Combining Pattern Classifiers

More documents

Recommendations

Info

30 FUNDAMENTALS OF PATTERN RECOGNITION Fig. 1.10 Rotated check-board data (100,000 points in each plot). The Matlab code for N data points is: function [d,labd]=gendatcb(N,a,alpha); d=rand(N,2); d_transformed=[d(:,1)*cos(alpha)-d(:,2)*sin(alpha), ... d(:,1)*sin(alpha)+d(:,2)*cos(alpha)]; s=ceil(d_transformed(:,1)/a)+floor(d_transformed(:,2)/a); labd=2-mod(s,2); 1.5.4 Discriminant Functions and Decision Boundaries The class with the highest posterior probability is the most natural choice for a given x. Therefore the posterior probabilities can be used directly as the discriminant functions, that is, g i (x) ¼ P(v i jx), i ¼ 1, ..., c (1:40) Hence we rewrite the maximum membership rule (1.3) as D(x) ¼ v i [ V , P(v i jx) ¼ max i¼1,...,c {P(v ijx)} (1:41) In fact, a set of discriminant functions leading to the same classification regions would be g i (x) ¼ P(v i )p(xjv i ), i ¼ 1, ..., c (1:42)
BAYES DECISION THEORY 31 because the denominator of Eq. (1.31) is the same for all i, and so will not change the ranking order of g i values. Another useful set of discriminant functions derived from the posterior probabilities is g i (x) ¼ log½P(v i )p(xjv i ) Š, i ¼ 1, ..., c (1:43) Example: Decision/Classification Boundaries. Let x [ R. Figure 1.11 shows two sets of discriminant functions for three normally distributed classes with P(v 1 ) ¼ 0:45, P(v 2 ) ¼ 0:35, P(v 3 ) ¼ 0:20, p(xjv 1 ) N 4, (2:0) 2 p(xjv 2 ) N 5, (1:2) 2 p(xjv 3 ) N 7, (1:0) 2 The first set (top plot) depicts a set of functions (1.42), P(v i )p(xjv i ), i ¼ 1, 2, 3. The classification boundaries are marked with bullets on the x-axis. The posterior probabilities (1.40) are depicted in the bottom plot. The classification regions specified by the boundaries are displayed with different shades of gray in the bottom plot. Note that the same regions are found in both plots. Fig. 1.11 (a) Plot of two equivalent sets of discriminant functions: Pðv 1 Þpðxjv 1 Þ (the thin line), Pðv 2 Þpðxjv 2 Þ (the dashed line), and Pðv 3 Þpðxjv 3 Þ (the thick line). (b) Plot of the three posterior probability functions Pðv 1 jxÞ (the thin line), Pðv 2 jxÞ (the dashed line), and Pðv 3 jxÞ (the thick line). In both plots x [ ½0; 10Š.
Page 1 and 2: Combining Pattern Classifiers
Page 3 and 4: Copyright # 2004 by John Wiley & So
Page 5 and 6: vi CONTENTS 1.5.2 Normal Distributi
Page 7 and 8: viii CONTENTS 5.1.2 Probabilities B
Page 9 and 10: x CONTENTS 9.1.1 Equivalence of MIN
Page 11 and 12: Preface Everyday life throws at us
Page 13 and 14: PREFACE xv employing it for buildin
Page 15 and 16: Notation and Acronyms CART LDC MCS
Page 17 and 18: 1 Fundamentals of Pattern Recogniti
Page 19 and 20: BASIC CONCEPTS: CLASS, FEATURE, AND
Page 21 and 22: CLASSIFIER, DISCRIMINANT FUNCTIONS,
Page 23 and 24: CLASSIFIER, DISCRIMINANT FUNCTIONS,
Page 25 and 26: CLASSIFICATION ERROR AND CLASSIFICA
Page 27 and 28: CLASSIFICATION ERROR AND CLASSIFICA
Page 29 and 30: EXPERIMENTAL COMPARISON OF CLASSIFI
Page 41 and 42: BAYES DECISION THEORY 25 4. Make su
Page 43 and 44: BAYES DECISION THEORY 27 Fig. 1.8 N
Page 45: BAYES DECISION THEORY 29 coming fro
Page 49 and 50: BAYES DECISION THEORY 33 The error
Page 51 and 52: TAXONOMY OF CLASSIFIER DESIGN METHO
Page 53 and 54: Holmström et al. [32] consider ano
Page 55 and 56: K-HOLD-OUT PAIRED t-TEST 39 Fig. 1.
