# Understanding Support Vector Machine algorithm from examples (along with code)

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Note: This article was originally published on Oct 6th, 2015 and updated on Sept 13th, 2017

## Introduction

Mastering machine learning algorithms isn’t a myth at all. Most of the beginners start by learning regression. It is simple to learn and use, but does that solve our purpose? Of course not! Because, you can do so much more than just Regression!

Think of machine learning algorithms as an armory packed with axes, sword, blades, bow, dagger etc. You have various tools, but you ought to learn to use them at the right time. As an analogy, think of ‘Regression’ as a sword capable of slicing and dicing data efficiently, but incapable of dealing with highly complex data. On the contrary, ‘Support Vector Machines’ is like a sharp knife – it works on smaller datasets, but on them, it can be much more stronger and powerful in building models.

By now, I hope you’ve now mastered Random ForestNaive Bayes Algorithm and Ensemble Modeling. If not, I’d suggest you to take out few minutes and read about them as well. In this article, I shall guide you through the basics to advanced knowledge of a crucial machine learning algorithm, support vector machines.

1. What is Support Vector Machine?
2. How does it work?
3. How to implement SVM in Python and R?
4. How to tune Parameters of SVM?
5. Pros and Cons associated with SVM

## What is Support Vector Machine?

“Support Vector Machine” (SVM) is a supervised machine learning algorithm which can be used for both classification or regression challenges. However,  it is mostly used in classification problems. In this algorithm, we plot each data item as a point in n-dimensional space (where n is number of features you have) with the value of each feature being the value of a particular coordinate. Then, we perform classification by finding the hyper-plane that differentiate the two classes very well (look at the below snapshot).

Support Vectors are simply the co-ordinates of individual observation. Support Vector Machine is a frontier which best segregates the two classes (hyper-plane/ line).

You can look at definition of support vectors and a few examples of its working here.

## How does it work?

Above, we got accustomed to the process of segregating the two classes with a hyper-plane. Now the burning question is “How can we identify the right hyper-plane?”. Don’t worry, it’s not as hard as you think!

Let’s understand:

• Identify the right hyper-plane (Scenario-1): Here, we have three hyper-planes (A, B and C). Now, identify the right hyper-plane to classify star and circle.
You need to remember a thumb rule to identify the right hyper-plane: “Select the hyper-plane which segregates the two classes better”. In this scenario, hyper-plane “B” has excellently performed this job.
• Identify the right hyper-plane (Scenario-2): Here, we have three hyper-planes (A, B and C) and all are segregating the classes well. Now, How can we identify the right hyper-plane?

Here, maximizing the distances between nearest data point (either class) and hyper-plane will help us to decide the right hyper-plane. This distance is called as Margin. Let’s look at the below snapshot:
Above, you can see that the margin for hyper-plane C is high as compared to both A and B. Hence, we name the right hyper-plane as C. Another lightning reason for selecting the hyper-plane with higher margin is robustness. If we select a hyper-plane having low margin then there is high chance of miss-classification.

• Identify the right hyper-plane (Scenario-3):Hint: Use the rules as discussed in previous section to identify the right hyper-plane

Some of you may have selected the hyper-plane B as it has higher margin compared to A. But, here is the catch, SVM selects the hyper-plane which classifies the classes accurately prior to maximizing margin. Here, hyper-plane B has a classification error and A has classified all correctly. Therefore, the right hyper-plane is A.

• Can we classify two classes (Scenario-4)?: Below, I am unable to segregate the two classes using a straight line, as one of star lies in the territory of other(circle) class as an outlier.
As I have already mentioned, one star at other end is like an outlier for star class. SVM has a feature to ignore outliers and find the hyper-plane that has maximum margin. Hence, we can say, SVM is robust to outliers.
• Find the hyper-plane to segregate to classes (Scenario-5): In the scenario below, we can’t have linear hyper-plane between the two classes, so how does SVM classify these two classes? Till now, we have only looked at the linear hyper-plane.
SVM can solve this problem. Easily! It solves this problem by introducing additional feature. Here, we will add a new feature z=x^2+y^2. Now, let’s plot the data points on axis x and z:

In above plot, points to consider are:

• All values for z would be positive always because z is the squared sum of both x and y
• In the original plot, red circles appear close to the origin of x and y axes, leading to lower value of z and star relatively away from the origin result to higher value of z.

In SVM, it is easy to have a linear hyper-plane between these two classes. But, another burning question which arises is, should we need to add this feature manually to have a hyper-plane. No, SVM has a technique called the kernel trick. These are functions which takes low dimensional input space and transform it to a higher dimensional space i.e. it converts not separable problem to separable problem, these functions are called kernels. It is mostly useful in non-linear separation problem. Simply put, it does some extremely complex data transformations, then find out the process to separate the data based on the labels or outputs you’ve defined.

