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Class Lecture Notes

LECTURE THEME : - Discriminant Functions

Discriminant Functions: one way of representing classifiers

Given the classes $ \omega_1, \cdots, \omega_k $

The discriminant functions $ g_1(x),\ldots, g_K(x) $ such that $ g_i(x) $ n-dim S space $ \rightarrow \Re $

which are used to make decisions as follows:

decide $ \omega_i $ if $ g_i(x) \ge g_j(x), \forall j $

Note that many different choices of $ g_i(x) $ will yield the same decision rule, because we are interested in the order of values of $ g_i(x) $ for each x, and not their exact values.

For example: $ g_i(x) \rightarrow 2(g_i(x)) $ or $ g_i(x) \rightarrow ln(g_i(x)) $

In other words, we can take $ g_i(x) \rightarrow f(g_i(x)) $ for any monotonically increasing function f.

Relation to Bayes Rule

e.g. We can take $ g_i(\mathbf(x)) = P(\omega_i|\mathbf(x)) $

then $ g_i(\mathbf(x)) > g_j(\mathbf(x)), \forall j \neq i $

$ \Longleftrightarrow P(w_i|\mathbf(X)) > P(w_j|\mathbf(X)), \forall j \neq i $

OR we can take

$ g_i(\mathbf(x)) = p(\mathbf(x)|\omega_i)P(\omega_i) $

then $ g_i(\mathbf(x)) > g_j(\mathbf(x)), \forall j \neq i $

$ \Longleftrightarrow g_i(\mathbf(x)) = ln(p(\mathbf(x)|\omega_i)P(\omega_i)) = ln(p(\mathbf(x)|\omega_i))+ln(P(\omega_i) $

OR we can take

$ g_i(\mathbf(x)) = ln(p(\mathbf(x)|\omega_i)P(\omega_i)) = ln(p(\mathbf(x)|\omega_i))+ln(P(\omega_i) $

We can take any $ g_i $ as long as they have the same ordering in value as specified by Bayes rule.

Some useful links:

- Bayes Rule in notes: https://engineering.purdue.edu/people/mireille.boutin.1/ECE301kiwi/Lecture4

- Bayesian Inference: http://en.wikipedia.org/wiki/Bayesian_inference

Relational Decision Boundary

Ex : take two classes $ \omega_1 $ and $ \omega_2 $

$ g(\vec x)=g_1(\vec x)-g_2(\vec x) $

decide $ \omega_1 $ when $ g(\vec x)>0 $

and $ \omega_2 $ when $ g(\vec x)<0 $

when $ g(\vec x) = 0 $, you are at the decision boundary ( = hyperplane)

$ \lbrace \vec x | \vec x \;\;s.t \;\;g(\vec x)=0\rbrace $ is a hypersurface in your feature space i.e a structure of co-dimension one less dimension than space in which $ \vec x $ lies

Figure 1

Discriminant function for the Normal Density

Suppose we assume that the distribution of the feature vectors is such that the density function p(X|w) is normal for all i.

Eg: Length of hair among men is a normal random variable. Same for hairlength in women. Now we have:

Figure 2

Figure 3

Figure 4

Figure 5

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