Assignment #8, Chapter 9
April Jacob
2022-11-10
Questions 5, 7, 8
0.0.1 Non-Linear Kernel
- We have seen that we can fit an SVM with a non-linear kernel in order to perform classification using a non-linear decision boundary.We will now see that we can also obtain a non-linear decision boundary by performing logistic regression using non-linear transformations of the features.
a.Generate a data set with n = 500 and p = 2, such that the observations belong to two classes with a quadratic decision boundary between them. For instance, you can do this as follows:
library(e1071)## Warning: package 'e1071' was built under R version 4.2.2library(ISLR2)
set.seed(13)
x1 = runif(500) - 0.5
x2 = runif(500) - 0.5
y = 1 * (x1^2 - x2^2 > 0)- Plot the observations, colored according to their class labels. Your plot should display X1 on the x-axis, and X2 on the yaxis.
plot(x1[y == 0],
x2[y == 0],
col = "darkorange",
xlab = "X1",
ylab = "X2",
pch = 20)
points(x1[y == 1],
x2[y == 1],
col = "blue",
pch = 4)
- Fit a logistic regression model to the data, using X1 and X2 as predictors.
Both seem to be insignificant.
lm.fit = glm(y ~ x1 + x2,
family = binomial)
summary(lm.fit)##
## Call:
## glm(formula = y ~ x1 + x2, family = binomial)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.287 -1.183 1.070 1.150 1.275
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 0.01963 0.08974 0.219 0.827
## x1 -0.46453 0.30375 -1.529 0.126
## x2 0.03207 0.30589 0.105 0.917
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 693.08 on 499 degrees of freedom
## Residual deviance: 690.71 on 497 degrees of freedom
## AIC: 696.71
##
## Number of Fisher Scoring iterations: 3- Apply this model to the training data in order to obtain a predicted class label for each training observation. Plot the observations, colored according to the predicted class labels. The decision boundary should be linear.
data = data.frame(x1 = x1,
x2 = x2,
y = y)
lm.prob = predict(lm.fit,
data,
type = "response")
lm.pred = ifelse(lm.prob > 0.52, 1, 0)
data.pos = data[lm.pred == 1, ]
data.neg = data[lm.pred == 0, ]
plot(data.pos$x1,
data.pos$x2,
col = "darkorange",
xlab = "X1",
ylab = "X2",
pch = 20)
points(data.neg$x1,
data.neg$x2,
col = "blue",
pch = 4)
- Now fit a logistic regression model to the data using non-linear functions of X1 and X2 as predictors (e.g. X2 1 , X1×X2, log(X2), and so forth).
lm.fit = glm(y ~ poly(x1, 2) + poly(x2, 2) + I(x1 * x2),
data = data,
family = binomial)## Warning: glm.fit: algorithm did not converge## Warning: glm.fit: fitted probabilities numerically 0 or 1 occurred- Apply this model to the training data in order to obtain a predicted class label for each training observation. Plot the observations, colored according to the predicted class labels. The decision boundary should be obviously non-linear. If it is not, then repeat (a)-(e) until you come up with an example in which the predicted class labels are obviously non-linear.
lm.prob = predict(lm.fit,
data,
type = "response")
lm.pred = ifelse(lm.prob > 0.5, 1, 0)
data.pos = data[lm.pred == 1, ]
data.neg = data[lm.pred == 0, ]
plot(data.pos$x1,
data.pos$x2,
col = "darkorange",
xlab = "X1",
ylab = "X2",
pch = 20)
points(data.neg$x1,
data.neg$x2,
col = "blue",
pch = 4)
- Fit a support vector classifier to the data with X1 and X2 as predictors. Obtain a class prediction for each training observation. Plot the observations, colored according to the predicted class labels.
svm.fit = svm(as.factor(y) ~ x1 + x2,
data,
kernel = "linear",
cost = 0.1)
svm.pred = predict(svm.fit, data)
data.pos = data[svm.pred == 1, ]
data.neg = data[svm.pred == 0, ]
plot(data.pos$x1,
data.pos$x2,
col = "darkorange",
xlab = "X1",
ylab = "X2",
pch = 20)
points(data.neg$x1,
data.neg$x2,
col = "blue",
pch = 4)
- Fit a SVM using a non-linear kernel to the data. Obtain a class prediction for each training observation. Plot the observations, colored according to the predicted class labels.
svm.fit = svm(as.factor(y) ~ x1 + x2,
data,
gamma = 1)
svm.pred = predict(svm.fit, data)
data.pos = data[svm.pred == 1, ]
data.neg = data[svm.pred == 0, ]
plot(data.pos$x1,
data.pos$x2,
col = "darkorange",
xlab = "X1",
ylab = "X2",
pch = 20)
points(data.neg$x1,
data.neg$x2,
col = "blue",
pch = 4)
- Comment on your results.
