Linear Discriminant Analysis: An Exploratory Guide
Linear Discriminant Analysis (LDA) is a statistical technique that is used to distinguish a set of objects or variables using linear combinations of their features. It finds its application in areas such as computer vision, image processing, and bioinformatics.
In this guide, we will discuss the fundamental concepts of LDA, the steps involved in the analysis, its practical uses, and examples of its applications.
Understanding Linear Discriminant Analysis
Linear Discriminant Analysis (LDA) is a multivariate statistical approach that aims to maximize the separation of classes in a data set. In other words, LDA is used to project high-dimensional data onto a lower-dimensional space, such that the separation of classes in the new space is maximized.
To understand better what LDA does, let us consider a simple example. Imagine we have two classes of points in two dimensions.
The goal of LDA is to find a line in this two-dimensional space that separates these two classes. The line should be such that it maximizes the distance between the two classes and minimizes the spread of data points within each class.
LDA is used to calculate the weights or coefficients that are multiplied to the original data to project the data onto the new line, which separates the two classes.
Steps involved in Linear Discriminant Analysis
There are three major steps involved in LDA. These are:
1. Computing class means and scatter matrices
2. Finding the linear discriminants
3. Projecting the data onto the new lower-dimensional space
Step 1: Computing class means and scatter matrices
The first step in LDA is to calculate the mean and scatter matrix for each class in the data set. The mean of each class is computed by taking the average of all features in that class.
The scatter matrix of each class is computed by calculating the covariance matrix of each class. The inverse of the scatter matrix is then calculated to be used later in the process.
Step 2: Finding the linear discriminants
The second step in LDA is to find the linear discriminants. Linear discriminants are computed by subtracting the mean of one class from the mean of the other class.
This difference is then multiplied by the inverse of the sum of both scatter matrices. The result is a vector that is perpendicular to the separating line between the two classes.
The number of linear discriminants is always one less than the number of classes.
Step 3: Projecting the data onto the new lower-dimensional space
The third and final step in LDA is to project the data onto the new lower-dimensional space.
The data is projected onto the line that is perpendicular to the separating line between the two classes. In this new lower-dimensional space, the distance between classes is maximized, and the spread of the data is minimized.
Applications of Linear Discriminant Analysis
LDA has a wide range of practical applications, including image recognition, character recognition, barcode scanning, and decision making. Below we discuss some of its applications.
Face Recognition
LDA is used in face recognition to project face images onto a lower-dimensional space, such that the distance between faces belonging to the same person is minimized, and the distance between faces belonging to different people is maximized. This method is effective in recognizing faces, and it is widely used in security systems.
Barcode Scanning
LDA is used in barcode scanning to separate the bars and spaces of different classes in barcode images. A linear discriminant is used to separate the bars from the spaces, allowing for effective scanning of the barcode.
Decision Making
LDA is used in decision making to classify data into different groups. The data is divided into training and test sets, and the linear discriminants are computed from the training set.
The test data is then projected onto the linear discriminants to see which class it belongs to.
Conclusion
In conclusion, LDA is a statistical technique used to find a separating line or plane between classes in a data set. It is commonly used in areas like computer vision and image processing to classify data and recognize patterns.
The three significant steps involved in LDA are computing class means and scatter matrices, finding linear discriminants, and projecting data onto the new lower-dimensional space. LDA has practical applications, such as face recognition, barcode scanning, and decision making.
Implementation of Linear Discriminant Analysis Algorithm in Python
Linear Discriminant Analysis (LDA) is a powerful statistical technique used for classification and discrimination analysis. In this section, we will explore how to implement LDA in Python by importing modules, loading datasets, assigning values for independent and dependent variables, splitting data into train and test sets, creating LDA models, checking accuracy, and plotting ROC curves for validation.
Importing Modules
The first step in implementing LDA in Python is to import necessary modules such as NumPy, Pandas, and Scikit-Learn. Numpy is used for scientific calculations; Pandas is used for data preparation, manipulation, and analysis, while Scikit-Learn is a machine learning library that provides efficient tools for data modeling, data preprocessing, and evaluation score functions.
import numpy as np
import pandas as pd
from sklearn.discriminant_analysis import LinearDiscriminantAnalysis as LDA
from sklearn.model_selection import train_test_split
from sklearn.metrics import roc_curve, auc
Loading Dataset
The second step is to load the dataset. In this example, we will use the preloaded breast cancer dataset from Scikit-Learn.
This dataset consists of 30 features and a binary target variable representing malignant or benign.
from sklearn.datasets import load_breast_cancer
data = load_breast_cancer()
X = data.data
Y = data.target
Assigning values for Independent and dependent variables respectively
The third step is to assign values for the independent variable (X) and dependent variable (Y). In this dataset, each feature represents an independent variable, and the target variable is the dependent variable.
Splitting data into Train and Test sets
The fourth step is to split the dataset into training and testing sets using Scikit-Learn’s train_test_split() function. We will use 80% of the dataset for training and 20% for testing.
X_train, X_test, Y_train, Y_test = train_test_split(X, Y, test_size=0.20, random_state=42)
Creating Linear Discriminant Analysis model and checking accuracy
The fifth step is to create the LDA model. We will fit the training data to the LDA model and predict the values for the test set.
lda = LDA()
lda.fit(X_train, Y_train)
Y_pred = lda.predict(X_test)
We can check the accuracy of the model using the score() function.
accuracy = lda.score(X_test, Y_test)
print("Accuracy:", accuracy)
Plotting ROC curve for validation
The last step of implementing LDA in Python is to plot the Receiver Operating Characteristics (ROC) curve for validation. The ROC curve displays the relationship between the true positive rate (TPR) and the false positive rate (FPR) of the classifier at varying thresholds.
fpr, tpr, thresholds = roc_curve(Y_test, Y_pred)
roc_auc = auc(fpr, tpr)
plt.figure(figsize=(6, 6))
plt.title("ROC Curve")
plt.plot(fpr, tpr, "b", label="AUC=%0.2f" % roc_auc)
plt.legend(loc="lower right")
plt.plot([0, 1], [0, 1], "r--")
plt.xlim([0, 1])
plt.ylim([0, 1])
plt.ylabel("True Positive Rate")
plt.xlabel("False Positive Rate")
plt.show()
Conclusion
In conclusion, we have provided a step-by-step guide on how to implement Linear Discriminant Analysis (LDA) in Python. We started by importing necessary modules, loading datasets, assigning values for independent and dependent variables, splitting data into train and test sets, creating LDA models, checking accuracy, and plotting ROC curves for validation.
By following these steps, data scientists and machine learning practitioners can develop accurate classification models using LDA and Scikit-Learn. In this article, we explored Linear Discriminant Analysis (LDA) – a statistical technique used for classification and discrimination analysis.
LDA is used to find a separating line or plane between classes in a data set and maximizes class separation. We provided an explanatory guide on the concept of LDA and also outlined steps involved in implementing LDA in Python for data analysis.
By following these steps, data scientists and machine learning practitioners can develop accurate classification models using LDA and Scikit-Learn. Overall, LDA is a powerful tool with practical applications in diverse fields like computer vision, image processing, and bioinformatics, among others, and it is vital for achieving better data classification performance.