Conditional Location Vector, Scatter Matrix, and Degrees of Freedom of Multivariate t Distribution

Description

These functions provide the conditional location vector, scatter matrix, and degrees of freedom of [Y given X], where Z = (X,Y) is the fully-joint multivariate t distribution with location vector equal to mean, scatter matrix sigma, and degrees of freedom df. For more details on the computation of the parameters and their respective formulae, see Roth (2013).

Usage

CondMVT(mean, sigma, df, dependent.ind, given.ind, X.given,
  check.sigma = TRUE)

Arguments

Value

Returns the conditional location vector (condMean), conditional scatter matrix (condVar), and the conditional degrees of freedom (cond_df) for the multvariate t distribution.

References

Roth, M. (2013). On the multivariate t-distribution, Tech Rep.

Examples

 # 10-dimensional multivariate normal distribution
 n <- 10
 df=3
 A <- matrix(rt(n^2,df), n, n)
 A <- tcrossprod(A,A) #A %*% t(A)
 CondMVT(mean=rep(1,n), sigma=A, df=df, dependent=c(2,3,5), given=c(1,4,7,9),X.given=c(1,1,0,-1))
 CondMVT(mean=rep(1,n), sigma=A, df=df, dep=3, given=c(1,4,7,9), X=c(1,1,0,-1))

Data Imputation Using EM (Multiple Iterations, Degrees of Freedom Unknown)

Description

This sub-package constitutes the subroutines for EM algorithm (for multiple iterations). It has 2 functions namely LIKE and EM_Umsteps. The function EM_Umsteps carries out missing data imputation as well as parameter estimation in multivariate t distribution in multiple iterations; assuming that the degrees of freedom are unknown. In addition to updating the location vector and the scatter matrix, therefore, the function also finds an estimate for the degrees of freedom. The bisection method is employed in the algorithm to iteratively update the degrees of freedom.The function LIKE (specifying the likelihood) facilitates the setting of tolerance level for convergence of the EM 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).Details of how EM works in light of unknown degrees of freedom can be found in Kinyanjui et al. (2020) and Liu and Rubin (1995).

Usage

EM_Umsteps(Y,mu,Sigma,df,K,e,error)

Arguments

Value

Completed dataset, updated location vector,scatter matrix, and degrees of freedom. 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.

Liu, C. and Rubin, D. B. (1995). ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statistica Sinica, 19-39.

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 (EM)
EMU1=EM_Uonestep (Y=Y8,mu=mu,Sigma= Sigma_stat,df= df_stat,e=0.00001)

#Multiple Iterations (EM)
EMU=EM_Umsteps(Y=Y8,mu=mu_stat,Sigma=Sigma_stat,df=df_stat,K=1000,e=0.00001,error=0.00001)

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

Data Imputation Using EM (Single Iteration, Degrees of Freedom Unknown)

Description

This sub-package constitutes the subroutines for EM algorithm (for a single iteration). It has 4 functions namely fun1, dfun1, Bisec, and EM_Uonestep. The function EM_Uonestep carries out missing data imputation as well as parameter estimation in multivariate t distribution in one iteration; assuming that the degrees of freedom are unknown. In addition to updating the location vector and the scatter matrix, therefore, the function also finds an estimate for the degrees of freedom. The bisection method is employed in the algorithm to iteratively update the degrees of freedom. In this respect, function fun1 specifies the degrees of freedom equation to be solved. dfun1 is its derivative. The two functions (fun1 and dfun1) are then solved numerically using the bisection method as specified in the function Bisec.Details of how EM works in light of unknown degrees of freedom can be found in Kinyanjui et al. (2020) and Liu and Rubin (1995).

Usage

EM_Uonestep(Y,mu,Sigma,df,e)

Arguments

Value

Completed dataset, updated location vector,scatter matrix, and degrees of freedom. 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.

Liu, C., & Rubin, D. B. (1995). ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statistica Sinica, 19-39.

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 (EM)
EMU1=EM_Uonestep (Y=Y8,mu=mu,Sigma= Sigma_stat,df= df_stat,e=0.00001)

#Results for Newly Completed Dataset (EM)
EMU1$Y2    #Newly completed Dataset (with imputed values)
EMU1$mu	   #updated location vector
EMU1$Sigma #updated scatter matrix
EMU1$df

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

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

Data Imputation Using EM (Single Iteration; 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 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.Arbitrary starting values are supplied to initiate the algorithm.