Page 57 and 58: 5 2cv PAIRED t-TEST 41 for i=1:K l
Page 59 and 60: DATA GENERATION: LISSAJOUS FIGURE D
Page 61 and 62: 46 BASE CLASSIFIERS where P(v i ) i
Page 63 and 64: 48 BASE CLASSIFIERS For the common
Page 65 and 66: 50 BASE CLASSIFIERS error was 42.8
Page 67 and 68: 52 BASE CLASSIFIERS Fig. 2.2 Classi
Page 69 and 70: 54 BASE CLASSIFIERS 2.2.2 Parzen Cl
Page 71 and 72: 56 BASE CLASSIFIERS 2.3 THE k-NEARE
Page 73 and 74: 58 BASE CLASSIFIERS Fig. 2.4 Illust
Page 75 and 76: 60 BASE CLASSIFIERS set are called
Page 77 and 78: 62 BASE CLASSIFIERS TABLE 2.2 Error
Page 79 and 80: 64 BASE CLASSIFIERS justification.
Page 81 and 82: 66 BASE CLASSIFIERS Fig. 2.7 Illust
Page 83 and 84: 68 BASE CLASSIFIERS For a ¼ 0 and
Page 85 and 86: 70 BASE CLASSIFIERS Fig. 2.8 (a) An
Page 87 and 88: 72 BASE CLASSIFIERS 2.4.2.3 Misclas
Page 89 and 90: 74 BASE CLASSIFIERS 2.4.3 Stopping
Page 91 and 92: 76 BASE CLASSIFIERS TABLE 2.4 A Tab
Page 93 and 94: 78 BASE CLASSIFIERS then calculate
Page 95 and 96: 80 BASE CLASSIFIERS pay off. Method
Page 97 and 98:
82 BASE CLASSIFIERS mation involved
Page 99 and 100:
84 BASE CLASSIFIERS . The threshold
Page 101 and 102:
86 BASE CLASSIFIERS Fig. 2.16 (a) U
Page 103 and 104:
88 BASE CLASSIFIERS Fig. 2.18 Possi
Page 105 and 106:
90 BASE CLASSIFIERS Eq. (2.90) or E
Page 107 and 108:
92 BASE CLASSIFIERS of E (the updat
Page 109 and 110:
94 BASE CLASSIFIERS For input-to-hi
Page 111 and 112:
96 BASE CLASSIFIERS chi2=tree_chi2(
Page 113 and 114:
98 BASE CLASSIFIERS ind=1; leaf=0;
Page 115 and 116:
100 BASE CLASSIFIERS % outputs of t
Page 117 and 118:
102 MULTIPLE CLASSIFIER SYSTEMS Fig
Page 119 and 120:
104 MULTIPLE CLASSIFIER SYSTEMS Fig
Page 121 and 122:
106 MULTIPLE CLASSIFIER SYSTEMS is
Page 123 and 124:
108 MULTIPLE CLASSIFIER SYSTEMS . u
Page 125 and 126:
110 MULTIPLE CLASSIFIER SYSTEMS Rus
Page 127 and 128:
112 FUSION OF LABEL OUTPUTS general
Page 129 and 130:
114 FUSION OF LABEL OUTPUTS TABLE 4
Page 131 and 132:
116 FUSION OF LABEL OUTPUTS and U m
Page 133 and 134:
Page 135 and 136:
120 FUSION OF LABEL OUTPUTS To rela
Page 137 and 138:
122 FUSION OF LABEL OUTPUTS where j
Page 139 and 140:
124 FUSION OF LABEL OUTPUTS Assigni
Page 141 and 142:
126 FUSION OF LABEL OUTPUTS 4.4 NAI
Page 143 and 144:
128 FUSION OF LABEL OUTPUTS s 2 ¼
Page 145 and 146:
130 FUSION OF LABEL OUTPUTS CI(v j
Page 147 and 148:
132 FUSION OF LABEL OUTPUTS first-o
Page 149 and 150:
134 FUSION OF LABEL OUTPUTS The mut
Page 151 and 152:
136 FUSION OF LABEL OUTPUTS Constru
Page 153 and 154:
Page 155 and 156:
140 FUSION OF LABEL OUTPUTS To labe
Page 157 and 158:
