Package 'rebmix'

Description

The adult dataset containing 48842 instances with 16 continuous, binary and discrete variables was extracted from the census bureau database. Extraction was done by Barry Becker from the 1994 census bureau database.

Usage

data(adult)

Format

adult is a data frame with 48842 cases (rows) and 16 variables (columns) named:

1. Type binary train or test.

2. Age continuous.

3. Workclass one of the 8 discrete values private, self-emp-not-inc, self-emp-inc, federal-gov, local-gov, state-gov, without-pay or never-worked.

4. Fnlwgt stands for continuous final weight.

5. Education one of the 16 discrete values bachelors, some-college, 11th, hs-grad, prof-school, assoc-acdm, assoc-voc, 9th, 7th-8th, 12th, masters, 1st-4th, 10th, doctorate, 5th-6th or preschool.

6. Education.Num continuous.

7. Marital.Status one of the 7 discrete values married-civ-spouse, divorced, never-married, separated, widowed, married-spouse-absent or married-af-spouse.

8. Occupation one of the 14 discrete values tech-support, craft-repair, other-service, sales, exec-managerial, prof-specialty, handlers-cleaners, machine-op-inspct, adm-clerical, farming-fishing, transport-moving, priv-house-serv, protective-serv or armed-forces.

9. Relationship one of the 6 discrete values wife, own-child, husband, not-in-family, other-relative or unmarried.

10. Race one of the 5 discrete values white, asian-pac-islander, amer-indian-eskimo, other or black.

11. Sex binary female or male.

12. Capital.Gain continuous.

13. Capital.Loss continuous.

14. Hours.Per.Week continuous.

15. Native.Country one of the 41 discrete values united-states, cambodia, england, puerto-rico, canada, germany, outlying-us(guam-usvi-etc), india, japan, greece, south, china, cuba, iran, honduras, philippines, italy, poland, jamaica, vietnam, mexico, portugal, ireland, france, dominican-republic, laos, ecuador, taiwan, haiti, columbia, hungary, guatemala, nicaragua, scotland, thailand, yugoslavia, el-salvador, trinadad&tobago, peru, hong or holand-netherlands.

16. Income binary <=50k or >50k.

Source

A. Asuncion and D. J. Newman. Uci machine learning repository, 2007. http://archive.ics.uci.edu/ml/.

References

A. Asuncion and D. J. Newman. Uci machine learning repository, 2007. http://archive.ics.uci.edu/ml/.

Examples

data(adult)

# Find complete cases.

adult <- adult[complete.cases(adult),]

# Show level attributes for binary and discrete variables.

levels(adult[["Type"]])
levels(adult[["Workclass"]])
levels(adult[["Education"]])
levels(adult[["Marital.Status"]])
levels(adult[["Occupation"]])
levels(adult[["Relationship"]])
levels(adult[["Race"]])
levels(adult[["Sex"]])
levels(adult[["Native.Country"]])
levels(adult[["Income"]])

Description

Returns the Akaike information criterion at pos.

Usage

## S4 method for signature 'REBMIX'
AIC(x = NULL, pos = 1, ...)
## S4 method for signature 'REBMIX'
AIC3(x = NULL, pos = 1, ...)
## S4 method for signature 'REBMIX'
AIC4(x = NULL, pos = 1, ...)
## S4 method for signature 'REBMIX'
AICc(x = NULL, pos = 1, ...)
## S4 method for signature 'REBMIX'
CAIC(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

H. Akaike. A new look at the statistical model identification. IEEE Transactions on Automatic Control, 19(51):716-723, 1974.

A. F. M. Smith and D. J. Spiegelhalter. Bayes factors and choice criteria for linear models. Journal of the Royal Statistical Society. Series B, 42(2):213-220, 1980. https://www.jstor.org/stable/2984964.

H. Bozdogan. Model selection and akaike's information criterion (aic): The general theory and its analytical extensions. Psychometrika, 52(3):345-370, 1987. doi:10.1007/BF02294361.

C. M. Hurvich and C.-L. Tsai. Regression and time series model selection in small samples. Biometrika, 76(2):297-307, 1989. https://www.jstor.org/stable/2336663.

Description

Returns the approximate weight of evidence criterion at pos.

Usage

## S4 method for signature 'REBMIX'
AWE(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

J. D. Banfield and A. E. Raftery. Model-based gaussian and non-gaussian clustering. Biometrics, 49(3):803-821, 1993. doi:10.2307/2532201.

Description

These data are the results of the extraction process from the vibrational data of healthy and faulty bearings. Different faults are considered: faultless (1), defect on outer race (2), defect on inner race (3) and defect on ball (4). The extracted features are: root mean square (RMS), square root of the amplitude (SRA), kurtosis value (KV), skewness value (SV), peak to peak value (PPV), crest factor (CF), impulse factor (IF), margin factor (MF), shape factor (SF), kurtosis factor (KF), frequency centre (FC), root mean square frequency (RMSF) and root variance frequency (RVF).

Usage

data(bearings)

Format

bearings is a data frame with 1906 cases (rows) and 14 variables (columns) named:

1. RMS continuous.

2. SRA continuous.

3. KV continuous.

4. SV continuous.

5. PPV continuous.

6. CF continuous.

7. IF continuous.

8. MF continuous.

9. SF continuous.

10. KF continuous.

11. FC continuous.

12. RMSF continuous.

13. RVF continuous.

14. Class discrete 1, 2, 3 or 4.

Source

Case Western Reserve University Bearing Data Center Website https://engineering.case.edu/bearingdatacenter/welcome.

References

B. Panic, J. Klemenc and M. Nagode. Gaussian mixture model based classification revisited: Application to the bearing fault classification. Journal of Mechanical Engineering, 66(4):215-226, 2020. doi:10.5545/sv-jme.2020.6563.

Examples

## Not run: 
data(bearings)

# Split dataset into train (75

set.seed(3)

Bearings <- split(p = 0.75, Dataset = bearings, class = 14)

# Estimate number of components, component weights and component
# parameters for train subsets.

bearingsest <- REBMIX(model = "REBMVNORM",
  Dataset = a.train(Bearings),
  Preprocessing = "histogram",
  cmax = 15,
  Criterion = "BIC")

# Classification.

bearingscla <- RCLSMIX(model = "RCLSMVNORM",
  x = list(bearingsest),
  Dataset = a.test(Bearings),
  Zt = a.Zt(Bearings))

bearingscla

summary(bearingscla)

## End(Not run)

Description

Returns as default the optimized RCLSMIX algorithm output for mixtures of conditionally independent normal, lognormal, Weibull, gamma, Gumbel, binomial, Poisson, Dirac, uniform or von Mises component densities. If model equals "RCLSMVNORM" optimized output for mixtures of multivariate normal component densities with unrestricted variance-covariance matrices is returned.

Usage

## S4 method for signature 'RCLSMIX'
BFSMIX(model = "RCLSMIX", x = list(), Dataset = data.frame(),
       Zt = factor(), ...)
## ... and for other signatures

Arguments

Value

Returns an optimized object of class RCLSMIX or RCLSMVNORM.

