Discriminant analysis based on Gaussian finite mixture modeling.

MclustDA(data, class, G = NULL, modelNames = NULL, 
         modelType = c("MclustDA", "EDDA"), 
         prior = NULL, 
         control = emControl(), 
         initialization = NULL, 
         warn = mclust.options("warn"), 
         verbose = interactive(),
         ...)

Arguments

data

A data frame or matrix giving the training data.

class

A vector giving the known class labels (either a numerical value or a character string) for the observations in the training data.

G

An integer vector specifying the numbers of mixture components (clusters) for which the BIC is to be calculated within each class. The default is G = 1:5.
A different set of mixture components for each class can be specified by providing this argument with a list of integers for each class. See the examples below.

modelNames

A vector of character strings indicating the models to be fitted by EM within each class (see the description in mclustModelNames). A different set of mixture models for each class can be specified by providing this argument with a list of character strings. See the examples below.

modelType

A character string specifying whether the models given in modelNames should fit a different number of mixture components and covariance structures for each class ("MclustDA", the default) or should be constrained to have a single component for each class with the same covariance structure among classes ("EDDA"). See Details section and the examples below.

prior

The default assumes no prior, but this argument allows specification of a conjugate prior on the means and variances through the function priorControl.

control

A list of control parameters for EM. The defaults are set by the call emControl().

initialization

A list containing zero or more of the following components:

hcPairs

A matrix of merge pairs for hierarchical clustering such as produced by function hc. The default is to compute a hierarchical clustering tree by applying function hc with modelName = "E" to univariate data and modelName = "VVV" to multivariate data or a subset as indicated by the subset argument. The hierarchical clustering results are used as starting values for EM.

subset

A logical or numeric vector specifying a subset of the data to be used in the initial hierarchical clustering phase.

warn

A logical value indicating whether or not certain warnings (usually related to singularity) should be issued when estimation fails. The default is controlled by mclust.options.

verbose

A logical controlling if a text progress bar is displayed during the fitting procedure. By default is TRUE if the session is interactive, and FALSE otherwise.

...

Further arguments passed to or from other methods.

Value

An object of class 'MclustDA' providing the optimal (according to BIC) mixture model. The details of the output components are as follows:

call

The matched call.

data

The input data matrix.

class

The input class labels.

type

A character string specifying the modelType estimated.

models

A list of Mclust objects containing information on fitted model for each class.

n

The total number of observations in the data.

d

The dimension of the data.

bic

Optimal BIC value.

loglik

Log-likelihood for the selected model.

df

Number of estimated parameters.

Details

The "EDDA" method for discriminant analysis is described in Bensmail and Celeux (1996), while "MclustDA" in Fraley and Raftery (2002).

References

Scrucca L., Fop M., Murphy T. B. and Raftery A. E. (2016) mclust 5: clustering, classification and density estimation using Gaussian finite mixture models, The R Journal, 8/1, pp. 289-317.

Fraley C. and Raftery A. E. (2002) Model-based clustering, discriminant analysis and density estimation, Journal of the American Statistical Association, 97/458, pp. 611-631.

Fraley C., Raftery A. E., Murphy T. B. and Scrucca L. (2012) mclust Version 4 for R: Normal Mixture Modeling for Model-Based Clustering, Classification, and Density Estimation. Technical Report No. 597, Department of Statistics, University of Washington.

Bensmail, H., and Celeux, G. (1996) Regularized Gaussian Discriminant Analysis Through Eigenvalue Decomposition.Journal of the American Statistical Association, 91, 1743-1748.

Author

Luca Scrucca

Examples

odd <- seq(from = 1, to = nrow(iris), by = 2)
even <- odd + 1
X.train <- iris[odd,-5]
Class.train <- iris[odd,5]
X.test <- iris[even,-5]
Class.test <- iris[even,5]