Non-linear kernel SVMs can find non-linear boundaries.
0.0.2 Vector Approaches
- In this problem, you will use support vector approaches in order to predict whether a given car gets high or low gas mileage based on the Auto data set.
- Create a binary variable that takes on a 1 for cars with gas mileage above the median, and a 0 for cars with gas mileage below the median.
gas.med = median(Auto$mpg)
new.var = ifelse(Auto$mpg > gas.med, 1, 0)
Auto$mpglevel = as.factor(new.var)- Fit a support vector classifier to the data with various values of cost, in order to predict whether a car gets high or low gas mileage. Report the cross-validation errors associated with different values of this parameter. Comment on your results. Note you will need to fit the classifier without the gas mileage variable to produce sensible results.
set.seed(13)
tune.out = tune(svm,
mpglevel ~ .,
data = Auto,
kernel = "linear",
ranges = list(cost = c(0.01, 0.1, 1, 5, 10, 100)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost
## 1
##
## - best performance: 0.01538462
##
## - Detailed performance results:
## cost error dispersion
## 1 1e-02 0.07416667 0.05988249
## 2 1e-01 0.05128205 0.06280743
## 3 1e+00 0.01538462 0.01324097
## 4 5e+00 0.01794872 0.02110955
## 5 1e+01 0.02307692 0.02549818
## 6 1e+02 0.03589744 0.02756327- Now repeat (b), this time using SVMs with radial and polynomial basis kernels, with different values of gamma and degree and cost. Comment on your results.
set.seed(83)
tune.out = tune(svm,
mpglevel ~ .,
data = Auto,
kernel = "polynomial",
ranges = list(cost = c(0.1, 1, 5, 10),
degree = c(2, 3, 4)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost degree
## 10 2
##
## - best performance: 0.5432051
##
## - Detailed performance results:
## cost degree error dispersion
## 1 0.1 2 0.5457051 0.03396838
## 2 1.0 2 0.5457051 0.03396838
## 3 5.0 2 0.5457051 0.03396838
## 4 10.0 2 0.5432051 0.05311555
## 5 0.1 3 0.5457051 0.03396838
## 6 1.0 3 0.5457051 0.03396838
## 7 5.0 3 0.5457051 0.03396838
## 8 10.0 3 0.5457051 0.03396838
## 9 0.1 4 0.5457051 0.03396838
## 10 1.0 4 0.5457051 0.03396838
## 11 5.0 4 0.5457051 0.03396838
## 12 10.0 4 0.5457051 0.03396838set.seed(1)
tune.out = tune(svm,
mpglevel ~ .,
data = Auto,
kernel = "radial",
ranges = list(cost = c(0.1, 1, 5, 10),
gamma = c(0.01, 0.1, 1, 5, 10, 100)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost gamma
## 10 0.01
##
## - best performance: 0.02557692
##
## - Detailed performance results:
## cost gamma error dispersion
## 1 0.1 1e-02 0.08929487 0.04382379
## 2 1.0 1e-02 0.07403846 0.03522110
## 3 5.0 1e-02 0.04852564 0.03303346
## 4 10.0 1e-02 0.02557692 0.02093679
## 5 0.1 1e-01 0.07903846 0.03874545
## 6 1.0 1e-01 0.05371795 0.03525162
## 7 5.0 1e-01 0.02820513 0.03299190
## 8 10.0 1e-01 0.03076923 0.03375798
## 9 0.1 1e+00 0.55115385 0.04366593
## 10 1.0 1e+00 0.06384615 0.04375618
## 11 5.0 1e+00 0.05884615 0.04020934
## 12 10.0 1e+00 0.05884615 0.04020934
## 13 0.1 5e+00 0.55115385 0.04366593
## 14 1.0 5e+00 0.49493590 0.04724924
## 15 5.0 5e+00 0.48217949 0.05470903
## 16 10.0 5e+00 0.48217949 0.05470903
## 17 0.1 1e+01 0.55115385 0.04366593
## 18 1.0 1e+01 0.51794872 0.05063697
## 19 5.0 1e+01 0.51794872 0.04917316
## 20 10.0 1e+01 0.51794872 0.04917316
## 21 0.1 1e+02 0.55115385 0.04366593
## 22 1.0 1e+02 0.55115385 0.04366593
## 23 5.0 1e+02 0.55115385 0.04366593
## 24 10.0 1e+02 0.55115385 0.04366593- Make some plots to back up your assertions in (b) and (c).
svm.linear = svm(mpglevel ~ .,
data = Auto,
kernel = "linear",
cost = 1)
svm.poly = svm(mpglevel ~ .,
data = Auto,
kernel = "polynomial",
cost = 10,
degree = 2)
svm.radial = svm(mpglevel ~ .,
data = Auto,
kernel = "radial",
cost = 10,
gamma = 0.01)
plotpairs = function(fit) {
for (name in names(Auto)[!(names(Auto) %in% c("mpg", "mpglevel", "name"))]) {
plot(fit, Auto, as.formula(paste("mpg~", name, sep = "")))
}
}
plotpairs(svm.linear)
plotpairs(svm.poly)
plotpairs(svm.radial)
0.0.3 OJ Data Set
- This problem involves the OJ data set which is part of the ISLR2 package.
