Index PF(A), 391 α-lower approximation, 340 α-lower bound, 339 α-reduct, 109 α-upper approximation, 340 α-upper bound, 339 δ-neighborhood consistent, 291 ε-approach nearness, 558 C, 443-2 system, 8 Cech distance, 549 abstract DIS-algebra, 389 IS-algebra, 394 NIS-algebra, 398 accuracy of approximation, 83 acyclic refinement, 347 affinity separation technique, 417 algorithm, 237 240 approach distance, 546 approach space, 549 approximation concept, 85 lower, 336 set, 80 upper, 336 approximation space, 90, 439, 442, 443, 459 generalized, 91 approximations, 229 232, 239, 241 association rule irreducible, 127 attribute, 80 condition, 83 in i.s.r. systems, 180 reduction, 183, 230, 294 selection, 110 significance, 111 Bayesian confirmation measure, 192 bisimulation, 531 bisimulation-based approximation, 540 Boolean reasoning, 87, 107 approximate, 131 boundary of X, 440 region, 80, 445 characteristic relation, 231 characterization, 608 class, 570 classification, 186, 202, 277 clustering function, 117 common attributes, 587, 591 complete, 559 composite property, 344 concept approximation, 85 exact in mereological model, 571 vague, 78 condition atttribute, 83 conflict, 138 conjunction function, 313 consecutive 1s property, 181 consistency assumption, 278 context inducing, 145
646 Index core, 86, 190 covering, 454, 459 covering-based rough sets, 438, 440 442, 459 criterion, 186, 206 cut, 112 on attribute, 137 data mining, 617 attribute, 83 class, 84 generalized, 84 rule, 84, 190, 206, 213 coverage, 214 irredundant, 213 length, 214 minimal, 121 true, 84 truth degree, 84 rule reduction, 435 system, 83 consistent, 84 inconsistent, 84 table, 83, 213 degenerated, 213 separable subtable, 213 deoxyribonucleic acid, 411 dependency of attributes, 85 description of a point, 547 deterministic information system, 382 deterministic information system algebra, 387 differentiation, 606 digraph, 424 discernibility function, 106, 132 decision relative, 108 matrix, 106, 133 decision relative, 108 relation, 87 discretization, 88, 113 of large data sets, 136 distance, 548 DNA molecular technique, 413 rough-set computing, 418 dominance, 188, 206 -based rough set approach, 186 cone, 188 relation, 231 233 Dow Jones Industrial Average, 501 DRSA, 186 dynamic programming approach, 215 E-approximation lower, 573 upper, 573 Ehrenfeucht and Pawlak seminar, 176 Ehrenfeucht, Andrzej, 176 elementary property, 338 encoding process, 424 equivalence relation, 230, 335 error approximation, 110 evaluation function of an i.s.r., 180 exclusive rule, 612 extension of approximation space, 98 external relation, 586 feature selection, 294 filter prime, 391 focusing mechanism, 608 framework, 446, 452 gaze tracking system, 479 gel electrophoresis technique, 418 graded ill-known set, 313 granular computing, 142 interactive rough, 145 granular concept, 587 description language, 592 hierarchy, 590 GCH, 587 hierarchy construction, 594 granule, 143 information, 143 of knowledge, 188 greatest definable subset, 183 hard coding the environment, 250 heterogeneous Euclidean-overlap metric function (HEOM), 281 higher order rule, 600 hydrogen bond, 412 i.s.r. systems, 179 ICHD-II
Index 647 International Classification of Headache ver 2.0, 606 IHD International Headache Society, 606 ill-known set, 311 implicant, 87 approximate, 88 prime, 87 implication function, 313 residual, 575 inclusion function, 91 inclusion property kernel, 347 inclusive rule, 614 incomplete information system, 383 indicator function, 250 indiscernibility relation, 80, 183, 263, 439, 440, 459 information granulation, 143 storage and retrieval systems, 175 system, 80, 181, 383 ingredient, 570 internal relation, 586 invariants, 249 inverted file, 180 jmaf, 185, 193 kernel functions, 493 Kleene connectives, 448 450 knowledge approximation algebra, 385 knowledge discovery, 229 231, 246 least definable superset, 183 LERS data mining system, 267 level-wise attribute selection, 595 linear separability, 492 Lipski, Witold, 178 lower approximation, 80, 188, 189, 206, 312, 313, 336, 445 lower approximation of a set X, 442 lower