Simulation of Gaussian Finite Mixture Models

Description

Simulation of Gaussian finite mixture models for prespecified levels of average or/and maximum overlap. Pairwise overlap is defined as the sum of two misclassification probabilities.

Details

Function 'MixSim' simulates a finite mixture model for a prespecified level of average or/and maximum overlap.

Function 'overlap' computes all misclassification probabilities for a finite mixture model.

Function 'pdplot' constructs a parallel distribution plot for a finite mixture model.

Function 'simdataset' simulates a dataset from a finite mixture model.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

Maintainer: Volodymyr Melnykov <vmelnykov@cba.ua.edu>

References

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

Davies, R. (1980) “The distribution of a linear combination of chi-square random variables”, Applied Statistics, 29, 323-333.

Meila, M. (2006) “Comparing clusterings - an information based distance”, Journal of Multivariate Analysis, 98, 873-895.

Examples

# Simulate parameters of a mixture model
A <- MixSim(BarOmega = 0.01, MaxOmega = 0.10, K = 10, p = 5)

# Display the mixture via the parallel distribution plot
pdplot(A$Pi, A$Mu, A$S, MaxInt = 0.5)

Classification Proportion

Description

Computes the agreement proportion between two classification vectors.

Usage

ClassProp(id1, id2)

Arguments

Value

Returns the value of the proportion of agreeing elements.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Meila, M. (2006) “Comparing clusterings - an information based distance”, Journal of Multivariate Analysis, 98, 873-895.

See Also

RandIndex, and VarInf.

Examples

id1 <- c(rep(1, 50), rep(2,100))
id2 <- rep(1:3, each = 50)
ClassProp(id1, id2)

Mixture Simulation based on generalized overlap of Maitra

Description

Generates a finite mixture model with Gaussian components for a prespecified level of goMega (generalized overlap of Maitra).

Usage

MixGOM(goMega = NULL, K, p, sph = FALSE, hom = FALSE,
       ecc = 0.90, PiLow = 1.0, int = c(0.0, 1.0), resN = 100,
       eps = 1e-06, lim = 1e06)

Arguments

Details

Returns mixture parameters satisfying the prespecified level of goMega.

Value

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. (2010) “A re-defined and generalized percent-overlap-of-activation measure for studies of fMRI reproducibility and its use in identifying outlier activation maps”, NeuroImage, 50, 124-135.

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

Davies, R. (1980) “The distribution of a linear combination of chi-square random variables”, Applied Statistics, 29, 323-333.

See Also

overlapGOM, MixSim, and simdataset.

Examples

set.seed(1234)

# controls average and maximum overlaps
(ex.1 <- MixGOM(goMega = 0.05, K = 4, p = 5))

# controls maximum overlap
(ex.2 <- MixGOM(goMega = 0.15, K = 4, p = 5, sph = TRUE))

Mixture Simulation

Description

Generates a finite mixture model with Gaussian components for prespecified levels of maximum and/or average overlaps.

Usage

MixSim(BarOmega = NULL, MaxOmega = NULL, K, p, sph = FALSE, hom = FALSE,
       ecc = 0.90, PiLow = 1.0, int = c(0.0, 1.0), resN = 100,
       eps = 1e-06, lim = 1e06)

Arguments

Details

If 'BarOmega' is not specified, the function generates a mixture solely based on 'MaxOmega'; if 'MaxOmega' is not specified, the function generates a mixture solely based on 'BarOmega'.

If 'hom' is TRUE, only one of 'BarOmega' or 'MaxOmega' can be specified.

Value

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

Davies, R. (1980) “The distribution of a linear combination of chi-square random variables”, Applied Statistics, 29, 323-333.

See Also

overlap, pdplot, and simdataset.

