自控第二章

d ( x ) = ω *' x * = ω ' x + ω0
(2-2a) ω = [ω0ω1 ⋯ω d ] (2*
ω = [ω1 ⋯ω d ]'
'
is the weight vector; is the augmented weight vector with the bias term;
x * = [1x1 ⋯ xd ]
(2-2b) (2(2-2c) (2-
is the augmented feature vector.
Figure 2.2. Two-dimensional linear decision function with normal Twovector n and data distance D0 from the origin.
d ( x) = ω1 x1 + ω2 x2 + ω0 = 0
d(x)=0. The generalization of the linear decision function for a dimensional feature space in Rd is straightforward:
(2-2) (2Where
d ( x ) = ω1 f1 ( x ) + ⋯ω k f k ( x ) + ω0 = ω *' y *
y * = [1, f1 ( x ), f 2 ( x ) ⋯ f k ( x )]'
Figure 2.4. A two-class discrimination problem in the original twofeature space (a) and in a transformed one-dimensional feature onespace (b).
ln(d ( x )) ≥ l
Figure 2.3. (a) Quadratic decision function, d(x) ;(b) Logarithmic decision function, .
Figure 2.3b illustrates this logarithmic decision function for the quadratic classifier example using the new threshold value ln(49)=3.89. It is sometimes convenient to express a generalized decision function as a functional linear combination: (2-4) (2with (2-4a) (2-
The roots set of d(x), the decision surface, or discriminant, is now a linear d- dimensional surface called a hyperplane that can be characterized by its distance Do from the coordinates origin and its unitary normal vector n in the positive direction (d(x)﹥0) as follows (d(x)﹥ (see e.g. Friedman and Kandel, 1999): ω ω0 n= D0 = , ω ω (2-2d) (2where ‖w‖ represents the vector w length. Notice also that ︱d(z)︱/‖w‖ is precisely the distance of any d(z)︱ point z to the hyperplane.
Figure 2.1. Two classes of patterns described by two-dimensional feature vectors (features x1, and x2).
In Figure 2.1 we used "0" to denote class u, patterns and "x" to denote class 02 patterns. The ellipses represent "boundaries" of the patterns. distributions. It also shows a straight line separating the two classes. The classes. equation of the straight line in terms of the coordinates (features)x1, x2 using coefficients or weights w1, and w2 and a bias term w0 as shown in equation (2-1). (2(2- 1) (2We say that d(x) is a linear decision function that divides (categorizes) Rd into two decision regions: the upper half plane corresponding to d(x)>0 where each feature vector is assigned to w1; the lower half plane corresponding to d(x)<0 where each feature vector is assigned to w2. The classification is arbitrary for
concepts, approaches about pattern • discrimination • Request: Master the basic approaches of pattern discrimination • Focus: Basic notions
Contents of the chapter
2.1 .1 Generalized Decision Functions
As long as the classes do not overlap one can always find a generalized decision function defined in Rd that separates a class wi, from a total of
certain threshold∆ for class discrimination: (2(2-3a) For instance, in a two-class one-dimensional classification problem with twoonea quadratic decision function d ( x) = x 2 , one would design the classifier by selecting an adequate threshold ∆ so that the following decision rule would If ( x) = x 2 ≥ ∆ then ω1 else ω2 (2(2-3b) d x∈ x∈ In this decision rule we chose to assign the equality case to class w,. w,. Figure 2.3a shows a two-class discrimination using a quadratic decision twofunction with a threshold ∆ =49, which will discriminate between the class 49, apply:
Exercises
2.1 Decision Regions and Functions
In the particular case of a classifier. The main goal is to divide classifier. the feature space into regions assigned to the classification classes. classes. These regions are called decision regions. Let us assume two classes w1,w2 of patterns described by twotwodimensional feature vectors (coordinates x1, and x2) as shown in Figure 2.1.
Pattern Recognition Chapter 2
Pattern Discrimination (Pattern Recognition Concepts, Methods and Applications J.P. Marques de Sa Springer)
• Teaching purpose: Introduce the basic purpose:
classes, so that the following decision rule applies:
d i ( x) > 0 x ∈ ωi d i;( x) < 0 x ∈ ω j j≠i if if with . (2(2-3) For some generalized decision functions we will establish a
2 Pattern Discrimination 2.1 Decision Regions and Functions
2.1.1 Generalized Decision Functions 2.1.2 Hyperplane Separability