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This project is about data preprocessing. Here are examples of different normalizatons and standardizations.
Neural Network is trained with and without principal component analysis. PCA reduces dimensionality in data by linear combinations.
load('data_PCA_E7.RData')
summary(X)
## Var1 Var2 Var3 Var4
## Min. :505.0 Min. :18900 Min. :0.073 Min. : 12000
## 1st Qu.:610.0 1st Qu.:26900 1st Qu.:0.145 1st Qu.:168000
## Median :645.0 Median :37200 Median :0.178 Median :219000
## Mean :635.3 Mean :36101 Mean :0.418 Mean :203439
## 3rd Qu.:665.0 3rd Qu.:41900 3rd Qu.:0.246 3rd Qu.:229000
## Max. :725.0 Max. :51200 Max. :1.950 Max. :756000
x.pca<-prcomp(X)
print(x.pca)
## Standard deviations (1, .., p=4):
## [1] 1.196981e+05 6.207397e+03 4.548899e+01 4.669683e-01
##
## Rotation (n x k) = (4 x 4):
## PC1 PC2 PC3 PC4
## Var1 -2.306577e-05 1.185786e-03 -9.999895e-01 -4.427279e-03
## Var2 4.770627e-02 9.988607e-01 1.183337e-03 2.167368e-06
## Var3 -3.746301e-07 -3.057227e-06 4.427278e-03 -9.999902e-01
## Var4 -9.988614e-01 4.770621e-02 7.960712e-05 5.808034e-07
plot(x.pca,type='l')
print(summary(x.pca))
## Importance of components:
## PC1 PC2 PC3 PC4
## Standard deviation 1.197e+05 6.207e+03 45.49 0.467
## Proportion of Variance 9.973e-01 2.680e-03 0.00 0.000
## Cumulative Proportion 9.973e-01 1.000e+00 1.00 1.000
z-score standardization
x.pca <- prcomp(X,center=TRUE,scale=TRUE)
print(x.pca)
## Standard deviations (1, .., p=4):
## [1] 1.3076326 1.1684277 0.7876379 0.5518154
##
## Rotation (n x k) = (4 x 4):
## PC1 PC2 PC3 PC4
## Var1 -0.1547026 -0.7010420 0.6670785 -0.1990313
## Var2 -0.6832255 0.1556693 0.2086951 0.6822143
## Var3 0.2508370 0.6504772 0.7078286 -0.1137494
## Var4 0.6680949 -0.2473593 0.1021336 0.6942847
plot(x.pca,type='l')
print(summary(x.pca))
## Importance of components:
## PC1 PC2 PC3 PC4
## Standard deviation 1.3076 1.1684 0.7876 0.55182
## Proportion of Variance 0.4275 0.3413 0.1551 0.07613
## Cumulative Proportion 0.4275 0.7688 0.9239 1.00000
NN without PCA
library(nnet)
rm(X)
rm(y)
load('data_NN_E7.RData')
summary(X)
## Var1 Var2 Var3 Var4
## Min. : 22.00 Min. :505.0 Min. :0.0000 Min. :0.1830
## 1st Qu.: 40.00 1st Qu.:605.0 1st Qu.:0.0000 1st Qu.:0.3230
## Median : 55.00 Median :630.0 Median :1.0000 Median :0.3680
## Mean : 63.63 Mean :628.9 Mean :0.6826 Mean :0.3562
## 3rd Qu.: 80.00 3rd Qu.:655.0 3rd Qu.:1.0000 3rd Qu.:0.4120
## Max. :200.00 Max. :725.0 Max. :1.0000 Max. :0.5120
## Var5 Var6 Var7 Var8
## Min. :0.00500 Min. :0.0730 Min. :0.0120 Min. :0.00400
## 1st Qu.:0.00900 1st Qu.:0.1510 1st Qu.:0.1720 1st Qu.:0.01000
## Median :0.01100 Median :0.1950 Median :0.2210 Median :0.01200
## Mean :0.01249 Mean :0.5973 Mean :0.2181 Mean :0.01515
## 3rd Qu.:0.01500 3rd Qu.:1.3710 3rd Qu.:0.2300 3rd Qu.:0.01600
## Max. :0.03000 Max. :2.6000 Max. :0.7640 Max. :0.06000
## Var9 Var10
## Min. :0.00000 Min. :0.0000
## 1st Qu.:0.00000 1st Qu.:0.0000
## Median :0.00000 Median :1.0000
## Mean :0.01709 Mean :0.5289
## 3rd Qu.:0.00000 3rd Qu.:1.0000
## Max. :1.00000 Max. :1.0000
min-max scaling
minx=apply(X,2,min)
maxx=apply(X,2,max)
miny=min(y)
maxy=max(y)
X_S=scale(X,minx,maxx-minx)
y_s=scale(y,miny,maxy-miny)
NN model
model_nn <- nnet(X_S, y_s, size=20,
maxit=300, decay=0.03, linout=TRUE, reltol=1.e-6, MaxNWts=100000)
## # weights: 241
## initial value 12074.926569
## iter 10 value 53.040354
## iter 20 value 39.697988
## iter 30 value 34.312506
## iter 40 value 31.351618
## iter 50 value 28.171835
## iter 60 value 26.879735
## iter 70 value 25.852764
## iter 80 value 25.353068
## iter 90 value 25.059211
## iter 100 value 24.893339
## iter 110 value 24.763462
## iter 120 value 24.611410
## iter 130 value 24.505750
## iter 140 value 24.425072
## iter 150 value 24.307079
## iter 160 value 24.157097
## iter 170 value 23.999051
## iter 180 value 23.881802
## iter 190 value 23.792141
## iter 200 value 23.710257
## iter 210 value 23.643505
## iter 220 value 23.589224
## iter 230 value 23.534901
## iter 240 value 23.494990
## iter 250 value 23.469023
## iter 260 value 23.439473
## iter 270 value 23.415806
## iter 280 value 23.392344
## iter 290 value 23.275909
## iter 300 value 23.187084
## final value 23.187084
## stopped after 300 iterations
produce the predictions
y_s.predict <- predict(model_nn, X_S)
print(cor(y_s.predict,y_s))
## [,1]
## [1,] 0.9388577
str(X)