142 FUSION OF LABEL OUTPUTS Fig. 4.
Page 159 and 160:
144 FUSION OF LABEL OUTPUTS Next we
Page 161 and 162:
146 FUSION OF LABEL OUTPUTS 3. None
Page 163 and 164:
148 FUSION OF LABEL OUTPUTS Similar
Page 165 and 166:
5 Fusion of Continuous- Valued Outp
Page 167 and 168:
HOW DO WE GET PROBABILITY OUTPUTS?
Page 169 and 170:
HOW DO WE GET PROBABILITY OUTPUTS?
Page 171 and 172:
CLASS-CONSCIOUS COMBINERS 157 As se
Page 173 and 174:
CLASS-CONSCIOUS COMBINERS 159 Fig.
Page 175 and 176:
CLASS-CONSCIOUS COMBINERS 161 Fig.
Page 177 and 178:
CLASS-CONSCIOUS COMBINERS 163 The c
Page 179 and 180:
The mean of the error of D 1 is 0:2
Page 181 and 182:
CLASS-CONSCIOUS COMBINERS 167 befor
Page 183 and 184:
CLASS-CONSCIOUS COMBINERS 169 The f
Page 185 and 186:
CLASS-INDIFFERENT COMBINERS 171 par
Page 187 and 188:
CLASS-INDIFFERENT COMBINERS 173 The
Page 189 and 190:
CLASS-INDIFFERENT COMBINERS 175 the
Page 191 and 192:
WHERE DO THE SIMPLE COMBINERS COME
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
COMMENTS 187 and the weighted produ
Page 203 and 204:
6 Classifier Selection 6.1 PRELIMIN
Page 205 and 206:
WHY CLASSIFIER SELECTION WORKS 191
Page 207 and 208:
ESTIMATING LOCAL COMPETENCE DYNAMIC
Page 209 and 210:
ESTIMATING LOCAL COMPETENCE DYNAMIC
Page 211 and 212:
PREESTIMATION OF THE COMPETENCE REG
Page 213 and 214:
SELECTION OR FUSION? 199 TABLE 6.2
Page 215 and 216:
BASE CLASSIFIERS AND MIXTURE OF EXP
Page 217 and 218:
7 Bagging and Boosting 7.1 BAGGING
Page 219 and 220:
BAGGING 205 Since there are only tw
Page 221 and 222:
BAGGING 207 and d j,1 (x) ¼ d j,2
Page 223 and 224:
BAGGING 209 5. Repeat steps (1) to
Page 225 and 226:
BAGGING 211 Fig. 7.4 Error rates fo
Page 227 and 228:
BOOSTING 213 HEDGE (b) Given: D¼f
Page 229 and 230:
BOOSTING 215 shows how the probabil
Page 231 and 232:
BOOSTING 217 Fig. 7.8 Testing error
Page 233 and 234:
BOOSTING 219 into account by the cl
Page 235 and 236:
BOOSTING 221 Fig. 7.10 example. Mar
Page 237 and 238:
BIAS-VARIANCE DECOMPOSITION 223 7.3
Page 239 and 240:
BIAS-VARIANCE DECOMPOSITION 225 the
Page 241 and 242:
BIAS-VARIANCE DECOMPOSITION 227 dis
Page 243 and 244:
WHICH IS BETTER: BAGGING OR BOOSTIN
Page 245 and 246:
PROOF OF THE ERROR FOR AdaBoost (TW
Page 247 and 248:
PROOF OF THE ERROR FOR AdaBoost (TW
Page 249 and 250:
PROOF OF THE ERROR FOR AdaBoost (C
Page 251 and 252:
238 MISCELLANEA for tree classifier
Page 253 and 254:
240 MISCELLANEA This greedy algorit
Page 255 and 256:
242 MISCELLANEA better than the fit