Methods

signature(model = "RCLSMIX")

a character giving the default class name "RCLSMIX" for mixtures of conditionally independent normal, lognormal, Weibull, gamma, Gumbel, binomial, Poisson, Dirac, uniform or von Mises component densities.

signature(model = "RCLSMVNORM")

a character giving the class name "RCLSMVNORM" for mixtures of multivariate normal component densities with unrestricted variance-covariance matrices.

Author(s)

Marko Nagode

References

R. Kohavi and G. H. John. Wrappers for feature subset selection, Artificial Intelligence, 97(1-2):273-324, 1997. doi:10.1016/S0004-3702(97)00043-X.

Description

Returns the Bayesian information criterion at pos.

Usage

## S4 method for signature 'REBMIX'
BIC(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

G. Schwarz. Estimating the dimension of the model. The Annals of Statistics, 6(2):461-464, 1978.

Description

Returns the list of data frames containing bin means $\bar{\bm{y}}_{1}, \ldots, \bar{\bm{y}}_{v}$ and frequencies $k_{1}, \ldots, k_{v}$ for the histogram preprocessing.

Usage

## S4 method for signature 'list'
bins(Dataset = list(), K = matrix(),
     ymin = numeric(), ymax = numeric(), ...)
## ... and for other signatures

Arguments

Methods

signature(x = "list")

a list of data frames.

Author(s)

Branislav Panic, Marko Nagode

References

M. Nagode. Finite mixture modeling via REBMIX. Journal of Algorithms and Optimization, 3(2):14-28, 2015. https://repozitorij.uni-lj.si/Dokument.php?id=127674&lang=eng.

Examples

# Generate multivariate normal datasets.

n <- c(7, 10)

Theta <- new("RNGMVNORM.Theta", c = 2, d = 2)

a.theta1(Theta, 1) <- c(8, 6)
a.theta1(Theta, 2) <- c(6, 8)
a.theta2(Theta, 1) <- c(8, 2, 2, 4)
a.theta2(Theta, 2) <- c(2, 1, 1, 4)

sim2d <- RNGMIX(model = "RNGMVNORM", 
  Dataset.name = paste("sim2d_", 1:2, sep = ""),
  rseed = -1,
  n = n,
  Theta = a.Theta(Theta))

# Calculate optimal numbers of bins.

opt.k <- optbins(Dataset = sim2d@Dataset,
  Rule = "Knuth equal",
  kmin = 1, 
  kmax = 20)

opt.k

Y <- bins(Dataset = sim2d@Dataset, K = opt.k)

Y

opt.k <- optbins(Dataset = sim2d@Dataset,
  Rule = "Knuth unequal",
  kmin = 1, 
  kmax = 20)

opt.k

Y <- bins(Dataset = sim2d@Dataset, K = opt.k)

Y

Description

Returns as default the boot output for mixtures of conditionally independent normal, lognormal, Weibull, gamma, Gumbel, binomial, Poisson, Dirac, uniform or von Mises component densities. If x is of class RNGMVNORM the boot output for mixtures of multivariate normal component densities with unrestricted variance-covariance matrices is returned.

Usage

## S4 method for signature 'REBMIX'
boot(x = NULL, rseed = -1, pos = 1, Bootstrap = "parametric",
     B = 100, n = numeric(), replace = TRUE, prob = numeric(), ...)
## ... and for other signatures
## S4 method for signature 'REBMIX.boot'
summary(object, ...)
## ... and for other signatures

Arguments

Value

Returns an object of class REBMIX.boot or REBMVNORM.boot.

Methods

signature(x = "REBMIX")

an object of class REBMIX for mixtures of conditionally independent normal, lognormal, Weibull, gamma, Gumbel, binomial, Poisson, Dirac, uniform or von Mises component densities.

signature(x = "REBMVNORM")

an object of class REBMVNORM for mixtures of multivariate normal component densities with unrestricted variance-covariance matrices.

signature(object = "REBMIX")

an object of class REBMIX.

signature(object = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

G. McLachlan and D. Peel. Finite Mixture Models. John Wiley & Sons, New York, 2000.

Examples

## Not run: 
data(weibull)

# Create object of class EM.Control.

EM <- new("EM.Control", strategy = "single", variant = "EM",
  acceleration = "fixed", acceleration.multiplier = 1.0, tolerance = 1.0E-4,
  maximum.iterations = 1000)

# Estimate number of components, component weights and component parameters.

weibullest <- REBMIX(Dataset = list(weibull),
  Preprocessing = "kernel density estimation",
  cmin = 2,
  cmax = 4,
  Criterion = "BIC",
  pdf = "Weibull",
  EMcontrol = EM)

# Plot finite mixture.

plot(weibullest, what = c("pdf", "marginal cdf", "IC", "logL", "D"),
  nrow = 3, ncol = 2, npts = 1000)

# Bootstrap finite mixture.

weibullboot <- boot(x = weibullest, Bootstrap = "nonparametric", B = 10)

weibullboot

## End(Not run)

Description

Returns an object of class Histogram. The method can be called recursively. This way more than one dataset can be binned into one histogram. The method is time consuming.

Usage

## S4 method for signature 'Histogram'
chistogram(x = NULL, Dataset = data.frame(),
           K = numeric(), ymin = numeric(), ymax = numeric(), ...)
## ... and for other signatures

Arguments

Methods

signature(x = "Histogram")

an object of class Histogram.

Author(s)

Marko Nagode

Examples

# Create three datasets.

set.seed(1)

n <- 15

Dataset1 <- as.data.frame(cbind(rnorm(n, 157, 8), rnorm(n, 71, 10)))
Dataset2 <- as.data.frame(cbind(rnorm(n, 244, 14), rnorm(n, 61, 29)))
Dataset3 <- as.data.frame(cbind(rnorm(n, 198, 8), rnorm(n, 252, 13)))

apply(Dataset1, 2, range)
apply(Dataset2, 2, range)
apply(Dataset3, 2, range)

# Bin the first dataset.

hist <- chistogram(Dataset = Dataset1, K = c(4, 5), ymin = c(100.0, 0.0), ymax = c(300.0, 300.0))

# Bin the second dataset.

hist <- chistogram(x = hist, Dataset = Dataset2)

# Bin the third dataset.

hist <- chistogram(x = hist, Dataset = Dataset3)

hist

Description

Returns (invisibly) the object containing train and test observations $\bm{x}_{1}, \ldots, \bm{x}_{n}$ as well as true class membership $\bm{\Omega}_{g}$ for the test dataset. Vectors $\bm{x}$ are subvectors of $\bm{y} = (y_{1}, \ldots, y_{d})^{\top}$.

Usage

## S4 method for signature 'RCLS.chunk'
chunk(x = NULL, variables = expression(1:d))
## ... and for other signatures

Arguments

Value

Returns an object of class RCLS.chunk.

Methods

signature(x = "RCLS.chunk")

an object of class RCLS.chunk.

Author(s)

Marko Nagode

Examples

data(iris)

# Split dataset into train (75%) and test (25%) subsets.

set.seed(5)

Iris <- split(p = 0.75, Dataset = iris, class = 5)

# Extract chunk from train and test datasets.

Iris14 <- chunk(x = Iris, variables = c(1,4))

Iris14

Description

Returns the classification likelihood criterion at pos.

