Dynamic programming initialization of 2D GMM EM

Description

Function for generating initial conditions of 2D GMM model by dynamic programming.

Usage

DP_init_2D(X, Y, KS)

Arguments

Value

Initial values for EM.

Expectation-maximization algorithm for 1D data

Description

The function performs the EM algorithm to find the local maximum likelihood for the estimated Gaussian mixture parameters.

Usage

EM_iter(X, alpha, mu, sig, Y = NULL, opts = NULL)

Arguments

Value

Returns a list of GMM parameter values that correspond to the local extremes for each component.

alpha

Vector of optimal alpha (weights) values.

mu

Vector of optimal mu (means) values.

sigma

Vector of optimal sigma (standard devations) values.

logLik

Log-likelihood statistic for the estimated number of components.

crit

Value of the selected information criterion in local extreme of likelihood function.

See Also

runGMM and gaussian_mixture_vector

Examples

data("example")
opts <- GMM_1D_opts
Y <- matrix(1, 1, length(example$Dist))
rcpt <- EM_iter(example$Dist, 1, mean(example$Dist), sd(example$Dist), Y, opts)

Expectation-maximization algorithm for 2D data

Description

The function performs the EM algorithm to find the local maximum likelihood for the estimated Gaussian mixture parameters.

Usage

EM_iter_2D(X, Y, init, opts = NULL)

Arguments

Value

Function returns a list of GMM parameters for tested number of components:

alpha

Weights (alpha) of each component.

center

Means of decomposition.

covar

Covariances of each component.

KS

Estimated number of components.

logL

Log-likelihood statistic for the estimated number of components.

IC

The value of the selected information criterion which was used to calculate the number of components.

See Also

runGMM2D

Examples

data("example2D")
X <- example2D[,1:2]
Y <- matrix(1, 1, nrow(X))

opts <- GMM_2D_opts

# It is necessary to define the initial conditions. Here we use random initialization.
alpha <- matrix(1, 1, opts$KS)/opts$KS
center <- as.matrix(X[sample(nrow(X), opts$KS),])
rownames(center) <- NULL
covar <- replicate(opts$KS, diag(apply(as.matrix(X), 2, sd)/opts$KS), simplify = "array")

init <- list(alpha = alpha,
             center = center,
             covar = covar,
             KS = opts$KS)

gmm <- EM_iter_2D(X, Y, init, opts)

Default configuration for 1D Gaussian Mixture decomposition

Description

A list with parameters customizing a GMM for 1D and binned data. Each component of the list is an effective argument for runGMM.

Usage

GMM_1D_opts

Format

A list with the following components:

KS

Maximum number of components of the model.

eps_change

Criterion for early stopping of EM (1e-7, by default) given by the following formula:

\sum{(|\alpha - \alpha_{old})|} + \frac{\sum{(\frac{|\sigma^2 - \sigma^2_{old}|}{\sigma^2})}}{length(\alpha)}

max_iter

Maximum number of iterations of EM algorithm. By default it is max_iter = 10 000.

SW

Parameter for calculating minimum variance of each Gaussian component (0.25, by default) using the following formula:

(\frac{SW*range(x)}{no.of.components)})^2

. Lower value means smaller component variance allowed.

IC

Information criterion used to select the number of model components. Possible methods are "AIC","AICc", "BIC" (default), "ICL-BIC" or "LR".

sigmas.dev

Parameter used to define close GMM components that needs to be merged. For each component, standard deviation is multiplied by sigmas.dev to estimate the distance from component mean. All other components within this distance are merged. By default it is sigmas.dev = 1. When sigmas.dev = 0 no components are merged.

quick_stop

Logical value. Determines if stop searching of the number of components earlier based on the Likelihood Ratio Test. Used to speed up the function (TRUE, by default).

signi

Significance level set for Likelihood Ratio Test (0.05, by default).

fixed

Logical value. Fit GMM for selected number of components given by KS (FALSE, by default).

plot

Logical value. If TRUE (default), the figure visualizing GMM decomposition will be displayed.

col.pal

Name of the RColorBrewer palette used in the figure. By default "Blues".

