Principles of Modeling Uncertainties in Spatial Data and Spatial Analyses

Cover
......Page 1
Principles of Modeling Uncertainties in Spatial Data and Spatial Analyses......Page 3
Contents......Page 5
Foreword......Page 18
Preface......Page 20
Acknowledgments......Page 24
Section I: Overview......Page 25
1.1.1 THE CONCEPT OF UNCERTAINTY......Page 26
1.1.2.2 Systematic Error......Page 28
1.2 UNIVERSALITY OF UNCERTAINTY......Page 29
1.3.1 DIMENSIONS OF SPATIAL DATA......Page 31
1.3.2.1.1 Accuracy......Page 32
1.3.2.2 Elements of Spatial Data Quality......Page 33
1.3.2.2.2 Attribute Accuracy......Page 34
REFERENCES......Page 35
2.1 INTRODUCTION......Page 37
2.2 UNCERTAINTIES INHERENT IN THE NATURAL WORLD......Page 39
2.4.1 SPATIAL DATA......Page 40
2.4.2 SPATIAL DATA MEASUREMENT METHODS......Page 41
2.4.3 UNCERTAINTIES IN SPATIAL DATA CAPTURE......Page 42
2.4.3.3 The Total Station......Page 43
2.4.3.4 Satellite Positioning Technology......Page 44
2.4.3.6 Remote Sensing Technology......Page 45
2.5.1 UNCERTAINTIES IN SPATIAL ANALYSES......Page 46
2.5.2.1 Uncertainty Introduced from Computation and Equipment......Page 47
REFERENCES......Page 48
3.2 PROBABILITY THEORY......Page 49
3.2.1.2 Definition of Probability......Page 50
3.2.1.4.2 Independence......Page 51
3.2.1.4.4 Bayes’ Formula......Page 52
3.2.1.4.6 Poisson Approximation......Page 53
3.2.2.3 The Distribution Function of a Function of the Random Variable......Page 54
3.2.3.1 Expectation and Variance......Page 55
3.2.3.3 Chebyshev’s Inequality......Page 56
3.2.4.1 Laws of Large Numbers......Page 57
3.2.4.2 The Central Limit Theorem......Page 58
3.3.2.1.1 Binomial Distribution......Page 59
3.3.2.1.3 Hypergeometric Distribution......Page 77
3.3.3.2 Statistic of the Population and the Sample......Page 78
3.3.3.3.1 Point Estimation with the Method of Moments......Page 80
3.3.3.3.2 Point Estimation with the Method of Maximum Likelihood......Page 82
3.3.3.3.4 Interval Estimation of a Population Parameter......Page 83
3.3.4.2.1 All Known Parameters......Page 84
3.3.5.2 Lattice Data......Page 85
3.4.1 TERMINOLOGIES AND SYMBOLS......Page 86
3.4.2 NONSPECIFICITY MEASURE......Page 88
3.4.3 DISCORD MEASURE......Page 90
3.4.5 APPLICATION OF THE EVIDENCE THEORY FOR SPATIAL DATA......Page 91
3.5.2.1 Fuzzy Set......Page 92
3.5.2.2 Operations of Fuzzy Sets......Page 94
3.5.3.1 Definition of Level Set......Page 95
3.5.4.1 Left-Shouldered Membership Function......Page 96
3.5.4.2 Right-Shouldered Membership Function......Page 97
3.5.4.3 Middle Membership Function......Page 98
3.5.5 FUZZY TOPOLOGY......Page 99
3.5.5.2 Fuzzy Topological Space......Page 100
3.5.5.2.5 Fuzzy Continuous Projection......Page 101
3.6.1 DEFINITIONS AND PROPERTIES......Page 102
3.6.2 APPROXIMATION AND MEMBERSHIP RELATION......Page 103