Examples

set.seed(1234)

# controls average and maximum overlaps
(ex.1 <- MixSim(BarOmega = 0.05, MaxOmega = 0.15, K = 4, p = 5))
summary(ex.1)

# controls average overlap
(ex.2 <- MixSim(BarOmega = 0.05, K = 4, p = 5, hom = TRUE))
summary(ex.2)

# controls maximum overlap
(ex.3 <- MixSim(MaxOmega = 0.15, K = 4, p = 5, sph = TRUE))
summary(ex.3)

Rand's Index

Description

Computes Rand, adjusted Rand, Fowlkes and Mallows, and Merkin indices.

Usage

RandIndex(id1, id2)

Arguments

Value

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Rand, W.M. (1971) “Objective criteria for the evaluation of clustering methods”, Journal of the American Statistical Association, 66:336, 846-850.

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The Journal of Computational and Graphical Statistics, 2:19, 354-376.

Meila, M. (2006) “Comparing clusterings - an information based distance”, Journal of Multivariate Analysis, 98, 873-895.

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.

See Also

MixSim, pdplot, simdataset, ClassProp, and VarInf.

Examples

id1 <- c(rep(1, 50), rep(2,100))
id2 <- rep(1:3, each = 50)
RandIndex(id1, id2)

Variation of Information

Description

Computes the variation of information for two classification vectors.

Usage

VarInf(id1, id2)

Arguments

Value

Returns the variation of information. It is equal to 0 if and only if two classification vectors are identical.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Meila, M. (2006) “Comparing clusterings - an information based distance”, Journal of Multivariate Analysis, 98, 873-895.

See Also

ClassProp, and RandIndex.

Examples

id1 <- c(rep(1, 50), rep(2,100))
id2 <- rep(1:3, each = 50)
VarInf(id1, id2)

Overlap

Description

Computes misclassification probabilities and pairwise overlaps for finite mixture models with Gaussian components. Overlap is defined as sum of two misclassification probabilities.

Usage

overlap(Pi, Mu, S, eps = 1e-06, lim = 1e06)

Arguments

Value

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

Davies, R. (1980) “The distribution of a linear combination of chi-square random variables”, Applied Statistics, 29, 323-333.

See Also

MixSim, pdplot, and simdataset.

Examples

data("iris", package = "datasets")
p <- ncol(iris) - 1
id <- as.integer(iris[, 5])
K <- max(id)

# estimate mixture parameters
Pi <- prop.table(tabulate(id))
Mu <- t(sapply(1:K, function(k){ colMeans(iris[id == k, -5]) }))
S <- sapply(1:K, function(k){ var(iris[id == k, -5]) })
dim(S) <- c(p, p, K)

overlap(Pi = Pi, Mu = Mu, S = S)

Generalized overlap of Maitra

Description

Computes the generalized overlap as defined by R. Maitra.

Usage

overlapGOM(Pi, Mu, S, eps = 1e-06, lim = 1e06)

Arguments

Value

Returns the value of goMega.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. (2010) “A re-defined and generalized percent-overlap-of-activation measure for studies of fMRI reproducibility and its use in identifying outlier activation maps”, NeuroImage, 50, 124-135.

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.

Davies, R. (1980) “The distribution of a linear combination of chi-square random variables”, Applied Statistics, 29, 323-333.

See Also

MixSim, MixGOM, and overlap.

Examples

data("iris", package = "datasets")
p <- ncol(iris) - 1
id <- as.integer(iris[, 5])
K <- max(id)

# estimate mixture parameters
Pi <- prop.table(tabulate(id))
Mu <- t(sapply(1:K, function(k){ colMeans(iris[id == k, -5]) }))
S <- sapply(1:K, function(k){ var(iris[id == k, -5]) })
dim(S) <- c(p, p, K)

overlapGOM(Pi = Pi, Mu = Mu, S = S)

Parallel Distribution Plot

Description

Constructs a parallel distribution plot for Gaussian finite mixture models.

Usage

pdplot(Pi, Mu, S, file = NULL, Nx = 5, Ny = 5, MaxInt = 1, marg = c(2,1,1,1))

Arguments

Details

If 'file' is specified, produced plot will be saved as a .pdf-file.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

See Also

MixSim, overlap, and simdataset.

