teknomo/SVM+in+Python.py

## SVM+in+Python.py

# coding: utf-8

# # SVM in Python
#
# by *Kardi Teknomo*
#
# last update: June 2017

# This is a practice/laboratory session of SVM tutorial using Python.
#
# First, you need to import the necessary modules.

# In[1]:

import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from sklearn import svm
import sklearn.metrics as metric
get_ipython().magic('matplotlib inline')


# # Linearly Separable Dataset
#
# Our first dataset can be uploaded. Put the CSV files in the same folder as the Jupyter notebook.

# ## Data

# In[2]:

filedata='SVM_Dataset1.csv'
data1=pd.read_csv(filedata)
data1


# We separate the X training data from the y training data

# In[3]:

X1=data1['X1']
X2=data1['X2']
X_training=np.array(list(zip(X1,X2)))
X_training


# In[4]:

y_training=data1['y']
y_training


# Let us plot this data. Can you imagine a line separating the two classes?

# In[5]:

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)
plt.xlabel('X1')
plt.ylabel('X2');
plt.savefig('chart0.png')


# ## SVM Training

# To use Vector Support Classification (SVC) algorithm, we need define the model **kernel**. Let us use  *linear kernel*. Then, we use the **fit()** function to train the model with our training data.
#

# In[6]:

svc = svm.SVC(kernel='linear').fit(X_training,y_training)
svc


# In[7]:

target_names=['-1','+1']
target_names


# The trained model can be plotted with specifying the decision_function() method.

# In[8]:

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(target_names,loc=2)

X,Y = np.mgrid[1:4:100j,2:8:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)
plt.contour(X,Y,Z,colors=['k'], linestyles=['-'],levels=[0])

plt.title('Linearly Separable')
plt.savefig('chart1.png')


# The following plot show the margin and the support vectors

# In[9]:

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(target_names,loc=2)

X,Y = np.mgrid[1:4:100j,2:8:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('Margin and Support Vectors')
plt.savefig('chart2.png')


# ## Training Performances

# To get the normalize accuracy, of the training, we can use **score(X,y)** function.

# In[10]:

svc.score(X_training, y_training)


# Alternatively, if you have test sample, you can also use the metric from sklearn. To use this on the training sample, we first need to define the y-prediction (which is based on the prediction of the model with X comes from the training sample) and the y-true value (which is based on the y of the training sample).

# In[11]:

y_pred=svc.predict(X_training)
y_pred


# In[12]:

y_true = y_training
y_true


# The absolute accuracy is measured as follow.

# In[13]:

metric.accuracy_score(y_true, y_pred, normalize=False)


# Confusion matrix is useful to see if there is misclassification. If there is no missclassification, then the corect values would be in the diagonal.

# In[14]:

cnf_matrix=metric.confusion_matrix(y_true, y_pred)
cnf_matrix


# We can also visualize the confusion matrix through the following function

# In[15]:

import itertools
def plot_confusion_matrix(cm, classes,
                          normalize=False,
                          title='Confusion matrix',
                          cmap=plt.cm.Blues):
    """
    This function prints and plots the confusion matrix.
    Normalization can be applied by setting `normalize=True`.
    """
    plt.imshow(cm, interpolation='nearest', cmap=cmap)
    plt.title(title)
    plt.colorbar()
    tick_marks = np.arange(len(classes))
    plt.xticks(tick_marks, classes, rotation=45)
    plt.yticks(tick_marks, classes)

    if normalize:
        cm = cm.astype('float') / cm.sum(axis=1)[:, np.newaxis]
        print("Normalized confusion matrix")
    else:
        print('Confusion matrix, without normalization')

    print(cm)

    thresh = cm.max() / 2.
    for i, j in itertools.product(range(cm.shape[0]), range(cm.shape[1])):
        plt.text(j, i, cm[i, j],
                 horizontalalignment="center",
                 color="white" if cm[i, j] > thresh else "black")

    plt.tight_layout()
    plt.ylabel('True label')
    plt.xlabel('Predicted label')


plt.figure()
plot_confusion_matrix(cnf_matrix, classes=target_names, normalize=False)
plt.show()


# ### SVM Prediction
#
# Now we can also use the trained SVM to predict something that is outside the training data. Let us predict the class y of the given test data [X1, X2] = [3, 6]

# In[16]:

svc.predict([[3,6]])


# The test data is now plotted.

# In[17]:

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.scatter(3,6,c='r',marker='s',s=90)
plt.legend(['-1','+1','prediction'],loc=2)