Examples

# display all default settings
GMM_1D_opts

# create a new settings object
custom.settings <- GMM_1D_opts
custom.settings$IC <- "AIC"
custom.settings

Default configuration for 2D Gaussian Mixture decomposition

Description

A list with parameters customizing a GMM_2D. Each component of the list is an effective argument for runGMM2D.

Usage

GMM_2D_opts

Format

A list with the following components:

eps_change

Criterion for early stopping of EM (1e-7, by default).

max_iter

Maximum number of iterations of EM algorithm. By default it is max_iter = 50 000.

SW

Regularizing coefficient for covariance.

max_var_ratio

Maximum dissimilarity between horizontal and vertical dispersion. By default it is max_var_ratio = 5.

IC

Information criterion used to select the number of model components. Possible methods are "AIC","AICc", "BIC" (default), "ICL-BIC" or "LR".

cov_type

Type of covariance defined for each model component. Possible "sphere","diag" or "full" (default).

init_nb

Number of random initial conditions. By default it is init_nb = 10.

KS

Maximum number of components of the model. By default it is KS = 5.

quick_stop

Logical value. Determines if stop searching of the number of components earlier based on the Likelihood Ratio Test. Used to speed up the function (TRUE, by default).

signi

Significance level set for Likelihood Ratio Test (0.05, by default).

init_con

Type of initial conditions. Could be "rand" (default),"DP" or "diag".

fixed

Logical value. Fit GMM for selected number of components given by KS (FALSE, by default).

plot

Logical value. If TRUE, the GMM decomposition figure will be displayed (FALSE, by default).

Examples

# display all default settings
GMM_2D_opts

# create a new settings object
custom.settings <- GMM_2D_opts
custom.settings$IC <- "AIC"
custom.settings

Data example of binned problem in mass spectrometry

Description

This data set is part of mass spectrometry measurements. First column represent X values. Second column represent counts of X.

Usage

data(binned)

Format

A matrix of X and Y (in histogram)

Log-Likelihood for 2D Gaussian Mixture Model

Description

Function calculate log-likelihood of 2D Gaussian distribution

Usage

calc_lLik2D(X, Y, gmm)

Arguments

Value

Function returns a list with:

lLik

Log-Likelihood for 2D GMM

cum_pdf

Cumulative probability distibution

2D plot support

Description

Supporting function for 2D GMM ploting.

Usage

coords_to_img(X, Y)

Arguments

Diagonal initialization of 2D GMM

Description

Function for generating initial conditions of 2D GMM model by diagonal.

Usage

diag_init_2D(X, KS)

Arguments

Value

Initial values for EM.

Dynamic programming split

Description

Supporting function for dynamic programming in 1D decomposition.

Usage

dyn_pr_split_w(xhist, yhist, K, aux_mx, s_corr)

Arguments

Value

List of parameters.

Dynamic programming split of aux

Description

Supporting function for dynamic programming in 1D decomposition.

Usage

dyn_pr_split_w_aux(xhist, yhist, s_corr)

Arguments

Ellipses for plot 2D GMM

Description

Function for defining ellipses in 2D GMM plot by the confidence interval.

Usage

ellips2D(center, covariance, cov_type, crit = 0.95)

Arguments

Data of 1D mixed-normal distributions

Description

This data set was randomly drown for 6 components GMM. The parameters of distributions are as follow:
means <- c(-14.56, -14.16, -11.80, -8.77, -2.89, 2.31);
sigma <- c(2.06, 4.49, 4.42, 2.39, 3.92, 1.36);
alpha <- c(0.2012, 0.2898, 0.0334, 0.0092, 0.4278, 0.0384)

Usage

data(example)

Format

A vector containing 1500 observations

Source

Randomly generated data

Data of 2D mixed-normal distributions

Description

This data set contain translated image information into X and Y coordinates and count for each pair X and Y.

Usage

data(example2D)

Format

data.frame of X and Y coordinates and counts

Class assignment for 2D Gaussian Mixture Model data

Description

Function which assign each point of 2D matrix data to a cluster by maximum probability.

Usage

find_class_2D(X, gmm)

Arguments

Value

Return a vector of cluster assignment of each point of X matrix.

Thresholds estimations for 1D from GMM distribution

Description

Function to calculate cutoffs between each component of mixture normal distributions using probability distribution function.

Usage

find_thr_by_dist(input, sigmas.dev = 2.5, alpha, mu, sigma)

Arguments

Value

Return a vector of thresholds.

