphical Statistics, 2:19, 354-376.

Melnykov, V., Chen, W.-C., and Maitra, R. (2012) “MixSim: An R Package for Simulating Data to Study Performance of Clustering Algorithms”, Journal of Statistical Software, 51:12, 1-25.
Examples set.seed(123) K <- 3; p <- 4 X <- as.matrix(iris[,-5]) id.true <- rep(1:K, each = 50) # Obtain initial memberships based on the K-means algorithm id.km <- kmeans(X, K)$cluster # Run the CEM algorithm for Manly K-means model la <- matrix(0.1, K, p) C <- Manly.Kmeans(X, id = id.km, la = la) # Run the EM algorithm for a Gaussian mixture model based on K-means solution G <- Manly.EM(X, id = id.km) id.G <- G$id # Run FORWARD SELECTION ('silent' is on) F <- Manly.select(X, model = G, method = "forward", silent = TRUE) # Run the EM algorithm for a full Manly mixture model based on Gaussian mixture solution la <- matrix(0.1, K, p) M <- Manly.EM(X, id = id.G, la = la) # Run BACKWARD SELECTION ('silent' is off) B <- Manly.select(X, model = M, method = "backward") BICs <- c(G$bic, M$bic, F$bic, B$bic) names(BICs) <- c("Gaussian", "Manly", "Forward", "Backward") BICs acidity Acidity data Description Acidity index measured in a sample of 155 lakes in the Northeastern United States.
The data are on the log scale.

Usage data(acidity) 4 Format A data vector with 155 observations on the acidity index.

Details The data was ﬁrst analysed by Crawford et al.

(1994).

References ais Crawford, S. L. (1994) An application of the Laplace method to ﬁnite mixture distribution, Journal of the American Statistical Association, 89, 259-267.
Examples data(acidity) ais Australian Institute of Sport data Description Data on 102 male and 100 female athletes collected at the Australian Institute of Sport, courtesy of Richard Telford and Ross Cunningham.

Usage data(ais) Format A data frame with 202 observations on the following 13 variables.
sex Factor with levels: female, male; sport Factor with levels: B_Ball, Field, Gym, Netball, Row Swim, T_400m, Tennis, T_Sprnt, W_Polo; RCC Red cell count; WCC White cell count; Hc Hematocrit; Hg Hemoglobin; Fe Plasma ferritin concentration; BMI Body Mass Index; SSF Sum of skin folds; Bfat Body fat percentage; LBM Lean body mass; Ht Height, cm; Wt Weight, kg bankruptcy Details 5 The data have been made publicly available in connection with the book by Cook and Weisberg (1994).
References Cook and Weisberg (1994) An Introduction to Regression Graphics, John Wiley & Sons, New York.

Examples data(ais) bankruptcy Bankruptcy data Description The data set contain the ratio of retained earnings (RE) to total assets, and the ratio of earnings before interests and taxes (EBIT) to total assets of 66 American ﬁrms recorded in the form of ratios.

Half of the selected ﬁrms had ﬁled for bankruptcy.


Usage data(bankruptcy) Format A data frame with the following variables: Y The status of the ﬁrm: 0 bankruptcy or 1 ﬁnancially sound; RE Ratio of retained earnings to total assets; EBIT Ratio of earnings before interests and taxes to total assets References Altman E.I.
(1968) Financial ratios, discriminant analysis and the prediction of corporate bankruptcy, J Finance 23(4): 589-609 Examples data(bankruptcy) 6 Manly.EM ClassAgree Calculates the confusion matrix and number of misclassiﬁcations Description Calculates the confusion matrix and number of misclassiﬁcations.
Usage ClassAgree(est.id, trueid) Arguments est.id trueid Value estimated membership vector true membership vector ClassificationTable confusion table between true and estimated partitions MisclassificationNum number of misclassiﬁcations Examples set.seed(123) K <- 3; p <- 4 X <- as.matrix(iris[,-5]) id.true <- rep(1:K, each = 50) # Obtain initial memberships based on the K-means algorithm id.km <- kmeans(X, K)$cluster ClassAgree(id.km, id.true) Manly.EM EM algorithm for Manly mixture model Description Runs the EM algorithm for a Manly mixture model with speciﬁed initial membership and transfor- mation parameters.
Manly.EM Usage 7 Manly.EM(X, id = NULL, la = NULL, tau = NULL, Mu = NULL, S = NULL, tol = 1e-5, max.iter = 1000) Arguments X id la tau Mu S tol dataset matrix (n x p) initial membership vector (length n) initial transformation parameters (K x p) initial vector of mixing proportions (length K) initial matrix of mean vectors (K x p) initial array of covariance matrices (p x p x K) tolerance level max.iter maximum number of iterations Details Runs the EM algorithm for a Manly mixture model for a provided dataset.
Manly mixture model assumes that a multivariate Manly transformation applied to each component allows to reach near- normality.