## 'data.frame': 7196 obs. of 10 variables:
## $ Var1 : num 90 90 90 90 38 38 38 38 65 90 ...
## $ Var2 : num 710 710 710 710 580 580 580 580 655 620 ...
## $ Var3 : num 1 1 1 1 1 1 1 1 0 1 ...
## $ Var4 : num 0.463 0.463 0.463 0.463 0.247 0.247 0.247 0.247 0.428 0.337 ...
## $ Var5 : num 0.011 0.011 0.011 0.011 0.013 0.013 0.013 0.013 0.011 0.008 ...
## $ Var6 : num 0.151 0.151 0.151 0.151 0.208 ...
## $ Var7 : num 0.027 0.027 0.027 0.027 0.219 0.219 0.219 0.219 0.172 0.226 ...
## $ Var8 : num 0.011 0.011 0.011 0.011 0.01 0.01 0.01 0.01 0.008 0.016 ...
## $ Var9 : num 1 1 1 1 0 0 0 0 0 0 ...
## $ Var10: num 0 0 0 0 1 1 1 1 1 0 ...
NN with PCA
remove binary variables and standardize X.
miny=min(y)
maxy=max(y)
y_s=scale(y,miny,maxy-miny)
x.pca <- prcomp(X[,c(1,2,4,5,6,7,8)],center=TRUE,scale=TRUE)
pred_pca <- predict(x.pca)
print(pred_pca[1:5,])
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## [1,] 2.769676 0.9674481 0.5816383 -1.7773290 -0.4168766 0.9762095 0.78995243
## [2,] 2.769676 0.9674481 0.5816383 -1.7773290 -0.4168766 0.9762095 0.78995243
## [3,] 2.769676 0.9674481 0.5816383 -1.7773290 -0.4168766 0.9762095 0.78995243
## [4,] 2.769676 0.9674481 0.5816383 -1.7773290 -0.4168766 0.9762095 0.78995243
## [5,] -1.412512 -0.0040778 -1.3465561 0.7394715 -1.0735162 0.2160057 0.03617177
print(summary(x.pca))
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 1.328 1.1657 1.1342 0.9897 0.8571 0.77783 0.52149
## Proportion of Variance 0.252 0.1941 0.1838 0.1399 0.1049 0.08643 0.03885
## Cumulative Proportion 0.252 0.4461 0.6299 0.7698 0.8747 0.96115 1.00000
binary variables are added back to the data
N=7
x_data <- data.frame(X[,c(3,9,10)],pred_pca[,1:N])
NN model with pca
model_pcann <- nnet(x_data, y_s, size=20,
maxit=300, decay=0.03, linout=TRUE, reltol=1.e-6, MaxNWts=100000)
## # weights: 241
## initial value 2548.425480
## iter 10 value 60.234397
## iter 20 value 29.770896
## iter 30 value 25.437188
## iter 40 value 23.035419
## iter 50 value 21.986917
## iter 60 value 20.768992
## iter 70 value 20.073042
## iter 80 value 19.625506
## iter 90 value 19.356206
## iter 100 value 19.097833
## iter 110 value 18.938737
## iter 120 value 18.784620
## iter 130 value 18.635220
## iter 140 value 18.516993
## iter 150 value 18.435954
## iter 160 value 18.361717
## iter 170 value 18.281183
## iter 180 value 18.209129
## iter 190 value 18.142622
## iter 200 value 18.071377
## iter 210 value 18.010881
## iter 220 value 17.943529
## iter 230 value 17.878284
## iter 240 value 17.816297
## iter 250 value 17.788753
## iter 260 value 17.768241
## iter 270 value 17.752451
## iter 280 value 17.736956
## iter 290 value 17.719340
## iter 300 value 17.698402
## final value 17.698402
## stopped after 300 iterations
predict and eval
y_s.pred <- predict(model_pcann, x_data)
print(cor(y_s.pred, y_s))
## [,1]
## [1,] 0.9514857
plot(y_s.pred,y_s)