Usage

EM_onestep(Y,mu,Sigma,df)

Arguments

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)

#Results for Newly Completed Dataset (EM)
EM1$Y2    #Newly completed Dataset (with imputed values)
EM1$mu	  #updated location vector
EM1$Sigma #updated scatter matrix

Creating Missing Values at Random in Multivariate Datasets

Description

This function randomly creates missing values in a multivariate dataset. The resultant missing data mechanism is missing at random (MAR). The percentage of missingness has to be specified. This percentage is computed as a proportion of the sample size. In addition, the function allows for more than one missing value in any given case. It is set such that in a p-variate dataset, for any i^{th} case, the maximum allowable number of missing values is p-1. This helps avoid a situation where a case has no observed value.

Usage

MISS (TT, Percent)

Arguments

Value

Data Y of size n×p with missing values (NA) created at random. The missing values are logical in nature.

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  

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

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

Data Imputation Using SEM and MCEM (Single Iteration; 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. It has 4 functions namely fun1, dfun1, Bisec, and SMCEM_Uonestep. The functions fun1 and dfun1 in the sub-package constitute the equation for the degrees of freedom and its derivative respectively. The Bisec function contains the bisection method subroutines to facilitate the iterative estimation of the degrees of freedom using fun1 and dfun1. The function SMCEM_Uonestep constitute the SEM and MCEM algorithms for single-iteration 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.Details of how SEM and MCEM impute missing values and estimate parameters in multivariate t context (unknown degrees of freedom) are explained by Kinyanjui et al. (2020).

Usage

SMCEM_Uonestep(Y,mu,Sigma,df,nob,e)

Arguments

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=0.00001)

#Single Iteration (MCEM)
MCEMU1=SMCEM_Uonestep(Y=Y8,mu=mu,Sigma=Sigma_stat,df= df_stat,nob=1000,e=0.00001)

#Results for Newly Completed Dataset (SEM)
SEMU1$Y2     #Newly completed Dataset (with imputed values)
SEMU1$mu	   #updated location vector
SEMU1$Sigma  #updated scatter matrix

#Results for Newly Completed Dataset (MCEM)
MCEMU1$Y2     #Newly completed Dataset (with imputed values)
MCEMU1$mu	    #updated location vector
MCEMU1$Sigma  #updated scatter matrix
MCEMU1$df     #updated degrees of freedom

Data Imputation Using SEM and MCEM (Multiple Iterations, Degrees of Freedom Known)

Description

This sub-package contains the subroutines for iterative imputation of missing values as well as parameter estimation (for the location vector and the scatter matrix) in multivariate t distribution using Stochastic EM (SEM) and Monte Carlo EM (MCEM). In this case, the degrees of freedom for the distribution are known or fixed a priori. SEM is implemented when the analyst specifies a single draw in the E-step. In case we have multiple draws in the E-step, the algorithm changes to MCEM. In both algorithms, the function SMCEM_onestep is run when we are only interested in the imputed values and the parameter updates in a single iteration. The function SMCEM_msteps is run when we are interested in multiple iterations (this is usually the case). Essentially, the first iterations (for instance, 10 percent of all iterations) is usually burnt-in in order to ward off the effects of initial values. Details of how SEM and MCEM operate can be found in among others Kinyanjui et al. (2021), Nielsen (2000), Levine and Casella (2001) Jank (2005) and Karimi et al. (2019).

Usage

SMCEM_msteps(Y,mu,Sigma,df, nob,K)

Arguments

Value

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

References

Karimi, B., Lavielle, M., and Moulines, É. (2019). On the Convergence Properties of the Mini-Batch EM and MCEM Algorithms.

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.

Levine, R. A. and Casella, G. (2001). Implementations of the Monte Carlo EM algorithm. Journal of Computational and Graphical Statistics, 10(3), 422-439.

Nielsen, S.F. (2000). The stochastic EM algorithm: estimation and asymptotic results. Bernoulli, 6(3), 457-489.