3.6.3.1 Numerical Characteristic......Page 104
3.6.3.2 Topological Characteristic......Page 105
3.6.4 APPROXIMATION FOR CLASSIFICATION......Page 106
3.6.5.2 Rough Inclusion of a Set......Page 107
3.7.1 MATHEMATICAL INFORMATION SOURCE MODEL......Page 108
3.7.2 AMOUNT OF SELF-INFORMATION......Page 109
3.7.3 ENTROPY OF INFORMATION......Page 110
3.7.4 ALGEBRAIC AND ANALYTICAL PROPERTIES OF ENTROPY......Page 111
REFERENCES......Page 114
Section II: Modeling Uncertainties in Spatial Data......Page 116
4.1 INTRODUCTION......Page 117
4.2.1 POSITIONAL ERROR FOR A POINT ON A LINE SEGMENT......Page 118
4.2.3 ERROR-BAND MODEL FOR LINE FEATURES......Page 119
4.3 DEFINITION OF SPATIAL FEATURES IN GEOGRAPHIC INFORMATION SCIENCE......Page 120
4.3.1 ONE-DIMENSIONAL SPATIAL FEATURES......Page 121
4.3.2 TWO-DIMENSIONAL SPATIAL FEATURES......Page 122
4.3.3 N-DIMENSIONAL SPATIAL FEATURES......Page 123
4.4 MODELING POSITIONAL UNCERTAINTY FOR A POINT......Page 125
4.5.1 CONFIDENCE INTERVAL MODEL FOR SPATIAL FEATURES IN ONE-DIMENSIONAL SPACE......Page 127
4.5.1.2 Confidence Interval of a Line Segment in a One-Dimensional Space......Page 128
4.5.1.3 Confidence Interval of a One-Dimensional Polyline......Page 129
4.5.2.1 Confidence Region Model for a Point in Two-Dimensional Space......Page 130
4.5.2.2 Confidence Region Model for a Line Segment in Two-Dimensional Space......Page 131
4.5.2.3 A Confidence Region Model for a Polyline in Two-Dimensional Space......Page 133
4.5.3.1 Confidence Space for a Point in an n-Dimensional Space......Page 136
4.5.3.3 Confidence Space for an n-Dimensional Polyline......Page 137
4.6 PROBABILITY DISTRIBUTION MODEL FOR LINE SEGMENT......Page 139
4.7.1 ERROR DISTRIBUTION AND PROBABILITY DENSITY FUNCTIONS......Page 143
4.7.3 THE G-BAND MODEL......Page 145
4.7.4.1 Directional Independence......Page 147
4.7.4.1.1 Homogeneity and Directional Independence......Page 148
4.8.1 POSITIONAL ERROR AT ANY POINT ON A CURVE......Page 151
4.8.2 THE εσ ERROR MODEL......Page 154
4.8.3 THE εm ERROR MODEL......Page 155
4.8.4 AN ERROR MODEL FOR IRREGULAR CURVE: THIRD-ORDER SPLINE CURVE......Page 158
4.8.5 CASE STUDY AND ANALYSIS......Page 161
4.8.6 NOTE ON CURVE ERROR MODELS......Page 163
4.9.1 ERROR MODELING BASED ON ITS COMPONENT VERTICES......Page 164
4.9.2 ERROR MODELING BASED ON ITS COMPOSING LINE SEGMENTS......Page 165
4.9.3 CASE STUDY AND ANALYSIS......Page 166
REFERENCES......Page 168
5.1.1 ATTRIBUTE AND ATTRIBUTE DATA......Page 170
5.1.3 SOURCES OF ATTRIBUTE UNCERTAINTY......Page 171
5.1.5 UNCERTAINTY IN CLASSIFICATION......Page 174
5.2.1 SAMPLING METHODS......Page 175
5.2.2 INTERNAL AND EXTERNAL TESTING......Page 176
5.2.3 ERROR MATRIX METHOD......Page 177
5.2.4.1 Maximum Likelihood Classification......Page 180