See Also

runGMM

Examples

data(example)

alpha <- c(0.45, 0.5, 0.05)
mu <- c(-14, -2, 5)
sigma <- c(2, 4, 1.5)

dist.plot <- generate_dist(example$Dist, alpha = alpha, mu = mu, sigma = sigma, 1e4)
thr <- find_thr_by_dist(dist.plot, 2.5, alpha = alpha, mu = mu, sigma = sigma)

Thresholds estimations for 1D from GMM parameters

Description

Function to calculate cutoffs between each component of mixture normal distributions based on the component parameters.

Usage

find_thr_by_params(alpha, mu, sigma, input, sigmas.dev = 2.5)

Arguments

Value

Return a vector of thresholds.

See Also

runGMM

Examples

data(example)

alpha <- c(0.45, 0.5, 0.05)
mu <- c(-14, -2, 5)
sigma <- c(2, 4, 1.5)

dist.plot <- generate_dist(example$Dist, alpha = alpha, mu = mu, sigma = sigma, 1e4)
thr <- find_thr_by_params(alpha = alpha, mu = mu, sigma = sigma, dist.plot)

Gaussian mixture decomposition for 2D data

Description

Function to choose the optimal number of components of a 2D mixture normal distributions, minimizing the value of the information criterion.

Usage

gaussian_mixture_2D(X, Y = NULL, opts = NULL)

Arguments

Value

Function returns a list of GMM parameters for the optimal number of components:

alpha

Weights (alpha) of each component.

center

Means of decomposition.

covar

Covariances of each component.

KS

Estimated number of components.

logL

Log-likelihood statistic for the estimated number of components.

IC

The value of the selected information criterion which was used to calculate the number of components.

cls

Assigment of point to the clusters.

See Also

runGMM2D, GMM_2D_opts

Examples

data(example2D)
custom.settings <- GMM_2D_opts
exp <- gaussian_mixture_2D(example2D[,1:2], example2D[,3], opts = custom.settings)

Gaussian mixture decomposition for 1D data

Description

Function to estimate number of components of a mixture normal distributions, minimizing the value of the information criterion.

Usage

gaussian_mixture_vector(X, Y = NULL, opts = NULL)

Arguments

Value

Function returns a list of GMM parameters for the estimated number of components:

model

A list of model component parameters - mean values (mu), standard deviations (sigma) and weights (alpha) for each component.

IC

The value of the selected information criterion which was used to calculate the number of components.

logLik

Log-likelihood statistic for the estimated number of components.

KS

Estimaged number of model components.

See Also

runGMM and generate_norm1D

Examples

data <- generate_norm1D(1000, alpha = c(0.2,0.4,0.4), mu = c(-15,0,15), sigma = c(1,2,3))

custom.settings <- GMM_1D_opts
custom.settings$IC <- "AIC"
custom.settings$KS <- 10

exp <- gaussian_mixture_vector(data$Dist, opts = custom.settings)

Generation of GMM data with high precision

Description

Function to generate PDF of GMM distributions and its cumulative results with high lincespacing.

Usage

generate_dist(X, alpha, mu, sigma, precision)

Arguments

Value

List with following elements:

x

Numeric vector with equaliy spread data of given precison.

dist

Matrix with PDF of each GMM component and cumulative distribution.

See Also

runGMM and generate_norm1D

Examples

data <- generate_norm1D(1000, alpha = c(0.2, 0.4, 0.4),
                               mu = c(-15, 0, 15), sigma = c(1, 2, 3))
dist <- generate_dist(data$Dist, alpha = c(0.2, 0.4, 0.4),
                                 mu = c(-15, 0, 15),
                                 sigma = c(1, 2, 3), precision = 1000)

Generator of multiple random 2D mixed-normal distributions

Description

Generator of multiple 2D mixed normal distribution with given model parameters ranges.

Usage

generate_dset2D(
  n = 1500,
  m = 1500,
  KS_range = 2:8,
  mu_range = c(-15, 15),
  cov_range = c(1, 5)
)

Arguments

Value

List with 2D GMM distributions where each list contains elements of generate_norm2D.

