SlideServe
Télécharger Obtenir Pro
Menu
Accueil Pro
Préférences
ApparenceLight Aide et commentaires
© SlideServe · v2026

Chapter 2 Decision Functions

grady-reyes avatar
grady-reyes
14 diapositives · 298 Views · Nov 18, 2014

This chapter explores the fundamentals of decision functions, focusing on linear and generalized methods. It presents basic concepts using simple examples and visual aids, discussing the separation of classes in both two-dimensional and n-dimensional spaces. The chapter details linear decision functions, including absolute and pairwise separability, and extends to nonlinear decision functions and polynomial classifiers. Additionally, geometric interpretations of decision functions and their properties are examined, highlighting the role of hyperplanes in classification tasks.

Chapter 2 Decision Functions - slide 1
01 / 14Chapter 2 Decision Functions
1 / 17

Télécharger la présentation

Chapter 2 Decision Functions
Une image / un lien ci-dessous est fourni (tel quel) pour télécharger la présentation

Politique de téléchargement : le contenu du site vous est fourni TEL QUEL pour votre information et votre usage personnel ; il ne peut être vendu, concédé sous licence ni partagé sur d’autres sites sans l’accord de son auteur.Le contenu vous est fourni TEL QUEL, pour votre information et votre usage personnel uniquement. Téléchargez la présentation en cliquant sur ce lien. Lors du téléchargement, si vous ne parvenez pas à télécharger une présentation, l’éditeur a peut-être supprimé le fichier de son serveur.Pendant le téléchargement, si vous ne parvenez pas à obtenir une présentation, le fichier a peut-être été supprimé par l’éditeur.

F I N

Plus de contenu associé

Transcription de la présentation

  1. Chapter 2 Decision Functions • Contents 2.1 Basic concepts 2.2 Linear decision functions 2.3 Generalized decision functions 2.4 Geometric discussions 2.5 Orthogonal functions .

  2. 2.1 Basic concepts • A simple example • Two classes C1 and C2 • Two-dimensional feature vector • X = (x1, x2)T • Figure 2.1.1 • Clearly separable by a straight line • d(x) = w1x1 + w2x2 + w3 = 0 • Decision rule • d(x) > 0  x C1 • d(x) < 0  x C2 • d(x) called the linear decision function

  3. 2.1 Basic concepts • n-dimensional Euclidean vector space (Rn) • Decision function represented by a hyperplane • n-dimensional feature vector • X = (x1, x2, …, xn)T • hyperplane • d(x) = w1x1 + w2x2 + … + wnxn + wn+1 = 0 • Decision rule • d(x) > 0  x C1 • d(x) < 0  x C2 • vector notation • X = (x1, x2, …, xn, 1)T • d(x) = wTx

  4. 2.1 Basic concepts • Nonlinear decision functions • Figure 2.1.2 • circumference • d(x) = 1 - x12 - x22 = 0 • Decision rule (Note it is the same as the previous ones.) • d(x) > 0  x C1 • d(x) < 0  x C2

  5. 2.1 Basic concepts • More than two classes • m pattern classes {C1, C2, …, Cm} in Rn • Definition 2.1.1 • If a surface d(x), xRn, separate Ci and the remaining Cj, ji • i.e, • d(x) > 0  x Ci • d(x) < 0  x Cj, ji • d(x) called a decision function of Ci • Example 2.1.2 • Figure 2.1.4

  6. 2.2 Linear decision functions • Two cases • Absolute separation • Pairwise separation • Absolute separation • If each class Ci has a linear decision function di(x) for 1im • i.e. • d(x) = wiTx > 0, x Ci • d(x) = wiTx < 0, otherwise • Then absolute separation exists between C1~Cm (absolutely separable) • Example 2.2.1 • Figure 2.2.1