- Create a training set containing a random sample of 800 observations, and a test set containing the remaining observations.
set.seed(13)
train = sample(dim(OJ)[1], 800)
OJ.train = OJ[train, ]
OJ.test = OJ[-train, ]- Fit a support vector classifier to the training data using cost = 0.01, with Purchase as the response and the other variables as predictors. Use the summary() function to produce summary statistics, and describe the results obtained.
svm.linear = svm(Purchase ~ .,
kernel = "linear",
data = OJ.train,
cost = 0.01)
summary(svm.linear)##
## Call:
## svm(formula = Purchase ~ ., data = OJ.train, kernel = "linear", cost = 0.01)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: linear
## cost: 0.01
##
## Number of Support Vectors: 439
##
## ( 219 220 )
##
##
## Number of Classes: 2
##
## Levels:
## CH MM- What are the training and test error rates?
train.pred = predict(svm.linear,
OJ.train)
table(OJ.train$Purchase,
train.pred)## train.pred
## CH MM
## CH 441 57
## MM 78 224(78 + 57)/(441 + 224 + 78 + 57)## [1] 0.16875test.pred = predict(svm.linear,
OJ.test)
table(OJ.test$Purchase,
test.pred)## test.pred
## CH MM
## CH 139 16
## MM 25 90(25 + 16)/(139 + 90 + 25 + 16)## [1] 0.1518519- Use the tune() function to select an optimal cost. Consider values in the range 0.01 to 10.
set.seed(83)
tune.out = tune(svm,
Purchase ~ .,
data = OJ.train,
kernel = "linear",
ranges = list(cost = 10^seq(-2, 1, by = 0.25)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost
## 5.623413
##
## - best performance: 0.16625
##
## - Detailed performance results:
## cost error dispersion
## 1 0.01000000 0.18000 0.04794383
## 2 0.01778279 0.17500 0.04330127
## 3 0.03162278 0.17625 0.03884174
## 4 0.05623413 0.17250 0.04322101
## 5 0.10000000 0.17250 0.04281744
## 6 0.17782794 0.17000 0.03917553
## 7 0.31622777 0.17250 0.04073969
## 8 0.56234133 0.17250 0.04281744
## 9 1.00000000 0.17125 0.04332131
## 10 1.77827941 0.17000 0.04090979
## 11 3.16227766 0.16875 0.04379958
## 12 5.62341325 0.16625 0.04528076
## 13 10.00000000 0.17125 0.04489571- Compute the training and test error rates using this new value for cost.
svm.linear = svm(Purchase ~ .,
kernel = "linear",
data = OJ.train,
cost = tune.out$best.parameters$cost)
train.pred = predict(svm.linear,
OJ.train)
table(OJ.train$Purchase,
train.pred)## train.pred
## CH MM
## CH 441 57
## MM 72 230(72 + 57)/(441 + 230 + 72 + 57)## [1] 0.16125test.pred = predict(svm.linear,
OJ.test)
table(OJ.test$Purchase,
test.pred)## test.pred
## CH MM
## CH 134 21
## MM 23 92(23 + 21)/(134 + 92 + 23 + 21)## [1] 0.162963- Repeat parts (b) through (e) using a support vector machine with a radial kernel. Use the default value for gamma.