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 (SEM)
SEM1=SMCEM_onestep(Y=Y8,mu= mu_stat,Sigma=Sigma_stat,df=df,nob=1)

#Single Iteration (MCEM)
MCEM1=SMCEM_onestep(Y=Y8,mu= mu_stat,Sigma=Sigma_stat,df=df,nob=100)

#Multiple Iterations (SEM)
SEM=SMCEM_msteps(Y=Y8,mu= mu_stat,Sigma= Sigma_stat,df=df,nob=1,K=500)

#Results for Newly Completed Dataset (Burning in first 50 iterations in SEM)
T_mu=rep(0,3)
T_Sigma=matrix(rep(0,3*3),nrow=3)
T_Data=matrix(rep(0,3*10), nrow =10)
for (l in 51:500){
  T_mu = T_mu + SEM$muchain[l,]
  T_Sigma = T_Sigma + SEM$SigmaChain[,,l]
  T_Data= T_Data+ SEM$YChain[,,l]
}
#updated location vector
round((T_mu/450),4) 
#updated scatter matrix  
round((T_Sigma/450),4)
#complete dataset as an average of (K-50) complete datasets for the various iterations.  
T_Data1=  T_Data/450	
T_Data1

#Multiple Iterations (MCEM)
MCEM=SMCEM_msteps(Y=Y8,mu=mu_stat,Sigma=Sigma_stat,df=df,nob=100,
K=500)

#Results for Newly Completed Dataset (Burning in first 50 iterations in MCEM)
T_mu=rep(0,3)
T_Sigma=matrix(rep(0,3*3),nrow=3)
T_Data=matrix(rep(0,3*10), nrow =10)
for (l in 51:500){
  T_mu = T_mu + MCEM$muchain[l,]
  T_Sigma = T_Sigma + MCEM$SigmaChain[,,l]
  T_Data= T_Data+ MCEM$YChain[,,l]
}
#updated location vector
round((T_mu/450),4) 
#updated scatter matrix
round((T_Sigma/450),4) 
#complete dataset as an average of (K-50) complete datasets for the various iterations.
T_Data1= T_Data/450 
T_Data1                    

Data Imputation Using SEM and MCEM (Single Iteration, Degrees of Freedom Known)

Description

This sub-package contains the subroutines for iterative imputation of missing values as well as parameter estimation (for the location vector and the scatter matrix) in multivariate t distribution using Stochastic EM (SEM) and Monte Carlo EM (MCEM). In this case, the degrees of freedom for the distribution are known or fixed a priori. SEM is implemented when the analyst specifies a single draw in the E-step. In case we have multiple draws in the E-step, the algorithm changes to MCEM. In both algorithms, the function SMCEM_onestep is run when we are only interested in the imputed values and the parameter updates in a single iteration.

Usage

SMCEM_onestep(Y,mu,Sigma,df,nob)

Arguments

Value

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

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.

#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 (SEM)
SEM1=SMCEM_onestep(Y=Y8,mu= mu_stat,Sigma=A,df=df,nob=1)

#Single Iteration (MCEM)
MCEM1=SMCEM_onestep(Y=Y8,mu= mu_stat,Sigma=A,df=df,nob=100)

#Results for Newly Completed Dataset (SEM)
SEM1$Y2    #Newly completed Dataset (with imputed values)
SEM1$mu	   #updated location vector
SEM1$Sigma #updated scatter matrix

#Results for Newly Completed Dataset (MCEM)
MCEM1$Y2    #Newly completed Dataset (with imputed values)
MCEM1$mu	   #updated location vector
MCEM1$Sigma #updated scatter matrix

Conditional Multivariate t Density and Random Deviates

Description

This function provides the density function for the conditional multivariate t distribution, [Y given X], where Z = (X,Y) is the fully-joint multivariate t distribution with location vector (or mode) equal to mean and covariance matrix sigma.

Usage

dcmvt(x, mean, sigma,df, dependent.ind, given.ind, X.given, check.sigma=TRUE, log = FALSE)

Arguments

Value

numeric

References

Genz, A. and Bretz, F. (2009), Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg.

S. Kotz and S. Nadarajah (2004), Multivariate t Distributions and Their Applications. Cambridge University Press. Cambridge.

Examples

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

# density of Z[c(2,5)] given Z[c(1,4,7,9)]=c(1,1,0,-1)
dcmvt(x=c(1.2,-1), mean=rep(1,n), sigma=A,dependent.ind=c(2,5),df=df, given.ind=c(1,4,7,9),
X.given=c(1,1,0,-1))

dcmvt(x=-1, mean=rep(1,n), sigma=A,df=df, dep=3, given=c(1,4,7,9,10), X=c(1,1,0,0,-1))
dcmvt(x=c(1.2,-1), mean=rep(1,n), sigma=A,df=df, dep=c(2,5))

