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<math>g_i(x) \rightarrow 2(g_i(x))</math> or <math>g_i(x) \rightarrow ln(g_i(x))</math> | <math>g_i(x) \rightarrow 2(g_i(x))</math> or <math>g_i(x) \rightarrow ln(g_i(x))</math> | ||
| − | In other words, we can take <math>g_i(x) \rightarrow f(g_i(x))</math> for any monotonically increasing function f. | + | In other words, we can take <math>g_i(x) \rightarrow f(g_i(x))</math> for any monotonically increasing function ''f''. |
| + | |||
| + | '''Relation to Bayes Rule''' | ||
| + | |||
| + | e.g. We can take <math>g_i(\mathbf(x)) = P(\omega_i|\mathbf(x))</math> | ||
| + | then <math>g_i(\mathbf(x)) > g_j(\mathbf(x)), \forall j \neq i</math> | ||
Revision as of 15:43, 10 March 2008
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 $