set.seed(8)
svm.radial = svm(Purchase ~ .,
data = OJ.train,
kernel = "radial")
summary(svm.radial)##
## Call:
## svm(formula = Purchase ~ ., data = OJ.train, kernel = "radial")
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: radial
## cost: 1
##
## Number of Support Vectors: 379
##
## ( 191 188 )
##
##
## Number of Classes: 2
##
## Levels:
## CH MMtrain.pred = predict(svm.radial,
OJ.train)
table(OJ.train$Purchase,
train.pred)## train.pred
## CH MM
## CH 453 45
## MM 84 218(84 + 45)/(453 + 218 + 84 + 45)## [1] 0.16125test.pred = predict(svm.radial,
OJ.test)
table(OJ.test$Purchase,
test.pred)## test.pred
## CH MM
## CH 141 14
## MM 25 90(25 + 14)/(141 + 90 + 25 + 14)## [1] 0.1444444set.seed(11)
tune.out = tune(svm, Purchase ~ .,
data = OJ.train,
kernel = "radial",
ranges = list(cost = 10^seq(-2, 1, by = 0.25)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost
## 1.778279
##
## - best performance: 0.18625
##
## - Detailed performance results:
## cost error dispersion
## 1 0.01000000 0.37750 0.06892024
## 2 0.01778279 0.37750 0.06892024
## 3 0.03162278 0.37375 0.07440738
## 4 0.05623413 0.22250 0.05027701
## 5 0.10000000 0.19500 0.04216370
## 6 0.17782794 0.19375 0.03691676
## 7 0.31622777 0.19250 0.04830459
## 8 0.56234133 0.18625 0.04427267
## 9 1.00000000 0.18750 0.04487637
## 10 1.77827941 0.18625 0.04619178
## 11 3.16227766 0.18875 0.04693746
## 12 5.62341325 0.19000 0.04706674
## 13 10.00000000 0.18875 0.04466309svm.radial = svm(Purchase ~ .,
data = OJ.train,
kernel = "radial",
cost = tune.out$best.parameters$cost)
train.pred = predict(svm.radial,
OJ.train)
table(OJ.train$Purchase,
train.pred)## train.pred
## CH MM
## CH 454 44
## MM 82 220(82 + 44)/(454 + 220 + 82 + 44)## [1] 0.1575test.pred = predict(svm.radial,
OJ.test)
table(OJ.test$Purchase,
test.pred)## test.pred
## CH MM
## CH 140 15
## MM 25 90(25 + 15)/(140 + 90 + 25 + 15)## [1] 0.1481481- Repeat parts (b) through (e) using a support vector machine with a polynomial kernel. Set degree = 2.
set.seed(10)
svm.poly = svm(Purchase ~ .,
data = OJ.train,
kernel = "poly",
degree = 2)
summary(svm.poly)##
## Call:
## svm(formula = Purchase ~ ., data = OJ.train, kernel = "poly", degree = 2)
##
##
## Parameters:
## SVM-Type: C-classification
## SVM-Kernel: polynomial
## cost: 1
## degree: 2
## coef.0: 0
##
## Number of Support Vectors: 446
##
## ( 219 227 )
##
##
## Number of Classes: 2
##
## Levels:
## CH MMtrain.pred = predict(svm.poly,
OJ.train)
table(OJ.train$Purchase,
train.pred)## train.pred
## CH MM
## CH 461 37
## MM 116 186(116 + 37)/(461 + 186 + 116 + 37)## [1] 0.19125test.pred = predict(svm.poly,
OJ.test)
table(OJ.test$Purchase,
test.pred)## test.pred
## CH MM
## CH 144 11
## MM 38 77(38 + 11)/(144 + 77 + 38 + 11)## [1] 0.1814815set.seed(19)
tune.out = tune(svm,
Purchase ~ .,
data = OJ.train,
kernel = "poly",
degree = 2,
ranges = list(cost = 10^seq(-2, 1, by = 0.25)))
summary(tune.out)##
## Parameter tuning of 'svm':
##
## - sampling method: 10-fold cross validation
##
## - best parameters:
## cost
## 10
##
## - best performance: 0.18
##
## - Detailed performance results:
## cost error dispersion
## 1 0.01000000 0.37750 0.03670453
## 2 0.01778279 0.35750 0.04609772
## 3 0.03162278 0.34625 0.04966904
## 4 0.05623413 0.32875 0.05622685
## 5 0.10000000 0.30625 0.06620937
## 6 0.17782794 0.25625 0.06515207
## 7 0.31622777 0.21125 0.07487258
## 8 0.56234133 0.21125 0.06958458
## 9 1.00000000 0.21375 0.06755913
## 10 1.77827941 0.19750 0.07090682
## 11 3.16227766 0.18625 0.07370408
## 12 5.62341325 0.18125 0.05929271
## 13 10.00000000 0.18000 0.05041494svm.poly = svm(Purchase ~ .,
data = OJ.train,
kernel = "poly",
degree = 2,
cost = tune.out$best.parameters$cost)
train.pred = predict(svm.poly,
OJ.train)
table(OJ.train$Purchase,
train.pred)## train.pred
## CH MM
## CH 456 42
## MM 85 217(85 + 42)/(456 + 217 + 85 + 42)## [1] 0.15875test.pred = predict(svm.poly,
OJ.test)
table(OJ.test$Purchase,
test.pred)## test.pred
## CH MM
## CH 140 15
## MM 30 85(30 + 15)/(140 + 85 + 30 + 15)## [1] 0.1666667- Overall, which approach seems to give the best results on this data?
Radial seems to be best with fewer errors.