See Also

generate_norm2D

Examples

dset <- generate_dset2D(n = 1500, m = 10,
                       KS_range = 2:5,
                       mu_range = c(-10, 10),
                       cov_range = c(1, 3))

Generator of 1D mixed-normal distributions

Description

Generator of mixed-normal distribution with given model parameters for certain points number.

Usage

generate_norm1D(n, alpha, mu, sigma)

Arguments

Value

List with following elements:

Dist

Numeric vector with generated data

Cls

Numeric vector with classification of each point to particular mixed distribution

Examples

data <- generate_norm1D(1000, alpha = c(0.2, 0.4, 0.4),
                               mu = c(-15, 0, 15), sigma = c(1, 2, 3))

Generator of 2D mixed-normal distributions

Description

Generator of 2D mixed normal distribution with given model parameters for certain points number.

Usage

generate_norm2D(n, alpha, mu, cov)

Arguments

Value

List with following elements:

Dist

Numeric marix with generated data.

Cls

Numeric vector with classification of each point to particular distribution.

Examples

data <- generate_norm2D(1500, alpha = c(0.2, 0.4, 0.4),
                              mu = matrix(c(1, 2, 1, 3, 2, 2), nrow = 2),
                              cov = c(0.01, 0.02, 0.03))

Merging of overlapping components

Description

Function merges distributions of overlapping components at distances defined in the argument sigmas.dev.

Usage

gmm_merge(alpha, mu, sigma, sigmas.dev = 2.5)

Arguments

Value

Function returns a list which contains:

model

data.frame of the GMM decomposition parameters after merging.

KS

number of components after merging.

See Also

runGMM

2D plot support

Description

Transform image into coordinates data

Usage

img_to_coords(img)

Arguments

Supporting function for spliters

Description

Supporting function used in spliters dyn_pr_split_w_aux and dyn_pr_split_w.

Usage

my_qu_ix_w(xinvec, yinvec, s_corr)

Arguments

Probability distribution of 2D normal distribution

Description

Function for calculation PDF of 2D normal model.

Usage

norm_pdf_2D(x, center, covar)

Arguments

QQplot of GMM decomposition for 1D data

Description

Function return ggplot object with fit diagnostic Quantile-Quantile plot for one normal distribution and fitted GMM. This plot is also return as regular output of runGMM.

Usage

plot_QQplot(X, alpha, mu, sigma)

Arguments

Value

An object extending ggplot that arranges two quantile-quantile plots into a single figure. One panel shows a QQ plot of the input data against a normal distribution, and the other shows a QQ plot against data simulated from the fitted Gaussian mixture model.

See Also

runGMM

Examples

data(example)

alpha <- c(0.45, 0.5, 0.05)
mu <- c(-14, -2, 5)
sigma <- c(2, 4, 1.5)

plot_QQplot(example$Dist, alpha, mu, sigma)

Plot of GMM decomposition for 1D data

Description

Function plot the decomposed distribution together with histogram of data. Moreover the cut-off are marked. This plot is also return as regular output of runGMM.

Usage

plot_gmm_1D(X, dist, Y = NULL, threshold = NA, pal = "Blues")

Arguments

Value

A ggplot object showing the histogram or density of the input data together with the Gaussian mixture model decomposition. Individual mixture components and the overall fitted density are displayed as line plots, and optional cut-off thresholds are marked as vertical dashed lines.

See Also

runGMM

Examples

data(example)

alpha <- c(0.45, 0.5, 0.05)
mu <- c(-14, -2, 5)
sigma <- c(2, 4, 1.5)

dist.plot <- generate_dist(example$Dist, alpha, mu, sigma, 1e4)
thr <- find_thr_by_params(alpha, mu, sigma, dist.plot)
plot_gmm_1D(example$Dist, dist.plot, Y = NULL, threshold = thr, pal="Dark2")

Plot of GMM decomposition for 2D binned data

Description

Function plot the heatmap of binned data with marked GMM decomposition. This plot is also return as regular output of runGMM2D.

Usage

plot_gmm_2D_binned(X, Y, gmm, opts)

Arguments

Value

A ggplot object showing the heatmap of binned two-dimensional data with an overlay of the Gaussian mixture model decomposition. Mixture component centers are indicated by points and covariance ellipses corresponding to selected probability contours are drawn around each component.

