In-depth analysis of linear discriminant analysis LDA

Linear Discriminant Analysis (LDA) is a classic pattern classification method that can be used for dimensionality reduction and feature extraction. In face recognition, LDA is often used for feature extraction. The main idea is to project the data into a low-dimensional subspace to achieve the maximum difference of different categories of data in the subspace and the minimum variance of the same category of data in the subspace. By calculating the eigenvectors of the inter-class scatter matrix and the intra-class scatter matrix, the optimal projection direction can be obtained, thereby achieving dimensionality reduction and feature extraction of the data. LDA has good classification performance and computational efficiency in practical applications, and is widely used in image recognition, pattern recognition and other fields.

The basic idea of ​​linear discriminant analysis (LDA) is to project high-dimensional data into a low-dimensional space so that the distribution of different categories of data in this space can maximize the difference. It improves the accuracy of classification by projecting the original data into a new space so that data of the same category are as close as possible and data of different categories are as far apart as possible. Specifically, LDA determines the projection direction by calculating the ratio between the intra-class divergence matrix and the inter-class divergence matrix, so that the projected data meets this goal as much as possible. In this way, in the projected low-dimensional space, data of the same category will be gathered together more closely, and data between different categories will be more dispersed, making classification easier.

Basic principles of linear discriminant analysis LDA

Linear discriminant analysis (LDA) is a common supervised learning algorithm, mainly used for dimensionality reduction and classification. The basic principle is as follows:

Suppose we have a set of labeled data sets, and each sample has multiple feature vectors. Our goal is to classify these data points into different labels. In order to achieve this goal, we can perform the following steps: 1. Calculate the mean vector of all sample feature vectors under each label to obtain the mean vector of each label. 2. Calculate the overall mean vector of all data points, which is the mean of all sample feature vectors in the entire data set. 3. Calculate the intra-class divergence matrix for each label. The intra-class divergence matrix is ​​the product of the difference between the feature vectors of all samples within each label and the mean vector for that label, and then the results for each label are summed. 4. Calculate the product of the inverse matrix of the within-class divergence matrix and the between-class divergence matrix to obtain the projection vector. 5. Normalize the projection vector to ensure that its length is 1. 6. Project the data points onto the projection vector to obtain a one-dimensional feature vector. 7. Use the set threshold to classify the one-dimensional feature vector into different labels. Through the above steps, we can project multi-dimensional data points into a one-dimensional feature space and classify them into corresponding labels based on thresholds. This method can help us achieve dimensionality reduction and classification of data.

The core idea of ​​LDA is to calculate the mean vector and divergence matrix to discover the internal structure and category relationships of the data. The data is dimensionally reduced by projecting vectors, and a classifier is used for classification tasks.

Linear discriminant analysis LDA calculation process

The calculation process of LDA can be summarized as the following steps:

Calculate the mean vector of each category, that is, the mean vector of all samples in each category Feature vectors are averaged and the overall mean vector is calculated.

When calculating the intra-class divergence matrix, the difference between the feature vector and the mean vector of the samples in each category needs to be multiplied and accumulated.

Calculate the inter-class dispersion matrix by multiplying the difference between the total mean vector in each category and the mean vector of each category, and then accumulating the results of all categories.

4. Calculate the projection vector, that is, project the feature vector to a vector on a one-dimensional space. This vector is the product of the inverse matrix of the intra-class divergence matrix and the between-class divergence matrix, and then normalize the vector change.

5. Project all samples to obtain one-dimensional feature vectors.

6. Classify samples according to one-dimensional feature vectors.

7. Evaluate classification performance.

Linear discriminant analysis LDA method advantages and disadvantages

Linear discriminant analysis LDA is a common supervised learning algorithm. Its advantages and disadvantages are as follows:

Advantages:

• LDA is a linear classification method that is simple to understand and easy to implement.
• LDA can not only be used for classification, but also for dimensionality reduction, which can improve the performance of the classifier and reduce the amount of calculations.
• LDA assumes that the data satisfies the normal distribution and has a certain degree of robustness to noise. For data with less noise, LDA has a very good classification effect.
• LDA takes into account the internal structure of the data and the relationship between categories, retains the discriminant information of the data as much as possible, and improves the accuracy of classification.

shortcoming:

• LDA assumes that the covariance matrices of each category are equal, but in practical applications, it is difficult to meet this assumption and may affect the classification effect.
• LDA has poor classification effect for non-linearly separable data.
• LDA is sensitive to outliers and noise, which may affect the classification effect.
• LDA needs to calculate the inverse matrix of the covariance matrix. If the feature dimension is too high, it may cause a very large amount of calculation and is not suitable for processing high-dimensional data.

In summary, linear discriminant analysis LDA is suitable for processing low-dimensional, linearly separable and data that satisfies the normal distribution, but it is not suitable for high-dimensional, non-linear separable or data that does not satisfy the normal distribution. For situations such as state distribution, other algorithms need to be selected.

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