# gives an error since `x' and `dep' are incompatibe
#dcmvt(x=-1, mean=rep(1,n), sigma=A,df=df, dep=c(2,3),
#      given=c(1,4,7,9,10), X=c(1,1,0,0,-1))

rcmvt(n=10, mean=rep(1,n), sigma=A,df=df, dep=c(2,5),
         given=c(1,4,7,9,10), X=c(1,1,0,0,-1),type="shifted",
         method="eigen")

rcmvt(n=10, mean=rep(1,n), sigma=A,df=df, dep=3,
         given=c(1,4,7,9,10), X=c(1,1,0,0,-1),type="Kshirsagar",
         method="chol")

Conditional Multivariate t Distribution

Description

Computes the distribution function of the conditional multivariate t, [Y given X], where Z = (X,Y) is the fully-joint multivariate t distribution with mean equal to location vector, df equal to degrees of freedom and scatter matrix sigma. Computations are based on algorithms by Genz and Bretz.

Usage

pcmvt(lower = -Inf, upper = Inf, mean, sigma, df, dependent.ind, given.ind, X.given,
check.sigma = TRUE, algorithm = GenzBretz(), ...)

Arguments

Details

This program involves the computation of multivariate t probabilities with arbitrary correlation matrices.

Value

The evaluated distribution function is returned with attributes

References

Genz, A. and Bretz, F. (1999), Numerical computation of multivariate t-probabilities with application to power calculation of multiple contrasts. Journal of Statistical Computation and Simulation, 63, 361–378.

Genz, A. and Bretz, F. (2002), Methods for the computation of multivariate t-probabilities. Journal of Computational and Graphical Statistics, 11, 950–971.

Genz, A. (2004), Numerical computation of rectangular bivariate and trivariate normal and t-probabilities, Statistics and Computing, 14, 251–260.

Genz, A. and Bretz, F. (2009), Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195. Springer-Verlag, Heidelberg.

See Also

dcmvt(),rcmvt(),pmvt(),GenzBretz()

Examples

n <- 10
df=3
A <- matrix(rt(n^2,df), n, n)
A <- tcrossprod(A,A) #A %*% t(A)

pcmvt(lower=-Inf, upper=1, mean=rep(1,n), sigma=A, df=df,	dependent.ind=3, 
      given.ind=c(1,4,7,9,10), X.given=c(1,1,0,0,-1))


pcmvt(lower=-Inf, upper=c(1,2), mean=rep(1,n),
      sigma=A,df=df, dep=c(2,5), given=c(1,4,7,9,10),
      X=c(1,1,0,0,-1))

pcmvt(lower=-Inf, upper=c(1,2), mean=rep(1,n), sigma=A,df=df,
      dep=c(2,5))

Conditional Multivariate t Density and Random Deviates

Description

This function provides the random number generator for the conditional multivariate t distribution, [Y given X], where Z = (X,Y) is the fully-joint multivariate t distribution with location vector equal to mean and scatter matrix sigma.

Usage

rcmvt(n, mean, sigma, df,dependent.ind, given.ind, X.given,
check.sigma = TRUE,type = c("Kshirsagar", "shifted"),
method = c("eigen", "svd", "chol"))

Arguments

Value

A 'vector' of length n, equal to the length of 'mean'

Examples

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

# density of Z[c(2,5)] given Z[c(1,4,7,9)]=c(1,1,0,-1)
dcmvt(x=c(1.2,-1), mean=rep(1,n), sigma=A, df=df,
         dependent.ind=c(2,5), given.ind=c(1,4,7,9),
         X.given=c(1,1,0,-1))

dcmvt(x=-1, mean=rep(1,n), sigma=A,df=df, dep=3, given=c(1,4,7,9,10), X=c(1,1,0,0,-1))

dcmvt(x=c(1.2,-1), mean=rep(1,n), sigma=A,df=df, dep=c(2,5))

# gives an error since `x' and `dep' are incompatibe
#dcmvt(x=-1, mean=rep(1,n), sigma=A,df=df, dep=c(2,3),
#given=c(1,4,7,9,10), X=c(1,1,0,0,-1))

rcmvt(n=10, mean=rep(1,n), sigma=A,df=df, dep=c(2,5),
         given=c(1,4,7,9,10), X=c(1,1,0,0,-1),type="shifted",
         method="eigen")

rcmvt(n=10, mean=rep(1,n), sigma=A,df=df, dep=3,
         given=c(1,4,7,9,10), X=c(1,1,0,0,-1),type="Kshirsagar",
         method="chol")