SMCEM_Umsteps {CondMVT}

Data Imputation Using SEM and MCEM (Multiple Iterations; Degrees of Freedom Unknown)

Description

This sub-package provides subroutines for implementation of SEM and MCEM techniques in imputing missing values as well as estimating multivariate t parameters when the degrees of freedom are unknown.The functions SMCEM_msteps constitute the SEM and MCEM algorithms for multiple-iterative data imputation and parameter estimation for multivariate t data with unknown degrees of freedom. The functions represent SEM when the number of draws in the E-step (denoted by nob) is 1 and MCEM when we have more than one draw in the E-step.More details on the implementation of SEM and MCEM techniques can be found in Kinyanjui et al. (2020).

Usage

SMCEM_Umsteps(Y,mu,Sigma,df,nob,K,e)

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.

nob

number of draws in the E-step

K

the number of iterations, which must be specified.

e

tolerance level for convergence of the bisection method for estimation of df.

Value

Completed dataset, updated location vector,scatter matrix, and degrees of freedom when employing the SEM and MCEM algorithms. All outputs are numeric.

References

Kinyanjui, P. K., Tamba, C. L., Orawo, L. A. O., & Okenye, J. O. (2020). Missing data imputation in multivariate t distribution with unknown degrees of freedom using expectation maximization algorithm and its stochastic variants. Model Assisted Statistics and Applications, 15(3), 263-272.

Examples

# 3-dimensional multivariate t distribution
n <- 25
p=3
df=3
mu=c(10,20,30)
A=matrix(c(14,10,12,10,13,9,12,9,18), 3,3)
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)
df_stat=6

#Imputing Missing Values and Updating Parameter Estimates

#Single Iteration (SEM)
SEMU1=SMCEM_Uonestep(Y=Y8,mu=mu,Sigma=Sigma_stat,df= df_stat,nob=1,e=0.0001)

#Single Iteration (MCEM)
MCEMU1=SMCEM_Uonestep(Y=Y8,mu=mu,Sigma=Sigma_stat,df= df_stat,nob=50,e=0.0001)

#Multiple Iterations (SEM)
SEMU=SMCEM_Umsteps(Y=Y8,mu=mu_stat,Sigma=Sigma_stat,df=df_stat,nob=1,K=100,e=0.0001)

#Results for Newly Completed Dataset (Burning in first 10 iterations in SEM)
T_mu=rep(0,3)
T_Sigma=matrix(rep(0,3*3),nrow=3)
T_Data=matrix(rep(0,3*25), nrow =25)
T_df=rep()
for (l in 11:100){
 T_mu = T_mu + SEMU$muchain[l,]
  T_Sigma = T_Sigma + SEMU$SigmaChain[,,l]
 T_Data= T_Data+ SEMU$YChain[,,l]
}
#updated location vector
round((T_mu/90),4)

#updated scatter matrix 
round((T_Sigma/90),4)

#updated degrees of freedom 
udfs=mean(SEMU$dfchain[11:100])

#complete dataset as an average of (K-10) complete datasets for the various iterations.
T_Data1=  T_Data/90	

#Multiple Iterations (MCEM)
MCEMU=SMCEM_Umsteps(Y=Y8,mu=mu_stat,Sigma=Sigma_stat,df=df_stat,nob=50,K=100,e=0.0001)

#Results for Newly Completed Dataset (Burning in first 10 iterations in MCEM)
T_mu=rep(0,3)
T_Sigma=matrix(rep(0,3*3),nrow=3)
T_Data=matrix(rep(0,3*25), nrow =25)
T_df=rep()
for (l in 11:100){
  T_mu = T_mu + MCEMU$muchain[l,]
  T_Sigma = T_Sigma + MCEMU$SigmaChain[,,l]
  T_Data= T_Data+ MCEMU$YChain[,,l]
}
#updated location vector
round((T_mu/90),4)

#updated scatter matrix 
round((T_Sigma/90),4)

#updated degrees of freedom  
udf=mean(MCEMU$dfchain[11:100])
udf

#complete dataset as an average of (K-10) complete datasets for the various iterations.  
T_Data1=  T_Data/90
T_Data1