## Discriminant Analysis

Discriminant analysis is a statistical technique used in research that aims to classify or predict a categorical dependent variable based on one or more continuous or binary independent variables. It is often used when the dependent variable is non-metric (categorical) and the independent variables are metric (continuous or binary).

### Discriminant Analysis Methodology

Here are the basic steps in the discriminant analysis methodology:

**Define the Problem and Collect the Data**

Firstly, clearly define the problem and the objectives of the analysis. Following this, collect the data for the dependent variable (the groups you want to predict or classify) and the independent variables (the predictors). The dependent variable should be categorical, and the independent variables are usually continuous.

**Data Preprocessing**

Clean and preprocess the data. This includes dealing with missing values, outliers, and ensuring that the data meets the assumptions of discriminant analysis. These assumptions include the independence of observations, normal distribution of predictor variables within each group of the dependent variable, and homogeneity of variances across groups.

**Estimate Discriminant Functions**

The next step is to estimate the discriminant functions, which are linear combinations of the predictor variables. These functions will differentiate the groups in the dependent variable. If there are two groups, only one discriminant function is created. If there are more than two groups, there can be more than one discriminant function.

**Evaluate the Discriminant Functions**

Check the significance of the discriminant functions using a Wilks’ lambda test. This test will tell you whether the discriminant functions significantly differentiate between the groups in your dependent variable.

**Classification of Cases**

Use the discriminant functions to classify the cases into groups. This is usually done by assigning a case to the group for which it has the highest discriminant score.

**Validation**

The final step is to validate the model by testing its classification accuracy. This can be done by splitting your data into a training set (to develop the discriminant functions) and a validation set (to test the accuracy of the classification). Alternatively, cross-validation or other out-of-sample validation techniques can be used.

**Interpretation**

Based on the discriminant function(s), interpret the results. The weights or coefficients of the predictor variables in the discriminant function can indicate which variables are most important for discriminating between the groups.

**Prediction**

Once the discriminant analysis model is built and validated, it can be used to predict group membership for new cases

### Types of Discriminant Analysis

Types of Discriminant Analysis are as follows:

**Linear Discriminant Analysis (LDA)**

This type of discriminant analysis is used when all the predictor variables are continuous and normally distributed, and the groups have equal covariance matrices. LDA seeks to find a linear combination of the predictors that separates the groups as much as possible.

**Quadratic Discriminant Analysis (QDA)**

QDA is similar to LDA, but it does not assume that the groups have equal covariance matrices. This means that it can model more complex group boundaries, but it also requires estimating more parameters than LDA and can be more prone to overfitting.

**Regularized Discriminant Analysis (RDA)**

This is a compromise between LDA and QDA that allows for the modeling of more complex group boundaries than LDA but is less prone to overfitting than QDA. It does this by “shrinking” the group-specific covariance matrices towards a common covariance matrix, with the degree of shrinkage determined by a tuning parameter.

**Flexible Discriminant Analysis (FDA)**

This is an extension of LDA that uses basis expansion methods to model non-linear boundaries between groups. It essentially applies LDA in a transformed space of the predictors.

**Multinomial Discriminant Analysis (MDA)**

This is used when you have more than two groups and you want to model the probability of group membership as a function of the predictors. MDA extends LDA and QDA to more than two groups.

**Canonical Discriminant Analysis (CDA)**

This type of discriminant analysis is used to identify and measure the associations among a set of variables and between that set of variables and a set of dummy variables that represent membership in the groups.

### Discriminant Analysis Formulas

Discriminant analysis involves several important formulas. I’ll describe the general form of these formulas for Linear Discriminant Analysis (LDA) as it is one of the most commonly used forms of discriminant analysis.

The goal of LDA is to project a feature space (a dataset n-dimensional sample) onto a smaller subspace k (where k ≤ n-1) while maintaining the class-discriminatory information. It does this by maximizing the ratio of between-class variance to the within-class variance in any particular data dataset to guarantee maximal separability.

**1. Within-Class Scatter Matrix (Sw)**

The within-class scatter matrix Sw is computed as:

Sw = Σ Si

where:

Si = Σ (x – mi)(x – mi)^T

and x runs over all N data points xi in class i, mi is the mean vector of class i, and the caret (^) indicates transposition.

**2. Between-Class Scatter Matrix (Sb)**

The between-class scatter matrix Sb is computed as:

Sb = Σ Ni (mi – m)(mi – m)^T

where:

Ni is the number of samples in each class, mi is the mean vector of class i, and m is the overall mean.