When we look at the hyper-plane in original input space it looks like a circle:

Now, let’s  look at the methods to apply SVM algorithm in a data science challenge.

## How to implement SVM in Python and R?

In Python, scikit-learn is a widely used library for implementing machine learning algorithms, SVM is also available in scikit-learn library and follow the same structure (Import library, object creation, fitting model and prediction). Let’s look at the below code:

```#Import Library
from sklearn import svm
#Assumed you have, X (predictor) and Y (target) for training data set and x_test(predictor) of test_dataset
# Create SVM classification object
model = svm.svc(kernel='linear', c=1, gamma=1)
# there is various option associated with it, like changing kernel, gamma and C value. Will discuss more # about it in next section.Train the model using the training sets and check score
model.fit(X, y)
model.score(X, y)
#Predict Output
predicted= model.predict(x_test)```

The e1071 package in R is used to create Support Vector Machines with ease. It has helper functions as well as code for the Naive Bayes Classifier. The creation of a support vector machine in R and Python follow similar approaches, let’s take a look now at the following code:

```#Import Library
require(e1071) #Contains the SVM
# there are various options associated with SVM training; like changing kernel, gamma and C value.

# create model
model <- svm(Target~Predictor1+Predictor2+Predictor3,data=Train,kernel='linear',gamma=0.2,cost=100)

#Predict Output
preds <- predict(model,Test)
table(preds)```

### How to tune Parameters of SVM?

Tuning parameters value for machine learning algorithms effectively improves the model performance. Let’s look at the list of parameters available with SVM.

`sklearn.svm.SVC(C=1.0, kernel='rbf', degree=3, gamma=0.0, coef0=0.0, shrinking=True, probability=False,tol=0.001, cache_size=200, class_weight=None, verbose=False, max_iter=-1, random_state=None)`

I am going to discuss about some important parameters having higher impact on model performance, “kernel”, “gamma” and “C”.

kernel: We have already discussed about it. Here, we have various options available with kernel like, “linear”, “rbf”,”poly” and others (default value is “rbf”).  Here “rbf” and “poly” are useful for non-linear hyper-plane. Let’s look at the example, where we’ve used linear kernel on two feature of iris data set to classify their class.

Example: Have linear kernel

```import numpy as np
import matplotlib.pyplot as plt
from sklearn import svm, datasets```
```# import some data to play with
X = iris.data[:, :2] # we only take the first two features. We could
# avoid this ugly slicing by using a two-dim dataset
y = iris.target```
```# we create an instance of SVM and fit out data. We do not scale our
# data since we want to plot the support vectors
C = 1.0 # SVM regularization parameter
svc = svm.SVC(kernel='linear', C=1,gamma=0).fit(X, y)```
```# create a mesh to plot in
x_min, x_max = X[:, 0].min() - 1, X[:, 0].max() + 1
y_min, y_max = X[:, 1].min() - 1, X[:, 1].max() + 1
h = (x_max / x_min)/100
xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
np.arange(y_min, y_max, h))```
```plt.subplot(1, 1, 1)
Z = svc.predict(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
plt.contourf(xx, yy, Z, cmap=plt.cm.Paired, alpha=0.8)```
```plt.scatter(X[:, 0], X[:, 1], c=y, cmap=plt.cm.Paired)
plt.xlabel('Sepal length')
plt.ylabel('Sepal width')
plt.xlim(xx.min(), xx.max())
plt.title('SVC with linear kernel')
plt.show()```

Example: Have rbf kernel

Change the kernel type to rbf in below line and look at the impact.

`svc = svm.SVC(kernel='rbf', C=1,gamma=0).fit(X, y)`

I would suggest you to go for linear kernel if you have large number of features (>1000) because it is more likely that the data is linearly separable in high dimensional space. Also, you can RBF but do not forget to cross validate for its parameters as to avoid over-fitting.

gamma: Kernel coefficient for ‘rbf’, ‘poly’ and ‘sigmoid’. Higher the value of gamma, will try to exact fit the as per training data set i.e. generalization error and cause over-fitting problem.

Example: Let’s difference if we have gamma different gamma values like 0, 10 or 100.

```svc = svm.SVC(kernel='rbf', C=1,gamma=0).fit(X, y)

```

C: Penalty parameter C of the error term. It also controls the trade off between smooth decision boundary and classifying the training points correctly.

We should always look at the cross validation score to have effective combination of these parameters and avoid over-fitting.