5.2.4.3 The Four Parameters......Page 181
5.2.5 THE INCIDENCE OF DEFECTS METHOD......Page 183
5.2.6 MODELING ERROR PROPAGATION FOR CONTINUOUS ATTRIBUTE DATA......Page 186
5.3 SUMMARY AND COMMENT......Page 189
REFERENCES......Page 190
6.1 INTRODUCTION......Page 192
6.2 PROBLEM DEFINITION......Page 194
6.4 PROBABILITY THEORY-BASED SOLUTION......Page 195
6.5.1 THE MYCIN CERTAINTY FACTOR......Page 196
6.5.2 PROBABILISTIC INTERPRETATION OF CERTAINTY FACTORS......Page 198
6.5.4 DERIVATION OF THE FORMULAS......Page 199
6.6 AN EXAMPLE OF MODELING INTEGRATED UNCERTAINTY......Page 201
6.6.1 MODELING POSITIONAL UNCERTAINTY......Page 202
6.6.2 MODELING ATTRIBUTE UNCERTAINTY......Page 203
6.6.4 VISUALIZATION OF UNCERTAINTIES......Page 204
REFERENCES......Page 207
Section III: Modeling Uncertainties in Spatial Model......Page 208
7.1 AN OVERVIEW OF TOPOLOGICAL RELATIONSHIP MODELS......Page 209
7.1.3 ALGEBRAIC MODEL......Page 210
7.1.5 A NINE-INTERSECTION MODEL BETWEEN SIMPLE FUZZY REGIONS......Page 211
7.2.1.1 N-Cell and N-Cell Complex......Page 212
7.2.1.2 The Definition of a Spatial Object......Page 213
7.2.2 TOPOLOGICAL RELATIONSHIPS BETWEEN SPATIAL OBJECTS......Page 214
7.2.3 A DECISION ALGORITHM FOR TOPOLOGICAL RELATIONSHIP......Page 215
7.3 MODELING UNCERTAIN TOPOLOGICAL RELATIONSHIPS......Page 216
7.3.2 QUASI COINCIDENCE......Page 217
7.3.3 QUASI DIFFERENCE......Page 220
7.3.4 TOPOLOGICAL RELATIONS BETWEEN TWO FUZZY SETS R2......Page 222
7.4.2.2 Effect of an Infected Region on Neighboring Regions......Page 224
REFERENCES......Page 226
8.2 SURFACE MODELING......Page 228
8.2.1 COMMON INTERPOLATION METHODS......Page 229
8.2.2 REGULAR GRID DIGITAL ELEVATION MODEL......Page 232
8.3 ERROR SOURCES OF A DEM MODEL......Page 233
8.4 ACCURACY ESTIMATION FOR A TIN MODEL......Page 235
8.4.1 MATHEMATICAL FORMULA OF TIN ACCURACY ESTIMATION......Page 236
8.4.2 EXPERIMENTAL VALIDATION OF THE FORMULA......Page 239
8.4.3 THEORETICAL PROOF FOR THE MATHEMATICAL FORMULA......Page 240
8.5 ACCURACY ESTIMATION OF A REGULAR GRID DEM......Page 243
8.5.1.1 Two-Dimensional Cubic Interpolation......Page 244
8.5.1.2 Three-Dimensional Cubic Interpolation......Page 246
8.5.2.1 The Average Elevation Error......Page 248
8.5.2.2 A Formula for the Average Elevation Error......Page 250
8.6 SUMMARY......Page 252
REFERENCES......Page 253
Section IV: Modeling Uncertainties in Spatial Analyses......Page 254
9.1 INTRODUCTION......Page 255
9.2 REVIEW OF THE EXISTING MODELS......Page 256
9.3.1.1 Covariance Matrix of an Intersection Point......Page 257
9.3.1.2 Covariance Matrix of Vertices of the Generated Polygon......Page 260
9.3.1.3 Variances of Measurements of the Generated Polygon......Page 261
9.3.1.4 Uncertainty Interval for the Vertices of the Generated Polygon......Page 262