A user has a choice to specify either initial id vector ’id’ and transformation parameters ’la’ or initial mode parameters ’la’, ’tau’, ’Mu’, and ’S’.

In the case when transformation param- eters are not provided, the function runs the EM algorithm without any transformations, i.e., it is equivalent to the EM algorithm for a Gaussian mixtuire model.
If some transformation parameters have to be excluded from the consideration, in the corresponding positions of matrix ’la’, the user has to specify value 0.

Notation: n - sample size, p - dimensionality of the dataset X, K - number of mixture components.
Value la tau Mu S gamma id ll bic iter flag See Also Manly.select matrix of the estimated transformation parameters (K x p) vector of mixing proportions (length K) matrix of the estimated mean vectors (K x p) array of the estimated covariance matrices (p x p x K) matrix of posterior probabilities (n x K) estimated membership vector (length n) log likelihood value Bayesian Information Criterion number of EM iterations run convergence ﬂag (0 - success, 1 - failure) 8 Examples set.seed(123) K <- 3; p <- 4 X <- as.matrix(iris[,-5]) id.true <- rep(1:K, each = 50) Manly.Kmeans # Obtain initial memberships based on the K-means algorithm id.km <- kmeans(X, K)$cluster # Run the EM algorithm for a Gaussian mixture model based on K-means solution A <- Manly.EM(X, id.km) id.Gauss <- A$id ClassAgree(id.Gauss, id.true) # Run the EM algorithm for a Manly mixture model based on Gaussian mixture solution la <- matrix(0.1, K, p) B <- Manly.EM(X, id.Gauss, la) id.Manly <- B$id ClassAgree(id.Manly, id.true) Manly.Kmeans k-means algorithm with Manly transformation Description Runs the CEM algorithm for k-means clustering with speciﬁed initial membership and transforma- tion parameters.
Usage Manly.Kmeans(X, id = NULL, la = NULL, Mu = NULL, S = NULL, initial = "k-means", K = NULL, nstart = 100, method = "ward.D", tol = 1e-5, max.iter = 1000) Arguments X id la Mu S dataset matrix (n x p) initial membership vector (length n) initial transformation parameters (K x p) initial matrix of mean vectors (K x p) initial vector of variances (K) Manly.Kmeans initial K nstart method tol 9 initialization strategy of the EM algorithm ("k-means" - partition obtained by k- means clustering, "hierarchical" - partition obtained by hierarchical clustering) number of clusters for the k-means initialization number of random starts for the k-means initialization linkage method for the hierarchical initialization tolerance level max.iter maximum number of iterations Details Runs the CEM algorithm for k-means clustering with Manly transformation for a provided dataset.
The model assumes that a multivariate Manly transformation applied to each component allows to reach near-normality.

A user has a choice to specify either initial id vector ’id’ and transformation parameters ’la’ or initial mode parameters ’la’, ’Mu’, and ’S’.

In the case when transformation parameters are not provided, the function runs the EM algorithm without any transformations, i.e., it is equivalent to the EM algorithm for a k-means model.
If some transformation parameters have to be excluded from the consideration, in the corresponding positions of matrix ’la’, the user has to specify value 0.

