Editing Linear discriminant analysis (section)

==Discriminant functions==
Discriminant analysis works by creating one or more linear combinations of predictors, creating a new [[latent variable]] for each function. These functions are called discriminant functions. The number of functions possible is either <math>N_g-1</math> where <math>N_g</math> = number of groups, or <math>p</math> (the number of predictors), whichever is smaller. The first function created maximizes the differences between groups on that function. The second function maximizes differences on that function, but also must not be correlated with the previous function. This continues with subsequent functions with the requirement that the new function not be correlated with any of the previous functions.

Given group <math>j</math>, with  <math>\mathbb{R}_j</math>  sets of sample space, there is a discriminant rule such that if <math>x \in\mathbb{R}_j</math>, then <math>x\in j</math>. Discriminant analysis then, finds “good” regions of  <math>\mathbb{R}_j</math> to minimize classification error, therefore leading to a high percent correct classified in the classification table.<ref>Hardle, W., Simar, L. (2007). ''Applied Multivariate Statistical Analysis''. Springer Berlin Heidelberg. pp.&nbsp;289–303.</ref>

Each function is given a discriminant score{{clarify|date=April 2019}} to determine how well it predicts group placement. 
*Structure Correlation Coefficients: The correlation between each predictor and the discriminant score of each function. This is a zero-order correlation (i.e., not corrected for the other predictors).<ref>Garson, G. D. (2008). Discriminant function analysis. https://web.archive.org/web/20080312065328/http://www2.chass.ncsu.edu/garson/pA765/discrim.htm.</ref>
*Standardized Coefficients: Each predictor's weight in the linear combination that is the discriminant function.  Like in a regression equation, these coefficients are partial (i.e., corrected for the other predictors). Indicates the unique contribution of each predictor in predicting group assignment.
*Functions at Group Centroids: Mean discriminant scores for each grouping variable are given for each function. The farther apart the means are, the less error there will be in classification.