X,Y = np.mgrid[1:4:100j,2:8:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)
plt.contour(X,Y,Z,colors=['k'], linestyles=['-'],levels=[0])

plt.title('Prediction')
plt.savefig('chart3.png')


# In[18]:

del X1, X2, X_training, y_training


# # Non Linearly Separable Dataset
#
# ## Data
# Now we upload the second dataset.

# In[19]:

filedata='SVM_Dataset2.csv'
data2=pd.read_csv(filedata)
data2


# We separate the X training data from the y training data

# In[20]:

X1=data2['x1']
X2=data2['x2']
X_training=np.array(list(zip(X1,X2)))
X_training


# In[21]:

y_training=data2['y']
y_training


# Let us plot this data. Can you imagine a line separating the two classes?

# In[22]:

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(target_names,loc=3)
plt.xlabel('X1')
plt.ylabel('X2');
plt.savefig('nl-chart0.png')


# ## Training

# Since the plot show that no line can separate the two classes. If we still want to use linear kernel, we can define a regularization cost parameter **C**. We use the **fit()** function to train the model with our training data. Feel free to change the regularization parameter to make such that the error of classification would be minimum.

# In[23]:

svc = svm.SVC(kernel='linear',C=1).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(target_names,loc=3)

X,Y = np.mgrid[2:18:100j,4:22:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)
plt.contour(X,Y,Z,colors=['k'], linestyles=['-'],levels=[0])

plt.title('Non-Linearly Separable')
plt.savefig('nl-chart1.png')


# In[24]:

svc = svm.SVC(kernel='linear',C=100).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)

X,Y = np.mgrid[2:18:100j,4:22:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('Linear Kernel with regularization')
plt.savefig('nl-chart-regularization.png')

svc.score(X_training,y_training)


# ## Training Performance
#

# In[25]:

svc.score(X_training,y_training)


# In[26]:

y_pred=svc.predict(X_training)
y_true = y_training
metric.accuracy_score(y_true, y_pred, normalize=False)


# Using linear kernel, we found 4 support vectors but one out of 20 data is in the wrong side. Thus the accuracy is 19 / 20 * 100 = 94.99%.
#
# It seems we are stuck with one misclassification.
#
#
# ## Training with Polynomial Kernel
#
# Now it is the time to change the kernel into non-linear kernel. Let us try to use polynomial kernel.
#
# There is no misclassification if the degree = 2 and above.
#
# You can play with the degree and regularization parameter C.
# + Will the number support vectors increase if you increase the degree?
# + Will the number support vectors increase if you increase the regularization parameter?
#

# In[27]:

svc = svm.SVC(kernel='poly',C=1, degree=2).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)

X,Y = np.mgrid[2:17:100j,0:21:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('Polynomial Kernel')
plt.savefig('nl-poly2-kernel.png')

svc.score(X_training,y_training)


# ## Overfitting
#
# Just to give rough idea of overfitting, now let us use RBF kernel.
#

# In[28]:

svc = svm.SVC(kernel='rbf',C=1, gamma=3).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)

X,Y = np.mgrid[2:17:100j,0:21:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('RBF Kernel gamma 1')
plt.savefig('nl-RBF1-kernel.png')

svc.score(X_training,y_training)


# In[29]:

y_pred=svc.predict(X_training)
y_true = y_training
metric.accuracy_score(y_true, y_pred, normalize=False)


# The accuracy of the training has reached 100% (20/20) due to the RBF kernel. However, look at the support vectors. The whole dataset becomes the support vectors and it is an indication that the RBF model overfit our data. Adding more data may need to change the model.
#
# Thus, in our example the Polynomial model with lower degree (degree 2) works better.

# # One Dimensional SVM
#
# This problem at first may seem to be simpler problem. We have X in one dimension. However, the challenge is the SVM training in scikit-learn does not accept 1D training data.
#
# ## Data

# In[30]:

filedata='SVM_Dataset3.csv'
data3=pd.read_csv(filedata)
data3


# The actual X data is just one dimension. The training and scatter plot will not work.
#
# To make it works, we need to add dummy X2 such that we can plot using scatter plot and train using SVM.

