i
discriminant.
When R = 2, the classifier called dichotomizer is simplified as below:
Discriminator
X
g i (X)
TLU 1
-1
Discriminant
i0 Class
Its discriminant function is g(X) = g 1(X) - g 2(X)
The membership in a class are determined based on the comparison of R discriminant functions g i (X), i =1, …, R, computed for the input pattern under consideration.
x1 x2 xn
Pattern
i 0(X) Classifier
1 or 2 or … or R Class
Geometric Explanation of Classification
Pattern -- an n-dimensional vector.
All n-dimensional patterns constitute an n-dimensional Euclidean space E n and is called pattern space.
is a straight line.
x2 g(X) > 0 g (X) < 0
-2 -1 (0,0)
0 (-1/2, -1) -1
(-1, -2)
-2
(2,0) x 1 12
(3/2,-1)
(1, -) = 0
g(X) = -2x 1 + x 2 +2
Infinite number of discrimminant functions may exist.
After each incorrect response, the classifier modifies its parameters by means of iterative, supervised learning based on comparing the targeted correct response with the actual response.
x1
w1
x 2 w2
x n wn
+ g(y)
xn+1 wn+1
TLU 1
0 = I0 =1 or -1
If g(X) > 0, then X belongs to Class 1; If g (X) < 0, then X belongs to Class 2.
The following figure is an example where 6 patterns
belong to one of the 2 classes and the decision surface
1 g 1(X)
X
i Maximum
i0
g i (X)
Selector Class
gR(X) R
Discriminators
For a given pattern, the i-th discriminator computes the value of the function g (X) called briefly the
gi-ti h(iXc)laasrseisffcaglia(rXv)a>lugesj(aXn)d, it,jh=e
pattern X 1, …, R, i
belongs to j. The
the
decision surface equation is g i(X) - g j(X) = 0.
Assuming that the discrimminant functions are known, the block diagram of a basic pattern classifier can be shown below:
Training and Classification
Consider neural network classifiers that derive their weights during the learning cycle.
The sample patterns, called the training sequence, are presented to the machine along with the correct response provided by the teacher.
A pattern classifier maps sets of patters in E n into one of the regions denoted by numbers i 0 = 1, 2, …, R.
Classifiers That Use The Discriminant Functions
If all patterns can be divided into R classes, then the region of the space containing only patterns of r-th class is called the r-th region, r = 1, …, R. Regions are separated from each other by decision surface.
AI - NN Lecture Notes
Chapter 8 Feed-forward Networks
§8.1 Introduction To Classification
The Classification Model
X = [x1 x2 … xn ]t -- the input patterns of classifier. i0 (X) -- decision function The response of the classifier is 1 or 2 or … or R.