See Also

runGMM2D

Examples

data(example2D)
custom.settings <- GMM_2D_opts
res <- runGMM2D(example2D[,1:2], example2D[,3], opts = custom.settings)

plot_gmm_2D_binned(example2D[,1:2], example2D[,3], res$model, custom.settings)

Plot of GMM decomposition for 2D data

Description

Function plot the decomposed distribution together with histogram of data. This plot is also return as regular output of runGMM.

Usage

plot_gmm_2D_orig(X, gmm, opts)

Arguments

Value

A ggplot object showing the scatter plot of two-dimensional data with an overlay of the Gaussian mixture model decomposition. Mixture component centers are indicated by points and covariance ellipses corresponding to selected probability contours are drawn around each component.

See Also

runGMM2D

Examples

custom.settings <- GMM_2D_opts
data <- generate_norm2D(1500, alpha = c(0.2, 0.4, 0.4),
                              mu = matrix(c(1, 2, 1, 3, 2, 2), nrow = 2),
                              cov = c(0.01, 0.02, 0.03))

res <- runGMM2D(data$Dist, opts = custom.settings)
plot_gmm_2D_orig(data$Dist, res$model, custom.settings)

Random initialization of 2D GMM EM

Description

Function for generating random initial conditions of 2D GMM model.

Usage

rand_init_2D(X, KS)

Arguments

Value

Initial values for EM.

Function to fit Gaussian Mixture Model (GMM) to 1D data

Description

Function fits GMM with initial conditions found using dynamic programming-based approach by using expectation-maximization (EM) algorithm. The function works on original and binned (e.g. obtained by creating histogram on 1D data) data. Additionally, threshold values that allows to assign data to individual Gaussian components are provided. Function allows to estimate the number of GMM components using five different information criteria and merging of similar components.

Usage

runGMM(X, Y = NULL, opts = NULL)

Arguments

Value

Function returns a list which contains:

model

A list of model component parameters - mean values (mu), standard deviations (sigma) and weights (alpha) for each component. Output of gaussian_mixture_vector.

KS

Estimaged number of model components.

IC

The value of the selected information criterion which was used to calculate the number of components.

logLik

Log-likelihood statistic for the estimated number of components.

threshold

Vector of thresholds between each component.

cluster

Assignment of original X values to individual components (clusters) by thresholds.

fig

ggplot object (output of the plot_gmm_1D function). It contains GMM decomposition together with a histogram of the data.

QQplot

ggplot object (output of the plot_QQplot function). It presents diagnostic Quantile-Quantile plot for a single normal distribution and fitted GMM.

See Also

gaussian_mixture_vector, EM_iter

Examples

data(example)

custom.settings <- GMM_1D_opts
custom.settings$sigmas.dev <- 1.5
custom.settings$max_iter <- 1000
custom.settings$KS <- 10

mix_test <- runGMM(example$Dist, opts = custom.settings)
mix_test$QQplot

#example for binned data
data(binned)

custom.settings <- GMM_1D_opts
custom.settings$quick_stop <- TRUE
custom.settings$KS <- 40
custom.settings$col.pal <- "Dark2"
custom.settings$plot <- FALSE

binned_test <- runGMM(X = binned$V1, Y = binned$V2, opts = custom.settings)
binned_test$fig

Function to fit Gaussian Mixture Model (GMM) to 2D data

Description

Main function to perform GMM on 2D data. Function choose the optimal number of components of a 2D mixture normal distributions by minimizing the value of the information criterion.

Usage

runGMM2D(X, Y = NULL, opts = NULL)

Arguments

Value

Function returns a list of GMM parameters for the estimated number of components:

model

alpha

Weights (alpha) of each component.

center

Means of decomposition.

covar

Covariances of each component.

KS

Estimated number of components.

logL

Log-likelihood statistic for the estimated number of components.

IC

The value of the selected information criterion which was used to calculate the number of components.

cls

Assigment of point to the clusters.

fig

Plot of decomposition.

Examples

data(example2D)
custom.settings <- GMM_2D_opts
custom.settings$fixed <- TRUE
custom.settings$KS <- 3
custom.settings$max_iter <- 5000
custom.settings$plot <- TRUE

res <- runGMM2D(example2D[,1:2], example2D[,3], opts = custom.settings)