Notation: n - sample size, p - dimensionality of the dataset X, K - number of mixture components.
matrix of the estimated transformation parameters (K x p) matrix of the estimated mean vectors (K x p) array of the estimated covariance matrices (K) estimated membership vector (length n) number of EM iterations run convergence ﬂag (0 - success, 1 - failure) Value la Mu S id iter flag See Also Manly.EM Examples set.seed(123) K <- 3; p <- 4 X <- as.matrix(iris[,-5]) id.true <- rep(1:K, each = 50) # Obtain initial memberships based on the traditional K-means algorithm id.km <- kmeans(X, K)$cluster # Run the CEM algorithm for k-means with Manly transformation based on traditional k-means solution la <- matrix(0.1, K, p) 10 Manly.model B <- Manly.Kmeans(X, id.km, la) id.Manly <- B$id ClassAgree(id.Manly, id.true) Manly.model Manly mixture model Description Runs all the functionality of a Manly mixture model.
Usage Manly.model(X, K = 1:5, Gaussian = FALSE, initial = "k-means", nstart = 100, method = "ward.D", select = "none", silent = TRUE, plot = FALSE, var1 = NULL, var2 = NULL, VarAssess = FALSE, conf.CI = NULL, overlap = FALSE, N = 1000, tol = 1e-5, max.iter = 1000, ...) short.iter = 5, Arguments X K Gaussian initial nstart method short.iter select silent plot var1 var2 VarAssess conf.CI overlap N tol max.iter ... dataset matrix (n x p) number of components tested whether Gaussian mixture models are run or not initialization strategy of the EM algorithm ("k-means" - partition obtained by k- means clustering, "hierarchical" - partition obtained by hierarchical clustering, "emEM" - parameters estimated by the emEM algorithm) number of random starts for the k-means or the emEM initialization linkage method for the hierarchical initialization number of short emEM iterations to run control to run Manly.select or not ("none" - do not run Manly.select , "forward" - run forward selection, "backward" - run backward selection) control the output from Manly.select control to construct the density or contour plot or not x-axis variable for contour plot or variable for density plot y-axis variable for contour plot run the variability assessment of the Manly mixture model or not specify the conﬁdence level of parameter estimates estimate the overlap of Manly mixture components or not number of Monte Carlo simulations to run in the Manly.overlap function tolerance level maximum number of iterations further arguments related to Manly.plot Manly.overlap Details 11 Wrapper function that incorporates all functionality associated with Manly mixture modeling.
best mixture model obtained estimated variance-covariance matrix for model parameter estimates estimated overlap of Manly mixture components Value Model VarAssess Overlap See Also Manly.EM Examples set.seed(123) K <- 3; p <- 4 X <- as.matrix(iris[,-5]) id.true <- rep(1:K, each = 50) Obj <- Manly.model(X, K = 1:5, initial = "emEM", nstart = 1, short.iter = 5) Manly.overlap Estimates the overlap for a Manly mixture Description Estimates the pairwise overlap matrix for a Manly mixture by simulating samples based on user- speciﬁed parameters.
Usage Manly.overlap(tau, Mu, S, la, N = 1000) Arguments la tau Mu S N matrix of transformation parameters (K x p) vector of mixing proportions (length K) matrix of mean vectors (K x p) array of covariance matrices (p x p x K) number of samples simulated 12 Details Manly.plot Estimates the pairwise overlap matrix for a Manly mixture.

Overlap is deﬁned as sum of two misclassiﬁcation probabilities.
Value OmegaMap BarOmega MaxOmega References matrix of misclassiﬁcation probabilities (K x K); OmegaMap[i,j] is the proba- bility that X coming from the i-th component is classiﬁed to the j-th component.

value of average overlap.

value of maximum overlap.

