Linear Discriminant Analysis (LDA)

Question 1

Linear Discriminant Analysis (LDA)

Answer 1

Linear Discriminant Analysis (LDA) is a method used in statistics, pattern recognition, and machine learning to find a linear combination of features that separates or characterizes two or more classes of objects or events. In summary, LDA is a robust and widely-used method for dimensionality reduction and classification that can be particularly effective when the assumptions of LDA are met.

Question 2

1. Definition

Answer 2

Linear Discriminant Analysis (LDA) is a method that seeks to find a linear combination of features that characterizes or separates two or more classes of objects or events. The resulting combination may be used as a linear classifier, or more commonly, for dimensionality reduction before classification.

Question 3

2. Goal

Answer 3

The primary goal of LDA is to project a feature space (a dataset n-dimensional samples) onto a smaller subspace k (where k ≤ n-1) while maintaining the class-discriminatory information.

Question 4

3. Procedure

Answer 4

LDA starts by computing the separability between different classes and within each class. The within-class scatter is computed as the sum of the scatter matrices for each individual class. The between-class scatter is computed as the scatter between the mean of each class and the overall mean. The goal is to find a vector that maximizes the between-class scatter and minimizes the within-class scatter.

Question 5

4. Assumptions

Answer 5

LDA assumes that the observations within each class are drawn from a multivariate Gaussian distribution with a class-specific mean vector and a covariance matrix that is common to all K classes.

Question 6

5. Multiclass LDA

Answer 6

While originally developed for binary classification problems, LDA can be extended to handle multiclass problems. This is done by transforming the multiclass problem into multiple binary classification problems.

Question 7

6. Benefits

Answer 7

LDA can be relatively simple to understand and implement, especially compared to some other dimensionality reduction techniques. It is a good technique to use when classes are well-separated, and the dataset is small compared to the number of features. It also performs well when the distribution of predictors is approximately normal.

Question 8

7. Limitations

Answer 8

If the assumptions of LDA (multivariate normality and equal covariance matrices) are not met, the method can produce inaccurate and misleading results. LDA can also be sensitive to outliers.

Question 9

8. Comparison to PCA

Answer 9

Unlike Principal Component Analysis (PCA), which encodes the data in a way that the total variance of the transformed data is maximized, LDA encodes the data in a way that maximizes the separability among the known categories (classes).

Question 10

9. Applications

Answer 10

LDA is used widely in pattern recognition, including computer vision and image recognition tasks. It’s also used in machine learning and statistics, especially in the pre-processing step for pattern classification and machine learning applications.