# common EEE covariance structure (which is essentially equivalent to linear discriminant analysis)
irisMclustDA <- MclustDA(X.train, Class.train, modelType = "EDDA", modelNames = "EEE")
summary(irisMclustDA, parameters = TRUE)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> EDDA model summary: 
#> 
#>  log-likelihood  n df       BIC
#>        -125.443 75 22 -345.8707
#>             
#> Classes       n     % Model G
#>   setosa     25 33.33   EEE 1
#>   versicolor 25 33.33   EEE 1
#>   virginica  25 33.33   EEE 1
#> 
#> Class prior probabilities:
#>     setosa versicolor  virginica 
#>  0.3333333  0.3333333  0.3333333 
#> 
#> Class = setosa
#> 
#> Means:
#>               [,1]
#> Sepal.Length 5.024
#> Sepal.Width  3.480
#> Petal.Length 1.456
#> Petal.Width  0.228
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.26418133  0.06244800   0.15935467  0.03141333
#> Sepal.Width    0.06244800  0.09630933   0.03326933  0.03222400
#> Petal.Length   0.15935467  0.03326933   0.18236800  0.04091733
#> Petal.Width    0.03141333  0.03222400   0.04091733  0.03891200
#> 
#> Class = versicolor
#> 
#> Means:
#>               [,1]
#> Sepal.Length 5.992
#> Sepal.Width  2.776
#> Petal.Length 4.308
#> Petal.Width  1.352
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.26418133  0.06244800   0.15935467  0.03141333
#> Sepal.Width    0.06244800  0.09630933   0.03326933  0.03222400
#> Petal.Length   0.15935467  0.03326933   0.18236800  0.04091733
#> Petal.Width    0.03141333  0.03222400   0.04091733  0.03891200
#> 
#> Class = virginica
#> 
#> Means:
#>               [,1]
#> Sepal.Length 6.504
#> Sepal.Width  2.936
#> Petal.Length 5.564
#> Petal.Width  2.076
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.26418133  0.06244800   0.15935467  0.03141333
#> Sepal.Width    0.06244800  0.09630933   0.03326933  0.03222400
#> Petal.Length   0.15935467  0.03326933   0.18236800  0.04091733
#> Petal.Width    0.03141333  0.03222400   0.04091733  0.03891200
#> 
#> Training confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          1        24
#> Classification error = 0.0267 
#> Brier score          = 0.0097 
summary(irisMclustDA, newdata = X.test, newclass = Class.test)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> EDDA model summary: 
#> 
#>  log-likelihood  n df       BIC
#>        -125.443 75 22 -345.8707
#>             
#> Classes       n     % Model G
#>   setosa     25 33.33   EEE 1
#>   versicolor 25 33.33   EEE 1
#>   virginica  25 33.33   EEE 1
#> 
#> Training confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          1        24
#> Classification error = 0.0267 
#> Brier score          = 0.0097 
#> 
#> Test confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          2        23
#> Classification error = 0.04 
#> Brier score          = 0.0243 

# common covariance structure selected by BIC
irisMclustDA <- MclustDA(X.train, Class.train, modelType = "EDDA")
summary(irisMclustDA, parameters = TRUE)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> EDDA model summary: 
#> 
#>  log-likelihood  n df       BIC
#>       -87.93758 75 36 -331.3047
#>             
#> Classes       n     % Model G
#>   setosa     25 33.33   VEV 1
#>   versicolor 25 33.33   VEV 1
#>   virginica  25 33.33   VEV 1
#> 
#> Class prior probabilities:
#>     setosa versicolor  virginica 
#>  0.3333333  0.3333333  0.3333333 
#> 
#> Class = setosa
#> 
#> Means:
#>               [,1]
#> Sepal.Length 5.024
#> Sepal.Width  3.480
#> Petal.Length 1.456
#> Petal.Width  0.228
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length  0.154450439 0.097646496  0.017347101 0.005327878
#> Sepal.Width   0.097646496 0.105230813  0.004066916 0.005599939
#> Petal.Length  0.017347101 0.004066916  0.041742267 0.003476241
#> Petal.Width   0.005327878 0.005599939  0.003476241 0.006454832
#> 
#> Class = versicolor
#> 
#> Means:
#>               [,1]
#> Sepal.Length 5.992
#> Sepal.Width  2.776
#> Petal.Length 4.308
#> Petal.Width  1.352
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.26653496  0.06976610   0.17015657  0.04336127
#> Sepal.Width    0.06976610  0.09861066   0.07509657  0.03823057
#> Petal.Length   0.17015657  0.07509657   0.19799871  0.06126251
#> Petal.Width    0.04336127  0.03823057   0.06126251  0.03627058
#> 
#> Class = virginica
#> 
#> Means:
#>               [,1]
#> Sepal.Length 6.504
#> Sepal.Width  2.936
#> Petal.Length 5.564
#> Petal.Width  2.076
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.37570480 0.025364426  0.280227270  0.03871029
#> Sepal.Width    0.02536443 0.080872957  0.006413281  0.05009229
#> Petal.Length   0.28022727 0.006413281  0.309059434  0.05805268
#> Petal.Width    0.03871029 0.050092291  0.058052679  0.07540425
#> 
#> Training confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          0        25
#> Classification error = 0.0133 
#> Brier score          = 0.0054 
summary(irisMclustDA, newdata = X.test, newclass = Class.test)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> EDDA model summary: 
#> 
#>  log-likelihood  n df       BIC
#>       -87.93758 75 36 -331.3047
#>             
#> Classes       n     % Model G
#>   setosa     25 33.33   VEV 1
#>   versicolor 25 33.33   VEV 1
#>   virginica  25 33.33   VEV 1
#> 
#> Training confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          0        25
#> Classification error = 0.0133 
#> Brier score          = 0.0054 
#> 
#> Test confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          2        23
#> Classification error = 0.04 
#> Brier score          = 0.0297 