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of ﬁnite mixture model- ing and clustering algorithms”, Journal of Computational and Graphical Statistics, 2:19, 354-376.
Melnykov, V., Chen, W.-C., and Maitra, R. (2012) “MixSim: An R Package for Simulating Data to Study Performance of Clustering Algorithms”, Journal of Statistical Software, 51:12, 1-25.
Examples set.seed(123) #sets the number of components, dimensionality and sample size K <- 3 p <- 2 #sets the mixture parameters tau <- c(0.25, 0.3, 0.45) Mu <- matrix(c(4.5,4,5,7,8,5.5),3) la <- matrix(c(0.2,0.5,0.3,0.25,0.35,0.4),3) S <- array(NA, dim = c(p,p,K)) S[,,1] <- matrix(c(0.4,0,0,0.4),2) S[,,2] <- matrix(c(1,-0.2,-0.2,0.6),2) S[,,3] <- matrix(c(2,-1,-1,2),2) #computes the overlap A <- Manly.overlap(tau, Mu, S, la) print(A) Manly.plot Density plot or contour plot for Manly mixture model Description Provides a contour plot or a density plot for the ﬁtted data with Manly mixture model.
Usage Manly.plot(X, var1 = NULL, var2 = NULL, model = NULL, x.slice = 100, y.slice = 100, x.mar = 1, y.mar = 1, col = "lightgrey", ...) 13 dataset matrix (n x p) x-axis variable for contour plot or variable for density plot y-axis variable for contour plot ﬁtted Manly mixture model number of slices in the ﬁrst variable sequence in the contour number of slices in the second variable sequence in the contour value to be subtracted/added to the smallest/largest observation in the x-axis value to be subtracted/added to the smallest/largest observation in the y-axis color of the contour lines further arguments related to contour and hist Manly.plot Arguments X var1 var2 model x.slice y.slice x.mar y.mar col ...
Details Provides the contour plot or density plot for the ﬁtted data by Manly mixture model.
See Also Manly.EM Examples set.seed(123) K <- 2; p <- 2 X <- as.matrix(faithful) # Obtain initial memberships based on the K-means algorithm id.km <- kmeans(X, K)$cluster # Run the EM algorithm for a Manly mixture model based on K-means solution la <- matrix(0.1, K, p) B <- Manly.EM(X, id.km, la) Manly.plot(X, model = B, var1 = 1, x.mar = 1, y.mar = 2, xaxs="i", yaxs="i", xaxt="n", yaxt="n", xlab="", ylab = "", nlevels = 10, drawlabels = FALSE, lwd = 3.2, col = "lightgrey", pch = 19) 14 Manly.select Manly.select Manly transformation selection Description Runs forward or backward model selection procedures for ﬁnding the optimal model in terms of BIC.
Usage Manly.select(X, model, method, tol = 1e-5, max.iter = 1000, silent = FALSE) Arguments X model method tol max.iter silent Details dataset matrix (n x p) list containing parameters of the initial model model selection method (options ’forward’ and ’backward’) tolerance level maximum number of iterations output control Runs Manly forward and backward model selection procedures for a provided dataset.

Forward and backward selection can be started from any ManlyMix object provided in ’model’.
Manly trans- formation parameters are provided in matrix ’model$la’.

If some transformations are not needed for speciﬁc components, zeros have to be speciﬁed in corresponding poisition.

When all transfor- mation parameters are set to zero, Manly mixture model degenerates to a Gaussian mixture model.

Notation: n - sample size, p - dimensionality of the dataset X, K - number of mixture components.
Value la tau Mu S gamma id ll bic iter flag matrix of the estimated transformation parameters (K x p) vector of mixing proportions (length K) matrix of the estimated mean vectors (K x p) array of the estimated covariance matrices (p x p x K) matrix of posterior probabilities (n x K) estimated membership vector (length n) log likelihood value Bayesian Information Criterion number of EM iterations run convergence ﬂag (0 - success, 1 - failure) 15 Manly.sim See Also Manly.EM Examples set.seed(123) K <- 3; p <- 4 X <- as.matrix(iris[,-5]) id.true <- rep(1:K, each = 50) # Obtain initial memberships based on the K-means algorithm id.km <- kmeans(X, K)$cluster # Run the EM algorithm for a Gaussian mixture model based on K-means solution G <- Manly.EM(X, id = id.km) id.G <- G$id # Run FORWARD SELECTION ('silent' is on) F <- Manly.select(X, model = G, method = "forward", silent = TRUE) # Run the EM algorithm for a full Manly mixture model based on Gaussian mixture solution la <- matrix(0.1, K, p) M <- Manly.EM(X, id = id.G, la = la) # Run BACKWARD SELECTION ('silent' is off) B <- Manly.select(X, model = M, method = "backward") BICs <- c(G$bic, M$bic, F$bic, B$bic) names(BICs) <- c("Gaussian", "Manly", "Forward", "Backward") BICs Manly.sim Simulates Manly mixture dataset Description Simulates Manly mixture dataset given the mixture parameters and sample size.
Usage Manly.sim(n, la, tau, Mu, S) Arguments n la sample size matrix of transformation parameters (K x p) 16 tau Mu S Details vector of mixing proportions (length K) matrix of mean vectors (K x p) array of covariance matrices (p x p x K) Manly.var Simulates a Manly mixture dataset.

