Linear Discriminant Analysis (LDA) is a powerful dimensionality reduction technique used for classification tasks. It aims to find the optimal linear combinations of features that best separate different classes. Here's a breakdown of the LDA procedure:
1. Data Preparation
- Gather your data: Collect a dataset containing features and corresponding class labels.
- Prepare your data: Ensure data is properly formatted, with features as columns and class labels as a separate column.
- Handle missing values: Replace missing values using appropriate methods like mean imputation or dropping rows.
- Standardize your data: Scale features to have zero mean and unit variance.
2. Calculate Within-Class Scatter Matrix (Sw)
- Compute the mean of each class: Calculate the average feature values for each class.
- Calculate the scatter matrix for each class: Subtract the class mean from each data point in that class, multiply the result by its transpose, and sum over all data points in that class.
- Sum the scatter matrices for all classes: This gives you the within-class scatter matrix (Sw), which measures the variability within each class.
3. Calculate Between-Class Scatter Matrix (Sb)
- Calculate the overall mean of all data points: Find the average feature values across all classes.
- Calculate the scatter matrix between classes: For each class, subtract the overall mean from the class mean, multiply the result by its transpose, and multiply by the number of data points in that class.
- Sum the scatter matrices for all classes: This gives you the between-class scatter matrix (Sb), which measures the variability between classes.
4. Find the Optimal Projection Matrix (W)
- Calculate the inverse of the within-class scatter matrix: Find the inverse of Sw.
- Multiply the inverse of Sw by Sb: Calculate the product of the inverse of Sw and Sb.
- Perform eigenvalue decomposition: Obtain the eigenvectors and eigenvalues of the resulting matrix.
- Select the eigenvectors corresponding to the largest eigenvalues: These eigenvectors represent the directions of maximum separation between classes.
- Construct the projection matrix (W): Combine the selected eigenvectors to form the projection matrix W.
5. Project Data onto Lower-Dimensional Space
- Multiply the original data by the projection matrix (W): This transforms the data into a lower-dimensional space, while preserving the information that best separates classes.
- Classify the data: Use a suitable classification algorithm (e.g., logistic regression) on the projected data.
6. Evaluate the Model
- Assess model performance: Use metrics like accuracy, precision, recall, and F1-score to evaluate the model's performance on unseen data.
- Tune hyperparameters: Adjust the number of dimensions to find the optimal balance between dimensionality reduction and classification performance.
Examples
- Image classification: LDA can be used to reduce the dimensionality of image features before applying a classifier.
- Text classification: LDA can be used to represent text documents as vectors in a lower-dimensional space, facilitating classification based on topic or sentiment.
