Key Points

• Linear discriminant analysis (LDA) is a supervised learning technique that can be used for classification, dimensionality reduction, feature extraction, clustering, or visualization.
• It assumes that the data follow a multivariate normal distribution with class-specific parameters and estimates the prior probabilities, the means, and the covariance matrices of each class from the data.
• It computes the linear discriminant function, a linear combination of features that maximizes the ratio of the between-class variance to the within-class variance.
• It projects the data onto the linear discriminant. It obtains the LDA score for each data point, which indicates its position and distance from the discriminant function and the likelihood of each data point belonging to each class.
• It classifies the data points based on the LDA score and the posterior probability of each class. It evaluates the model’s performance using various metrics, such as the confusion matrix, the accuracy, the error rate, or the ROC curve.

Table of Contents

What is Linear Discriminant Analysis (LDA), and how does it work?

Linear Discriminant Analysis (LDA) is a linear model for classification and dimensionality reduction. The algorithm finds a linear combination of features that best separates the classes in a dataset, a key step in solving the binary classification problem. 

It maximizes the ratio of between-class variance to within-class variance and works well for linearly separable data. LDA is a generalization of Fisher’s linear discriminant (FLD) method developed by Sir Ronald Fisher in 1936. LDA extends FLD to handle multiple classes of data.

LDA is useful for various purposes, such as feature extraction, data preprocessing, face detection, pattern recognition, machine learning, and data visualization. LDA can also be combined with other methods, such as eigenfaces, kernel methods, neural networks, or ensemble methods, to improve the analysis’s performance and flexibility.

How does LDA work?

Suppose we have a dataset of two class points in a two-dimensional space. Our goal is to find a line that can separate the two classes as well as possible. One way to do this is to use FLD, a method developed by Sir Ronald Fisher, and finds a line that maximizes the distance between the means of the two classes.

What if we have more than two classes?

However, FLD has a limitation: it can only handle two data classes. What if we have more than two classes? It is where LDA comes in. LDA is a generalization of FLD, which extends it to handle multiple data classes. LDA finds a line that maximizes the between-class and within-class variance ratio, as shown in the figure below. 

The between-class variance measures how far apart the means of the classes are, while the within-class variance measures how spread out the points within each class are. By maximizing the ratio of between-class variance to within-class variance, LDA tries to find a line that separates the classes as much as possible while minimizing the overlap and dispersion within each class.

Linear Discriminant Analysis Mathematical Formula

Mathematically, LDA can be formulated as follows. Let X be an n×p matrix of n observations with p features, and y be a vector of n class labels. Let K be the number of classes and nk​ be the number of observations in class k. Let μk​ be the mean vector of class k, and let μ be the mean vector of the data. Let SW​ be the within-class scatter matrix, and let SB​ be the between-class scatter matrix, defined as:

It can be solved by finding the eigenvectors and eigenvalues of SW-1 Sb and choosing the eigenvector corresponding to the largest eigenvalue. This eigenvector is the optimal w that maximizes the objective function. In principal component analysis, the data projection onto the discriminant vector is denoted as the LDA score; this can be used for classification or as a dimensionality reduction technique.

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LDA Assumptions 

These assumptions are important to ensure that LDA can produce accurate and reliable results. However, LDA can also perform well even if some assumptions are violated, depending on the data and the problem.

• The values of each predictor variable follow a normal distribution. If we created a histogram to visualize the data distribution for a particular predictor, it would essentially follow the “bell shape.”
• All predictor variables have the same variance. It is rarely the case in real-world data, thus we usually scale each variable to have the same mean and variance before building an LDA model.
• The features are independent of each other. It means that there is no correlation or dependence between the features and that the covariance matrix of the data is diagonal.
• The classes are linearly separable. It means that a linear decision boundary can accurately classify the different classes.

These assumptions are important to ensure that analysis can produce accurate and reliable results. However, it can also perform well even if some assumptions are violated, depending on the data and the problem. It can also be modified to handle cases where the number of features exceeds the number of observations, called the small sample size (SSS) problem. It can be done using regularization techniques, such as shrinkage or penalization.

What are the applications of LDA?

LDA can be used for various purposes, such as feature extraction, data preprocessing, face detection, pattern recognition, machine learning, and data visualization. LDA can also be combined with other methods, such as eigenfaces, kernel methods, neural networks, or ensemble methods, to improve the analysis’s performance and flexibility. Quadratic discriminant analysis, or LDA, can be used for data analysis in various real-world examples and scenarios, including classification problems.

Feature extraction

It can be used to reduce the dimensionality of the data while preserving the class information. It can improve the subsequent analysis’s efficiency and accuracy, such as classification or clustering. It can extract the most discriminative features from a high-dimensional dataset of images, texts, or sounds and use them for recognition or classification tasks.

