Package 'EMMIXSSL'

Classifier based on Bayes rule

Description

A classifier based on Bayes rule, that is maximum a posterior probabilities of class membership

Usage

Classifier_Bayes(dat, n, p, g, pi, mu, sigma, ncov = 2)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
n	Number of observations.
p	Dimension of observation vecor.
g	Number of classes.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.

Details

The posterior probability can be expressed as

$\tau_i(y_j;\theta)=Prob\{z_{ij}=1|y_j\}=\frac{\pi_i\phi(y_j;\mu_i,\Sigma_i)}{\sum_{h=1}^g\pi_h\phi(y_j;\mu_h,\Sigma_h) },$

where $\phi$ is a normal probability function with mean $\mu_i$ and covariance matrix $\Sigma_i$, and $z_{ij}$ is is a zero-one indicator variable denoting the class of origin. The Bayes' Classifier of allocation assigns an entity with feature vector $y_j$ to Class $C_k$ if

$k= arg max_i \tau_i(y_j;\theta).$

Value

cluster

A vector of the class membership.

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)
cluster<-Classifier_Bayes(dat=dat$Y,n=150,p=3,g=4,mu=mu,sigma=sigma,pi=pi,ncov=2)

---

Transform a variance matrix into a vector

Description

Transform a variance matrix into a vector i.e., Sigma=R^T*R

Usage

cov2vec(sigma)

Arguments

sigma

A variance matrix

Details

The variance matrix is decomposed by computing the Choleski factorization of a real symmetric positive-definite square matrix. Then, storing the upper triangular factor of the Choleski decomposition into a vector.

Value

par A vector representing a variance matrix

---

Discriminant function

Description

Discriminant function in the particular case of g=2 classes with an equal-covariance matrix

Usage

discriminant_beta(pi, mu, sigma)

Arguments

pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.

Details

Discriminant function in the particular case of g=2 classes with an equal-covariance matrix can be expressed

$d(y_i,\beta)=\beta_0+\beta_1 y_i,$

where $\beta_0=\log\frac{\pi_1}{\pi_2}-\frac{1}{2}\frac{\mu_1^2-\mu_2^2}{\sigma^2}$ and $\beta_1=\frac{\mu_1-\mu_2}{\sigma^2}$.

Value

beta0	An intercept of discriminant function
beta	A coefficient of discriminant function

---

Fitting Gaussian mixture models

Description

Fitting Gaussian mixture model to a complete classified dataset or a incomplete classified dataset with/without the missing-data mechanism.

Usage

EMMIXSSL(
  dat,
  zm,
  pi,
  mu,
  sigma,
  ncov,
  xi = NULL,
  type,
  iter.max = 500,
  eval.max = 500,
  rel.tol = 1e-06,
  sing.tol = 1e-20
)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
zm	An n-dimensional vector containing the class labels including the missing-label denoted as NA.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.
xi	A 2-dimensional vector containing the initial values of the coefficients in the logistic function of the Shannon entropy.
type	Three types of Gaussian mixture models, 'ign' indicates fitting the model to a partially classified sample on the basis of the likelihood that ignores the missing label mechanism, 'full' indicates fitting the model to a partially classified sample on the basis of the full likelihood, taking into account the missing-label mechanism, and 'com' indicate fitting the model to a completed classified sample.
iter.max	Maximum number of iterations allowed. Defaults to 500
eval.max	Maximum number of evaluations of the objective function allowed. Defaults to 500
rel.tol	Relative tolerance. Defaults to 1e-15
sing.tol	Singular convergence tolerance; defaults to 1e-20.

Value

objective	Value of objective likelihood
convergence	Value of convergence
iteration	Number of iteration
pi	Estimated vector of the mixing proportions.
mu	Estimated matrix of the location parameters.
sigma	Estimated covariance matrix
xi	Estimated coefficient vector for a logistic function of the Shannon entropy

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)
xi<-c(-0.5,1)
m<-rlabel(dat=dat$Y,pi=pi,mu=mu,sigma=sigma,xi=xi,ncov=2)
zm<-dat$clust
zm[m==1]<-NA
inits<-initialvalue(g=4,zm=zm,dat=dat$Y,ncov=2)
## Not run: 
fit_pc<-EMMIXSSL(dat=dat$Y,zm=zm,pi=inits$pi,mu=inits$mu,sigma=inits$sigma,xi=xi,type='full',ncov=2)

## End(Not run)

---

Error rate of the Bayes rule for two-class Gaussian homoscedastic model

Description

The optimal error rate of Bayes rule for two-class Gaussian homoscedastic model

Usage

errorrate(beta0, beta, pi, mu, sigma)

Arguments

beta0	An $n\times p$ matrix where each row represents an individual observation
beta	Number of observations.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.