9.3.1.5 A Note on the Analytical Model......Page 263
9.3.2 SIMULATION APPROACH......Page 264
9.3.2.1 A Simulated Sample Covariance Matrix for the Generated Polygon......Page 266
9.3.2.2 Simulated Sample Variances of Measurements of the Generated Polygon......Page 267
9.3.2.3 Simulated Uncertainty Intervals of the Generated Polygon......Page 268
REFERENCES......Page 269
10.1 INTRODUCTION......Page 270
10.2 EXISTING BUFFER ANALYSIS ERROR MODEL......Page 271
10.3.1 ERROR OF COMMISSION AND ERROR OF OMISSION......Page 272
10.3.1.1 Buffer Around a Point......Page 274
10.3.1.2 Buffer Around a Straight-Line Segment......Page 275
10.3.1.3 Buffer Around a Polyline......Page 277
10.3.1.4 Buffer Around a Polygon......Page 279
10.3.1.5 Discussion of the Error of Commission and the Error of Omission......Page 280
10.3.2 NORMALIZED DISCREPANT AREA......Page 281
REFERENCES......Page 286
11.1 INTRODUCTION......Page 288
11.2 UNCERTAINTIES IN LINE SIMPLIFICATION......Page 289
11.3.1 MODELING MEASUREMENT ERROR OF THE INITIAL LINE......Page 290
11.3.2 MODELING LINE SIMPLIFICATION ERROR......Page 291
11.3.3 MODELING THE TOTAL ERROR......Page 293
11.3.4 AN EXAMPLE......Page 294
REFERENCES......Page 296
Section V: Quality Control of Spatial Data......Page 298
12.1 INTRODUCTION......Page 299
12.2.1 CONDITIONAL EQUATIONS OF THE PARCEL AREA......Page 300
12.2.1.1 The Rectangular Conditional Equation......Page 301
12.2.1.2 Conditional Equation for Arcs......Page 302
12.2.2 ESTIMATION OF VARIANCE COMPONENTS......Page 305
12.2.3.1 Area Conditional Equation with Scale Parameter......Page 307
12.2.3.2 The Effect of Scale Error and Its Significance Test......Page 308
12.3 QUALITY CONTROL METHODS FOR CADASTRAL DATA......Page 310
12.4.1 AREA ADJUSTMENT NEGLECTING ERROR OF REGISTERED PARCEL AREA......Page 313
12.4.2 ESTIMATION OF VARIANCE COMPONENTS WITH RECTANGULAR CONSTRAINTS......Page 315
12.4.3 AREA ADJUSTMENT WITH SCALE PARAMETER AND RECTANGULAR CONSTRAINTS......Page 316
12.5 SUMMARY......Page 319
REFERENCES......Page 320
13.1 INTRODUCTION......Page 321
13.2.1 THE GENERIC TWO-DIMENSIONAL POLYNOMIAL TRANSFORMATION MODEL......Page 322
13.2.3 THE TWO-DIMENSIONAL AFFINE TRANSFORMATION MODEL......Page 323
13.3.1 GENERIC THREE-DIMENSIONAL POLYNOMIAL MODEL......Page 324
13.3.3 RATIONAL FUNCTION MODEL......Page 325
13.4.1 RELATIONSHIP DEFINED BY UNIT VECTOR......Page 326
13.4.2 AFFINE LBTM......Page 328
13.5.1 DATA SET......Page 329
13.5.2 EXPERIMENTS USING THE TWO-DIMENSIONAL TRANSFORMATION MODELS......Page 330
13.5.3 EXPERIMENTS ON THE THREE-DIMENSIONAL TRANSFORMATION MODEL......Page 331
13.6 AN EXPERIMENTAL ON LINE-BASED TRANSFORMATION MODEL......Page 332
13.6.1 DATA SETS......Page 333
13.6.2 RESULTS AND ANALYSIS......Page 334