Data preprocessing

It can transform the data into a more suitable form for the analysis, such as scaling, centering, or normalizing. It can help remove the noise and the outliers and enhance the signal and the data structure. It can be used to preprocess the data before applying other methods, such as PCA, logistic regression, or neural networks, and to improve the performance and stability of these methods.

Face detection

It can detect and recognize faces in images or videos by finding the optimal projection that separates the faces from the background and the faces from each other. It can help identify and verify the identity of persons and perform tasks such as face recognition, face verification, face tracking, or face alignment. It detects and recognizes faces in security systems, social media, or biometric applications.

Pattern recognition

It can recognize and classify patterns in data, such as shapes, colors, textures, or motions. It can help discover and understand the underlying structure and the meaning of the data and perform tasks such as object recognition, scene recognition, gesture recognition, or activity recognition. It recognizes and classifies patterns in computer vision, natural language processing, or speech processing applications.

Machine learning

It can be used to learn and train models from data, such as classifiers, regressors, or predictors. It can help predict and estimate the outcomes and behaviors of the data and perform tasks such as classification, regression, prediction, or recommendation. It trains models in machine, deep, or reinforcement learning applications.

Quadratic discriminant analysis

It can project the input data onto a lower-dimensional space for simplified viewing, like a line, plane, or sphere. It can help to explore and analyze the data and to present and convey the results and the insights of the analysis, such as the key findings, the patterns, or the trends. It can visualize and communicate data in data science, business intelligence, or data journalism applications.

Advantages and disadvantages of LDA

LDA has advantages and disadvantages over other statistical techniques, such as ANOVA, regression analysis, logistic regression, probit regression, PCA, and factor analysis. 

Advantages

• It is simple and intuitive to understand and implement and has a solid theoretical foundation and interpretation.
• It is robust, efficient, and can handle large, complex, noisy, and missing data.
• It is flexible and versatile and can be applied to various types and domains of data, as well as different tasks and objectives of analysis.
• It is powerful and effective and can achieve high performance and accuracy, as well as low error rate and complexity.

Disadvantages

• It is sensitive and dependent on the model’s assumptions and parameters, such as normal distribution, identical covariance matrices, independence of features, prior probabilities, scaling, or regularization.
• It may produce inaccurate or misleading results if these assumptions or parameters are not met or chosen properly.
• It is linear and parametric and may not be able to capture the non-linear and non-parametric relationships and patterns in the data.
• It may also suffer from the curse of dimensionality, which means that the performance and complexity may deteriorate as the number of features or classes increases.
• It is supervised and discriminative and requires labeled data and class information for the analysis. LDA may not be suitable for unsupervised or generative tasks, such as clustering, density estimation, or anomaly detection.

I have used LDA for data analysis many times and found it to be a very useful and effective tool. I have also faced challenges and difficulties with LDA, such as choosing the right assumptions and parameters, dealing with non-linear and high-dimensional data, or finding enough labeled data. However, I have also learned some solutions and strategies to overcome these challenges, such as using regularization, kernel methods, neural networks, or ensemble methods.

Comparison of LDA with other models

Model	Accuracy	Complexity	Interpretability	Scalability	Robustness
LDA	High if the assumptions are met, moderate otherwise	Low, linear model with few parameters	High, coefficients and scores can be easily interpreted	High, fast and efficient algorithm	Low, sensitive to outliers and noise
Logistic regression	High for binary problems, moderate for multiclass problems	Low to moderate, depending on the number of features and regularization	Moderate, coefficients can be interpreted as log-odds ratios	High, fast and efficient algorithm	Moderate, can handle some outliers and noise
Support vector machines	High for both linear and non-linear problems	Moderate to high, depending on the kernel and the number of support vectors	Low, coefficients and kernels are hard to interpret	Moderate to low, slow and memory-intensive algorithm	High, can handle outliers and noise
Decision trees	Moderate to high, depending on the depth and the pruning	Low to high, depending on the depth and the number of nodes	High, rules and splits can be easily interpreted	Moderate, fast but memory-intensive algorithm	Moderate, can handle some outliers and noise
Neural networks	High for both linear and non-linear problems	High, depending on the number of layers, nodes, and weights	Low, weights and activations are hard to interpret	Low, slow and memory-intensive algorithm	High, can handle outliers and noise

How do you implement LDA using different software tools?

LDA can be implemented using various software tools like MATLAB, SPSS, SAS, R, and Python and offers libraries to run both quadratic discriminant and linear regression analysis. These tools provide functions and packages that can perform LDA and related tasks, such as lda(), MASS, sklearn, caret, mclust, or e1071. These tools also provide different options and parameters to customize the analysis, such as prior probabilities, scaling, cross-validation, plotting, or diagnostics.