Usage

## S4 method for signature 'REBMIX'
CLC(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

C. Biernacki and G. Govaert. Using the classification likelihood to choose the number of clusters. In E. J. Wegman and S. P. Azen, editors, Computing Science and Statistics, 1997.

Description

Returns the data frame containing observations $\bm{x}_{1}, \ldots, \bm{x}_{n}$ and empirical densities $f_{1}, \ldots, f_{n}$ for the kernel density estimation or k-nearest neighbour or bin means $\bar{\bm{x}}_{1}, \ldots, \bar{\bm{x}}_{v}$ and empirical densities $f_{1}, \ldots, f_{v}$ for the histogram preprocessing. Vectors $\bm{x}$ and $\bar{\bm{x}}$ are subvectors of $\bm{y} = (y_{1}, \ldots, y_{d})^{\top}$ and $\bar{\bm{y}} = (\bar{y}_{1}, \ldots, \bar{y}_{d})^{\top}$.

Usage

## S4 method for signature 'REBMIX'
demix(x = NULL, pos = 1, variables = expression(1:d), ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

M. Nagode and M. Fajdiga. The rebmix algorithm for the univariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(5):876-892, 2011a. doi:10.1080/03610920903480890.

M. Nagode and M. Fajdiga. The rebmix algorithm for the multivariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(11):2022-2034, 2011b. doi:10.1080/03610921003725788.

M. Nagode. Finite mixture modeling via REBMIX. Journal of Algorithms and Optimization, 3(2):14-28, 2015. https://repozitorij.uni-lj.si/Dokument.php?id=127674&lang=eng.

Examples

# Generate simulated dataset.

n <- c(15, 15)

Theta <- new("RNGMIX.Theta", c = 2, pdf = rep("normal", 3))

a.theta1(Theta, 1) <- c(10, 20, 30)
a.theta1(Theta, 2) <- c(3, 4, 5)
a.theta2(Theta, 1) <- c(3, 2, 1)
a.theta2(Theta, 2) <- c(15, 10, 5)

simulated <- RNGMIX(Dataset.name = paste("simulated_", 1:4, sep = ""),
  rseed = -1,
  n = n,
  Theta = a.Theta(Theta))

# Create object of class EM.Control.

EM <- new("EM.Control", strategy = "best")

# Estimate number of components, component weights and component parameters.

simulatedest <- REBMIX(model = "REBMVNORM",
  Dataset = a.Dataset(simulated),
  Preprocessing = "h",
  cmax = 8,
  Criterion = "BIC",
  EMcontrol = NULL)

# Preprocess simulated dataset.

f <- demix(simulatedest, pos = 3, variables = c(1, 3))

f

# Plot finite mixture.

opar <- plot(simulatedest, pos = 3, nrow = 3, ncol = 1)

par(usr = opar[[2]]$usr, mfg = c(2, 1))

text(x = f[, 1], y = f[, 2], labels = format(f[, 3], digits = 3), cex = 0.8, pos = 1)

Description

Returns the data frame containing observations $\bm{x}_{1}, \ldots, \bm{x}_{n}$ and predictive marginal densities $f(\bm{x} | c, \bm{w}, \bm{\Theta})$. Vectors $\bm{x}$ are subvectors of $\bm{y} = (y_{1}, \ldots, y_{d})^{\top}$. If $\bm{x} = \bm{y}$ the method returns the data frame containing observations $\bm{y}_{1}, \ldots, \bm{y}_{n}$ and the corresponding predictive mixture densities $f(\bm{y} | c, \bm{w}, \bm{\Theta})$.

Usage

## S4 method for signature 'REBMIX'
dfmix(x = NULL, Dataset = data.frame(), pos = 1, variables = expression(1:d), ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

M. Nagode and M. Fajdiga. The rebmix algorithm for the univariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(5):876-892, 2011a. doi:10.1080/03610920903480890.

M. Nagode and M. Fajdiga. The rebmix algorithm for the multivariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(11):2022-2034, 2011b. doi:10.1080/03610921003725788.

M. Nagode. Finite mixture modeling via REBMIX. Journal of Algorithms and Optimization, 3(2):14-28, 2015. https://repozitorij.uni-lj.si/Dokument.php?id=127674&lang=eng.

Examples

# Generate simulated dataset.

n <- c(15, 15)

Theta <- new("RNGMIX.Theta", c = 2, pdf = rep("normal", 3))

a.theta1(Theta, 1) <- c(10, 20, 30)
a.theta1(Theta, 2) <- c(3, 4, 5)
a.theta2(Theta, 1) <- c(3, 2, 1)
a.theta2(Theta, 2) <- c(15, 10, 5)

simulated <- RNGMIX(Dataset.name = paste("simulated_", 1:4, sep = ""),
  rseed = -1,
  n = n,
  Theta = a.Theta(Theta))

# Number of classes or nearest neighbours to be processed.

K <- c(as.integer(1 + log2(sum(n))), # Minimum v follows Sturges rule.
  as.integer(10 * log10(sum(n)))) # Maximum v follows log10 rule.

# Estimate number of components, component weights and component parameters.

simulatedest <- REBMIX(model = "REBMVNORM",
  Dataset = a.Dataset(simulated),
  Preprocessing = "h",
  cmax = 4,
  Criterion = "BIC")

# Preprocess simulated dataset.

Dataset <- data.frame(c(-7, 1), NA, c(3, 7))

f <- dfmix(simulatedest, Dataset = Dataset, pos = 3, variables = c(1, 3))

f

# Plot finite mixture.

opar <- plot(simulatedest, pos = 3, nrow = 3, ncol = 1,
  contour.drawlabels = TRUE, contour.labcex = 0.6)

par(usr = opar[[2]]$usr, mfg = c(2, 1))

points(x = f[, 1], y = f[, 2])

text(x = f[, 1], y = f[, 2], labels = format(f[, 3], digits = 3), cex = 0.8, pos = 4)

Description

Object of class EM.Control.

Objects from the Class

Objects can be created by calls of the form new("EM.Control", ...). Accessor methods for the slots are a.strategy(x = NULL), a.variant(x = NULL), a.acceleration(x = NULL), a.tolerance(x = NULL), a.acceleration.multiplier(x = NULL), a.maximum.iterations(x = NULL), a.K(x = NULL), a.eliminate.zero.components(x = NULL), a.likelihood.tolerance.check(x = NULL) and a.likelihood.estimation.rule(x = NULL) where x stands for an object of class EM.Control. Setter methods a.strategy(x = NULL), a.variant(x = NULL), a.acceleration(x = NULL), a.tolerance(x = NULL), a.acceleration.multiplier(x = NULL), a.maximum.iterations(x = NULL), a.K(x = NULL), a.eliminate.zero.components(x = NULL), a.likelihood.tolerance.check{x = NULL} and a.likelihood.estimation.rule{x = NULL} are provided to write to strategy, variant, acceleration, tolerance, acceleration.multiplier, maximum.iterations, eliminate.zero.components, likelihood.tolerance.check and likelihood.estimation.rule slot respectively.

