# Support Vector Machine

A support vector machine (hereinafter, SVM) is a supervised machine learning algorithm in that it is trained by a set of data and then classifies any new input data depending on what it learned during the training phase. SVM can be used both for classification and regression problems but here we focus on its use for classification.

The idea is to separate two distinct groups by maximizing the distance between those points that are most hard to classify. To put it more formally, it maximizes the distance or margin between support vectors around the separating hyperplane. Support vectors here imply the data points that lie closest to the hyperplane. Hyperplanes are decision boundaries that are represented by a line (in two dimensional space) or a plane (in three dimensional space) that separate the two groups.

Suppose a hypothetical problem of classifying apples from lemons. Support vectors in this case are apples that look closest to lemons and lemons that look closest to apples. They are the most difficult ones to classify. SVM draws a separating line or hyperplane that maximizes the distance or margin between support vectors, in this case the apples that look closest to the lemons and lemons that look closest to apples. Therefore support vectors are critical in determining the position as well as the slope of the hyperplane.

For additional information about the support vector regression or support vector machine, refer to Wikipedia: Support-vector machine.

# Keep in Mind

• Note that optimization problem to solve for a linear separator is maximizing the margin which could be calculated as $$\frac{2}{\lVert w \rVert}$$. This could then be rewritten as minimizing $$\lVert w \rVert$$, or minimizing a monotonic transformation version of it expressed as $$\frac{1}{2}\lVert w \rVert^2$$. Additional constraint of $$y_i(w^T x_i + b) \geq 1$$ needs to be imposed to ensure that the data points are still correctly classified. As such, the constrained optimization problem for SVM looks as the following:
$\text{min} \frac{\lVert w \rVert ^2}{2}$

s.t. $$y_i(w^T x_i + b) \geq 1$$,

where $$w$$ is a weight vector, $$x_i$$ is each data point, $$b$$ is bias, and $$y_i$$ is each data point’s corresponding label that takes the value of either $$\{-1, 1\}$$. For detailed information about derivation of the optimization problem, refer to MIT presentation slides, The Math Behind Support Vector Machines, and Demystifying Maths of SVM - Part1.

• If data points are not linearly separable, non-linear SVM introduces higher dimensional space that projects data points from original finite-dimensional space to gain linearly separation. Such process of mapping data points into a higher dimensional space is known as the Kernel Trick. There are numerous types of Kernels that can be used to create higher dimensional space including linear, polynomial, Sigmoid, and Radial Basis Function.

• Setting the right form of Kernel is important as it determines the structure of the separator or hyperplane.

# Implementations

## Python

In this example, we will use scikit-learn, which is a very popular Python library for machine learning. We will look at two support vector machine models: LinearSVC, which performs linear support vector classification (example 1); and SVC, which can accept several different kernels (including non-linear ones). For the latter case, we’ll use the non-linear radial basis function kernel (example 2 below). The last part of the code example plots the decision boundary, ie the support vectors, for the second example.

from sklearn.datasets import make_classification, make_gaussian_quantiles
from sklearn.svm import LinearSVC, SVC
from sklearn.model_selection import train_test_split
import matplotlib.pyplot as plt
import numpy as np

###########################
# Example 1: Linear SVM ###
###########################

# Generate linearly separable data:
X, y = make_classification(n_features=2, n_redundant=0, n_informative=1,
n_clusters_per_class=1)
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.2)

# Train linear SVM model
svm = LinearSVC(tol=1e-5)
svm.fit(X_train, y_train)

# Test model
test_score = svm.score(X_test, y_test)
print(f'The test score is {test_score}')

###############################
# Example 2: Non-linear SVM ###
###############################

# Generate non-linearly separable data
X, y = make_gaussian_quantiles(n_features=2, n_classes=2)
X_train, X_test, y_train, y_test = train_test_split(
X, y, test_size=0.2)

# Train non-linear SVM model
nl_svm = SVC(kernel='rbf', C=50)
nl_svm.fit(X_train, y_train)

# Test model
test_score = nl_svm.score(X_test, y_test)
print(f'The non-linear test score is {test_score}')

####################################
# Plot non-linear SVM boundaries ###
####################################
plt.figure()
decision_function = nl_svm.decision_function(X)
support_vector_indices = np.where(
np.abs(decision_function) <= 1 + 1e-15)
support_vectors = X[support_vector_indices]
plt.scatter(X[:, 0], X[:, 1], c=y, s=30, cmap=plt.cm.Paired)
ax = plt.gca()
xlim = ax.get_xlim()
ylim = ax.get_ylim()
xx, yy = np.meshgrid(np.linspace(xlim, xlim, 50),
np.linspace(ylim, ylim, 50))
Z = nl_svm.decision_function(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
plt.contour(xx, yy, Z, colors='k', levels=[-1, 0, 1], alpha=0.5,
linestyles=['--', '-', '--'])
plt.scatter(support_vectors[:, 0], support_vectors[:, 1], s=100,
linewidth=1, facecolors='none', edgecolors='k')
plt.tight_layout()
plt.show()


## R

There are a couple of ways to implement SVM in R. Here we’ll demonstrate using the e1071 package. To learn more about the package, check out its CRAN page, as well as this vignette. Note that we’ll also load the tidyverse to help with some data wrangling and plotting.