R

R is a free and open-source software environment for statistical computing and graphics. R has a rich and diverse collection of functions and packages for data analysis and machine learning, including LDA and related tasks. One of the most commonly used functions for LDA in R is lda(), part of the MASS package. The MASS package also contains other functions for LDA, such as predict.lda(), plot.lda(), or qda().

Read More how to perform Linear Discriminant Analysis (LDA) in R Programming.

Python

Python is a high-level, general-purpose, and interpreted programming language. Python has a large and diverse collection of libraries and modules for data analysis and machine learning, including LDA and related tasks. One of the most commonly used modules for LDA in Python is sklearn, which is part of the scikit-learn package. The scikit-learn package also contains other modules and classes for LDA, such as LinearDiscriminantAnalysis, QuadraticDiscriminantAnalysis, or cross_val_score.

Conclusion

LDA is a powerful and versatile technique that can be used for various purposes, such as classification, dimensionality reduction, feature extraction, clustering, or visualization. It can handle both binary and multiclass problems and deal with both linearly and nonlinearly separable classes, depending on the model’s assumptions and parameters.

LDA can be implemented using software tools like R and Python with similar results and outputs. However, there may be some differences and nuances in the syntax, the parameters, the options, and the features of these tools, which may affect the analysis’s performance and interpretation. Therefore, it is important to understand and compare the advantages and disadvantages of these tools and to choose the most suitable tool for your data and goals.

However, LDA also has some limitations and challenges that need to be considered and addressed, such as the assumptions of normality, homoscedasticity, and independence of the features, the sensitivity to outliers and noise, the choice of the number of components, the scalability and the computational complexity of the algorithm, or the interpretation and the validation of the results. Therefore, it is important to understand and evaluate LDA’s suitability and performance for your problem or dataset and to compare and contrast it with other techniques, such as logistic regression, support vector machines, decision trees, or neural networks.

FAQS

Is linear discriminant analysis supervised or unsupervised?

Linear discriminant analysis is a supervised learning technique, as it requires the class labels of the training data to learn the discriminant function.

Is linear discriminant analysis a generative model?

Linear discriminant analysis is a generative model, assuming the data is generated from a multivariate normal distribution with class-specific parameters. It estimates the prior probabilities, the means, and the covariance matrices of each class from the data.

Is linear discriminant analysis a form of clustering?

Linear discriminant analysis is not a form of clustering, as it does not group the data into clusters based on similarity or distance but rather separates the data into classes based on a linear combination of features that maximizes the between-class variance and minimizes the within-class variance.

What is linear discriminant analysis in machine learning?

Linear discriminant analysis in machine learning is a method that can be used to train a classifier that can predict the class label of a new data point based on its LDA score and the posterior probability of each class. It can also reduce the dimensionality of the feature space and extract the most discriminative features for the classification task.

What is linear discriminant analysis used for?

Linear discriminant analysis can be used for various purposes, such as face recognition, text classification, gene expression analysis, or image segmentation. It can be applied to any problem or dataset where there is a need to separate or group data points into different classes based on some features or variables.

What is linear discriminant analysis in SPSS?

Linear discriminant analysis in SPSS is a procedure that can be used to perform LDA on a dataset and obtain the results and outputs, such as the coefficients, the scores, the loadings, the classification, the confusion matrix, the accuracy, the error rate, or the ROC curve. It can be accessed from the Analyze menu, under Classify, and then Linear Discriminant.

What does linear discriminant analysis do?

Linear discriminant analysis does the following steps: It assumes that the data is generated from a multivariate normal distribution with class-specific parameters, and it estimates the prior probabilities, the means, and the covariance matrices of each class from the data.

• It computes the linear discriminant function, which is a linear combination of features that maximizes the ratio of the between-class variance to the within-class variance.
• It projects the data onto the linear discriminant. It obtains the LDA score for each data point, which indicates its position and distance from the discriminant function.
• It classifies the data points based on the LDA score and the posterior probability of each class. It evaluates the model’s performance using various metrics, such as the confusion matrix, the accuracy, the error rate, or the ROC curve.
• Linear discriminant analysis explained: Linear discriminant analysis is a technique that can help you explore, understand, and classify your data and discover the hidden patterns and structures that underlie the data. It is a technique that can help you to solve your problems and to achieve your goals. It is a technique that can help you to learn from your data and to make better decisions.

What does linear discriminant analysis show?

Linear discriminant analysis shows the following results and outputs: The linear discriminants’ coefficients indicate the direction and magnitude of the discriminant function and the importance of each feature for the discrimination.

What is non-linear discriminant analysis?

Non-linear discriminant analysis is a technique that extends the linear discriminant analysis to handle nonlinearly separable classes by using a non-linear transformation of the features, such as a kernel function, a neural network, or a polynomial function. Non-linear discriminant analysis can improve the model’s performance when the data is not linearly separable. Still, it may also increase the algorithm’s complexity and computational cost.