**3. Linear Discriminants**

The linear discriminants for the new subspace are the eigenvectors of Sw^-1 * Sb. That is, we want to solve the generalized eigenvalue problem for (Sw^-1 * Sb) * v = λv, where v are the eigenvectors we are looking for.

The discriminant function, which is used to classify a given new sample x into a class, is given by:

D(x) = x * W

where W is the matrix of eigenvectors.

**4. Score of a Case for a Group**

Once the discriminant functions have been calculated, the discriminant score of a case for a group is given by substituting the case’s values for the predictors into the discriminant function for that group.

5. QDA Formula

The discriminant function used in QDA is given by:

D_k(x) = -0.5 * log|Σ_k| – 0.5 * (x – μ_k)^T * Σ_k^-1 * (x – μ_k) + log(P(C_k))

where:

- D_k(x) is the discriminant function for class k.
- x is the feature vector of a sample.
- Σ_k is the covariance matrix for class k.
- μ_k is the mean vector for class k.
- (x – μ_k)^T is the transpose of the difference between the feature vector and the mean vector.
- Σ_k^-1 is the inverse of the covariance matrix for class k.
- |Σ_k| is the determinant of the covariance matrix for class k.
- P(C_k) is the prior probability of class k.

This function measures the distance from a sample to the center of a class, taking into account the spread or dispersion of the class. When a new sample is classified, it is assigned to the class that gives the highest value of the discriminant function. Note that in contrast to LDA, the quadratic term (x – μ_k)^T * Σ_k^-1 * (x – μ_k) allows QDA to model a more complex (i.e., non-linear) relationship between the features and the class labels.

### Examples of Discriminant Analysis

Discriminant analysis is often used in various fields such as marketing, finance, and medicine. Here are a few practical examples of its applications:

**Marketing Research**: Suppose a company wants to know what factors influence whether customers buy their product or a competitor’s. They may conduct a survey and ask respondents about their age, income, gender, and education. They can then use discriminant analysis to determine which of these variables are the best predictors of the brand of product purchased. The results may reveal, for example, that income and education level are significant discriminants, which can help the company target its marketing more effectively.**Finance and Credit Scoring**: A bank might use discriminant analysis to predict whether or not a loan applicant will default. The bank would use data from past customers, such as loan amount, income, credit score, and employment status, as predictor variables. The dependent variable would be whether the customer defaulted or not. The discriminant analysis can then help the bank make more informed loan approval decisions.**Medical Diagnostics**: Discriminant analysis can be used to classify patients into different categories based on symptoms or test results. For example, a researcher might use discriminant analysis to classify patients into those with and without a particular disease based on a range of symptoms or biomarkers.**Human Resource Management**: In HR, discriminant analysis can be used to predict job success based on a set of predictors like years of education, experience, skill level, and personality test scores. The dependent variable could be a binary measure of job success (successful or not) or a multi-category measure (like low, medium, or high performer).**Psychology**: A psychological study could use discriminant analysis to predict the success of therapy methods. For example, the dependent variable could be the type of therapy (e.g., cognitive-behavioral, psychodynamic, person-centered), and the independent variables could include demographic variables (like age or gender), psychometric scores, symptom severity, and the presence of any comorbid disorders.**Education**: Discriminant analysis could be used in educational research to predict the likelihood of students dropping out based on variables like attendance, grade point average, engagement in extracurricular activities, and socioeconomic status.

### When to use Discriminant Analysis

Here are several situations when discriminant analysis can be particularly useful:

**Multiclass Classification**

Discriminant analysis is often used when the dependent variable is categorical and has more than two categories. While other techniques like logistic regression can handle binary outcomes, discriminant analysis is particularly suitable for multiclass classification problems.

**Predictive Modeling**

f you are interested in predicting group membership based on a set of predictors, discriminant analysis can be a good choice. For example, it can be used to predict whether a loan applicant will default or not based on their financial characteristics.

**Understanding Group Differences**

Discriminant analysis can also be used when you are interested in understanding which variables discriminate between two or more naturally occurring groups. For instance, a company might use discriminant analysis to understand which characteristics differentiate customers who make a repeat purchase from those who do not.

**Assumption Fulfillment**

Discriminant analysis assumes that the predictors are normally distributed and that the groups have equal covariance matrices. If your data meet these assumptions, discriminant analysis can be a particularly effective method.