Slots

strategy:

a character containing the EM and REBMIX strategy. One of "none", "exhaustive", "best" and "single". The default value is "none".

variant:

a character containing the type of the EM algorithm to be used. One of "EM", "ECM", "SEM", "SEM-EM", "ECM-EM". The default value is "EM".

acceleration:

a character containing the type of acceleration of the EM iteration increment. One of "fixed", "line", "golden", "stem1", "stem2", "stem3", "square1", "square2" or "square3". The default value is "fixed".

tolerance:

tolerance value for the EM convergence criteria. The default value is 1.0E-4.

acceleration.multiplier:

acceleration.multiplier $a_{\mathrm{EM}}$, $1.0 \leq a_{\mathrm{EM}} \leq 2.0$. acceleration.multiplier for the EM step increment. The default value is 1.0. Only used when acceleration == "fixed".

maximum.iterations:

a positive integer containing the maximum allowed number of iterations of the EM algorithm. The default value is 1000.

K:

an integer containing the number of bins for the histogram based EM algorithm. This option can reduce computational time drastically if the datasets contain a large number of observations $n$ and K is set to the value $\ll n$. The default value of 0 means that the EM algorithm runs over all $n$.

eliminate.zero.components:

a logical indicating if the componenets with $w_{l} = 0$ should be eliminated from output. Only used with EMMIX-methods.

likelihood.tolerance.check:

a character containing the type of log likelihood convergence check. One of "absolute", "normalised" and "percentage". The default value is "normalised".

likelihood.estimation.rule:

a character containing the type of log likelihood estimate. One of "standard", "aitken-lindsay", "aitken-bohning" and "aitken-nicholas". The default value is "standard".

Author(s)

Branislav Panic

References

B. Panic, J. Klemenc, M. Nagode. Improved initialization of the EM algorithm for mixture model parameter estimation. Mathematics, 8(3):373, 2020. doi:10.3390/math8030373.

A. P. Dempster et al. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society. Series B, 39(1):1-38, 1977. https://www.jstor.org/stable/2984875.

G. Celeux and G. Govaert. A classification EM algorithm for clustering and two stochastic versions, Computational Statistics & Data Analysis, 14(3):315:332, 1992. doi:10.1016/0167-9473(92)90042-E.

R. Varadhan and C. Roland. Simple and Globally Convergent Methods for Accelerating the Convergence of Any EM Algorithm. Scandinavian Journal of Statistics, 35(2):335:353, 2008. doi:10.1111/j.1467-9469.2007.00585.x.

P. D. McNicholas et al. Serial and parallel implementations of model-based clustering via parsimonious Gaussian mixture models. Computational Statistics & Data Analysis, 54(3):711:723. 2010. doi:10.1016/j.csda.2009.02.011

Examples

# Inline creation by new call.

EM <- new("EM.Control", strategy = "exhaustive", 
  variant = "EM", acceleration = "fixed", 
  tolerance = 1e-4, acceleration.multiplier = 1.0, 
  maximum.iterations = 1000, K = 0, likelihood.tolerance.check = "absolute",
  likelihood.estimation.rule = "standard")

EM

# Creation of EM object with setter method.

EM <- new("EM.Control")

a.strategy(EM) <- "exhaustive"
a.variant(EM) <- "EM"
a.acceleration(EM) <- "fixed"
a.tolerance(EM) <- 1e-4
a.acceleration.multiplier(EM) <- 1.0
a.maximum.iterations(EM) <- 1000
a.K(EM) <- 256
a.likelihood.tolerance.check(EM) <- "normalised"
a.likelihood.estimation.rule(EM) <- "standard"

EM

Description

Returns as default the EM algorithm output for mixtures of conditionally independent normal, lognormal, Weibull, gamma, Gumbel, binomial, Poisson, Dirac or von Mises component densities. If model equals "REBMVNORM" output for mixtures of multivariate normal component densities with unrestricted variance-covariance matrices is returned.

Usage

## S4 method for signature 'REBMIX'
EMMIX(model = "REBMIX", Dataset = list(), 
       Theta = NULL, EMcontrol = NULL, ...)
## ... and for other signatures

Arguments

Value

Returns an object of class REBMIX or REBMVNORM.

Methods

signature(model = "REBMIX")

a character giving the default class name "REBMIX" for mixtures of conditionally independent normal, lognormal, Weibull, gamma, Gumbel, binomial, Poisson, Dirac or von Mises component densities.

signature(model = "REBMVNORM")

a character giving the class name "REBMVNORM" for mixtures of multivariate normal component densities with unrestricted variance-covariance matrices.

Author(s)

Branislav Panic

References

B. Panic, J. Klemenc, M. Nagode. Improved initialization of the EM algorithm for mixture model parameter estimation. Mathematics, 8(3):373, 2020. doi:10.3390/math8030373.

Examples

## Not run: 
devAskNewPage(ask = TRUE)

# Load faithful dataset.

data(faithful)

# Plot faithfull dataset.

plot(faithful)

# Number of dimensions.

d <- ncol(faithful)

# Obtain 2 component solution with Gaussian mixtures.

c <- 2

# Create EMMVNORM.Theta object with new call.

Theta <- new("EMMVNORM.Theta", d = d, c = c)

# Set parameters of Theta.
# Weights.

a.w(Theta) <- c(0.5, 0.5)

# Means.

a.theta1.all(Theta) <- c(2.0, 55.0, 4.5, 80.0)

# Covariances.

a.theta2.all(Theta) <- c(1, 0, 0, 1, 1, 0, 0, 1)

# Run EMMIX method.

model <- EMMIX(model = "REBMVNORM", Dataset = list(faithful), Theta = Theta)

# show.

model

# summary.

summary(model)

# plot.

plot(model, nrow = 3, ncol = 2, what = c("pdf", "marginal pdf", "marginal cdf"))

# Create EMMIX.Theta object with new call.

Theta <- new("EMMIX.Theta", c = c, pdf = c("normal", "normal"))

# Set parameters of Theta.
# Weights.

a.w(Theta) <- c(0.5, 0.5)

# Means.

a.theta1.all(Theta) <- c(2.0, 55.0, 4.5, 80.0)

# Covariances.

a.theta2.all(Theta) <- c(1, 1, 1, 1)

# Run EMMIX method.

model <- EMMIX(Dataset = list(faithful), Theta = Theta)

# show.

model

# summary.

summary(model)

# plot.

plot(model, nrow = 3, ncol = 2, what = c("pdf", "marginal pdf", "marginal cdf"))

## End(Not run)

Description

Object of class EMMIX.Theta.

Objects from the Class

Objects can be created by calls of the form new("EMMIX.Theta", ...). Accessor methods for the slots are a.c(x = NULL), a.d(x = NULL), a.pdf(x = NULL) and a.Theta(x = NULL), where x stands for an object of class EMMIX.Theta. Setter methods a.theta1(x = NULL, l = numeric()), a.theta2(x = NULL, l = numeric()), a.theta3(x = NULL, l = numeric()), a.theta1.all(x = NULL), a.theta2.all(x = NULL), a.theta3.all(x = NULL) and a.w(x = NULL) are provided to write to Theta slot, where $l = 1, \ldots, c$.

