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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.

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1 Answer 1

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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.

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