In R, SVMs can be tuned in a similar fashion as they are in Python. Mentioned below are the respective parameters for e1071 package:

• The kernel parameter can be tuned to take “Linear”,”Poly”,”rbf” etc.
• The gamma value can be tuned by setting the “Gamma” parameter.
• The C value in Python is tuned by the “Cost” parameter in R.

## Pros and Cons associated with SVM

• Pros:
• It works really well with clear margin of separation
• It is effective in high dimensional spaces.
• It is effective in cases where number of dimensions is greater than the number of samples.
• It uses a subset of training points in the decision function (called support vectors), so it is also memory efficient.
• Cons:
• It doesn’t perform well, when we have large data set because the required training time is higher
• It also doesn’t perform very well, when the data set has more noise i.e. target classes are overlapping
• SVM doesn’t directly provide probability estimates, these are calculated using an expensive five-fold cross-validation. It is related SVC method of Python scikit-learn library.

## Practice Problem

Find right additional feature to have a hyper-plane for segregating the classes in below snapshot:

## End Notes

In this article, we looked at the machine learning algorithm, Support Vector Machine in detail.  I discussed its concept of working, process of implementation in python, the tricks to make the model efficient by tuning its parameters, Pros and Cons, and finally a problem to solve. I would suggest you to use SVM and analyse the power of this model by tuning the parameters. I also want to hear your experience with SVM, how have you tuned parameters to avoid over-fitting and reduce the training time?

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Author

I am a Business Analytics and Intelligence professional with deep experience in the Indian Insurance industry. I have worked for various multi-national Insurance companies in last 7 years.

• nishant says:

hi,

gr8 articles..explaining the nuances of SVM…hope u can reproduce the same with R…..it would be gr8 help to all R junkies like me

• ASHISH says:

NEW VARIABLE (Z) = SQRT(X) + SQRT (Y)

• Rishabh says:

Given problem Data points looks like y=x^2+c. So i guess z=x^2-y OR z=y-x^2.

• dam van tai says:

i think x coodinates must increase after sqrt

• Mahmood A. Sheikh says:

Kernel

• Mahmood A. Sheikh says:

I mean kernel will add the new feature automatically.

• Sanjay says:

Nicely Explained .

The hyperplane to separate the classes for the above problem can be imagined as 3-D Parabola.

z=ax^2 + by^2 + c

• FrankSauvage says:

Thanks a lot for this great hands-on article!

• Harsha says:

Really impressive content. Simple and effective. It could be more efficient if you can describe each of the parameters and practical application where you faced non-trivial problem examples.

• Aman Srivastava says:

kernel

How does the python code look like if we are using LSSVM instead of SVM?

• Janpreet Singh says:

Polynomial kernel function?!
for exzmple : Z= A(x^2) + B(y^2) + Cx + Dy + E

• Krishna Kalaparti says:

Hi Sunil.

Great Article.

However, there’s an issue in the code you’ve provided. When i compiled the code, i got the following error:

Name error: name ‘h’ is not defined.

I’ve faced this error at line 16, which is: “xx, yy = np.meshgrid(np.arange(x_min, x_min, h), …).

Could you look into it and let me know how to fix it?

• Shikha says:

great explanation 🙂 I think new variable Z should be x^2 + y.

• VEERAMANI NATARAJAN says:

Nice Articlel

• Carlos says:

The solution is analogue to scenario-5 if you replace y by y-k

• K.Krithiga Lakshmi says:

Your SVM explanation and kernel definition is very simple, and easy to understand. Kudos for that effort.

• pfcohen says:

Most intuitive explanation of multidimensional svm I have seen. Thank you!

• yc says:

what is ‘h’ in the code of SVM .
xx, yy = np.meshgrid(np.arange(x_min, x_max, h),
np.arange(y_min, y_max, h))

• Iresh says:

z = (x^2 – y)
z > 0, red circles

• LI JENG HUANG says:

very neat explaination to SVC.
For the proposed problem, my answers are:
(1) z = a* x^2 + b y + c, a parabola.
(2) z = a (x-0)^2 + b (y- y0)^2 – R^2, a circle or an ellipse enclosing red stars.

• Hari says:

Great article.. I think the below formula would give a new variable that help to separate the points in hyper plane

z = y – |x|

THANKS FOR EASY EXPLANATION

• Raghu says:

Useful article for Machine learners.. Why can’t you discuss about effect of kernel functions.

• Yamani says:

The explanation is really impressive. Can you also provide some information about how to determine the theoretical limits for the parameter’s optimal accuracy.

• harshel jain says:

how can we use SVM for regression? can someone please explain..

• asmae says:

hi please if you have an idiea about how it work for regression can you help me ?