# In[31]:

X1=data3['x']
X2=np.ones((len(X),1),int)
X_training=np.array(list(zip(X1,X2)))
X_training


# In[32]:

y_training=data3['y']
y_training


# In[33]:

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=3)
plt.xlabel('X1')
plt.ylabel('X2');
plt.savefig('data3-chart0.png')


# ## SVM Training and Performance
#
# Now let us do the SVM training to this dataset and plot

# In[34]:

svc = svm.SVC(kernel='poly',C=1, degree=2).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)

X,Y = np.mgrid[-2:14:100j,0.98:1.02:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('Polynomial Kernel')
plt.savefig('data3-Poly-kernel.png')

svc.score(X_training,y_training)


# In[35]:

svc = svm.SVC(kernel='poly',C=1, degree=3).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)

X,Y = np.mgrid[-2:14:100j,0.98:1.02:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('Polynomial Kernel')
plt.savefig('data3-Poly-kernel.png')

svc.score(X_training,y_training)


# We have three support vector for Polynomial degree 3 and training accuracy of 100%. This should be the best model.

# ## Overfitting
#
# Just for curiosity, we can also try to use RBF kernel

# In[36]:

svc = svm.SVC(kernel='rbf',C=1, gamma=3).fit(X_training,y_training)

idxPlus=y_training[y_training<0].index
idxMin=y_training[y_training>0].index
plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
plt.legend(['-1','+1'],loc=2)

X,Y = np.mgrid[-2:14:100j,0.98:1.02:100j]
Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
Z = Z.reshape(X.shape)
plt.contourf(X,Y,Z > 0,alpha=0.4)

plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

plt.title('RBF Kernel')
plt.savefig('data3-RBF-kernel.png')

svc.score(X_training,y_training)


# Just as before, the accuracy is 100% but the whole data sets now become support vectors. It is a clear sign of overfitting. That means the model memorize the data rather than generalizing it. Thus, we should avoid to use RBF for this dataset.
#
# The support vectors in SVM are the quality data that we can use to generate the decision boundary (of the same model). Non-support vector data can be ignored, regardless how many data that you have. That is one of the strengths of SVM.

# Visit www.Revoledu.com for more tutorials in Data Science

	# coding: utf-8

	# # SVM in Python
	#
	# by Kardi Teknomo
	#
	# last update: June 2017

	# This is a practice/laboratory session of SVM tutorial using Python.
	#
	# First, you need to import the necessary modules.

	# In[1]:

	import numpy as np
	import pandas as pd
	import matplotlib.pyplot as plt
	from sklearn import svm
	import sklearn.metrics as metric
	get_ipython().magic('matplotlib inline')


	# # Linearly Separable Dataset
	#
	# Our first dataset can be uploaded. Put the CSV files in the same folder as the Jupyter notebook.

	# ## Data

	# In[2]:

	filedata='SVM_Dataset1.csv'
	data1=pd.read_csv(filedata)
	data1


	# We separate the X training data from the y training data

	# In[3]:

	X1=data1['X1']
	X2=data1['X2']
	X_training=np.array(list(zip(X1,X2)))
	X_training


	# In[4]:

	y_training=data1['y']
	y_training


	# Let us plot this data. Can you imagine a line separating the two classes?

	# In[5]:

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)
	plt.xlabel('X1')
	plt.ylabel('X2');
	plt.savefig('chart0.png')


	# ## SVM Training

	# To use Vector Support Classification (SVC) algorithm, we need define the model kernel. Let us use linear kernel. Then, we use the fit() function to train the model with our training data.
	#

	# In[6]:

	svc = svm.SVC(kernel='linear').fit(X_training,y_training)
	svc


	# In[7]:

	target_names=['-1','+1']
	target_names


	# The trained model can be plotted with specifying the decision_function() method.

	# In[8]:

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(target_names,loc=2)

	X,Y = np.mgrid[1:4:100j,2:8:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)
	plt.contour(X,Y,Z,colors=['k'], linestyles=['-'],levels=[0])

	plt.title('Linearly Separable')
	plt.savefig('chart1.png')


	# The following plot show the margin and the support vectors

	# In[9]:

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(target_names,loc=2)

	X,Y = np.mgrid[1:4:100j,2:8:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('Margin and Support Vectors')
	plt.savefig('chart2.png')


	# ## Training Performances

	# To get the normalize accuracy, of the training, we can use score(X,y) function.

	# In[10]:

	svc.score(X_training, y_training)


	# Alternatively, if you have test sample, you can also use the metric from sklearn. To use this on the training sample, we first need to define the y-prediction (which is based on the prediction of the model with X comes from the training sample) and the y-true value (which is based on the y of the training sample).

	# In[11]:

	y_pred=svc.predict(X_training)
	y_pred


	# In[12]:

	y_true = y_training
	y_true


	# The absolute accuracy is measured as follow.

	# In[13]:

	metric.accuracy_score(y_true, y_pred, normalize=False)


	# Confusion matrix is useful to see if there is misclassification. If there is no missclassification, then the corect values would be in the diagonal.