# general covariance structure selected by BIC
irisMclustDA <- MclustDA(X.train, Class.train)
summary(irisMclustDA, parameters = TRUE)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> MclustDA model summary: 
#> 
#>  log-likelihood  n df       BIC
#>       -71.74193 75 48 -350.7233
#>             
#> Classes       n     % Model G
#>   setosa     25 33.33   VEI 2
#>   versicolor 25 33.33   VEE 2
#>   virginica  25 33.33   XXX 1
#> 
#> Class prior probabilities:
#>     setosa versicolor  virginica 
#>  0.3333333  0.3333333  0.3333333 
#> 
#> Class = setosa
#> 
#> Mixing probabilities: 0.7229143 0.2770857 
#> 
#> Means:
#>                   [,1]      [,2]
#> Sepal.Length 5.1761949 4.6269248
#> Sepal.Width  3.6366552 3.0712877
#> Petal.Length 1.4777585 1.3992323
#> Petal.Width  0.2441875 0.1857668
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length     0.120728    0.000000   0.00000000 0.000000000
#> Sepal.Width      0.000000    0.046461   0.00000000 0.000000000
#> Petal.Length     0.000000    0.000000   0.04892923 0.000000000
#> Petal.Width      0.000000    0.000000   0.00000000 0.006358681
#> [,,2]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.03044364  0.00000000   0.00000000 0.000000000
#> Sepal.Width    0.00000000  0.01171594   0.00000000 0.000000000
#> Petal.Length   0.00000000  0.00000000   0.01233835 0.000000000
#> Petal.Width    0.00000000  0.00000000   0.00000000 0.001603451
#> 
#> Class = versicolor
#> 
#> Mixing probabilities: 0.2364317 0.7635683 
#> 
#> Means:
#>                  [,1]     [,2]
#> Sepal.Length 6.736465 5.761483
#> Sepal.Width  3.000982 2.706336
#> Petal.Length 4.669933 4.195931
#> Petal.Width  1.400893 1.336861
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length  0.030012918 0.008262520   0.02533959 0.008673053
#> Sepal.Width   0.008262520 0.020600060   0.01200205 0.008400168
#> Petal.Length  0.025339590 0.012002053   0.03924151 0.013788157
#> Petal.Width   0.008673053 0.008400168   0.01378816 0.007666627
#> [,,2]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length   0.16630011  0.04578222   0.14040543  0.04805696
#> Sepal.Width    0.04578222  0.11414392   0.06650279  0.04654492
#> Petal.Length   0.14040543  0.06650279   0.21743528  0.07639950
#> Petal.Width    0.04805696  0.04654492   0.07639950  0.04248041
#> 
#> Class = virginica
#> 
#> Mixing probabilities: 1 
#> 
#> Means:
#>               [,1]
#> Sepal.Length 6.504
#> Sepal.Width  2.936
#> Petal.Length 5.564
#> Petal.Width  2.076
#> 
#> Variances:
#> [,,1]
#>              Sepal.Length Sepal.Width Petal.Length Petal.Width
#> Sepal.Length     0.349184    0.019056     0.272144    0.040896
#> Sepal.Width      0.019056    0.079104     0.011296    0.048064
#> Petal.Length     0.272144    0.011296     0.285504    0.049536
#> Petal.Width      0.040896    0.048064     0.049536    0.074624
#> 
#> Training confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         25         0
#>   virginica       0          0        25
#> Classification error = 0 
#> Brier score          = 0.0041 
summary(irisMclustDA, newdata = X.test, newclass = Class.test)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> MclustDA model summary: 
#> 
#>  log-likelihood  n df       BIC
#>       -71.74193 75 48 -350.7233
#>             
#> Classes       n     % Model G
#>   setosa     25 33.33   VEI 2
#>   versicolor 25 33.33   VEE 2
#>   virginica  25 33.33   XXX 1
#> 
#> Training confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         25         0
#>   virginica       0          0        25
#> Classification error = 0 
#> Brier score          = 0.0041 
#> 
#> Test confusion matrix:
#>             Predicted
#> Class        setosa versicolor virginica
#>   setosa         25          0         0
#>   versicolor      0         24         1
#>   virginica       0          1        24
#> Classification error = 0.0267 
#> Brier score          = 0.0159 

