# What is Linear discriminant analysis

##### What is Linear Discriminant Analysis?

Linear Discriminant Analysis (LDA) is a statistical technique that is used to classify data into predetermined categories or classes. It is a widely used technique in machine learning and is particularly useful for classification problems where the number of classes is greater than two. The essence of LDA is to find a linear combination of features that best separates the different classes while preserving the overall variance of the data.

LDA is a supervised learning technique, meaning that it requires labeled data (i.e., data that is already classified into different categories). The aim of LDA is to find the best possible linear decision boundary that separates the different classes.

The decision boundary is found by computing the linear discriminant function of each data point and assigning it to the category that corresponds to the highest discriminant value.

##### The LDA Model

The LDA model assumes that each class has its own multivariate normal distribution with a mean vector and covariance matrix. The goal of LDA is to estimate these parameters from the data and then use them to find the best possible linear discriminant function.

To compute the linear discriminant function, LDA first calculates the between-class scatter matrix (Sb) and the within-class scatter matrix (Sw) as follows:

• Calculate the mean vector of each class (mj) and the overall mean vector (m).
• Calculate the sample covariance matrix of each class (Sj) and the overall sample covariance matrix (S).
• Calculate the between-class scatter matrix (Sb) as: Sb = Σ (mj-m) (mj-m)'.
• Calculate the within-class scatter matrix (Sw) as: Sw = Σ Sj.

Once the scatter matrices have been computed, LDA then finds the linear discriminant function by solving the following generalized eigenvalue problem:

Sw^-1 Sb w = λ w

where Sw^-1 Sb is the generalized eigenmatrix, w is the eigenvector, and λ is the eigenvalue. The eigenvectors corresponding to the largest eigenvalues are chosen as the linear discriminants because they capture the most variance between the classes.

The LDA model assumes that the covariance matrices of each class are equal. This is known as the homoscedasticity assumption. If the covariance matrices are unequal, then the LDA model may not be appropriate and alternative techniques such as Quadratic Discriminant Analysis (QDA) may be used.

##### Applications of LDA

LDA has many practical applications in various fields such as finance, biology, and marketing. Some examples of how LDA is used in practice include:

• Image recognition – LDA can be used to classify images into different categories such as animals, vehicles, and buildings.
• Sentiment analysis – LDA can be used to classify text into positive, negative, or neutral categories.
• Breast cancer diagnosis – LDA can be used to classify tissue samples as being benign or malignant.
• Stock market prediction – LDA can be used to classify stocks as being undervalued, overvalued, or fairly priced.
##### Limitations of LDA

Although LDA is a powerful tool for classification, it has certain limitations that should be taken into account when implementing it:

• Linear boundary – LDA assumes a linear decision boundary which may not be appropriate for all datasets.
• Homoscedasticity – LDA assumes that the covariance matrices of each class are equal which may not be true for all datasets.
• Number of features – LDA can only handle a limited number of features relative to the number of observations. When the number of features is large, LDA may become computationally expensive or yield poor classification results.
• Training time – LDA requires training data to estimate the parameters of the model which can be time-consuming for large datasets.
##### Conclusion

LDA is a powerful statistical technique for classification that can be used in a variety of applications. It is particularly useful for datasets with multiple classes and a limited number of features. When using LDA, it is important to take into account the assumptions and limitations of the technique to obtain accurate and reliable classification results.