distance, 549 mapping lower-semi-continuous, 580 upper-semi-continuous, 580 measure of similarity Jaccard-Needham, 375 median of a decision class, 137 medical diagnosis, 606 minimum description length, 183 principle, 83 MIR, 463, 464, 466 mixed approximation, 346 MLEM2 rule induction algorithm, 267 model mereological for rough sets, 571 rough mereological for rough sets, 574 MPEG 7, 464, 466 multi-classification, 494 1-v-1, 495 multiple criteria sorting, 186, 205 music genre classification, 476 music information retrieval MIR, 463 music recommendation, 465, 468 music social networking systems, 467 NEAR system, 551 nearness, 547 necessity measure, 321 negative region, 445 rule, 612 neighborhood, 281 entropy, 289 mutual information, 290 rough sets, 280 neighbourhood of points, 552 nitrogen-containing base, 411 non-deterministic information system, 383 non-deterministic information system algebra, 395 non-linear transformation, 492 NP-hard problem, 410, 435 ontology approximation, 103 ordinal classification with monotonicity constraints, 186, 205 overlap, 570 parameterized approximations, 264 part, 570 partial order, 335 Pawlak and physicians, 178 approximations, 263
648 Index machine, 3 Zdzisław Ignacy, 2 Pawlak s approximation quality, 253 rough sets, 440, 442, 459 perception based computing, 151 perceptional vector analysis, 250 Polish school of Artificial Intelligence, 2 positive region, 445 region of X, 440 rule, 611 possibility distribution, 313 measure, 321 precision, 265 preference, 186, 187, 205, 206 prime filter, 391 implicant, 87 probabilistic approximations, 264 rule, 611 probe function, 550 process mining, 149 property of the n-ary relation, 338 property-driven approximation space, 338 pseudo-random number generator, 5 quadtrees, 257 quality approximation space, 98 concept approximation, 98 decision rule, 85 of approximation, 189 recall, 265 recursive granulations, 594 reduct, 86, 106, 190 -α, 109 approximate, 110 decision relative, 107 minimal, 133 region boundary, 80, 264 negative, 264 positive, 85, 264 region-of-interest, 546 neighbourhood, 546 relation discernibility, 87 indiscernibility, 80 relational roughification, 523 RHINOS, 607, 618 rough approximation quality, 253 inclusion, 574 membership function, 88 mereology, 92 patterns, 500 real functions, 500 representation of ill-known set, 319 set, 82, 206 based classifiers, 94 based logics, 139 theory, 3, 229 sets, 175, 439, 440, 448, 460 support vector clustering, 507 machines, 492 rule boundary, 265 certain, 267 extraction, 298 induction, 614 positive, 265 possible, 265 sample selection, 291 saturated sequent, 452 454 scalability, 131 semantics of a granular concept, 592 set approximation, 80 crisp, 81 rough, 82 set-valued information system, 229 231 similarity relation, 91, 230 similarity-based roughification, 520 social choice functions, 364 subordination relation, 439, 441, 443, 445, 448, 459 C, 444 sufficiently near, 559 supervaluationism, 626 support vector machines, 492
Index 649 clustering, 507 regression, 500 symbolic value grouping, 88, 117 symmetric kernel, 351 syntax of a granular concept, 592 system information, 80 t norm Łukasiewicz, 575 Archimedean, 575 minimum, 575 product, 575 terminological roughification, 533 the first mathematical model of Crick and Watson s DNA encoding, 3 the Pawlak approach to conflict analysis, 3 three-valued Kleene connectives, 439 logic, 438, 440, 443, 445, 459 non-deterministic matrix, 440 threshold, 553 tolerance relation, 91, 230 transitive closure, 336 translations, 252 uncertainty function, 91 upper approximation, 80, 188, 189, 206, 312, 313, 336, 445 of a set X, 442 vague concept, 78 vagueness, 79, 624 higher order, 101 value set reduction, 112 variable consistency, 186, 193 visual neighbourhood, 553 perception, 249 voting procedure, 364 amendment, 364 approval, 365 Black, 365 Borda, 364 Coombs, 365 Copeland, 364 Dodgson, 364 Hare, 365 max-min, 364 Nanson, 365 plurality, 364 runoff, 365 voting procedures agreement between, 374 comparison criteria, 365 distance between, 374 wisdom technology (Wistech), 154