  7. 2.2 Linear decision functions • Absolute separation (Continued) • How do we classify an incoming pattern x ? • Classify x into C1 if • d1(x) > 0 • d2(x) < 0 • d3(x) < 0 • Definition 2.2.1 (decision region) • Di = {x| di(x) > 0; dj(x) < 0, ji}, 1im • Example 2.2.2 • Figure 2.2.2 • A case of no absolute separation • Figure 2.2.3

  8. 2.2 Linear decision functions • Pairwise separation • Each pair of classes separable by linear function • Pair of Ci and Cj separable by dij if • dij(x) > 0 for all x Ci • dij(x) < 0 for all x Cj • Consequently, for all x Ci • dij(x) > 0 for all ji • Decision rule • classify x into Ci if • dij(x) > 0 for all ji • Example 2.2.4 • Figure 2.2.4

  9. 2.2 Linear decision functions • Pairwise separation (Continued) • Definition 2.2.2 (decision region) • Di = {x| dij(x) > 0, ji}, 1im • Example 2.2.5 • Figure 2.2.5 • Union of decision regions • not the whole space • rejection region

  10. 2.3 Generalized Decision Functions • Generalized decision functions • high complexity of boundaries  nonlinear surfaces needed • d(x) = w1f1(x) + w2f2(x) + … + wnfn(x) + wn+1 • fi(x), 1in : scalar functions of the pattern x, x Rn • vector notation • d(x) = i=1,n+1wifi(x) = wTx* where x*= (f1(x), f2(x), …,fn(x), fn+1(x))T and wT = (w1, w2, …, wn, wn+1) • polynomial classifier is popularly used • fi(x) are polynomials • eg) f1(x) = x1, f2(x) = x12, f3(x)=x1x2, …..

  11. 2.3 Generalized Decision Functions • Quadratic decision functions • 2-nd order polynomial classifier • eg) 2-D patterns (n=2), x=(x1,x2) • d(x) = w1x12 + w2x1x2 + w3x22 + w4x1 + w5x2 + w6 • for patterns x Rn • d(x) = i=1,nwiixi2 + i=1,n-1 j=i+1,nwijxixj + i=1,nwixi + wn+1 • number of terms =(n+1)(n+2)/2 • eg) n=2  6 terms, n=3  10 terms, .., n=10  65 terms, …

  12. 2.3 Generalized Decision Functions • Quadratic decision functions (Continued) • in case of order m • fi(x)=xi1e1 xi2e2 ….ximem • Theorem 2.3.1 • dm(x) = i1=1,n j2=i1,n …. im=im-1,n wi1i2…imxi1xj2….xjm + dm-1(x) where d0(x) = wn+1 • proof by mathematical induction • Example 2.3.1 • Example 2.3.2 • number of terms = (n+m)!/(n!m!) • matrix notation • d(x) = xTAx + xTb + c

  13. 2.4 Geometric Discussion • Importance of geometric interpretation of decision function’s properties • hyperplanes • dichotomies • Hyperplanes • linear decision functions • in 2-D, straight line • in 3-D, plane • in n-D where n>3, hyperplane • Figure 2.4.1 • hyperplane H • unit normal vector n • point on hyperplane, P, Q • vector associated with P and Q, y, x • normal vector n • n = w0/|w0|  equation 2.4.7 • distance between an arbitrary point R from H • Dz = | (w0T/|w0|)(z-y)| = | (w0Tz + wn+1) / |w0||  equation 2.4.11

  14. (1,2) 5/4 5/3 2.4 Geometric Discussion • Hyperplanes (Continued) • Example 2.4.1 • 3x1 + 4x2 – 5 = 0 in R2 • |w0| = 5 • n = (3/5, 4/5)T • D(1,2) = 1.2 • Example 2.4.2 • 2x1 - x2 + 2x3 - 7 = 0 in R3 • excluding the patterns whose distance from hyperplane is less than 0.01 • by |(2y1-y2+2y3-7) /|w0|| = |(2y1-y2+2y3-7) / 3| <0.01