	# In[14]:

	cnf_matrix=metric.confusion_matrix(y_true, y_pred)
	cnf_matrix


	# We can also visualize the confusion matrix through the following function

	# In[15]:

	import itertools
	def plot_confusion_matrix(cm, classes,
	normalize=False,
	title='Confusion matrix',
	cmap=plt.cm.Blues):
	"""
	This function prints and plots the confusion matrix.
	Normalization can be applied by setting `normalize=True`.
	"""
	plt.imshow(cm, interpolation='nearest', cmap=cmap)
	plt.title(title)
	plt.colorbar()
	tick_marks = np.arange(len(classes))
	plt.xticks(tick_marks, classes, rotation=45)
	plt.yticks(tick_marks, classes)

	if normalize:
	cm = cm.astype('float') / cm.sum(axis=1)[:, np.newaxis]
	print("Normalized confusion matrix")
	else:
	print('Confusion matrix, without normalization')

	print(cm)

	thresh = cm.max() / 2.
	for i, j in itertools.product(range(cm.shape[0]), range(cm.shape[1])):
	plt.text(j, i, cm[i, j],
	horizontalalignment="center",
	color="white" if cm[i, j] > thresh else "black")

	plt.tight_layout()
	plt.ylabel('True label')
	plt.xlabel('Predicted label')


	plt.figure()
	plot_confusion_matrix(cnf_matrix, classes=target_names, normalize=False)
	plt.show()


	# ### SVM Prediction
	#
	# Now we can also use the trained SVM to predict something that is outside the training data. Let us predict the class y of the given test data [X1, X2] = [3, 6]

	# In[16]:

	svc.predict([[3,6]])


	# The test data is now plotted.

	# In[17]:

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.scatter(3,6,c='r',marker='s',s=90)
	plt.legend(['-1','+1','prediction'],loc=2)

	X,Y = np.mgrid[1:4:100j,2:8:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)
	plt.contour(X,Y,Z,colors=['k'], linestyles=['-'],levels=[0])

	plt.title('Prediction')
	plt.savefig('chart3.png')


	# In[18]:

	del X1, X2, X_training, y_training


	# # Non Linearly Separable Dataset
	#
	# ## Data
	# Now we upload the second dataset.

	# In[19]:

	filedata='SVM_Dataset2.csv'
	data2=pd.read_csv(filedata)
	data2


	# We separate the X training data from the y training data

	# In[20]:

	X1=data2['x1']
	X2=data2['x2']
	X_training=np.array(list(zip(X1,X2)))
	X_training


	# In[21]:

	y_training=data2['y']
	y_training


	# Let us plot this data. Can you imagine a line separating the two classes?

	# In[22]:

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(target_names,loc=3)
	plt.xlabel('X1')
	plt.ylabel('X2');
	plt.savefig('nl-chart0.png')


	# ## Training

	# Since the plot show that no line can separate the two classes. If we still want to use linear kernel, we can define a regularization cost parameter C. We use the fit() function to train the model with our training data. Feel free to change the regularization parameter to make such that the error of classification would be minimum.

	# In[23]:

	svc = svm.SVC(kernel='linear',C=1).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(target_names,loc=3)

	X,Y = np.mgrid[2:18:100j,4:22:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)
	plt.contour(X,Y,Z,colors=['k'], linestyles=['-'],levels=[0])

	plt.title('Non-Linearly Separable')
	plt.savefig('nl-chart1.png')


	# In[24]:

	svc = svm.SVC(kernel='linear',C=100).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)

	X,Y = np.mgrid[2:18:100j,4:22:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('Linear Kernel with regularization')
	plt.savefig('nl-chart-regularization.png')

	svc.score(X_training,y_training)


	# ## Training Performance
	#

	# In[25]:

	svc.score(X_training,y_training)


	# In[26]:

	y_pred=svc.predict(X_training)
	y_true = y_training
	metric.accuracy_score(y_true, y_pred, normalize=False)


	# Using linear kernel, we found 4 support vectors but one out of 20 data is in the wrong side. Thus the accuracy is 19 / 20 * 100 = 94.99%.
	#
	# It seems we are stuck with one misclassification.
	#
	#
	# ## Training with Polynomial Kernel
	#
	# Now it is the time to change the kernel into non-linear kernel. Let us try to use polynomial kernel.
	#
	# There is no misclassification if the degree = 2 and above.
	#
	# You can play with the degree and regularization parameter C.
	# + Will the number support vectors increase if you increase the degree?
	# + Will the number support vectors increase if you increase the regularization parameter?
	#

