stat841

Course Note for Sept.30th (Classfication_by Liang Jiaxi)

1.

2. Classification
Classification is a function between two discrete random varialbe [math]\displaystyle{ \,X }[/math] and [math]\displaystyle{ \,Y }[/math]. Given n pairs of data [math]\displaystyle{ \,(X_{1},Y_{1}), (X_{2},Y_{2}, \dots , (X_{n},Y_{n})) }[/math], where [math]\displaystyle{ \,X_{i}= \{ X_{i1}, X_{i2}, \dots , (X_{id}, \} \in \mathcal{X} \subset \Re^{d} }[/math]
is a d-dimensional vector and [math]\displaystyle{ \,Y_{i} }[/math] takes values in a finite set [math]\displaystyle{ \, \mathcal{Y} }[/math]

3. Error data
Definition:
True error rate of a classifier(h) is defined as the probability that [math]\displaystyle{ \overline{Y} }[/math] predicted from [math]\displaystyle{ \,X }[/math] by classifier [math]\displaystyle{ \,h }[/math] does not actually equal to [math]\displaystyle{ \,Y }[/math], namely, [math]\displaystyle{ \, L(h)=P(h(X) \neq Y) }[/math].
Empirical error rate(training error rate) of a classifier(h) is defined as the frequency of event that [math]\displaystyle{ \overline{Y} }[/math] predicted from [math]\displaystyle{ \,X }[/math] by [math]\displaystyle{ \,h }[/math] does not equal to [math]\displaystyle{ \,Y }[/math] in total n prediction. The mathematical expression is as below:
[math]\displaystyle{ \, L_{h}= \frac{1}{n} \sum_{i=1}^{n} I(h(X_{i} \neq Y_{i})) }[/math], where [math]\displaystyle{ \,I }[/math] is an indicator that [math]\displaystyle{ \, I= }[/math].

4. Bayes Classifier