Package 'RMBC'

Title: Robust Model Based Clustering
Description: A robust clustering algorithm (Model-Based) similar to Expectation Maximization for finite mixture normal distributions is implemented, its main advantage is that the estimator is resistant to outliers, that means that results of parameter estimation are still correct when there are atypical values in the sample (see Gonzalez, Maronna, Yohai and Zamar (2021) <https://arxiv.org/abs/2102.06851>).
Authors: Juan Domingo Gonzalez [cre, aut], Victor J. Yohai [aut], Ruben H. Zamar [aut] Ricardo Maronna [aut]
Maintainer: Juan Domingo Gonzalez <[email protected]>
License: GPL (>= 2)
Version: 0.1.0
Built: 2025-01-20 05:26:39 UTC
Source: https://github.com/jdgonzalezwork/rmbc

Help Index

is_in_gr

Description

Given Y data and a set of mixture parameters, this function return a boolean vector B whose lentght is equal than Y length. B[i] is TRUE if only if Y[i] not belong to the union of confidence ellipsoids of level given by the level

Usage

is_in_gr(Y, cutoff = 0.999, theta.mu, theta.sigma)

Arguments

Y

A matrix of size n x p.
cutoff

quantiles of chi-square to be used as a threshold for outliers detection, defaults to 0.999
theta.mu

The estimated centers: A list with K elements, each of them is an array of length p.
theta.sigma

The estimated scatter matrices: A list with K matrices, each of them has dimension p x p

Value

A n x K matrix, where each row has the values of the quadratic discriminant with regarding to the j-th mixture component, j = 1,...,K

klfor2normals Compute the Kullback-Leibler divergence for 2 normal multivariate distributions

Description

klfor2normals Compute the Kullback-Leibler divergence for 2 normal multivariate distributions

Usage

klfor2normals(theta1.mu, theta1.sigma, theta2.mu, theta2.sigma)

Arguments

theta1.mu

the location parameter of the first distribution
theta1.sigma

the covariance matrix of the first distribution
theta2.mu

the location parameter of the second distribution
theta2.sigma

the covariance matrix of the second distribution

Value

the K-L divergence.

Phytoplankton_acoustic_data

Description

Data obtained by taking laboratory measurements of ultrasonic acoustic signals: a pulse is emitted by a transducer, this pulse interacts with phytoplankton suspended in the water and produces an acoustic dispersion (scattering), which is recorded by an electronic acquisition device. A filtering process of the signal is performed in a first stage. Portions of the signal belong o one of the two main cases:

  • (a) Signals corresponding to the acoustic response of phytoplankton

  • (b) Signals corresponding to spurious dispersers, such as bubbles or particles in suspension, whose intensity is greater than in case (a).

To classify a signal in one of these two groups biologists create a vector (X1, X2) defined as follows:

  • X1 = ratio of filtered to non-filtered signal power

  • X2 = filtered signal power expressed in dB.

The available data consists of 375 such measurements. These data is particularly useful to compare robust procedures because 20 to be outliers produced by a communication failure between the electronic device (digital oscilloscope) and the software for acquiring the acoustic signal. This failure occurs once every 5 microseconds, which allows the scientists to identify the outliers. The outliers appear as a separated group in the region X1 < 0.5 and X2 > 20.

Usage

phytoplankton_acoustic_data

Format

a list of length 2, where its elements are

  • Y: A matrix of dimension 375 x 2, each row contains X1 and X2 values

  • outliers_index: An array with the outliers index-locations

References

  • [1] Cinquini, M., Bos, P., Prario, I and Blanc, S. (2016), “Advances on modelling, simulation and signal processing of ultrasonic scattering responses from phytoplankton cultures,” in Proceedings of Meetings on Acoustics 22ICA, 28, American Society of Acoustics.