	# In[27]:

	svc = svm.SVC(kernel='poly',C=1, degree=2).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)

	X,Y = np.mgrid[2:17:100j,0:21:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('Polynomial Kernel')
	plt.savefig('nl-poly2-kernel.png')

	svc.score(X_training,y_training)


	# ## Overfitting
	#
	# Just to give rough idea of overfitting, now let us use RBF kernel.
	#

	# In[28]:

	svc = svm.SVC(kernel='rbf',C=1, gamma=3).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)

	X,Y = np.mgrid[2:17:100j,0:21:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('RBF Kernel gamma 1')
	plt.savefig('nl-RBF1-kernel.png')

	svc.score(X_training,y_training)


	# In[29]:

	y_pred=svc.predict(X_training)
	y_true = y_training
	metric.accuracy_score(y_true, y_pred, normalize=False)


	# The accuracy of the training has reached 100% (20/20) due to the RBF kernel. However, look at the support vectors. The whole dataset becomes the support vectors and it is an indication that the RBF model overfit our data. Adding more data may need to change the model.
	#
	# Thus, in our example the Polynomial model with lower degree (degree 2) works better.

	# # One Dimensional SVM
	#
	# This problem at first may seem to be simpler problem. We have X in one dimension. However, the challenge is the SVM training in scikit-learn does not accept 1D training data.
	#
	# ## Data

	# In[30]:

	filedata='SVM_Dataset3.csv'
	data3=pd.read_csv(filedata)
	data3


	# The actual X data is just one dimension. The training and scatter plot will not work.
	#
	# To make it works, we need to add dummy X2 such that we can plot using scatter plot and train using SVM.

	# In[31]:

	X1=data3['x']
	X2=np.ones((len(X),1),int)
	X_training=np.array(list(zip(X1,X2)))
	X_training


	# In[32]:

	y_training=data3['y']
	y_training


	# In[33]:

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=3)
	plt.xlabel('X1')
	plt.ylabel('X2');
	plt.savefig('data3-chart0.png')


	# ## SVM Training and Performance
	#
	# Now let us do the SVM training to this dataset and plot

	# In[34]:

	svc = svm.SVC(kernel='poly',C=1, degree=2).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)

	X,Y = np.mgrid[-2:14:100j,0.98:1.02:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('Polynomial Kernel')
	plt.savefig('data3-Poly-kernel.png')

	svc.score(X_training,y_training)


	# In[35]:

	svc = svm.SVC(kernel='poly',C=1, degree=3).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)

	X,Y = np.mgrid[-2:14:100j,0.98:1.02:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('Polynomial Kernel')
	plt.savefig('data3-Poly-kernel.png')

	svc.score(X_training,y_training)


	# We have three support vector for Polynomial degree 3 and training accuracy of 100%. This should be the best model.

	# ## Overfitting
	#
	# Just for curiosity, we can also try to use RBF kernel

	# In[36]:

	svc = svm.SVC(kernel='rbf',C=1, gamma=3).fit(X_training,y_training)

	idxPlus=y_training[y_training<0].index
	idxMin=y_training[y_training>0].index
	plt.scatter(X_training[idxPlus,0],X_training[idxPlus,1],c='b',s=50)
	plt.scatter(X_training[idxMin,0],X_training[idxMin,1],c='r',s=50)
	plt.legend(['-1','+1'],loc=2)

	X,Y = np.mgrid[-2:14:100j,0.98:1.02:100j]
	Z = svc.decision_function(np.c_[X.ravel(),Y.ravel()])
	Z = Z.reshape(X.shape)
	plt.contourf(X,Y,Z > 0,alpha=0.4)

	plt.contour(X,Y,Z,colors=['k','k','k'], linestyles=['--','-','--'],levels=[-1,0,1])
	plt.scatter(svc.support_vectors_[:,0],svc.support_vectors_[:,1],s=120,facecolors='none')
	plt.scatter(X_training[:,0],X_training[:,1],c=y_training,s=50,alpha=0.95);

	plt.title('RBF Kernel')
	plt.savefig('data3-RBF-kernel.png')

	svc.score(X_training,y_training)


	# Just as before, the accuracy is 100% but the whole data sets now become support vectors. It is a clear sign of overfitting. That means the model memorize the data rather than generalizing it. Thus, we should avoid to use RBF for this dataset.
	#
	# The support vectors in SVM are the quality data that we can use to generate the decision boundary (of the same model). Non-support vector data can be ignored, regardless how many data that you have. That is one of the strengths of SVM.

	# Visit www.Revoledu.com for more tutorials in Data Science