Details

The optimal error rate of Bayes rule for two-class Gaussian homoscedastic model can be expressed as

$err(y_j;\theta)=\pi_1\phi\{-\frac{\beta_0+\beta_1^T\mu_1}{(\beta_1^T\Sigma\beta_1)^{\frac{1}{2}}}\}+\pi_2\phi\{\frac{\beta_0+\beta_1^T\mu_2}{(\beta_1^T\Sigma\beta_1)^{\frac{1}{2}}}\}$

where $\phi$ is a normal probability function with mean $\mu_i$ and covariance matrix $\Sigma_i$.

Value

errval

A vector of error rate.

---

Gastrointestinal binary labels

Description

A panel of seven endoscopists viewed the videos and determined which patient needs resection (malignant) or no-resection (benign).

References

http://www.depeca.uah.es/colonoscopy_dataset/

---

Gastrointestinal trinary labels

Description

Gastrointestinal trinary ground truth (Adenoma, Serrated, and Hyperplastic)

References

http://www.depeca.uah.es/colonoscopy_dataset/

---

Gastrointestinal dataset

Description

The collected dataset is composed of 76 colonoscopic videos (recorded with both White Light (WL) and Narrow Band Imaging (NBI)), the histology (classification ground truth), and the endoscopist's opinion (including 4 experts and 3 beginners). There are $n=76$ observations, and each observation consists of 698 features extracted from colonoscopic videos on patients with gastrointestinal lesions.

References

http://www.depeca.uah.es/colonoscopy_dataset/

---

Posterior probability

Description

Get posterior probabilities of class membership

Usage

get_clusterprobs(dat, n, p, g, pi, mu, sigma, ncov = 2)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
n	Number of observations.
p	Dimension of observation vecor.
g	Number of multivariate normal classes.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.

Details

The posterior probability can be expressed as

$\tau_i(y_j;\theta)=Prob\{z_{ij}=1|y_j\}=\frac{\pi_i\phi(y_j;\mu_i,\Sigma_i)}{\sum_{h=1}^g\pi_h\phi(y_j;\mu_h,\Sigma_h) },$

where $\phi$ is a normal probability function with mean $\mu_i$ and covariance matrix $\Sigma_i$, and $z_{ij}$ is is a zero-one indicator variable denoting the class of origin.

Value

clusprobs

Posterior probabilities of class membership for the ith entity

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)
tau<-get_clusterprobs(dat=dat$Y,n=150,p=3,g=4,mu=mu,sigma=sigma,pi=pi,ncov=2)

---

Shannon entropy

Description

Shannon entropy

Usage

get_entropy(dat, n, p, g, pi, mu, sigma, ncov = 2)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
n	Number of observations.
p	Dimension of observation vecor.
g	Number of multivariate normal classes.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.

Details

The concept of information entropy was introduced by shannon1948mathematical. The entropy of $y_j$ is formally defined as

$e_j( y_j; \theta)=-\sum_{i=1}^g \tau_i( y_j; \theta) \log\tau_i(y_j;\theta).$

Value

clusprobs

The posterior probabilities of the i-th entity that belongs to the j-th group.

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)
en<-get_entropy(dat=dat$Y,n=150,p=3,g=4,mu=mu,sigma=sigma,pi=pi,ncov=2)

---

Initial values for ECM

Description

Inittial values for claculating the estimates based on solely on the classified features.

Usage

initialvalue(dat, zm, g, ncov = 2)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
zm	An n-dimensional vector containing the class labels including the missing-label denoted as NA.
g	Number of multivariate normal classes.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.