Manly mixture data points are computed from back-transforming Gaussian distributed data points using user-speciﬁed transformation parameters ’la’.
Value X id Examples set.seed(123) the simulated Manly mixture dataset the simulated membership of the data #sets the number of components, dimensionality and sample size K <- 3 p <- 2 n <- 1000 #sets the parameters to simulate data from tau <- c(0.25, 0.3, 0.45) Mu <- matrix(c(12,4,4,12,4,10),3) la <- matrix(c(1.2,0.5,1,0.5,0.5,0.7),3) S <- array(NA, dim = c(p,p,K)) S[,,1] <- matrix(c(4,0,0,4),2) S[,,2] <- matrix(c(5,-1,-1,3),2) S[,,3] <- matrix(c(2,-1,-1,2),2) #use function Manly.sim to simulate dataset with membership A <- Manly.sim(n, la, tau, Mu, S) #plot the data plot(A$X, col = A$id) Manly.var Variability assessment of Manly mixture model Description Runs the variability assessment for a Manly mixture model.
Usage Manly.var(X, model = NULL, conf.CI = NULL) seeds Arguments X model conf.CI Details dataset matrix (n x p) Manly mixture model conﬁdence level, say 95 percent conﬁdence 17 Returns the estimated variance-covariance matrix and conﬁdence intervals for model parameter estimates.

variance-covariance matrix.

conﬁdence intervals for each parameter.
Value V CI See Also Manly.EM Examples set.seed(123) #Use iris dataset K <- 3; p <- 4 X <- as.matrix(iris[,-5]) #Use k-means clustering result #all skewness parameters set to be 0.1 as the initialization of the EM algorithm id.km <- kmeans(X, K)$cluster la <- matrix(0.1, K, p) #Run the EM algorithm with Manly mixture model M.EM <- Manly.EM(X, id.km, la) # Run the variability assessment Manly.var(X, M.EM, conf.CI = 0.95) seeds Wheat kernel Data Description The examined group comprised kernels belonging to three different varieties of wheat: Kama, Rosa and Canadian, 70 elements each, randomly selected for the experiment.
High quality visualization of the internal kernel structure was detected using a soft X-ray technique.

Studies were conducted using combine harvested wheat grain originating from experimental ﬁelds, explored at the Institute of Agrophysics of the Polish Academy of Sciences in Lublin.

18 Usage data(seeds) Format A data frame with 210 observations on the following 7 variables.
seeds V1 Area A; V2 Perimeter P; V3 Compactness; V4 Length of kernel; V5 Width of kernel; V6 Asymmetry coefﬁcient; V7 Length of kernel groove; V8 Seed species: 1, 2, 3 References M. Charytanowicz, J. Niewczas, P. Kulczycki, P.A.

Kowalski, S. Lukasik, S. Zak (2010), A Com- plete Gradient Clustering Algorithm for Features Analysis of X-ray Images.

Information Technolo- gies in Biomedicine, Ewa Pietka, Jacek Kawa, Springer-Verlag, Berlin-Heidelberg.
Examples data(seeds) Index ∗ CEM algorithm Manly.Kmeans, 8 ∗ Contour plot Manly.plot, 12 ∗ Data simulation Manly.sim, 15 ∗ EM algorithm ClassAgree, 6 Manly.EM, 6 Manly.select, 14 ∗ Forward and backward selection ClassAgree, 6 Manly.EM, 6 Manly.select, 14 ∗ Manly mixture ClassAgree, 6 Manly.EM, 6 Manly.model, 10 Manly.overlap, 11 Manly.plot, 12 Manly.select, 14 Manly.sim, 15 Manly.var, 16 ∗ Manly transformation Manly.Kmeans, 8 ∗ Overlap Manly.overlap, 11 ∗ Variability assessment Manly.var, 16 ∗ datasets acidity, 3 ais, 4 bankruptcy, 5 seeds, 17 ∗ k-means Manly.Kmeans, 8 acidity, 3 ais, 4 bankruptcy, 5 ClassAgree, 6 contour, 13 hist, 13 Manly.EM, 6 Manly.Kmeans, 8 Manly.model, 10 Manly.overlap, 11 Manly.plot, 10, 12 Manly.select, 14 Manly.sim, 15 Manly.var, 16 ManlyMix-package, 2 seeds, 17 19