Two examples are shown below that use linear SVM and non-linear SVM respectively. The first example shows how to implement linear SVM. We start by constructing data, separating them into training and test set. Using the training set, we fit the data using the svm() function. Notice that kernel argument for svm() function is specified as linear for our first example. Next, we predict the test data based on the model estimates using the predict() function. The first example result suggests that only one out of 59 data points is incorrectly classified.

The second example shows how to implement non-linear SVM. The data in example two is generated in a way to have data points of one class centered around the middle whereas data points of the other class spread on two sides. Notice that kernel argument for the svm() function is specified as radial for our second example, based on the shape of the data. The second example result suggests that only two out of 58 data points are incorrectly classified.

# Install and load the packages
if (!require("tidyverse")) install.packages("tidyverse")
if (!require("e1071")) install.packages("e1071")
library(tidyverse) # package for data manipulation
library(e1071)     # package for SVM

###########################
# Example 1: Linear SVM ###
###########################

# Construct a completely separable data set
## Set seed for replication
set.seed(0715)
## Make variable x
x = matrix(rnorm(200, mean = 0, sd = 1), nrow = 100, ncol = 2)
## Make variable y that labels x by either -1 or 1
y = rep(c(-1, 1), c(50, 50))
## Make x to have unilaterally higher value when y equals 1
x[y == 1,] = x[y == 1,] + 3.5
## Construct data set
d1 = data.frame(x1 = x[,1], x2 = x[,2], y = as.factor(y))
## Split it into training and test data
flag = sample(c(0,1), size = nrow(d1), prob=c(0.5,0.5), replace = TRUE)
d1 = setNames(split(d1, flag), c("train", "test"))

# Plot
ggplot(data = d1$train, aes(x = x1, y = x2, color = y, shape = y)) + geom_point(size = 2) + scale_color_manual(values = c("darkred", "steelblue")) # SVM classification svmfit1 = svm(y ~ ., data = d1$train, kernel = "linear", cost = 10, scale = FALSE)
print(svmfit1)
plot(svmfit1, d1$train) # Predictability pred.d1 = predict(svmfit1, newdata = d1$test)
table(pred.d1, d1$test$y)

###############################
# Example 2: Non Linear SVM ###
###############################

# Construct less separable data set
## Make variable x
x = matrix(rnorm(200, mean = 0, sd = 1), nrow = 100, ncol = 2)
## Make variable y that labels x by either -1 or 1
y <- rep(c(-1, 1) , c(50, 50))
## Make x to have extreme values when y equals 1
x[y == 1, ][1:25,] = x[y==1,][1:25,] + 3.5
x[y == 1, ][26:50,] = x[y==1,][26:50,] - 3.5
## Construct data set
d2 = data.frame(x1 = x[,1], x2 = x[,2], y = as.factor(y))
## Split it into training and test data
d2 = setNames(split(d2, flag), c("train", "test"))

# Plot data
ggplot(data = d2$train, aes(x = x1, y = x2, color = y, shape = y)) + geom_point(size = 2) + scale_color_manual(values = c("darkred", "steelblue")) # SVM classification svmfit2 = svm(y ~ ., data = d2$train, kernel = "radial", cost = 10, scale = FALSE)
print(svmfit2)
plot(svmfit2, d2$train) # Predictability pred.d2 = predict(svmfit2, newdata = d2$test)
table(pred.d2, d2$test$y)



## Stata

clear all
set more off

*Install svmachines
ssc install svmachines

*Import Data with a binary outcome for classification
use http://www.stata-press.com/data/r16/fvex.dta, clear

*First try logistic regression to benchmark the prediction quality of SVM against
logit outcome group sex arm age distance y // Run the regression
predict outcome_predicted // Generate predictions from the regression

gen log_loss = outcome*log(outcome_predicted)+(1-outcome)*log(1-outcome_predicted)

*Run SVM
svmachines outcome group sex arm age distance y, prob // Specifiying the prob option to generate predicted probabilities in the next line
predict sv_outcome_predicted, probability


Next we will Calculate the log loss (or cross-entropy loss) for SVM.

Note: Predictions following svmachines generate three variables from the stub you provide in the predict command (in this case sv_outcome_predicted). The first is just the same as the stub and stores the best-guess classification (the group with the highest probability out of the possible options). The next n variables store the probability that the given observation will fall into each of the possible classes (in the binary case, this is just n=2 possible classes). These new variables are the stub + the value of each class. In the case below, the suffixes are _0 and _1. We use sv_outcome_predicted_1 because it produces probabilities that are equivalent in their intepretation (probability of having a class of 1) to the probabilities produced by the logit model and that can be used in calculating the log loss. Calculating loss functions for multi-class classifiers is more complicated, and you can read more about that at the link above.

gen log_loss_svm = outcome*log(sv_outcome_predicted_1)+(1-outcome)*log(1-sv_outcome_predicted_1)

*Show log loss for both logit and SVM, remember lower is better
sum log_loss log_loss_svm