plot(irisMclustDA)




plot(irisMclustDA, dimens = 3:4)




plot(irisMclustDA, dimens = 4)





plot(irisMclustDA, what = "classification")

plot(irisMclustDA, what = "classification", newdata = X.test)

plot(irisMclustDA, what = "classification", dimens = 3:4)

plot(irisMclustDA, what = "classification", newdata = X.test, dimens = 3:4)

plot(irisMclustDA, what = "classification", dimens = 4)

plot(irisMclustDA, what = "classification", dimens = 4, newdata = X.test)


plot(irisMclustDA, what = "train&test", newdata = X.test)

plot(irisMclustDA, what = "train&test", newdata = X.test, dimens = 3:4)

plot(irisMclustDA, what = "train&test", newdata = X.test, dimens = 4)


plot(irisMclustDA, what = "error")

plot(irisMclustDA, what = "error", dimens = 3:4)

plot(irisMclustDA, what = "error", dimens = 4)

plot(irisMclustDA, what = "error", newdata = X.test, newclass = Class.test)

plot(irisMclustDA, what = "error", newdata = X.test, newclass = Class.test, dimens = 3:4)

plot(irisMclustDA, what = "error", newdata = X.test, newclass = Class.test, dimens = 4)


# \donttest{
# simulated 1D data
n <- 250 
set.seed(1)
triModal <- c(rnorm(n,-5), rnorm(n,0), rnorm(n,5))
triClass <- c(rep(1,n), rep(2,n), rep(3,n))
odd <- seq(from = 1, to = length(triModal), by = 2)
even <- odd + 1
triMclustDA <- MclustDA(triModal[odd], triClass[odd])
summary(triMclustDA, parameters = TRUE)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> MclustDA model summary: 
#> 
#>  log-likelihood   n df       BIC
#>       -942.4306 375  6 -1920.423
#>        
#> Classes   n     % Model G
#>       1 125 33.33     X 1
#>       2 125 33.33     X 1
#>       3 125 33.33     X 1
#> 
#> Class prior probabilities:
#>         1         2         3 
#> 0.3333333 0.3333333 0.3333333 
#> 
#> Class = 1
#> 
#> Mixing probabilities: 1 
#> 
#> Means:
#> [1] -4.951981
#> 
#> Variances:
#> [1] 0.8268339
#> 
#> Class = 2
#> 
#> Mixing probabilities: 1 
#> 
#> Means:
#> [1] -0.01814707
#> 
#> Variances:
#> [1] 1.201987
#> 
#> Class = 3
#> 
#> Mixing probabilities: 1 
#> 
#> Means:
#> [1] 4.875429
#> 
#> Variances:
#> [1] 1.200321
#> 
#> Training confusion matrix:
#>      Predicted
#> Class   1   2   3
#>     1 124   1   0
#>     2   0 124   1
#>     3   0   0 125
#> Classification error = 0.0053 
#> Brier score          = 0.0073 
summary(triMclustDA, newdata = triModal[even], newclass = triClass[even])
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> MclustDA model summary: 
#> 
#>  log-likelihood   n df       BIC
#>       -942.4306 375  6 -1920.423
#>        
#> Classes   n     % Model G
#>       1 125 33.33     X 1
#>       2 125 33.33     X 1
#>       3 125 33.33     X 1
#> 
#> Training confusion matrix:
#>      Predicted
#> Class   1   2   3
#>     1 124   1   0
#>     2   0 124   1
#>     3   0   0 125
#> Classification error = 0.0053 
#> Brier score          = 0.0073 
#> 
#> Test confusion matrix:
#>      Predicted
#> Class   1   2   3
#>     1 124   1   0
#>     2   1 122   2
#>     3   0   1 124
#> Classification error = 0.0133 
#> Brier score          = 0.0089 
plot(triMclustDA, what = "scatterplot")

