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What are the most effective methods for selecting discriminant functions in LDA?

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Linear discriminant analysis (LDA) is a popular dimensionality reduction technique that can also be used for classification. It projects the data onto a lower-dimensional space that maximizes the separation between classes while minimizing the within-class variance. However, to perform LDA, you need to select the discriminant functions that define the projection axes. How can you choose the best discriminant functions for your data? Here are some methods that can help you.

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Muhammad Alothman

Advisory Board Member @ Mercy Corps | Monitoring and Evaluation Advisor
Zahra Amini

Machine Learning Lecturer | Interested in AI | Data Science
Andres Felipe Diaz-Delgadillo

Molecular Diagnostics and Tropical Virology

1 Eigenvalue criterion

One way to select the discriminant functions is to use the eigenvalue criterion. This method ranks the discriminant functions by their eigenvalues, which represent the amount of variance explained by each function. The higher the eigenvalue, the more important the function is for discriminating between classes. You can then choose the number of discriminant functions that explain a certain percentage of the total variance, or use a scree plot to identify the point where the eigenvalues drop sharply.

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Andres Felipe Diaz-Delgadillo

Molecular Diagnostics and Tropical Virology
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To select the appropriate method for Linear Discriminant Analysis, one has to consider the homocedasticity of the sample and design factors. One may utilize Kruskal tests for example to assess normality. Another option depends on how the data is structured and whether it is non parametric or even qualitative. Other necessary evaluations are the same-size groups or class comparisons. The general idea is to be sure whether your data is suited for linear discrimination that is if there is a direct relationship within the data that supports your null hypothesis robustly.

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Muhammad Alothman

Advisory Board Member @ Mercy Corps | Monitoring and Evaluation Advisor
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the eigenvalue criterion in LDA provides a principled approach for selecting discriminant functions by leveraging information about data spread within and between classes. It is a valuable tool for dimensionality reduction and feature selection in the context of linear classification problems.

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2 Wilks' lambda criterion

Another way to select the discriminant functions is to use the Wilks' lambda criterion. This method measures the ratio of the within-class variance to the total variance for each discriminant function. The lower the Wilks' lambda, the better the function is for separating the classes. You can then use a significance test to determine which functions have a Wilks' lambda that is significantly different from 1, or use a stepwise procedure to add or remove functions based on their contribution to the overall Wilks' lambda.

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Muhammad Alothman

Advisory Board Member @ Mercy Corps | Monitoring and Evaluation Advisor
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Wilks' Lambda criterion is often used in stepwise variable selection procedures in discriminant analysis. It plays a crucial role in discriminant analysis, helping us to make informed decisions about the effectiveness of discriminant functions in separating groups based on their multivariate (multi-dimensionl) characteristics.

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3 Cross-validation criterion

A third way to select the discriminant functions is to use the cross-validation criterion. This method evaluates the performance of each discriminant function on a subset of the data that is not used for training. The idea is to select the functions that minimize the classification error or maximize the accuracy on the validation data. You can then use different methods to perform cross-validation, such as k-fold, leave-one-out, or bootstrap, and compare the results of different numbers of discriminant functions.

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4 Information criterion

A fourth way to select the discriminant functions is to use the information criterion. This method balances the trade-off between the complexity and the fit of the discriminant functions. The complexity is measured by the number of parameters or functions, while the fit is measured by the likelihood or the log-likelihood of the data. You can then use different criteria, such as the Akaike information criterion (AIC) or the Bayesian information criterion (BIC), to select the functions that minimize the information loss or maximize the information gain.

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Muhammad Alothman

Advisory Board Member @ Mercy Corps | Monitoring and Evaluation Advisor
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Information criteria, such as the Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC), serve as guiding principles in statistical model selection. Based on testing, their primary objective is to strike a balance between the goodness of fit to the data and the complexity of the model. These criteria offer a quantitative means to compare competing models, considering both their performance and the number of parameters involved.

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5 Domain knowledge criterion

A fifth way to select the discriminant functions is to use the domain knowledge criterion. This method relies on your prior knowledge or intuition about the data and the problem. You can then use different sources of information, such as the literature, the experts, or the visualizations, to select the functions that make sense or are relevant for your context. For example, you might want to choose the functions that correspond to certain features, factors, or dimensions that are meaningful for your analysis.

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6 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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Zahra Amini

Machine Learning Lecturer | Interested in AI | Data Science
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Selecting discriminant functions in LDA effectively involves: 1?? Ensuring data features are normally distributed with equal covariance matrices across classes, as LDA assumes this. 2?? Scaling features since LDA is sensitive to feature scaling. 3?? The number of discriminant functions is one less than the number of classes. 4?? Maximizing class separation by choosing a direction that distinguishes classes as clearly as possible. ?? These steps are crucial for the effective application of LDA in classification tasks.

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Muhammad Alothman

Advisory Board Member @ Mercy Corps | Monitoring and Evaluation Advisor
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Utilizing other techniques, such as k-fold cross-validation, to assess how well the model performs on unseen data. This helps detect overfitting and provides a more robust estimate of model performance.

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What are the most effective methods for selecting discriminant functions in LDA?

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2

3

4

5

6

1 Eigenvalue criterion

2 Wilks' lambda criterion

3 Cross-validation criterion

4 Information criterion

5 Domain knowledge criterion

6 Here’s what else to consider

Statistics

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What are the most effective methods for selecting discriminant functions in LDA?

1

2

3

4

5

6

1 Eigenvalue criterion

2 Wilks' lambda criterion

3 Cross-validation criterion

4 Information criterion

5 Domain knowledge criterion

6 Here’s what else to consider

Statistics

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