Page 257 and 258:
244 MISCELLANEA 8.2 ERROR CORRECTIN
Page 259 and 260:
246 MISCELLANEA Exhaustive Codes. D
Page 261 and 262:
248 MISCELLANEA Fig. 8.2 Three impl
Page 263 and 264:
250 MISCELLANEA TABLE 8.3 Possible
Page 265 and 266:
252 MISCELLANEA to be the sum of al
Page 267 and 268:
254 MISCELLANEA 8.3.1.3 Jaccard Ind
Page 269 and 270:
256 MISCELLANEA tive definition of
Page 271 and 272:
258 MISCELLANEA . Randomizing. Some
Page 273 and 274:
260 MISCELLANEA Fig. 8.7 Stacked cl
Page 275 and 276:
262 MISCELLANEA The generic cluster
Page 277 and 278:
264 MISCELLANEA The ultimate goal o
Page 279 and 280:
266 MISCELLANEA logicalA=A(i)==A(j)
Page 281 and 282:
268 THEORETICAL VIEWS AND RESULTS T
Page 283 and 284:
270 THEORETICAL VIEWS AND RESULTS F
Page 285 and 286:
Page 287 and 288:
Page 289 and 290:
Page 291 and 292:
278 THEORETICAL VIEWS AND RESULTS C
Page 293 and 294:
Page 295 and 296:
282 THEORETICAL VIEWS AND RESULTS 9
Page 297 and 298:
Page 299 and 300:
286 THEORETICAL VIEWS AND RESULTS t
Page 301 and 302:
288 THEORETICAL VIEWS AND RESULTS s
Page 303 and 304:
290 THEORETICAL VIEWS AND RESULTS 9
Page 305 and 306:
Page 307 and 308:
10 Diversity in Classifier Ensemble
Page 309 and 310:
WHAT IS DIVERSITY? 297 Fig. 10.1 Di
Page 311 and 312:
WHAT IS DIVERSITY? 299 Clearly orac
Page 313 and 314:
MEASURING DIVERSITY IN CLASSIFIER E
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
RELATIONSHIP BETWEEN DIVERSITY AND
Page 321 and 322:
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
USING DIVERSITY 315 Breiman [214] d
Page 329 and 330:
USING DIVERSITY 317 a member of the
Page 331 and 332:
USING DIVERSITY 319 A Matlab functi
Page 333 and 334:
USING DIVERSITY 321 Fig. 10.13 meth
Page 335 and 336:
EQUIVALENCE BETWEEN DIVERSITY MEASU
Page 337 and 338:
MATLAB CODE FOR SOME OVERPRODUCE AN
Page 339 and 340:
MATLAB CODE FOR SOME OVERPRODUCE AN
Page 341 and 342:
330 REFERENCES 13. G. T. Toussaint.
Page 343 and 344:
332 REFERENCES 55. D. L. Wilson. As
Page 345 and 346:
334 REFERENCES 97. D. W. Ruck, S. K
Page 347 and 348:
336 REFERENCES 132. X. Lin, S. Yaco
Page 349 and 350:
338 REFERENCES 169. R. A. Jacobs. M
Page 351 and 352:
340 REFERENCES 206. R. Avnimelech a
Page 353 and 354:
342 REFERENCES 244. E. Pȩkalska, M
Page 355 and 356:
344 REFERENCES 279. K. Tumer and J.
Page 357 and 358:
Index Bagging, 166, 203 nice, 211 p
Page 359 and 360:
INDEX 349 codeword, 244 dichotomy,
show all

Combining Pattern Classifiers

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?