Slots

c:

number of components $c > 0$. The default value is 1.

d:

number of dimensions.

pdf:

a character vector of length $d$ containing continuous or discrete parametric family types. One of "normal", "lognormal", "Weibull", "gamma", "Gumbel", "binomial", "Poisson", "Dirac" or "vonMises".

Theta:

a list containing $c$ parametric family types pdfl. One of "normal", "lognormal", "Weibull", "gamma", "Gumbel", "binomial", "Poisson", "Dirac" or circular "vonMises" defined for $0 \leq y_{i} \leq 2 \pi$. Component parameters theta1.l follow the parametric family types. One of $\mu_{il}$ for normal, lognormal, Gumbel and von Mises distributions and $\theta_{il}$ for Weibull, gamma, binomial, Poisson and Dirac distributions. Component parameters theta2.l follow theta1.l. One of $\sigma_{il}$ for normal, lognormal and Gumbel distributions, $\beta_{il}$ for Weibull and gamma distributions, $p_{il}$ for binomial distribution, $\kappa_{il}$ for von Mises distribution. Component parameters theta3.l follow theta2.l. One of $\xi_{il} \in \{-1, 1\}$ for Gumbel distribution.

w:

a vector of length $c$ containing component weights $w_{l}$ summing to 1.

Author(s)

Branislav Panic

Examples

Theta <- new("EMMIX.Theta", c = 2, pdf = c("normal", "Gumbel"))  

a.w(Theta) <- c(0.4, 0.6)

a.theta1(Theta, l = 1) <- c(2, 10)
a.theta2(Theta, l = 1) <- c(0.5, 2.3)
a.theta3(Theta, l = 1) <- c(NA, 1.0)
a.theta1(Theta, l = 2) <- c(20, 50)
a.theta2(Theta, l = 2) <- c(3, 4.2)
a.theta3(Theta, l = 2) <- c(NA, -1.0)

Theta

Theta <- new("EMMIX.Theta", c = 2, pdf = c("normal", "Gumbel", "Poisson"))  

a.w(Theta) <- c(0.4, 0.6)

a.theta1.all(Theta) <- c(2, 10, 30, 20, 50, 60)
a.theta2.all(Theta) <- c(0.5, 2.3, NA, 3, 4.2, NA)
a.theta3.all(Theta) <- c(NA, 1.0, NA, NA, -1.0, NA)

Theta

Theta <- new("EMMVNORM.Theta", c = 2, d = 3)

a.w(Theta) <- c(0.4, 0.6)

a.theta1(Theta, l = 1) <- c(2, 10, -20)
a.theta2(Theta, l = 1) <- c(9, 0, 0, 0, 4, 0, 0, 0, 1)
a.theta1(Theta, l = 2) <- c(-2.4, -15.1, 30)
a.theta2(Theta, l = 2) <- c(4, -3.2, -0.2, -3.2, 4, 0, -0.2, 0, 1)

Theta

Theta <- new("EMMVNORM.Theta", c = 2, d = 3)

a.w(Theta) <- c(0.4, 0.6)

a.theta1.all(Theta) <- c(2, 10, -20, -2.4, -15.1, 30)

a.theta2.all(Theta) <- c(9, 0, 0, 0, 4, 0, 0, 0, 1, 
  4, -3.2, -0.2, -3.2, 4, 0, -0.2, 0, 1)

Theta

Description

Returns an object of class Histogram. The method can be called recursively. This way more than one dataset can be binned into one histogram. Set shrink to TRUE only when the method is called for the last time to optimize the size of the object. The method is memory consuming.

Usage

## S4 method for signature 'Histogram'
fhistogram(x = NULL, Dataset = data.frame(),
           K = numeric(), ymin = numeric(), ymax = numeric(), 
           shrink = FALSE, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "Histogram")

an object of class Histogram.

Author(s)

Marko Nagode

Examples

# Create three datasets.

set.seed(1)

n <- 15

Dataset1 <- as.data.frame(cbind(rnorm(n, 157, 8), rnorm(n, 71, 10)))
Dataset2 <- as.data.frame(cbind(rnorm(n, 244, 14), rnorm(n, 61, 29)))
Dataset3 <- as.data.frame(cbind(rnorm(n, 198, 8), rnorm(n, 252, 13)))

apply(Dataset1, 2, range)
apply(Dataset2, 2, range)
apply(Dataset3, 2, range)

# Bin the first dataset.

hist <- fhistogram(Dataset = Dataset1, K = c(4, 5), ymin = c(100.0, 0.0), ymax = c(300.0, 300.0))

# Bin the second dataset.

hist <- fhistogram(x = hist, Dataset = Dataset2)

# Bin the third dataset and shrink the hist object.

hist <- fhistogram(x = hist, Dataset = Dataset3, shrink = TRUE)

hist

Description

The unfilled survey of the Corona Borealis region contains the velocities of 82 galaxies from 6 well separated conic sections of space.

Usage

data(galaxy)

Format

galaxy is a data frame with 82 cases (rows) and 1 continuous variable (columns) called Velocity.

Source

K. Roeder. Density estimation with confidence sets exemplified by superclusters and voids in the galaxies. Journal of American Statistical Association, 85(411):617-624, 1990. https://www.jstor.org/stable/2289993.

References

S. Richardson and P. J. Green. On bayesian analysis of mixtures with an unknown number of components. Journal of the Royal Statistical Society B, 59(4):731-792, 1997. https://www.jstor.org/stable/2985194.

G. McLachlan and D. Peel. Contribution to the discussion of paper by s. richardson and p.j. green. Journal of the Royal Statistical Society B, 59(4):779-780, 1997. https://www.jstor.org/stable/2985194.

M. Stephens. Bayesian analysis of mixture models with an unknown number of components - an alternative to reversible jump methods. The Annals of Statistics, 28(1):40-74, 2000. https://www.jstor.org/stable/2673981.

Description

Object of class Histogram.

Objects from the Class

Objects can be created by calls of the form new("Histogram", ...). Accessor methods for the slots are a.Y(x = NULL), a.K(x = NULL), a.ymin(x = NULL), a.ymax(x = NULL), a.y0(x = NULL), a.h(x = NULL), a.n(x = NULL) and a.ns(x = NULL).

Slots

Y:

a data frame of size $v \times (d + 1)$ containing d-dimensional histogram. Each of the first $d$ columns represents one random variable and contains bin means $\bar{\bm{y}}_{1}, \ldots, \bar{\bm{y}}_{v}$. Column $d + 1$ contains frequencies $k_{1}, \ldots, k_{v}$.

K:

an integer or a vector of length $d$ containing numbers of bins $v$.

ymin:

a vector of length $d$ containing minimum observations.

ymax:

a vector of length $d$ containing maximum observations.

y0:

a vector of length $d$ containing origins.

h:

a vector of length $d$ containing bin widths.

n:

an integer containing total number $n$ of observations.

ns:

an integer containing number $n_{\mathrm{s}}$ of samples.

