# Find corresponding linear discriminant function in a two-class, three-dimensional classification

I am new to Patter Recognition and I am kind of stuck at a homework assignment. Any help regarding the issue will be appreciated. Thank you very much.

In a two-class, three-dimensional classification problem, the feature vectors in each class are normally distributed with covariance matrix:

|0.3 0.1 0.1 |
|0.1 0.3 0.1 | = ∑
|0.1 0.1 0.3 |

The respective mean vectors are [0, 0, 0]t and [0.5, 0.5, 0.5]t.

Derive the corresponding linear discriminant functions and the equation describing the decision surface.

There doesn't seem to be any Bayesian component to what you are describing. Write down the multivariate normal densities $f_0(x,y,z)$ and $f_1(x,y,z)$ under the hypotheses $H_0: \mu = (0,0,0)$ and $H_1: \mu = (0.5,0.5,0.5)$. The decision surface is the set of all $(x,y,z)$ for which $f_0(x,y,z) = f_1(x,y,z)$. You will find that it helps to equate $\ln f_0(x,y,z)$ and $\ln f_1(x,y,z)$ and proceed from there, and the answer might surprise you a little. Don't compute the numerical values of entries of the matrix inverse of $\Sigma$ before you actually need numerical values: that is, simply assume that the inverse is a symmetric matrix $A = [a_{i,j}]$ and write the densities in terms of symbols $a_{i,j}$ rather than numbers such as $0.25$.
A two-dimensional version of this question but with $f_0$ and $f_1$ having different covariance matrices (which complicates matters a lot) is answered here.