EM_msteps {CondMVT}

Data Imputation Using EM (Multiple Iterations; Degrees of Freedom Known)

Description

The sub-package contains subroutines for imputation of missing values as well as parameter estimation (for the location vector and the scatter matrix) in multivariate t distribution using the Expectation Maximization (EM) algorithm when the degrees of freedom are known. EM algorithm iteratively imputes the missing values and computes the estimates for the multivariate t parameters in two steps (E-step and M-step) as explained in Kinyanjui et al. (2021). For a single iteration, the function EM_onestep is run. For multiple iterations, the function EM_msteps is run. The function LIKE (specifying the likelihood) facilitates the setting of tolerance level for convergence of the algorithm (that is L(\theta^{t+1})-L(\theta^{t})\leq{\delta}, where \delta is a set tolerance level and t denotes the number of iterations).

Usage

EM_msteps(Y,mu,Sigma,df,K,error)

Arguments

Y

the multivariate t dataset

mu

the location vector, which must be specified. In cases where it is unknown, starting values are provided.

Sigma

scatter matrix, which must be specified. In cases where it is unknown, starting values are provided.

df

degrees of freedom, which must be specified.

K

the number of iterations, which must be specified.

error

tolerance level for convergence of the EM algorithm.

Value

Completed dataset (with imputed values), updated location vector, and scatter matrix. All outputs are numeric.

References

Kinyanjui, P.K., Tamba, C.L., & Okenye, J.O. (2021). Missing Data Imputation in a t -Distribution with Known Degrees of Freedom Via Expectation Maximization Algorithm and Its Stochastic Variants. International Journal of Applied Mathematics and Statistics.

Examples

# 3-dimensional multivariate t distribution
n <- 10
p=3
df=3
mu=c(1:3)
A <- matrix(rt(p^2,df), p, p)
A <- tcrossprod(A,A) #A %*% t(A)

Y7 <-mvtnorm::rmvt(n, delta=mu, sigma=A, df=df)
Y7
TT=Y7 #Complete Dataset

#Introduce MAR Data
Y8= MISS(TT,20) #The newly created incomplete dataset.
Y8

#Initializing Values
mu_stat=c(0.5,1,2)
Sigma_stat=matrix(c(0.33,0.31,0.3,0.31,0.335,0.295,0.3,0.295,0.32),3,3)

#Imputing Missing Values and Updating Parameter Estimates
#Single Iteration (EM)
EM1=EM_onestep(Y=Y8,mu=mu_stat,Sigma=Sigma_stat,df=df)

#Multiple Iterations (EM)
EM=EM_msteps(Y=Y8,mu=mu_stat,Sigma=Sigma_stat,df=3,K=1000,error=0.00001)

#Results for Newly Completed Dataset (EM)
EM$IMP   #Newly completed Dataset (with imputed values)
EM$mu	   #updated location vector
EM$Sigma #updated scatter matrix
EM$K1	#number of iterations the algorithm takes to converge