  • [2] Gonzalez J.D, Maronna R., Yohai V., & and Zamar . (2021). Robust Model-Based Clustering. arXiv preprint <https://arxiv.org/abs/2102.06851>

Examples

################################
# upload matrix ################
################################

Y <- phytoplankton_acoustic_data$Y

outliers_index <- phytoplankton_acoustic_data$outliers_index

Yclean=Y[-outliers_index,]

trueOutliers=Y[outliers_index,]

################################
# plot results ################
################################

plot(Y, main = "Phytoplankton acoustic data", cex.main = 3, lwd = 1,pch = 19, cex = 1, 
     type = "n", xlab = "x1", ylab = "x2",  xlim = c(0,1.1), ylim = c(0,43)
     )
     
points(trueOutliers,lwd=2,cex=1,pch=4)

points(Yclean,col=1,lwd=1.5,pch=21, bg=4, cex=1)

quad_disc

Description

Computes the quadratic discriminant of each mixture component,

Usage

quad_disc(Y, theta.alpha, theta.mu, theta.sigma)

Arguments

Y

A matrix of size n x p.
theta.alpha

The alpha values: An array of K positive real numbers they must verify the condition sum(thetaOld.mu)== 1.
theta.mu

The estimated centers: A list with K elements, each of them is an array of length p.
theta.sigma

The estimated scatter matrices: A list with K matrices, each of them has dimension p x p

Value

A n x K matrix, where each row has the values of the quadratic discriminant with regarding to the j-th mixture component, j = 1,...,K

Robust Model Base Clustering a robust and efficient version of EM algorithm.

Description

Robust Model Base Clustering a robust and efficient version of EM algorithm.

Usage

RMBC(Y, K, max_iter = 80, tolerance = 1e-04)

Arguments

Y

A matrix of size n x p.
K

The number of clusters.
max_iter

a maximum number of iterations used for the algorithm stopping rule
tolerance

tolerance parameter used for the algorithm stopping rule

Value

A list including the estimated mixture distribution parameters and cluster-label for the observations

  • alpha: K numeric values representing the convex combination coefficients.

  • mu: a list of length K with the location initial estimators.

  • sigma: a list of length K with the location scatter matrix estimators.

  • nonoutliers: an array of indices that contains the estimated nonoutliers observations

  • outliers: an array of indices that contains the estimated outliers observations

Examples

# Generate Sintetic data (three normal cluster in two dimension)
# clusters have different shapes and orentation.
# The data is contaminated uniformly (level 20%).
################################################
#### Start data generating process ############
##############################################

# generates base clusters

Z1 <- c(rnorm(100,0),rnorm(100,0),rnorm(100,0))
Z2 <- rnorm(300);
X <-  matrix(0, ncol=2,nrow=300);
X[,1]=Z1;X[,2]=Z2
true.cluster= c(rep(1,100),rep(2,100),rep(3,100))
# rotate, expand and translate base clusters
theta=pi/3;
aux1=matrix(c(cos(theta),-sin(theta),sin(theta),cos(theta)),nrow=2)

aux2=sqrt(4)*diag(c(1,1/4))

B=aux1%*%aux2%*%t(aux1)

X[true.cluster==3,]=X[true.cluster==3,]%*%aux2%*%aux1 + 
matrix(c(15,2), byrow = TRUE,nrow=100,ncol=2)
X[true.cluster==2,2] = X[true.cluster==2,2]*4
X[true.cluster==1,2] = X[true.cluster==1,2]*0.1
X[true.cluster==1, ] = X[true.cluster==1,]+ 
matrix(c(-15,-1),byrow = TRUE,nrow=100,ncol=2)

### Generate 60 sintetic outliers (contamination level 20%)

outliers=sample(1:300,60)
X[outliers, ] <- matrix(runif( 40, 2 * min(X), 2 * max(X) ),
                        ncol = 2, nrow = 60)

###############################################
#### END data generating process ############
#############################################

### APLYING RMBC ALGORITHM 

ret = RMBC(Y=X, K=3,max_iter = 82)

cluster = ret$cluster
#############################################
### plotting results ########################
#############################################
oldpar=par(mfrow=c(1,2))
plot(X,  main="actual clusters" )
for (j in 1:3){
  points(X[true.cluster==j,],pch=19, col=j+1)
}
points(X[outliers,],pch=19,col=1)

plot(X,main="clusters estimation")
for (j in 1:3){
  points(X[cluster==j,],pch=19, col=j+1)
}
points(X[ret$outliers,],pch=19,col=1)
par(oldpar)

RMBCaux

Description

Robust Model Base Clustering algorithm based on centers, a robust and efficient version of EM algorithm.

