predict.FAMILY: predict.FAMILY In FAMILY: A Convex Formulation for Modeling Interactions with Strong Heredity

Description

Similar to `R`'s generic `predict` function which predicts the model for new data for different values of α and λ.

Usage

 ```1 2``` ```## S3 method for class 'FAMILY' predict(object, new.X, new.Z, Bias.corr = FALSE, XequalZ = FALSE, ...) ```

Arguments

 `object` The fitted object as the output from the main function `FAMILY`. `new.X` Matrix of covariates `X`. Must have the same number of columns used for fitting the model. `new.Z` Matrix of covariates `Z`. Must have the same number of columns used for fitting the model. `Bias.corr` A logical variable indicating if we wish to re-fit the selected variables using `glm` or `lm`. `XequalZ` A logical variable indicating if X = Z or if we have two different sets of covariates. `...` Extra arguments for the generic S3 `predict` function

Value

The function returns an array of dimensions `[n, length(alphas), length(lambdas)]` where `n = nrow(new.X)`. This array contains one the following:

 `yhat` The fitted values using the data given `phat` The fitted estimated probabilities for logistic regression

References

Haris, Witten and Simon (2014). Convex Modeling of Interactions with Strong Heredity. Available on ArXiv at http://arxiv.org/abs/1410.3517

`FAMILY`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294``` ```library(FAMILY) library(pROC) library(pheatmap) ##################################################################################### ##################################################################################### ############################# EXAMPLE - CONTINUOUS RESPONSE ######################### ##################################################################################### ##################################################################################### ############################## GENERATE DATA ######################################## #Generate training set of covariates X and Z set.seed(1) X.tr<- matrix(rnorm(10*100),ncol = 10, nrow = 100) Z.tr<- matrix(rnorm(15*100),ncol = 15, nrow = 100) #Generate test set of covariates X and Z X.te<- matrix(rnorm(10*100),ncol = 10, nrow = 100) Z.te<- matrix(rnorm(15*100),ncol = 15, nrow = 100) #Scale appropiately meanX<- apply(X.tr,2,mean) meanY<- apply(Z.tr,2,mean) X.tr<- scale(X.tr, scale = FALSE) Z.tr<- scale(Z.tr, scale = FALSE) X.te<- scale(X.te,center = meanX,scale = FALSE) Z.te<- scale(Z.te,center = meanY,scale = FALSE) #Generate full matrix of Covariates w.tr<- c() w.te<- c() X1<- cbind(1,X.tr) Z1<- cbind(1,Z.tr) X2<- cbind(1,X.te) Z2<- cbind(1,Z.te) for(i in 1:16){ for(j in 1:11){ w.tr<- cbind(w.tr,X1[,j]*Z1[,i]) w.te<- cbind(w.te, X2[,j]*Z2[,i]) } } #Generate response variables with signal from #First 5 X features and 5 Z features. #We construct the coefficient matrix B. #B[1,1] contains the intercept #B[-1,1] contains the main effects for X. # For instance, B[2,1] is the main effect for the first feature in X. #B[1,-1] contains the main effects for Z. # For instance, B[1,10] is the coefficient for the 10th feature in Z. #B[i+1,j+1] is the coefficient of X_i Z_j B<- matrix(0,ncol = 16,nrow = 11) rownames(B)<- c("inter" , paste("X",1:(nrow(B)-1),sep = "")) colnames(B)<- c("inter" , paste("Z",1:(ncol(B)-1),sep = "")) # First, we simulate data as follows: # The first five features in X, and the first five features in Z, are non-zero. # And given the non-zero main effects, all possible interactions are involved. # We call this "high strong heredity" B_high_SH<- B B_high_SH[1:6,1:6]<- 1 #View true coefficient matrix pheatmap(as.matrix(B_high_SH), scale="none", cluster_rows=FALSE, cluster_cols=FALSE) Y_high_SH <- as.vector(w.tr%*%as.vector(B_high_SH))+rnorm(100,sd = 2) Y_high_SH.te <- as.vector(w.te%*%as.vector(B_high_SH))+rnorm(100,sd = 2) # Now a new setting: # Again, the first five features in X, and the first five features in