Examples

data("iris", package = "datasets")
p <- ncol(iris) - 1
id <- as.integer(iris[, 5])
K <- max(id)

# estimate mixture parameters
Pi <- prop.table(tabulate(id))
Mu <- t(sapply(1:K, function(k){ colMeans(iris[id == k, -5]) }))
S <- sapply(1:K, function(k){ var(iris[id == k, -5]) })
dim(S) <- c(p, p, K)

pdplot(Pi = Pi, Mu = Mu, S = S)

Permutations

Description

Returns all possible permutations given the number of elements.

Usage

perms(n)

Arguments

Value

Returns a matrix containing all possible permutations.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

See Also

ClassProp.

Examples

perms(3)

Functions for Printing or Summarizing Objects

Description

A MixSim and MixGOM classes are declared, and these are functions to print and summarize objects.

Usage

## S3 method for class 'MixSim'
print(x, ...)
## S3 method for class 'MixSim'
summary(object, ...)
## S3 method for class 'MixGOM'
print(x, ...)

Arguments

Details

These are useful functions for summarizing and debugging.

For other functions, they only show summaries of objects. Use names or str to explore the details.

Value

The results will cat or print on the STDOUT by default.

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

See Also

MixSim.

Examples

## Not run: 
# Functions applied by directly type the names of objects.

## End(Not run)

Dataset Simulation

Description

Simulates a datasets of sample size n given parameters of finite mixture model with Gaussian components.

Usage

simdataset(n, Pi, Mu, S, n.noise = 0, n.out = 0, alpha = 0.001,
           max.out = 100000, int = NULL, lambda = NULL)

Arguments

Details

The function simulates a dataset of n observations from a mixture model with parameters 'Pi' (mixing proportions), 'Mu' (mean vectors), and 'S' (covariance matrices). Mixture component sample sizes are produced as a realization from a multinomial distribution with probabilities given by mixing proportions. To make a dataset more challenging for clustering, a user might want to simulate noise variables or outliers. Parameter 'n.noise' specifies the desired number of noise variables. If an interval 'int' is specified, noise will be simulated from a Uniform distribution on the interval given by 'int'. Otherwise, noise will be simulated uniformly between the smallest and largest coordinates of mean vectors. 'n.out' specifies the number of observations outside (1 - 'alpha') ellipsoidal contours for the weighted component distributions. Outliers are simulated on a hypercube specified by the interval 'int'. A user can apply an inverse Box-Cox transformation providing a vector of coefficients 'lambda'. The value 1 implies that no transformation is needed for the corresponding coordinate.

Value

Author(s)

Volodymyr Melnykov, Wei-Chen Chen, and Ranjan Maitra.

References

Maitra, R. and Melnykov, V. (2010) “Simulating data to study performance of finite mixture modeling and clustering algorithms”, The 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.

See Also

MixSim, overlap, and pdplot.

Examples

## Not run: 
set.seed(1234)

repeat{
   Q <- MixSim(BarOmega = 0.01, K = 4, p = 2)
   if (Q$fail == 0) break
}

# simulate a dataset of size 300 and add 10 outliers simulated on (0,1)x(0,1)
A <- simdataset(n = 500, Pi = Q$Pi, Mu = Q$Mu, S = Q$S, n.out = 10, int = c(0, 1))
colors <- c("red", "green", "blue", "brown", "magenta")
plot(A$X, xlab = "x1", ylab = "x2", type = "n")
for (k in 0:4){
   points(A$X[A$id == k, ], col = colors[k+1], pch = 19, cex = 0.5)
}

repeat{
   Q <- MixSim(MaxOmega = 0.1, K = 4, p = 1)
   if (Q$fail == 0) break
}

# simulate a dataset of size 300 with 1 noise variable
A <- simdataset(n = 300, Pi = Q$Pi, Mu = Q$Mu, S = Q$S, n.noise = 1)
plot(A$X, xlab = "x1", ylab = "x2", type = "n")
for (k in 1:4){
   points(A$X[A$id == k, ], col = colors[k+1], pch = 19, cex = 0.5)
}

## End(Not run)