Data were collected on the genus of flea beetle Chaetocnema, which contains three species: concinna (Con), heikertingeri (Hei), and heptapotamica (Hep). Measurements were made on the width and angle of the aedeagus of each beetle. There are 74 measurements: 21 (Con), 31 (Hei), 22 (Hep). http://lib.stat.cmu.edu/DASL/Stories/FleaBeetles.html.

Fitting a multi-logistic regression, we obtain:

##     (Intercept)       Width      Angle
## Hei   109.87495 -0.80640574 -0.1201425
## Hep    82.66292  0.03581382 -7.0596853

Hence the hyperplanes that determine the decision boundaries are \(109.87459-0.80640574 x_1 - 0.1201425 x_2 = 0\) or equivalently \(x_2 = 914.536 -6.71208 x_1\). Similarly the other two boundaries are \(x_2=11.7092 +0.00507301 x_1\) and \(x_2 = 3.92125+0.121365 x_1\).

The 3 hyperplanes and the associated regions are shown below for the multinomial regression classifier. One ‘Con’ example is on the boundary has been mis-classified.

Next we compute the mis-classification rates for the LDA and QDA classfiers. We see they also both produce 1 mis-classification.

##                       NumberMisclassified
## MultinomialLogitModel                   1
## LDA                                     1
## QDA                                     1