plot(triMclustDA, what = "classification")

plot(triMclustDA, what = "classification", newdata = triModal[even])

plot(triMclustDA, what = "train&test", newdata = triModal[even])

plot(triMclustDA, what = "error")

plot(triMclustDA, what = "error", newdata = triModal[even], newclass = triClass[even])


# simulated 2D cross data
data(cross)
odd <- seq(from = 1, to = nrow(cross), by = 2)
even <- odd + 1
crossMclustDA <- MclustDA(cross[odd,-1], cross[odd,1])
summary(crossMclustDA, parameters = TRUE)
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> MclustDA model summary: 
#> 
#>  log-likelihood   n df     BIC
#>       -1381.404 250  9 -2812.5
#>        
#> Classes   n  % Model G
#>       1 125 50   XXX 1
#>       2 125 50   XXI 1
#> 
#> Class prior probabilities:
#>   1   2 
#> 0.5 0.5 
#> 
#> Class = 1
#> 
#> Mixing probabilities: 1 
#> 
#> Means:
#>           [,1]
#> X1  0.03982835
#> X2 -0.55982893
#> 
#> Variances:
#> [,,1]
#>          X1       X2
#> X1 1.030302  1.81975
#> X2 1.819750 72.90428
#> 
#> Class = 2
#> 
#> Mixing probabilities: 1 
#> 
#> Means:
#>          [,1]
#> X1  0.2877120
#> X2 -0.1231171
#> 
#> Variances:
#> [,,1]
#>         X1        X2
#> X1 87.3583 0.0000000
#> X2  0.0000 0.8995777
#> 
#> Training confusion matrix:
#>      Predicted
#> Class   1   2
#>     1 112  13
#>     2  13 112
#> Classification error = 0.104 
#> Brier score          = 0.0563 
summary(crossMclustDA, newdata = cross[even,-1], newclass = cross[even,1])
#> ------------------------------------------------ 
#> Gaussian finite mixture model for classification 
#> ------------------------------------------------ 
#> 
#> MclustDA model summary: 
#> 
#>  log-likelihood   n df     BIC
#>       -1381.404 250  9 -2812.5
#>        
#> Classes   n  % Model G
#>       1 125 50   XXX 1
#>       2 125 50   XXI 1
#> 
#> Training confusion matrix:
#>      Predicted
#> Class   1   2
#>     1 112  13
#>     2  13 112
#> Classification error = 0.104 
#> Brier score          = 0.0563 
#> 
#> Test confusion matrix:
#>      Predicted
#> Class   1   2
#>     1 118   7
#>     2   5 120
#> Classification error = 0.048 
#> Brier score          = 0.0298 
plot(crossMclustDA, what = "scatterplot")

plot(crossMclustDA, what = "classification")

plot(crossMclustDA, what = "classification", newdata = cross[even,-1])

plot(crossMclustDA, what = "train&test", newdata = cross[even,-1])

plot(crossMclustDA, what = "error")

plot(crossMclustDA, what = "error", newdata =cross[even,-1], newclass = cross[even,1])

# }