Principal Component Analysis On Matrix Using Python

Machine learning algorithms may take a lot of time working with large datasets. To overcome this a new dimensional reduction technique was introduced. If the input dimension is high Principal Component Algorithm can be used to speed up our machines. It is a projection method while retaining the features of the original data.

In this article, we will discuss the basic understanding of Principal Component(PCA) on matrices with implementation in python. Further, we implement this technique by applying one of the classification techniques.

Dataset

The dataset can be downloaded from the following link. The dataset gives the details of breast cancer patients. It has 32 features with 569 rows.

Let’s get started.Import all the libraries required for this project.

import matplotlib.pyplot as plt
import pandas as pd
import numpy as np
import seaborn as sns
%matplotlib inline

Loading the dataset 

dataset = pd.read_csv('cancerdataset.csv') 
dataset["diagnosis"]=dataset["diagnosis"].map({'M': 1, 'B': 0})
data=dataset.iloc[:,0:-1]
data.head()

We need to store the independent and dependent variables by using the iloc method.

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X = data.iloc[:, 2:].values 
y = data.iloc[:, 1].values 

Split the training and testing data in the 80:20 ratio.

from sklearn.model_selection import train_test_split 
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size = 0.2, random_state = 0) 

PCA Standardization

PCA can only be applied to numerical data. So,it is important to convert all the data into numerical format. We need to standardize data for converting features of different units to the same unit.

from sklearn.preprocessing import StandardScaler
from sklearn.preprocessing import StandardScaler 
sc = StandardScaler()   
X_train = sc.fit_transform(X_train) 
X_test = sc.transform(X_test) 

Covariance Matrix

Based on standardized data we will build the covariance matrix. It gives the variance between each feature in our original dataset. The negative value in the result below represents are inversely dependent on each other.

mean_vec=np.mean(X_train,axis=0)
cov_mat=(X_train-mean_vec).T.dot((X_train-mean_vec))/(X_train.shape[0]-1)
mean_vect=np.mean(X_test,axis=0)
cov_matt=(X_test-mean_vec).T.dot((X_test-mean_vec))/(X_test.shape[0]-1)
print(cov_mat)

Eigen Decomposition on Covariance Matrix

Each eigenvector will have an eigenvalue and sum of the eigenvalues represent the variance in the dataset. We can get the location of maximum variance by calculating eigenvalue. The eigenvector with lowest eigenvalue will give the lowest amount of variation in the dataset. These values need to be dropped off.

cov_mat=np.cov(X_train.T)
eig_vals,eig_vecs=np.linalg.eig(cov_mat)
cov_matt=np.cov(X_test.T)
eig_vals,eig_vecs=np.linalg.eig(cov_mat)
print(eig_vals)
print(eig_vecs)

We need to specify how many components we want to keep. The result gives a reduction of dimension from 32 to 2 features. The first and second PCA will capture the most variance in the original dataset.

from sklearn.decomposition import PCA
from sklearn.decomposition import PCA 
pca = PCA(n_components = 2) 
X_train = pca.fit_transform(X_train) 
X_test = pca.transform(X_test) 
X_train.shape

pca.components_

In this matrix array, each column represents the original data, and each row represents a PCA.

Fitting DecisionTree Regression To the training set 

As we are solving a classification problem, we can use the Decision Tree Classifier for model prediction.

from sklearn.tree import DecisionTreeClassifier   
# Create Decision Tree classifier object
clf = DecisionTreeClassifier()
# Train Decision Tree Classifier
clf = clf.fit(X_train,y_train)
#Predict the response for test dataset
y_pred = clf.predict(X_test)

Evaluating the Algorithm

For classification tasks, we will use a confusion matrix to check the accuracy of our machine learning model.

from sklearn.metrics import confusion_matrix 
confusion = pd.crosstab(y_test, y_pred, rownames=['Actual'], colnames=['Predicted'], margins=True)
confusion

Plot the training set  

from matplotlib.colors import ListedColormap 
  X1, y1 = X_train, y_train 
a, b = np.meshgrid(np.arange(start = X1[:, 0].min() - 1, 
                     stop = X1[:, 0].max() + 1, step = 0.01), 
                     np.arange(start = X1[:, 1].min() - 1, 
                     stop = X1[:, 1].max() + 1, step = 0.01)) 
plt.contourf(a, b, clf.predict(np.array([a.ravel(), 
             b.ravel()]).T).reshape(a.shape), alpha = 0.75, 
             cmap = ListedColormap(('white'))) 
plt.xlim(a.min(), a.max()) 
plt.ylim(X2.min(), X2.max()) 
for i, j in enumerate(np.unique(y_set)): 
    plt.scatter(X1[y1 == j, 0], X1[y1 == j, 1], 
                c = ListedColormap(('red','blue'))(i), label = j) 
plt.title('Decision Tree') 
plt.xlabel('PC1') # for Xlabel 
plt.ylabel('PC2') # for Ylabel 
plt.legend() # to show legend 
# show scatter plot 
plt.show()  

Final Thoughts

In the above article, we discussed how PCA is used for dimension reduction of large dataset. Further we have explored concepts like covariance matrix and eigen decomposition for calculating a principal component. Hope this article is useful to you.