Discriminant Analysis

There is a well-known concept of sample space when we study probability. Along with sample space, we have studied some more concepts namely events and classes. When we have to figure out a way to separate two different events and classes, the process is called discriminant analysis. 

If the method is a linear combination then it is called linear discriminant analysis and if the method is a quadratic combination then the process is called quadratic discriminant analysis. The discriminant analysis is just that in statistics and probability. In this article, we will study discriminant analysis with a special focus on linear and quadratic discriminant analysis. 

Linear Discriminant Analysis 

There are a lot of equations in this type of analysis and most of them involve different types of variables. With every observation that is made, we have to make sure that we note and observe the change in those variables. 

Whenever we go through different equations, there is a lot of data that we have to go through and if we are to use the methods given by the linear discriminant analysis then we will have to deal with all the data individually and cannot merge one equation with another one. Another catch in this concept is that the observations have to be made continuously and not a single break can be taken as if a break is taken, a mistake can easily be done in the calculations which can lead to mistakes in the observations. 

Important terms regarding the discriminant analysis

While discussing discriminant analysis, some important terms are worth discussing. Some of those terms are independence, multicollinearity, homogeneity, and multivariate homogeneity. Now in this section, we will learn about all these terms in detail. It is to be noted that knowing in depth about all these terms is imperative. 

Firstly, let us talk about multicollinearity. This concept involves the topic of exponents and powers. It is important to know that at the moment powers are involved in this concept, the relation between the variables of the equations decreases. The power of variables should be predictable. 

Secondly, let us talk about independence. The first step to solving any equation is using different variables independently. All the equations have to be randomly solved and not much information is to be derived from the other equations as all of them are completely independent and if the information is derived from the previous equations then it might lead to some miscalculations also. 

Thirdly, let us talk about multivariate normality. For every variable that is grouped, we need to have a variable that is always independent of the variable that was used in the previous equation. 

Lastly, let us talk about the homogeneity of the variances. This process is also called the process of covariance. This is by far the most important way. If the process of linear discriminant analysis is to be applied then the value of the covariances has to be equal. Another important fact is that the process of quadratic discriminant analysis can be used only when the values of the covariances are not equal. 

So, to sum up, linear discriminant analysis can be used when the values of covariances are equal,` and quadratic discriminant analysis is used when covariances are not the same. 

ANOVA 

The process of linear discriminant analysis is also related to ANOVA which stands for Analysis of Variance. In this process, we express any random variable as a linear combination of different types of features of measurement. The fun fact about ANOVA is that it uses different combinations of variables and expressions that are different from each other in all aspects. 

For ANOVA to exist we have to have all the variables and all the other related aspects completely different from each other. If this is not the case, then usually this rule will not apply. The other name for ANOVA is regression analysis. In fact, in most cases, regression analysis is preferred over ANOVA. 

Difference 

The major point of difference between the process of linear discriminant analysis and the process of ANOVA is the fact that ANOVA involves both a linearly dependent variable and a linearly independent variable whereas the other process involves only linearly independent variables. 

So, the involvement of a linearly dependent variable makes this process entirely different from the process of linear discriminant analysis. Furthermore, we can also say that the process of linear discriminant analysis is closely related to the process of principal component analysis also known as factor analysis. 

The only difference again between these two processes is the fact that in principal component analysis, we can clearly distinguish between the concepts of class and objects whereas in the case of linear discriminant analysis we do not have a clear demarcation as to where we have to differentiate between class and objects. 

Principal Component Analysis

In the process of principal component analysis or factor analysis, we do not have any rule regarding the distinction between an independent variable or the dependent variable, so if the existence of this rule is necessary, a rule of such sort has to be made. 

Conclusion

Discriminant Analysis is one of the most important and complicated parts of mathematics. In it, there are a lot of rules and if we do not follow even a single one of them, we end up making a lot of mistakes in solving them.