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Intuitive Idea To Linear Discriminant Analysis

Linear Discriminate Analysis (LDA) is a supervised classification method of qualitative variables in which two or more groups are known a priori and new observations are classified into one of them depending on their characteristics. Using Bayes theorem, LDA estimates the probability that an observation, given a certain value of the predictors, belongs to each of the classes of the qualitative variable, P (And= k | X= x). Finally, the observation is assigned to class k for which the predicted probability is higher.

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It is an alternative to logistic regression when the qualitative variable has more than two levels. While there are extensions of logistic regression for multiple classes, the LDA has several advantages:

  • If the classes are well separated, the estimated parameters in the logistic regression model are unstable. The LDA method does not suffer from this problem.
  • If the number of observations is low and the distribution of the predictors is almost normal in every class, LDA is more stable than the logistic regression.
  • When it comes to a classification problem with only two levels, both methods usually reach similar results.

Description Of The Linear Discriminant Analysis

The Discriminant Analysiswas proposed by R. Fisher, whose purpose is to analyze whether there are significant differences between groups of objects concerning a set of variables measured on them, if they exist, explain in what sense they are given and provide procedures for Systematic classification of new observations of unknown origin in one of the analyzed groups, is a classification technique where the objective is to obtain a function capable of classifying a new individual from the knowledge of the values of certain discriminating variables.

According to our Statistics Quiz help experts, the Discriminant Analysisallows to describe, select the variables that most influence the problem, build a function from these variables and predict in which group a new individual is classified, which has been evaluated in said function. This procedure is seen as a prediction model of a categorical response variable (group variable) from p generally continuous explanatory variables (classificatory variables). The steps to follow to carry out a Discriminant Analysisinclude:

  1. Raise the problem to be solved by the Discriminant Analysis.
  2. Analyze if there are noteworthy differences between the clusters.
  3. Establish the number and composition of the dimensions of discrimination between the groups analyzed.
  4. Determine which classifying variables explain most of the observed differences.
  5. Construct systematic procedures for classifying objects of unknown provenance in the analyzed groups.
  6. Evaluate the statistical and practical significance of the results obtained in the classification process.
  7. Like any other statistical technique, the application of the Discriminant Analysismust be preceded by a check of the assumptions assumed by the model, the Discriminant Analysisis based on the following assumptions: a)Multivariatenormalityb) Equality of variance-covariance matrices c) Linearity d) Absence of multi-collinearity and e) Uniqueness.
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The Conditions That Must Be Met For A Linear Discriminant Analysis To Be Valid Are:

  • Each predictor that is part of the model is normally distributed in each of the classes of the response variable.In the case of multiple predictors, the observations follow a multivariate normal distribution in all classes.
  • The variance of the predictor is the same in all classes of the response variable.In the case of multiple predictors, the covariance matrix is the same in all classes.If this is not fulfilled, we resort to Quadratic Discriminant Analysis (QDA). When the normal condition is not met, the LDA loses accuracy but can still reach relatively good ratings.

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