Value

pi	A g-dimensional initial vector of the mixing proportions.
mu	A initial $p \times g$ matrix of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)
xi<-c(-0.5,1)
m<-rlabel(dat=dat$Y,pi=pi,mu=mu,sigma=sigma,xi=xi,ncov=2)
zm<-dat$clust
zm[m==1]<-NA
inits<-initialvalue(g=4,zm=zm,dat=dat$Y,ncov=2)

---

Transfer a list into a vector

Description

Transfer a list into a vector

Usage

list2par(
  p,
  g,
  pi,
  mu,
  sigma,
  ncov = 2,
  xi = NULL,
  type = c("ign", "full", "com")
)

Arguments

p	Dimension of observation vecor.
g	Number of multivariate normal classes.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.
xi	A 2-dimensional vector containing the initial values of the coefficients in the logistic function of the Shannon entropy.
type	Three types to fit to the model, 'ign' indicates fitting the model on the basis of the likelihood that ignores the missing label mechanism, 'full' indicates that the model to be fitted on the basis of the full likelihood, taking into account the missing-label mechanism, and 'com' indicate that the model to be fitted to a completed classified sample.

Value

par	a vector including all list information

---

Full log-likelihood function

Description

Full log-likelihood function with both terms of ignoring and missing

Usage

loglk_full(dat, zm, pi, mu, sigma, ncov = 2, xi)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
zm	An n-dimensional vector containing the class labels including the missing-label denoted as NA.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.
xi	A 2-dimensional vector containing the initial values of the coefficients in the logistic function of the Shannon entropy.

Details

The full log-likelihood function can be expressed as

$\log L_{PC}^{({full})}(\boldsymbol{\Psi})=\log L_{PC}^{({ig})}(\theta)+\log L_{PC}^{({miss})}(\theta,\boldsymbol{\xi}),$

where$\log L_{PC}^{({ig})}(\theta)$is the log likelihood function formed ignoring the missing in the label of the unclassified features, and $\log L_{PC}^{({miss})}(\theta,\boldsymbol{\xi})$ is the log likelihood function formed on the basis of the missing-label indicator.

Value

lk	Log-likelihood value

---

Log likelihood for partially classified data with ingoring the missing mechanism

Description

Log likelihood for partially classified data with ingoring the missing mechanism

Usage

loglk_ig(dat, zm, pi, mu, sigma, ncov = 2)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
zm	An n-dimensional vector containing the class labels including the missing-label denoted as NA.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.

Details

The log-likelihood function for partially classified data with ingoring the missing mechanism can be expressed as

$\log L_{PC}^{({ig})}(\theta)=\sum_{j=1}^n \left[ (1-m_j)\sum_{i=1}^g z_{ij}\left\lbrace \log\pi_i+\log f_i(y_j;\omega_i)\right\rbrace +m_j\log \left\lbrace \sum_{i=1}^g\pi_i f_i(y_j;\omega_i)\right\rbrace \right],$

where $m_j$ is a missing label indicator, $z_{ij}$ is a zero-one indicator variable defining the known group of origin of each, and $f_i(y_j;\omega_i)$ is a probability density function with parameters $\omega_i$.

Value

lk	Log-likelihood value.

---

Log likelihood function formed on the basis of the missing-label indicator

Description

Log likelihood for partially classified data based on the missing mechanism with the Shanon entropy

Usage

loglk_miss(dat, zm, pi, mu, sigma, ncov = 2, xi)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
zm	An n-dimensional vector containing the class labels including the missing-label denoted as NA.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.
xi	A 2-dimensional vector containing the initial values of the coefficients in the logistic function of the Shannon entropy.

Details

The log-likelihood function formed on the basis of the missing-label indicator can be expressed by

$\log L_{PC}^{({miss})}(\theta,\boldsymbol{\xi})=\sum_{j=1}^n\big[ (1-m_j)\log\left\lbrace 1-q(y_j;\theta,\boldsymbol{\xi})\right\rbrace +m_j\log q(y_j;\theta,\boldsymbol{\xi})\big],$

where $q(y_j;\theta,\boldsymbol{\xi})$ is a logistic function of the Shannon entropy $e_j(y_j;\theta)$, and $m_j$ is a missing label indicator.

Value

lk	loglikelihood value

---

log summation of exponential function

Description

log summation of exponential variable vector.

Usage

logsumexp(x)

Arguments

x	A variable vector.

Value

val	log summation of exponential variable vector.

---

Label matrix

Description

Convert class indicator into a label maxtrix.

Usage

makelabelmatrix(clust)

Arguments

clust

An n-dimensional vector of class partition.

Value

Z	A matrix of class indicator.

