# Model robust optimal subsampling for A- and L- optimality criteria under linear regression Using this function sample from big data under linear regression when there are more than one model to describe the data. Subsampling probabilities are obtained based on the A- and L- optimality criteria. ```r modelRobustLinSub(r0,rf,Y,X,N,Apriori_probs,All_Combinations,All_Covariates) ``` ## Arguments - `r0`: sample size for initial random sample - `rf`: final sample size including initial(r0) and optimal(r) samples - `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 - `Apriori_probs`: vector of a priori model probabilities that are used to obtain the model robust subsampling probabilities - `All_Combinations`: list of possible models that can describe the data - `All_Covariates`: all the covariates in the models ## Returns The output of `modelRobustLinSub` gives a list of `Beta_Estimates` estimated model parameters for each model in a list after subsampling `Variance_Epsilon_Estimates` matrix of estimated variance for epsilon for each model after subsampling `Sample_A-Optimality` list of indexes for the initial and optimal samples obtained based on A-Optimality criteria `Sample_A-Optimality_MR` list of indexes for the initial and model robust optimal samples obtained based on A-Optimality criteria `Sample_L-Optimality` list of indexes for the initial and optimal samples obtained based on L-Optimality criteria `Sample_L-Optimality_MR` list of indexes for the initial and model robust optimal samples obtained based on L-Optimality criteria `Subsampling_Probability` matrix of calculated subsampling probabilities for A- and L- optimality criteria ## Details Two stage subsampling algorithm for big data under linear regression for multiple models that can describe the big data. First stage is to obtain a random sample of size $r_0$ and estimate the model parameters for all models. Using the estimated parameters subsampling probabilities are evaluated for A-, L-optimality criteria and model averaging A-, L-optimality subsampling methods. Through the estimated subsampling probabilities a sample of size $r \ge r_0$ is obtained. Finally, the two samples are combined and the model parameters are estimated for all the models. NOTE : If input parameters are not in given domain conditions necessary error messages will be provided to go further. If $r \ge r_0$ 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. If $0 < \alpha_{q} < 1$ for the a priori model probabilities are not satisfied an error message will be produced, where $q=1,\ldots,Q$ and $Q$ is the number of models in the model set. ## Examples ```r indexes<-1:ceiling(nrow(Electric_consumption)*0.005) Original_Data<-cbind(Electric_consumption[indexes,1],1, Electric_consumption[indexes,-1]) colnames(Original_Data)<-c("Y",paste0("X",0:ncol(Original_Data[,-c(1,2)]))) for (j in 3:5) { Original_Data[,j]<-scale(Original_Data[,j]) } No_of_Variables<-ncol(Original_Data[,-c(1,2)]) Squared_Terms<-paste0("X",1:No_of_Variables,"^2") term_no <- 2 All_Models <- list(c("X0",paste0("X",1:No_of_Variables))) Original_Data<-cbind(Original_Data,Original_Data[,-c(1,2)]^2) colnames(Original_Data)<-c("Y","X0",paste0("X",1:No_of_Variables), paste0("X",1:No_of_Variables,"^2")) for (i in 1:No_of_Variables){ x <- as.vector(combn(Squared_Terms,i,simplify = FALSE)) for(j in 1:length(x)){ All_Models[[term_no]] <- c("X0",paste0("X",1:No_of_Variables),x[[j]]) term_no <- term_no+1 } } All_Models<-All_Models[c(1,12:16)] names(All_Models)<-paste0("Model_",1:length(All_Models)) r0<-300; rf<-rep(100*c(6,9),50); modelRobustLinSub(r0 = r0, rf = rf, Y = as.matrix(Original_Data[,1]), X = as.matrix(Original_Data[,-1]),N = nrow(Original_Data), Apriori_probs = rep(1/length(All_Models),length(All_Models)), All_Combinations = All_Models, All_Covariates = colnames(Original_Data)[-1])->Results Beta_Plots<-plot_Beta(Results) ``` ## References \insertRef mahendran2023modelNeEDS4BigData

modelRobustLinSub function