Z, are involved. # But this time, only a subset of the possible interactions are involved. # Strong heredity is still maintained. # We call this "low strong heredity" B_low_SH<- B_high_SH B_low_SH[2:6,2:6]<-0 B_low_SH[3:4,3:5]<- 1 #View true coefficient matrix pheatmap(as.matrix(B_low_SH), scale="none", cluster_rows=FALSE, cluster_cols=FALSE) Y_low_SH <- as.vector(w.tr%*%as.vector(B_low_SH))+rnorm(100,sd = 1.5) Y_low_SH.te <- as.vector(w.te%*%as.vector(B_low_SH))+rnorm(100,sd = 1.5) ############################## FIT SOME MODELS ######################################## #Define alphas and lambdas #Define 3 different alpha values #Low alpha values penalize groups more #High alpha values penalize individual Interactions more alphas<- c(0.01,0.5,0.99) lambdas<- seq(0.1,1,length = 50) #high Strong heredity with l2 norm fit_high_SH<- FAMILY(X.tr, Z.tr, Y_high_SH, lambdas , alphas, quad = TRUE,iter=500, verbose = TRUE ) yhat_hSH<- predict(fit_high_SH, X.te, Z.te) mse_hSH <-apply(yhat_hSH,c(2,3), "-" ,Y_high_SH.te) mse_hSH<- apply(mse_hSH^2,c(2,3),sum) #Find optimal model and plot matrix im<- which(mse_hSH==min(mse_hSH),TRUE) plot(fit_high_SH\$Estimate[[im[2] ]][[im[1]]]) #Plot some matrices for different alpha values #Low alpha, higher penalty on groups plot(fit_high_SH\$Estimate[[ 1 ]][[ 25 ]]) #Medium alpha, equal penalty on groups and individual interactions plot(fit_high_SH\$Estimate[[ 2 ]][[ 25 ]]) #High alpha, more penalty on individual interactions plot(fit_high_SH\$Estimate[[ 3 ]][[ 40 ]]) #View Coefficients coef(fit_high_SH)[[im[2]]][[im[1]]] ############################## Uncomment code for EXAMPLE ########################### # #high Strong heredity with l_infinity norm norm # fit_high_SH<- FAMILY(X.tr, Z.tr, Y_high_SH, lambdas , # alphas, quad = TRUE,iter=500, verbose = TRUE, # norm = "l_inf") # yhat_hSH<- predict(fit_high_SH, X.te, Z.te) # mse_hSH <-apply(yhat_hSH,c(2,3), "-" ,Y_high_SH.te) # mse_hSH<- apply(mse_hSH^2,c(2,3),sum) # # #Find optimal model and plot matrix # im<- which(mse_hSH==min(mse_hSH),TRUE) # plot(fit_high_SH\$Estimate[[im[2] ]][[im[1]]]) # # # #Plot some matrices for different alpha values # #Low alpha, higher penalty on groups # plot(fit_high_SH\$Estimate[[ 1 ]][[ 30 ]]) # #Medium alpha, equal penalty on groups and individual interactions # plot(fit_high_SH\$Estimate[[ 2 ]][[ 10 ]]) # #High alpha, more penalty on individual interactions # plot(fit_high_SH\$Estimate[[ 3 ]][[ 20 ]]) # # # #View Coefficients # coef(fit_high_SH)[[im[2]]][[im[1]]] ############################## Uncomment code for EXAMPLE ########################### # #Redefine lambdas # lambdas<- seq(0.1,0.5,length = 50) # # #low Strong heredity with l_2 norm # fit_low_SH<- FAMILY(X.tr, Z.tr, Y_low_SH, lambdas , # alphas, quad = TRUE,iter=500, verbose = TRUE ) # yhat_lSH<- predict(fit_low_SH, X.te, Z.te) # mse_lSH <-apply(yhat_lSH,c(2,3), "-" ,Y_low_SH.te) # mse_lSH<- apply(mse_lSH^2,c(2,3),sum) # # #Find optimal model and plot matrix # im<- which(mse_lSH==min(mse_lSH),TRUE) # plot(fit_low_SH\$Estimate[[im[2] ]][[im[1]]]) # # # #Plot some matrices for different alpha values # #Low alpha, higher penalty on groups # plot(fit_low_SH\$Estimate[[ 1 ]][[ 25 ]]) # #Medium alpha, equal penalty on groups and individual interactions # plot(fit_low_SH\$Estimate[[ 2 ]][[ 10 ]]) # #High alpha, more penalty on individual interactions # plot(fit_low_SH\$Estimate[[ 3 ]][[ 10 ]]) # # # #View Coefficients # coef(fit_low_SH)[[im[2]]][[im[1]]] ##################################################################################### ##################################################################################### ############################### EXAMPLE - BINARY RESPONSE ########################### ##################################################################################### ##################################################################################### ############################## GENERATE DATA ######################################## #Generate data for logistic regression Yp_high_SH<- as.vector((w.tr)%*%as.vector(B_high_SH)) Yp_high_SH.te<- as.vector((w.te)%*%as.vector(B_high_SH)) Yprobs_high_SH<- 1/(1+exp(-Yp_high_SH)) Yprobs_high_SH.te<- 1/(1+exp(-Yp_high_SH.te)) Yp_high_SH<- rbinom(100, size = 1, prob = Yprobs_high_SH) Yp_high_SH.te<- rbinom(100, size = 1, prob = Yprobs_high_SH.te) lambdas<- seq(0.01,0.15,length = 50) ############################## FIT SOME MODELS ######################################## #Fit glm via l_2 norm fit_high_SH<- FAMILY(X.tr, Z.tr, Yp_high_SH, lambdas , alphas, quad = TRUE,iter=500, verbose = TRUE, family = "binomial") yhp_hSH<- predict(fit_high_SH, X.te, Z.te) mse_high_SH <-apply(yhp_hSH,c(2,3), "-" ,Yp_high_SH.te) mse_hSH<- apply(mse_high_SH^2,c(2,3),sum) im<- which(mse_hSH==min(mse_hSH),TRUE) plot(fit_high_SH\$Estimate[[im[2] ]][[im[1]]]) roc( Yp_high_SH.te,yhp_hSH[,im[1],im[2]],plot = TRUE) #View Coefficients coef(fit_high_SH)[[im[2]]][[im[1]]] ############################## Uncomment code for EXAMPLE ########################### # #Fit glm via l_infinity norm # fit_high_SH<- FAMILY(X.tr, Z.tr, Yp_high_SH, lambdas , norm = "l_inf", # alphas, quad = TRUE,iter=500, verbose = TRUE, # family = "binomial") # yhp_hSH<- predict(fit_high_SH, X.te, Z.te) # mse_high_SH <-apply(yhp_hSH,c(2,3), "-" ,Yp_high_SH.te) # mse_hSH<- apply(mse_high_SH^2,c(2,3),sum) # im<- which(mse_hSH==min(mse_hSH),TRUE) # plot(fit_high_SH\$Estimate[[im[2] ]][[im[1]]]) # roc( Yp_high_SH.te,yhp_hSH[,im[1],im[2]],plot = TRUE) # # #View Coefficients # coef(fit_high_SH)[[im[2]]][[im[1]]] ##################################################################################### ##################################################################################### ############################## EXAMPLE WHERE X=Z #################################### ######################## Uncomment Code for EXAMPLE ################################# ##################################################################################### ############################## GENERATE DATA ######################################## # #Redefine Lambdas # lambdas<- seq(0.01,0.3,length = 50) # # # #We consider the case X=Z now # w.tr<- c() # w.te<- c() # X1<- cbind(1,X.tr) # X2<- cbind(1,X.te) # # for(i in 1:11){ # for(j in 1:11){ # w.tr<- cbind(w.tr,X1[,j]*X1[,i]) # w.te<- cbind(w.te, X2[,j]*X2[,i]) # } # } # # B<- matrix(0,ncol = 11,nrow = 11) # rownames(B)<- c("inter" , paste("X",1:(nrow(B)-1),sep = "")) # colnames(B)<- c("inter" , paste("X",1:(ncol(B)-1),sep = "")) # # # B_high_SH<- B # B_high_SH[1:6,1:6]<- 1 # #We exclude quadratic terms in this example # diag(B_high_SH)[-1]<-0 # #View true coefficient matrix # pheatmap(as.matrix(B_high_SH), scale="none", # cluster_rows=FALSE, cluster_cols=FALSE) # # #With high Strong heredity: all possible interactions # Y_high_SH <- as.vector(w.tr%*%as.vector(B_high_SH))+rnorm(100) # Y_high_SH.te <- as.vector(w.te%*%as.vector(B_high_SH))+rnorm(100) # # ############################## FIT SOME MODELS ######################################## # # #high Strong heredity with l_2 norm # fit_high_SH<- FAMILY(X.tr, X.tr, Y_high_SH, lambdas , # alphas, quad = FALSE,iter=500, verbose = TRUE ) # yhat_hSH<- predict(fit_high_SH, X.te, X.te) # mse_hSH <-apply(yhat_hSH,c(2,3), "-" ,Y_high_SH.te) # mse_hSH<- apply(mse_hSH^2,c(2,3),sum) # # #Find optimal model and plot matrix # im<- which(mse_hSH==min(mse_hSH),TRUE) # plot(fit_high_SH\$Estimate[[im[2] ]][[im[1]]]) # # # #Plot some matrices for different alpha values # #Low alpha, higher penalty on groups # plot(fit_high_SH\$Estimate[[ 1 ]][[ 50 ]]) # #Medium alpha, equal penalty on groups and individual interactions # plot(fit_high_SH\$Estimate[[ 2 ]][[ 50 ]]) # #High alpha, more penalty on individual interactions # plot(fit_high_SH\$Estimate[[ 3 ]][[ 50 ]]) # # # #View Coefficients # coef(fit_high_SH,XequalZ = TRUE)[[im[2]]][[im[1]]] ```