LDA, or linear discriminant analysis, is a dimensionality reduction technique that can also be used to classify data. It works by finding a linear combination of features that maximizes the separation between different classes of data. In this article, you will learn how LDA can help you reduce the complexity and noise of your data, and how to apply it to your own classification problems.
LDA is a method that transforms the original features of your data into a lower-dimensional space, where the classes are more distinguishable. LDA assumes that the data follows a multivariate normal distribution, and that the covariance matrices of each class are equal. LDA tries to find the direction that maximizes the ratio of the between-class variance to the within-class variance. This direction is called the discriminant, and it can be used to project the data onto a new space, where the classes are more separated.
LDA or Fisher's discriminant analysis is one of the least known but amazing and practically effective algorithms in the field. The idea is simple yet genius : defining a classification loss function to find optimal linear weights where we are minimizing the distances between the same group members whilst maximizing the distance between groups themselves, through the nominator/denominator.
Discriminant analysis is a multivariate statistical method used to identify variables that contribute the most to the reconstruction of pre-established groups of observations. It also serves to predict to which group a new observation will be classified. Discriminant analysis is also referred to as ? Fisher’s Discriminant Analysis ? or ? Linear Discriminant Analysis ?.
LDA can be useful for several reasons. First, it can reduce the dimensionality of your data, which can improve the computational efficiency and avoid the curse of dimensionality. Second, it can reduce the noise and redundancy of your data, which can enhance the performance and interpretability of your classification models. Third, it can capture the most relevant information for discriminating between classes, which can improve the accuracy and robustness of your classification models.
LDA (Linear Discriminant Analysis) is effective for classifying data as it focuses on maximizing class separability. It's particularly useful in scenarios where dimensionality reduction is needed without losing critical classification information. For example, in facial recognition, LDA can distill key features from high-dimensional pixel data, enabling efficient and accurate classification of different individuals. By reducing noise and redundancy, LDA enhances model performance, making it a go-to method for problems where understanding the distinction between classes is crucial.
The objective in discriminant analysis is in some way the ? reverse ? of that used in cluster analysis. In discriminant analysis, there are pre-determined groups of objects and a set object characteristics. The objective is to relate the object characteristics to the groups. More precisely, the goal is to establish which variables contribute the most to discriminating or differentiating between the groups and to derive mathematical rules/equations. In other words: o Individuals or objects are classified into two (2) or more groups ? Explain? group membership using a set of characteristics (variables) Identify the most important characteristics (variables) Discriminant analysis is often realized after a cluster analysis has been performed.
Using LDA involves calculating the mean vector and covariance matrix for each class of data, followed by calculating the within-class scatter matrix and between-class scatter matrix. After that, you need to compute the eigenvalues and eigenvectors of the matrix product of the inverse of the within-class scatter matrix and the between-class scatter matrix. The next step is to select the k largest eigenvalues and their corresponding eigenvectors, where k is the number of dimensions you want to reduce your data to. Finally, you form a matrix with the selected eigenvectors as columns and multiply it with your original data to obtain the transformed data.
Scikit-learn's LDA is quite practical and works well as well. Spend some time to get acquinted with it and make sure to read the doc specs. At the same time there is also QDA (Quadratic Discriminant Analysis), as a separate model class, the quadratic version as the name suggests.
The choice of k, or the number of dimensions to reduce your data to, depends on your goal and your data. If you want to use LDA for classification, you can choose k to be one less than the number of classes, which is the maximum possible value. This will ensure that you preserve as much information as possible for discriminating between classes. However, if you want to use LDA for visualization or exploration, you can choose k to be two or three, which will allow you to plot your data in a 2D or 3D space. You can also use some criteria, such as the percentage of explained variance or the scree plot, to determine the optimal value of k.
To evaluate LDA, you can use some metrics, such as the classification accuracy, the confusion matrix, or the ROC curve, to measure how well your transformed data can be classified by different models. You can also compare LDA with other dimensionality reduction techniques, such as PCA or t-SNE, to see which one performs better for your data and your task. You can also visualize your transformed data and inspect how well the classes are separated and clustered.
LDA is a powerful and versatile technique, yet it has some limitations. It assumes that the data follows a multivariate normal distribution, which may not be the case for some real-world data. Additionally, it assumes that the covariance matrices of each class are equal, which may not be true for imbalanced or heterogeneous data. Furthermore, it is sensitive to outliers and missing values, which can influence the calculation of the mean vectors and the covariance matrices. Lastly, it may not capture nonlinear relationships and interactions between features, which could be important for complex data.
Well, as you can get from the name, it is a linear method at the end of the day : finding the optimal weights for a linear sum of features. However, though this may be very limiting for certain use cases, the power of well discriminant formulation of joint intra-class/inter-class distance minimization/maximization may be helpful for classification even very non-linear settings.
It is important to be aware of the strong underlying assumptions required when performing linear discriminant analysis. ? Linearity ? The groups are normally distributed ? Equal variability within each group (homogeneity of variance) ? No presence of multicollinearity ? Discrimination can be optimized using one or more of the variables in the dataset
Linear Discriminant Analysis (LDA) is a supervised machine learning algorithm, which is primarily used for dimensionality reduction and classification. LDA constructs a decision boundary by maximizing the distance between class means and minimizing the scatter within each class. It also assumes that classes have identical covariance matrices and that features are normally distributed within each class. If there are 2 classes and their prior probabilities are equal then we compute the Mahalanabis distance of an observation from the second population and the Mahalanabis distance of that observation from the first population. Then we assign the observation to the class which have a smaller Mahalanbis distance.