**Dimensionality Reduction**

Linear Discriminant Analysis (LDA), a type of discriminant analysis, can also be used for dimensionality reduction. That is, it can be used to reduce the number of variables in a dataset while preserving as much information as possible.

### Applications of Discriminant Analysis

Applications of Discriminant Analysis area s follows:

**Medical Research**: Discriminant Analysis can be used to classify patients into different groups based on their symptoms, medical history, or response to treatments. For instance, it can help distinguish between different types of diseases or predict patient outcomes.**Psychological Research**: In the field of psychology, Discriminant Analysis can be employed to identify which factors (such as personality traits, environmental factors, or genetic factors) predict different outcomes, such as the success of different therapeutic approaches or the development of certain behavioral patterns.**Educational Research**: Researchers in education may use Discriminant Analysis to predict academic success based on variables like previous academic achievement, socioeconomic status, and learning strategies.**Marketing**: Discriminant Analysis can be used to identify the most important factors that influence the choice of a particular product over another, allowing businesses to more effectively target their marketing strategies.**Medical Diagnostics**: It’s often used in medical fields to classify patients’ conditions based on symptoms or test results. This could include differentiating between different types of tumors, stages of a disease, or responses to different treatments.**Finance**: In the banking sector, Discriminant Analysis is used in credit scoring models to predict the probability of a borrower defaulting on a loan based on their financial information.**Human Resources**: It can be used to predict job performance or success in job applicants based on characteristics such as education level, years of experience, or personality test scores.**Ecology**: Discriminant Analysis can be used to classify different environments based on a set of features, such as climate conditions, soil properties, or vegetation types. This is especially useful in determining the habitats of various species or predicting the impact of climate change.**Customer Segmentation**: Businesses can use Discriminant Analysis to classify customers into different segments based on their buying behavior, demographic characteristics, and other attributes. This helps businesses understand their customers better and deliver more personalized offerings.**Face Recognition**: In computer vision, Linear Discriminant Analysis (LDA) is often used to enhance facial recognition technology by reducing dimensionality and improving classification accuracy.

### Advantages of Discriminant Analysis

Discriminant analysis offers several advantages that make it a valuable tool in a researcher’s statistical toolkit:

**Multiclass Classification**: Discriminant analysis can handle situations where there are more than two classes in the dependent variable, which is a limitation for some other methods such as logistic regression.**Understanding Group Differences**: Discriminant analysis does not just predict group membership; it also provides information on which variables are important discriminators between groups. This makes it a useful tool for exploratory research to understand the differences between groups.**Efficient with Large Variables**: Discriminant analysis can handle a large number of predictor variables efficiently. It becomes useful when the number of variables is very large, potentially exceeding the number of observations.**Dimensionality Reduction**: Linear Discriminant Analysis (LDA) can be used for dimensionality reduction – it can reduce the number of variables in a dataset while preserving as much information as possible.**Prior Probabilities**: Discriminant analysis allows for the inclusion of prior probabilities, meaning that researchers can incorporate prior knowledge about the proportions of observations in each group.**Model Interpretability**: The model produced by discriminant analysis is relatively interpretable compared to some other machine learning models, such as neural networks. The weights of the features in the model can provide an indication of their relative importance.

### Disadvantages of Discriminant Analysis

While discriminant analysis offers numerous benefits, there are also some limitations and disadvantages associated with its use:

**Assumption of Normality**: Discriminant analysis assumes that the predictors are normally distributed. If this assumption is violated, the performance of the model may be affected.**Assumption of Equal Covariance Matrices**: Discriminant analysis, particularly Linear Discriminant Analysis (LDA), assumes that the groups being compared have equal covariance matrices. If this assumption is not met, it may lead to inaccuracies in classification.**Multicollinearity**: Discriminant analysis may not work well if there is high multicollinearity among the predictor variables. This situation can lead to unstable estimates of the coefficients and difficulties in interpreting the results.**Outliers**: Discriminant analysis is sensitive to outliers, which can have a large influence on the classification function.**Overfitting**: Like many statistical techniques, discriminant analysis can result in overfitting if the model is too complex. Overfitting happens when the model fits the training data very well but performs poorly on new, unseen data.**Limited to Linear Relationships**: Linear Discriminant Analysis (LDA) assumes a linear relationship between predictor variables and the log-odds of the dependent variable. This limits its utility in scenarios where relationships are complex or nonlinear. In such cases, Quadratic Discriminant Analysis (QDA) or other non-linear methods might be more appropriate.