• Diana says:

That was a really good explanation! thanks a lot. I read many explanations about SVM but this one help me to understand the basics which I really needed it.

• Manjunath GS says:

• Diptesh says:

This is very useful for understanding easily.

• Dan says:

just substitude x with |x|

• Min says:

Same goes with Diana. This really help me a lot to figure out things from basic. I hope you would also share any computation example using R provided with simple dataset, so that anyone can practice with their own after referring to your article.

I have a question, if i have time-series dataset containing mixed linear and nonlinear data, (for example oxygen saturation data ; SaO2), by using svm to do classification for diseased vs health subjects, do i have to separate those data into linear and non-linear fisrt, or can svm just performed the analysis without considering the differences between the linearity of those data?

Thanks a lot!

• anu says:

z

• Renny Varghese says:

Could you please explain how SVM works for multiple classes? How would it work for 9 classes? I used a function called multisvm here: http://www.mathworks.com/matlabcentral/fileexchange/39352-multi-class-svm
but I’m not sure how it’s working behind the scenes. Everything I’ve read online is rather confusing.

• lubna says:

NEW VARIABLE (Z) = SQRT(X) + SQRT (Y)

• Frank says:

Thanks for the great article. There are even cool shirts for anyone who became SVM fan 😉
http://www.redbubble.com/de/people/perceptron/works/24728522-support-vector-machines?grid_pos=2&p=t-shirt&style=mens

• bilashi says:

great explanation!! Thanks for posting it.

• arun says:

I think this is |X|

It is very nicely written and understandable.
Thanks a lot…

• Walter says:

z=ax^2 + by^2

nice explanations with scenarios and margin values

• lishanth says:

wow!!! excellent explanation..
only now i understood the concepts clearly
thanks a lot..

• anwar says:

(Z) = SQRT(X) + SQRT (Y)

• Kresla Matty says:

thanks, and well done for the good article

• Jonathan benitez says:

• Aishwarya Jangam says:

Great Explanation..Thanks..

• Hams says:

simple and refreshed the core concepts in just 5 mins! kudos Mr.Sunil

• Shashi says:

Best starters material for SVM, really appreciate the simple and comprehensive writing style. Expecting more such articles from you

• Ravindar says:

Z= square(x)

• Narasimha says:

Hey Sunil, Nice job of explaining it concisely and intuitively! Easy to follow and covers many aspects in a short space. Thanks!

• John Doe says:

Very well written – concise, clear, well-organized. Thank you.

Excellent explanation..Can you please also tell what are the parameter values one should start with – like C, gamma ..Also, again a very basic question.. Can we say that lesser the % of support vectors (count of SVs/total records) better my model/richer my data is- assuming the datasize to be the same.. Waiting for more on parameter tuning..Really appreciate the knowledge shared..

• Kirana says:

Hi could you please explain why SVM perform well on small dataset?

• Chris says:

Another nice kernel for the problem stated in the article is the radial basis kernel.

• […] 资源：阅读这篇文章来理解SVM support vector machines。 […]

• chiru says:

wow excellent

• Zhen Zhang says:

very appreciating for explaining

• Andrey says:

Nice tutorial. The new feature to separate data would be something like z = y – x^2 as most dots following the parabola will have lower z than stars.

Very intuitive explanation. Thank you!
Good to add SVM for Regression of Continuous variables.

• neha says:

this is so simple method that anyone can get easily thnx for that
but also explain the 4 senario of svm.

• Nirav Pingle says:

Great article for understanding of SVM:

But, When and Why do we use the SVM algorithm can anyone make that help me understand because until this thing is clear there may not be use of this article.

• Mostafa says:

It is one of best explanation of machine learning technique that i have seen!
and new variable:
i think Z=|x| and new Axis are Z and Y

• venkat says:

higher degree polynomial will separate the points in the problem,

• Tirthankar says:

I guess the required feature is z = x^2 / y^2
For the red points, z will be close to 1 but for the blue points z values will be significantly more than 1

• murtaza ali says:

amazing article no doubt! It makes me clear all the concept and deep points regarding SVM.
many thanks.

• katherine says:

The best explanation ever! Thank you!

• Rahul says:

z = x^2+y^2

• […] [1] Naive Bayes and Text Classification. [2]Naive Bayes by Example. [3] Andrew Ng explanation of Naive Bayes video 1 and video 2 [4] Please explain SVM like I am 5 years old. [5] Understanding Support Vector Machines from examples. […]

• roshan says:

new variable = ABS(Y)

• Robert says:

Man, I was looking for definition of SVM for my diploma, but I got interested in explanation part of this article. Keep up good work!

• Aman Goel says:

we can use ‘poly’ kernel with degree=2