Author(s)

Marko Nagode

Examples

Y <- as.data.frame(matrix(1.0, nrow = 8, ncol = 3))

hist <- new("Histogram", Y = Y, K = c(4, 2), ymin = c(2, 1), ymax = c(10, 8))

a.Y(hist)
a.K(hist)
a.ymin(hist)
a.ymax(hist)
a.y0(hist)
a.h(hist)
a.n(hist)
a.ns(hist)

# Multiplay Y[ , d + 1] by 0.1.

a.Y(hist) <- 0.1

Description

Returns the Hannan-Quinn information criterion at pos.

Usage

## S4 method for signature 'REBMIX'
HQC(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

E. J. Hannan and B. G. Quinn. The determination of the order of an autoregression. Journal of the Royal Statistical Society. Series B, 41(2):190-195, 1979. https://www.jstor.org/stable/2985032.

Description

Returns the integrated classification likelihood criterion at pos.

Usage

## S4 method for signature 'REBMIX'
ICL(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

C. Biernacki, G. Celeux and G. Govaert. Assessing a mixture model for clustering with the integrated classification likelihood. Technical Report 3521, INRIA, Rhone-Alpes, 1998.

Description

Returns the approximate integrated classification likelihood criterion at pos.

Usage

## S4 method for signature 'REBMIX'
ICLBIC(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

C. Biernacki, G. Celeux and G. Govaert. Assessing a mixture model for clustering with the integrated classification likelihood. Technical Report 3521, INRIA, Rhone-Alpes, 1998.

Description

This is perhaps the best known database to be found in the pattern recognition literature. Fisher's paper is a classic in the field and is referenced frequently to this day. The data set contains 3 classes of 50 instances each, where each class refers to a type of iris plant. One class is linearly separable from the other 2; the latter are NOT linearly separable from each other.

Usage

data(iris)

Format

iris is a data frame with 150 cases (rows) and 5 variables (columns) named:

1. Sepal.Length continuous.

2. Sepal.Width continuous.

3. Petal.Length continuous.

4. Petal.Width continuous.

5. Class discrete iris-setosa, iris-versicolour or iris-virginica.

Source

A. Asuncion and D. J. Newman. Uci machine learning repository, 2007. http://archive.ics.uci.edu/ml/.

References

R. A. Fisher. The use of multiple measurements in taxonomic problems. Annals of Eugenics, 7(2):179-188, 1936.

Examples

## Not run: 
devAskNewPage(ask = TRUE)

data(iris)

# Show level attributes.

levels(iris[["Class"]])

# Split dataset into train (75

set.seed(5)

Iris <- split(p = 0.6, Dataset = iris, class = 5)

# Estimate number of components, component weights and component
# parameters for train subsets.

n <- range(a.ntrain(Iris))

irisest <- REBMIX(model = "REBMVNORM",
  Dataset = a.train(Iris),
  Preprocessing = "histogram",
  cmax = 10,
  Criterion = "ICL-BIC",
  EMcontrol = new("EM.Control", strategy = "single"))

plot(irisest, pos = 1, nrow = 3, ncol = 2, what = c("pdf"))
plot(irisest, pos = 2, nrow = 3, ncol = 2, what = c("pdf"))
plot(irisest, pos = 3, nrow = 3, ncol = 2, what = c("pdf"))

# Selected chunks.

iriscla <- RCLSMIX(model = "RCLSMVNORM",
  x = list(irisest),
  Dataset = a.test(Iris),
  Zt = a.Zt(Iris))

iriscla

summary(iriscla)

# Plot selected chunks.

plot(iriscla, nrow = 3, ncol = 2)

## End(Not run)

Description

Returns (invisibly) a vector containing numbers of bins $v$ for the histogram and the kernel density estimation or numbers of nearest neighbours $k$ for the k-nearest neighbour.

Usage

kseq(from = NULL, to = NULL, f = 0.05, ...)

Arguments

Author(s)

Marko Nagode

Examples

# Generate numbers of bins.

n <- 10000

Sturges <- as.integer(1 + log2(n)) # Minimum v follows Sturges rule.
Log10 <- as.integer(10 * log10(n)) # Maximum v follows Log10 rule.
RootN <- as.integer(2 * n^0.5) # Maximum v follows RootN rule.

K <- kseq(from = Sturges, to = Log10, f = 0.05)

K

K <- kseq(from = Sturges, to = RootN, f = 0.03)

K

Description

Returns the list with the data frame Mij containing the cluster levels $l$, the numbers of pixels $n$ and the cluster moments $\bm{M} = (M_{\mathrm{10}}, M_{\mathrm{01}}, M_{\mathrm{11}})^{\top}$ for 2D images or the data frame Mijk containing the cluster levels $l$, the numbers of voxels $n$ and the cluster moments $\bm{M} = (M_{\mathrm{100}}, M_{\mathrm{010}}, M_{\mathrm{001}}, M_{\mathrm{111}})^{\top}$ for 3D images and the adjacency matrix A of size $c_{\mathrm{max}} \times c_{\mathrm{max}}$. It may have some NA rows and columns. To calculate the adjacency matrix $A(i,j) = \exp{(-\|\bm{M}_{i} - \bm{M}_{j}\|^2 / 2 \sigma^2)}$, the raw cluster moments are first converted into z-scores.

Usage

## S4 method for signature 'array'
labelmoments(Zp = array(), cmax = integer(), Sigma = 1.0, ...)
## ... and for other signatures

Arguments

Methods

signature(Zp = "array")

an array.

Author(s)

Marko Nagode, Branislav Panic

References

A. Ng, M. Jordan and Y. Weiss. On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems 14 (NIPS 2001).

Examples

Zp <- matrix(rep(0, 100), nrow = 10, ncol = 10)

Zp[2, 2:4] <- 1; 
Zp[2:4, 5] <- 2; 
Zp[8, 7:10] <- 3; 
Zp[9, 6] <- 4; Zp[10, 5] <- 4
Zp[10, 1:4] <- 5
Zp[6:9, 1] <- 6

labelmoments <- labelmoments(Zp, cmax = 6, Sigma = 1.0)

set.seed(12)

mergelabels <- mergelabels(list(labelmoments$A), w = 1.0, k = 2, nstart = 3)

Zp

mergelabels

Description

Returns the log likelihood at pos.

Usage

## S4 method for signature 'REBMIX'
logL(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

G. McLachlan and D. Peel. Finite Mixture Models. John Wiley & Sons, New York, 2000.

Description

Returns a factor of predictive cluster membership for dataset.

Usage

## S4 method for signature 'RCLRMIX'
mapclusters(x = NULL, Dataset = data.frame(),
            s = expression(c), ...)
## ... and for other signatures

Arguments

Methods

signature(x = "RCLRMIX")

an object of class RCLRMIX.

signature(x = "RCLRMVNORM")

an object of class RCLRMVNORM.