Usage

RMBCaux(
  Y,
  K,
  thetaOld.alpha,
  thetaOld.mu,
  thetaOld.sigma,
  max_iter,
  niterFixedPoint,
  tolerance,
  cutoff = 1 - 0.001
)

Arguments

Y

A matrix of size n x p.
K

The number of clusters.
thetaOld.alpha

The initial alpha: An array of K positive real numbers they must verify the condition sum(thetaOld.mu)== 1.
thetaOld.mu

The initial centers: A list with K elements, each of them is an array of length p.
thetaOld.sigma

The initial stcatter matrix: A list with K matrix, each of them has dimension p x p
max_iter

a maximum number of iterations used for the algorithm stopping rule
niterFixedPoint

the maximum number of iteration in the internal loop which computes sigma an mu separately. The default value is niterFixedPoint=1
tolerance

tolerance parameter used for the algorithm stopping rule
cutoff

optional argument for outliers detection - quantiles of chi-square to be used as a threshold for outliers detection, defaults to 0.999

Value

A list including the estimated K centers and labels for the observations

  • centers: matrix of size K x p, with the estimated K centers.

  • cluster: array of size n x 1 integers labels between 1 and K.

  • tauPath: sequence of tau scale values at each iterations.

  • Wni: numeric array of size n x 1 indicating the weights associated to each observation.

  • emptyClusterFlag: a boolean value. True means that in some iteration there were clusters totally empty

  • niter: number of iterations until convergence is achived or maximum number of iteration is reached

  • didistance of each observation to its assigned cluster-center

robustINIT

Description

Robust Initializer for RMBC algorithm, it depends on the package ktaucenters

Usage

robustINIT(Y, K, nstart = 10)

Arguments

Y

A matrix of size n x p.
K

The number of groups
nstart

the number of starting points to the algorithm, defaults to 10

Value

A list including the initial parameters of the mixture distribution, namely

  • alphaINIT: K numeric values representing the convex combination coefficients.

  • muINIT: a list of length K with the location initial estimators.

  • sigmaINIT: a list of length K with the location scatter matrix estimators.

  • indicesINIT: indices with initial clusters

sumkl The sum of K-L divergence measure between two successive iterations for each component of a mixture distribution,

Description

sumkl The sum of K-L divergence measure between two successive iterations for each component of a mixture distribution,

Usage

sumkl(thetaNew.mu, thetaNew.sigma, thetaOld.mu, thetaOld.sigma)

Arguments

thetaNew.mu

the location parameters of the first distribution
thetaNew.sigma

the covariance matrix of the first distribution
thetaOld.mu

the location parameter of the second distribution
thetaOld.sigma

the covariance matrix of the second distribution

Value

the K-L divergence.

weightedMscale the M scale of an univariate sample (see reference below)

Description

weightedMscale the M scale of an univariate sample (see reference below)

Usage

weightedMscale(u, b = 0.5, weights, c, initialsc = 0)

Arguments

u

an univariate sample of size n.
b

the desired break down point
weights

the weights of each observation.
c

a tuning constant, if consistency to standard normal distribution is desired use normal_consistency_constants
initialsc

the initial scale value, defaults to 0

Value

the weighted-Mscale value

References

Maronna, R. A., Martin, R. D., Yohai, V. J., & Salibián-Barrera, M. (2018). Robust statistics: theory and methods (with R). Wiley.

weightedSestimator

Description

Computes the weighted location and scatter matrix estimators of the j-th mixture component , where the weights are calculated in the expectation-step.

Usage

weightedSestimator(
  Y,
  mu_init,
  sigma_init,
  max_iterFP = 1,
  weights,
  fixed_alpha
)

Arguments

Y

A matrix of size n x p.
mu_init

The previously computed center: an numerical array of length p.
sigma_init

The previously computed scatter matrix: an array of numeric values p x p
max_iterFP

the maximum number of fixed point iterations used for the algorithm, defaults to 1
weights

The weights that contain the probability membership of each observation (related to the overall mixture components)
fixed_alpha

the fixed alpha value for the corresponding mixture component

Value

A list including the estimated K centers and labels for the observations list(cov=matrixSigma,covAux1=covAux1,mu=muk,s=sk)

  • cov:the computed weithted scatter matrix

  • mu: the computed weithted center

  • s: the weighted scale factor s.

weightW

Description

Weight function ktaucenters

Usage

weightW(arg, p)

Arguments

arg

An 1-D array containing the distances.
p

the dimension of the element

Value

an array of the same size of arg with the value of the weights