A-optimality criteria based subsampling under Gaussian Linear Models

Using this function sample from big data under Gaussian linear regression models to describe the data. Subsampling probabilities are obtained based on the A-optimality criteria.

Usage

AoptimalGauLMSub(r1,r2,Y,X,N)

Arguments

r1: sample size for initial random sampling
r2: sample size for optimal sampling
Y: response data or Y
X: covariate data or X matrix that has all the covariates (first column is for the intercept)
N: size of the big data

Value

The output of AoptimalGauLMSub gives a list of

Beta_Estimates estimated model parameters in a data.frame after subsampling

Variance_Epsilon_Estimates matrix of estimated variance for epsilon in a data.frame after subsampling

Sample_A-Optimality list of indexes for the initial and optimal samples obtained based on A-Optimality criteria

Subsampling_Probability matrix of calculated subsampling probabilities for A-optimality criteria

Details

Two stage subsampling algorithm for big data under Gaussian Linear Model.

First stage is to obtain a random sample of size \(r_1\) and estimate the model parameters. Using the estimated parameters subsampling probabilities are evaluated for A-optimality criteria.

Through the estimated subsampling probabilities an optimal sample of size \(r_2 \ge r_1\) is obtained. Finally, the two samples are combined and the model parameters are estimated.

NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further.

If \(r_2 \ge r_1\) is not satisfied then an error message will be produced.

If the big data \(X,Y\) has any missing values then an error message will be produced.

The big data size \(N\) is compared with the sizes of \(X,Y\) and if they are not aligned an error message will be produced.

References

Lee J, Schifano ED, Wang H (2021). “Fast optimal subsampling probability approximation for generalized linear models.” Econometrics and Statistics.

Examples

Dist<-"Normal"; Dist_Par<-list(Mean=0,Variance=1,Error_Variance=0.5)
No_Of_Var<-2; Beta<-c(-1,2,1); N<-10000; Family<-"linear"
Full_Data<-GenGLMdata(Dist,Dist_Par,No_Of_Var,Beta,N,Family)

r1<-300; r2<-rep(100*c(6,12),50); Original_Data<-Full_Data$Complete_Data;

AoptimalGauLMSub(r1 = r1, r2 = r2,Y = as.matrix(Original_Data[,1]),
                 X = as.matrix(Original_Data[,-1]),
                 N = nrow(Original_Data))->Results
#> Step 1 of the algorithm completed.
#> Step 2 of the algorithm completed.

plot_Beta(Results)
#> Picking joint bandwidth of 0.00601
#> Picking joint bandwidth of 0.00517
#> Picking joint bandwidth of 0.00529