Author(s)

Marko Nagode, Branislav Panic

Examples

devAskNewPage(ask = TRUE)

# Generate normal dataset.

n <- c(50, 20, 40)

Theta <- new("RNGMVNORM.Theta", c = 3, d = 2)

a.theta1(Theta, 1) <- c(3, 10)
a.theta1(Theta, 2) <- c(8, 6)
a.theta1(Theta, 3) <- c(12, 11)
a.theta2(Theta, 1) <- c(3, 0.3, 0.3, 2)
a.theta2(Theta, 2) <- c(5.7, -2.3, -2.3, 3.5)
a.theta2(Theta, 3) <- c(2, 1, 1, 2)

normal <- RNGMIX(model = "RNGMVNORM", Dataset.name = paste("normal_", 1:10, sep = ""),
  n = n, Theta = a.Theta(Theta))

# Convert all datasets to single histogram.

hist <- NULL

n <- length(normal@Dataset)

hist <- fhistogram(Dataset = normal@Dataset[[1]], K = c(10, 10), 
  ymin = a.ymin(normal), ymax = a.ymax(normal))

for (i in 2:n) {
  hist <- fhistogram(x = hist, Dataset = normal@Dataset[[i]], shrink = i == n)
}

# Estimate number of components, component weights and component parameters.

normalest <- REBMIX(model = "REBMVNORM",
  Dataset = list(hist),
  Preprocessing = "histogram",
  cmax = 6,
  Criterion = "BIC")

summary(normalest)

# Plot finite mixture.

plot(normalest)

# Cluster dataset.

normalclu <- RCLRMIX(model = "RCLRMVNORM", x = normalest)

# Plot clusters.

plot(normalclu)

summary(normalclu)

# Map clusters.

Zp <- mapclusters(x = normalclu, Dataset = a.Dataset(normal, 4))

Zt <- a.Zt(normal)

Zp

Zt

Description

Returns the minimum desription length at pos.

Usage

## S4 method for signature 'REBMIX'
MDL2(x = NULL, pos = 1, ...)
## S4 method for signature 'REBMIX'
MDL5(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

M. H. Hansen and B. Yu. Model selection and the principle of minimum description length. Journal of the American Statistical Association, 96(454):746-774, 2001. https://www.jstor.org/stable/2670311.

Description

Returns the list with the normalised adjacency matrix L of size $c \times c$. The normalised adjacency matrix $L = D^{-1/2} P D^{-1/2}$ depends on the probability adjacency matrix $P(i,j) = \sum_{l = 1}^{n} p_{l} A_{l}(i,j)$, where $p_{l} = w_{l} / \sum_{i = 1}^{c}\sum_{j = i + 1}^{c} A_{l}(i,j)$ and the degree matrix $D(i,i) = \sum_{j = 1}^{c} P(i,j)$. The $A_{l}$ matrices may contain some NA rows and columns, which are eliminated by the method. The list also contains the vector of integers cluster of length $k$, which indicates the cluster to which each label is assigned.

Usage

## S4 method for signature 'list'
mergelabels(A = list(), w = numeric(), k = 2, ...)
## ... and for other signatures

Arguments

Methods

signature(A = "list")

a list.

Author(s)

Marko Nagode, Branislav Panic

References

A. Ng, M. Jordan and Y. Weiss. On spectral clustering: Analysis and an algorithm. Advances in Neural Information Processing Systems 14 (NIPS 2001).

Examples

Zp <- array(0, dim = c(10, 10, 2))

Zp[ , ,1][10, 1:4] <- 1
Zp[ , ,1][1:4, 10] <- 2

Zp[ , ,2][9, 1:5] <- 3
Zp[ , ,2][1:6, 9] <- 4

labelmoments <- labelmoments(Zp, cmax = 4, Sigma = 1.0)

labelmoments

set.seed(3)

mergelabels <- mergelabels(list(labelmoments$A), w = 1.0, k = 2, nstart = 5)

Zp

mergelabels

Description

Returns the matrix of size $n_{\mathrm{D}} \times d$ containing optimal numbers of bins $v_{1}, \ldots, v_{d}$ for all processed datasets.

Usage

## S4 method for signature 'list'
optbins(Dataset = list(), Rule = "Knuth equal",
        ymin = numeric(), ymax = numeric(), kmin = numeric(),
        kmax = numeric(), ...)
## ... and for other signatures

Arguments

Methods

signature(x = "list")

a list of data frames.

Author(s)

Branislav Panic, Marko Nagode

References

K. K. Knuth. Optimal data-based binning for histograms and histogram-based probability density models. Digital Signal Processing, 95:102581, 2019. doi:10.1016/j.dsp.2019.102581.

B. Panic, J. Klemenc, M. Nagode. Improved initialization of the EM algorithm for mixture model parameter estimation. Mathematics, 8(3):373, 2020. doi:10.3390/math8030373.

Examples

# Generate multivariate normal datasets.

n <- c(750, 1000)

Theta <- new("RNGMVNORM.Theta", c = 2, d = 2)

a.theta1(Theta, 1) <- c(8, 6)
a.theta1(Theta, 2) <- c(6, 8)
a.theta2(Theta, 1) <- c(8, 2, 2, 4)
a.theta2(Theta, 2) <- c(2, 1, 1, 4)

sim2d <- RNGMIX(model = "RNGMVNORM", 
  Dataset.name = paste("sim2d_", 1:5, sep = ""),
  rseed = -1,
  n = n,
  Theta = a.Theta(Theta))

# Calculate optimal numbers of bins.

opt.k <- optbins(Dataset = sim2d@Dataset,
  Rule = "Knuth equal",
  ymin = sim2d@ymin,
  ymax = sim2d@ymax,
  kmin = 2, 
  kmax = 20)

opt.k

# Create object of class EM.Control.

EM <- new("EM.Control", strategy = "exhaustive", variant = "EM",
  acceleration = "fixed", acceleration.multiplier = 1.0, tolerance = 1.0E-4,
  maximum.iterations = 1000)

# Estimate number of components, component weights and component parameters.

sim2dest <- REBMIX(model = "REBMVNORM", 
  Dataset = a.Dataset(sim2d),
  Preprocessing = "h",
  cmax = 10,
  ymin = a.ymin(sim2d),
  ymax = a.ymax(sim2d),
  K = opt.k,
  Criterion = "BIC",
  EMcontrol = EM)

# Plot finite mixture.

plot(sim2dest, pos = 3, nrow = 4, what = c("pdf", "marginal pdf", "IC"))

# Estimate number of components, component weights and component 
# parameters for well known Iris dataset.

Dataset <- list(iris[, c(1:4)])

# Calculate optimal numbers of bins using non-equal number of bins in each dimension.

opt.k <- optbins(Dataset = Dataset,
  Rule = "Knuth unequal",
  kmin = 2, 
  kmax = 20)

opt.k

# Estimate number of components, component weights and component parameters.

irisest <- REBMIX(model = "REBMVNORM", 
  Dataset = Dataset,
  Preprocessing = "h",
  cmax = 10,
  K = opt.k,
  Criterion = "BIC",
  EMcontrol = EM)
  
irisest

Description

Returns the partition coefficient of Bezdek at pos.

Usage

## S4 method for signature 'REBMIX'
PC(x = NULL, pos = 1, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

G. McLachlan and D. Peel. Finite Mixture Models. John Wiley & Sons, New York, 2000.

Description

Returns the data frame containing observations $\bm{x}_{1}, \ldots, \bm{x}_{n}$ and empirical distribution functions $F_{1}, \ldots, F_{n}$. Vectors $\bm{x}$ are subvectors of $\bm{y} = (y_{1}, \ldots, y_{d})^{\top}$.