Examples

cluster<-c(1,1,2,2,3,3)
label_maxtrix<-makelabelmatrix(cluster)

---

Negative objective function for EMMIXSSL

Description

Negative objective function for EMMIXSSL

Usage

neg_objective_function(
  dat,
  zm,
  g,
  par,
  ncov = 2,
  type = c("ign", "full", "com")
)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation
zm	An n-dimensional vector of group partition including the missing-label, denoted as NA.
g	Number of multivariate Gaussian groups.
par	An informative vector including mu, pi,sigma and xi.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.
type	Three types to fit to the model, 'ign' indicates fitting the model on the basis of the likelihood that ignores the missing label mechanism, 'full' indicates that the model to be fitted on the basis of the full likelihood, taking into account the missing-label mechanism, and 'com' indicate that the model to be fitted to a completed classified sample.

Value

val	Value of negatvie objective function.

---

Normalize log-probability

Description

Normalize log-probability.

Usage

normalise_logprob(x)

Arguments

x	A variable vector.

Value

val	A normalize log probability of variable vector.

---

Transfer a vector into a list

Description

Transfer a vector into a list

Usage

par2list(par, g, p, ncov = 2, type = c("ign", "full"))

Arguments

par	A vector with list information.
g	Number of multivariate normal classes.
p	Dimension of observation vecor.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix that sigma is a $p\times p$ matrix. ncov = 2 for the unequal covariance/scale matrices that sigma represents a list of g matrices with dimension $p\times p \times g$.
type	Three types to fit to the model, 'ign' indicates fitting the model on the basis of the likelihood that ignores the missing label mechanism, 'full' indicates that the model to be fitted on the basis of the full likelihood, taking into account the missing-label mechanism, and 'com' indicate that the model to be fitted to a completed classified sample.

Value

parlist

Return a list including mu, pi, sigma and xi.

---

Transfer a probability vector into a vector

Description

Transfer a probability vector into an informative vector

Usage

pro2vec(pro)

Arguments

pro	An propability vector

Value

y An informative vector

---

Generation of a missing-data indicator

Description

Generate the missing label indicator

Usage

rlabel(dat, pi, mu, sigma, ncov = 2, xi)

Arguments

dat	An $n\times p$ matrix where each row represents an individual observation.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.
xi	A 2-dimensional coefficient vector for a logistic function of the Shannon entropy.

Value

m	A n-dimensional vector of missing label indicator. The element of outputs m represents its label indicator is missing if m equals 1, otherwise its label indicator is available if m equals to 0.

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)
xi<-c(-0.5,1)
m<-rlabel(dat=dat$Y,pi=pi,mu=mu,sigma=sigma,xi=xi,ncov=2)

---

Normal mixture model generator.

Description

Generate random observations from the normal mixture distributions.

Usage

rmix(n, pi, mu, sigma, ncov = 2)

Arguments

n	Number of observations.
pi	A g-dimensional vector for the initial values of the mixing proportions.
mu	A $p \times g$ matrix for the initial values of the location parameters.
sigma	A $p\times p$ covariance matrix if ncov=1, or a list of g covariance matrices with dimension $p\times p \times g$ if ncov=2.
ncov	Options of structure of sigma matrix; the default value is 2; ncov = 1 for a common covariance matrix; ncov = 2 for the unequal covariance/scale matrices.

Value

Y	An $n\times p$ numeric matrix with samples drawn in rows.
Z	An $n\times g$ numeric matrix; each row represents zero-one indicator variables defining the known class of origin of each.
clust	An n-dimensional vector of class partition.

Examples

n<-150
pi<-c(0.25,0.25,0.25,0.25)
sigma<-array(0,dim=c(3,3,4))
sigma[,,1]<-diag(1,3)
sigma[,,2]<-diag(2,3)
sigma[,,3]<-diag(3,3)
sigma[,,4]<-diag(4,3)
mu<-matrix(c(0.2,0.3,0.4,0.2,0.7,0.6,0.1,0.7,1.6,0.2,1.7,0.6),3,4)
dat<-rmix(n=n,pi=pi,mu=mu,sigma=sigma,ncov=2)

---

Transform a vector into a matrix

Description

Transform a vector into a matrix i.e., Sigma=R^T*R

Usage

vec2cov(par)

Arguments

par	A vector representing a variance matrix

Details

The variance matrix is decomposed by computing the Choleski factorization of a real symmetric positive-definite square matrix. Then, storing the upper triangular factor of the Choleski decomposition into a vector.

Value

sigma A variance matrix

---

Transfer an informative vector to a probability vector

Description

Transfer an informative vector to a probability vector

Usage

vec2pro(vec)

Arguments

vec	An informative vector

Value

pro A probability vector

---