# Fisher Discriminant Analysis

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The 30 round points are data. The 15 red points were generated from a normal distribution with mean , the 15 blue ones with mean , and in both cases the covariance matrix was the identity matrix. The problem is to classify or predict the color using the inputs and .

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Contributed by: Ian McLeod (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The canonical direction is given by

,

where and are the between- and within-classes covariance matrices. Hastie, Tibshirani and Friedman (2009, §4.3.3) [3] show that is given by the largest eigenvalue of .

The more general case where the number of inputs is greater than 2 is also considered in [3], but the basic principle of finding the canonical direction is the same. In our illustrative problem we have inputs as well as classes. In general, there are orthogonal canonical directions with the first canonical direction as defined above. Sometimes, as in [2], it is sufficient just to use just the first canonical component. For extensions, see [3].

[1] Wikipedia, "Linear Discriminant Analysis."

[2] R. A. Fisher, "The Use of Multiple Measurements in Taxonomic Problems," *Annals of Eugenics,* 7, 1936 pp. 179–188.

[3] T. Hastie, R. Tibshirani, and J. Friedman, *The Elements of Statistical Learning: Data Mining, Inference, and Prediction,* 2nd ed., New York: Springer, 2009.

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