Usage

## S4 method for signature 'REBMIX'
pemix(x = NULL, pos = 1, variables = expression(1:d),
      lower.tail = TRUE, log.p = FALSE, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

M. Nagode and M. Fajdiga. The rebmix algorithm for the univariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(5):876-892, 2011a. doi:10.1080/03610920903480890.

M. Nagode and M. Fajdiga. The rebmix algorithm for the multivariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(11):2022-2034, 2011b. doi:10.1080/03610921003725788.

M. Nagode. Finite mixture modeling via REBMIX. Journal of Algorithms and Optimization, 3(2):14-28, 2015. https://repozitorij.uni-lj.si/Dokument.php?id=127674&lang=eng.

Examples

# Generate simulated dataset.

n <- c(15, 15)

Theta <- new("RNGMIX.Theta", c = 2, pdf = rep("normal", 3))

a.theta1(Theta, 1) <- c(10, 20, 30)
a.theta1(Theta, 2) <- c(3, 4, 5)
a.theta2(Theta, 1) <- c(3, 2, 1)
a.theta2(Theta, 2) <- c(15, 10, 5)

simulated <- RNGMIX(Dataset.name = paste("simulated_", 1:4, sep = ""),
  rseed = -1,
  n = n,
  Theta = a.Theta(Theta))

# Create object of class EM.Control.

EM <- new("EM.Control", strategy = "exhaustive", variant = "ECM",
  acceleration = "fixed", acceleration.multiplier = 1.0, tolerance = 1.0E-4,
  maximum.iterations = 1000)

# Estimate number of components, component weights and component parameters.

simulatedest <- REBMIX(Dataset = a.Dataset(simulated),
  Preprocessing = "kernel density estimation",
  cmax = 4,
  pdf = c("n", "n", "n"),
  EMcontrol = EM)

# Preprocess simulated dataset.

f <- pemix(simulatedest, pos = 3, variables = c(1))

f

Description

Returns the data frame containing observations $\bm{x}_{1}, \ldots, \bm{x}_{n}$ and predictive marginal distribution functions $F(\bm{x} | c, \bm{w}, \bm{\Theta})$. Vectors $\bm{x}$ are subvectors of $\bm{y} = (y_{1}, \ldots, y_{d})^{\top}$. If $\bm{x} = \bm{y}$ the method returns the data frame containing observations $\bm{y}_{1}, \ldots, \bm{y}_{n}$ and the corresponding predictive mixture distribution function $F(\bm{y} | c, \bm{w}, \bm{\Theta})$.

Usage

## S4 method for signature 'REBMIX'
pfmix(x = NULL, Dataset = data.frame(), pos = 1,
      variables = expression(1:d), lower.tail = TRUE, log.p = FALSE, ...)
## ... and for other signatures

Arguments

Methods

signature(x = "REBMIX")

an object of class REBMIX.

signature(x = "REBMVNORM")

an object of class REBMVNORM.

Author(s)

Marko Nagode

References

M. Nagode and M. Fajdiga. The rebmix algorithm for the univariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(5):876-892, 2011a. doi:10.1080/03610920903480890.

M. Nagode and M. Fajdiga. The rebmix algorithm for the multivariate finite mixture estimation. Communications in Statistics - Theory and Methods, 40(11):2022-2034, 2011b. doi:10.1080/03610921003725788.

M. Nagode. Finite mixture modeling via REBMIX. Journal of Algorithms and Optimization, 3(2):14-28, 2015. https://repozitorij.uni-lj.si/Dokument.php?id=127674&lang=eng.

Examples

# Generate simulated dataset.

n <- c(15, 15)

Theta <- new("RNGMIX.Theta", c = 2, pdf = rep("normal", 3))

a.theta1(Theta, 1) <- c(10, 20, 30)
a.theta1(Theta, 2) <- c(3, 4, 5)
a.theta2(Theta, 1) <- c(3, 2, 1)
a.theta2(Theta, 2) <- c(15, 10, 5)

simulated <- RNGMIX(Dataset.name = paste("simulated_", 1:4, sep = ""),
  rseed = -1,
  n = n,
  Theta = a.Theta(Theta))

# Number of classes or nearest neighbours to be processed.

K <- c(as.integer(1 + log2(sum(n))), # Minimum v follows Sturges rule.
  as.integer(10 * log10(sum(n)))) # Maximum v follows log10 rule.

# Estimate number of components, component weights and component parameters.

simulatedest <- REBMIX(Dataset = a.Dataset(simulated),
  Preprocessing = "h",
  cmax = 4,
  Criterion = "BIC",
  pdf = c("n", "n", "n"))

# Preprocess simulated dataset.

Dataset <- data.frame(c(25, 5, -20), NA, c(31, 20, 20))

f <- pfmix(simulatedest, Dataset = Dataset, pos = 3, variables = c(1, 3))

f

# Plot finite mixture.

opar <- plot(simulatedest, pos = 3, nrow = 3, ncol = 1,
  what = "pdf", contour.drawlabels = TRUE, contour.labcex = 0.6)

par(usr = opar[[2]]$usr, mfg = c(2, 1))

points(x = f[, 1], y = f[, 2])

text(x = f[, 1], y = f[, 2], labels = format(f[, 3], digits = 3), cex = 0.8, pos = 4)

Description

Plots true clusters if x equals "RNGMIX". Plots the REBMIX output depending on what argument if x equals "REBMIX". Plots predictive clusters if x equals "RCLRMIX". Wrongly clustered observations are plotted only if x@Zt is available. Plots predictive classes and wrongly classified observations if x equals "RCLSMIX".

Usage

## S4 method for signature 'RNGMIX,missing'
plot(x, y, pos = 1, nrow = 1, ncol = 1, cex = 0.8,
     fg = "black", lty = "solid", lwd = 1, pty = "m", tcl = 0.5,
     plot.cex = 0.8, plot.pch = 19, ...)
## S4 method for signature 'REBMIX,missing'
plot(x, y, pos = 1, what = c("pdf"),
     nrow = 1, ncol = 1, npts = 200, n = 200, cex = 0.8, fg = "black",
     lty = "solid", lwd = 1, pty = "m", tcl = 0.5,
     plot.cex = 0.8, plot.pch = 19, contour.drawlabels = FALSE,
     contour.labcex = 0.8, contour.method = "flattest",
     contour.nlevels = 12, log = "", ...)
## S4 method for signature 'RCLRMIX,missing'
plot(x, y, s = expression(c), nrow = 1, ncol = 1, cex = 0.8,
     fg = "black", lty = "solid", lwd = 1, pty = "m", tcl = 0.5,
     plot.cex = 0.8, plot.pch = 19, ...)
## S4 method for signature 'RCLSMIX,missing'
plot(x, y, nrow = 1, ncol = 1, cex = 0.8,
     fg = "black", lty = "solid", lwd = 1, pty = "m", tcl = 0.5,
     plot.cex = 0.8, plot.pch = 19, ...)
## ... and for other signatures