13.7 SUMMARY AND COMMENT......Page 335
REFERENCES......Page 336
14.2 HYBRID INTERPOLATION METHOD......Page 338
14.2.1 MATHEMATICAL EXPRESSION OF HYBRID INTERPOLATION......Page 339
14.2.2 THEORETICAL INTERPRETATION OF ρ......Page 340
14.2.3.1 Data Set A......Page 342
14.3 BIDIRECTIONAL INTERPOLATION METHOD......Page 345
14.3.1 MATHEMATICAL FORMULA FOR BIDIRECTIONAL INTERPOLATION......Page 346
14.3.2 SLOPE ESTIMATION......Page 347
14.3.3 EXPERIMENTAL RESULTS......Page 348
14.3.3.1 Data Set A......Page 349
14.3.3.2 Data Set B......Page 351
REFERENCES......Page 353
Section VI: Presentation of Data Quality Information......Page 354
15.1 INTRODUCTION......Page 355
15.2 ERROR ELLIPSE APPROACH......Page 356
15.2.1 BASIC EQUATION......Page 357
15.3 ARROW-BASED APPROACH......Page 358
15.3.1.1 Conformal Transformation......Page 359
15.3.1.2 Inverse Distance Interpolation......Page 362
15.3.2 ANALYSIS OF THE ARROW-BASED APPROACH......Page 363
15.4 GRAY-SCALE MAP......Page 364
15.4.2 SINGLE CLASS MAPPING......Page 365
15.5 COLOR MAP......Page 366
15.5.1 RGB COLOR MODEL......Page 367
15.5.1.2 Color Mixture......Page 368
15.5.2 HSI COLOR MODEL......Page 369
15.5.3 ANALYSIS OF THE COLOR APPROACH......Page 371
15.7 THREE-DIMENSIONAL APPROACH......Page 372
15.7.1 THE 3D APPROACH FOR VECTOR DATA......Page 373
15.7.2 THE 3D APPROACH FOR RASTER DATA......Page 374
15.8.2 RASTER IMAGE ANIMATION APPROACH......Page 375
REFERENCES......Page 376
16.1 INTRODUCTION......Page 378
16.1.1 THE FEDERAL GEOGRAPHIC DATA COMMITTEE METADATA STANDARD......Page 379
16.1.2 THE ANZLIC STANDARD......Page 380
16.1.4 THE ISO METADATA STANDARD......Page 381
16.2.1.1 Lineage......Page 383
16.2.1.2 Quality Information Report......Page 384
16.2.2 COMPLETENESS......Page 385
16.2.3.1 Measures for Conceptual Consistency......Page 386
16.2.3.4 Measures for Topological Consistency......Page 387
16.2.4.1.1 Point Measurement......Page 388
16.2.4.1.2 Line Measurement......Page 389
16.2.4.2 Relative or Internal Accuracy......Page 391
16.2.5.2 Temporal Consistency......Page 392
16.2.6.1 Classification Correctness......Page 393
16.3 AN OBJECT-ORIENTED METADATA SYSTEM......Page 394
16.3.1 DATABASE DESIGN......Page 395
16.4 SUMMARY AND COMMENT......Page 396
REFERENCES......Page 398
17.1 INTRODUCTION......Page 399
17.2.1 DATA TRANSFORMATION SERVICE......Page 400
17.2.3 THE CLIENT DESIGN......Page 402
17.3.1 QUALITY INFORMATION OF POINT OBJECTS......Page 403
17.3.1.1 Definition of Interfaces......Page 404
17.3.2 QUALITY INFORMATION OF LINE OBJECTS......Page 405
17.3.3 QUALITY INFORMATION OF AREA OBJECTS......Page 406
17.3.4 QUALITY INFORMATION OF BUFFER SPATIAL ANALYSIS......Page 407
17.3.5 QUALITY INFORMATION ON OVERLAY SPATIAL ANALYSIS......Page 409
17.4 SUMMARY AND COMMENT......Page 411
Section VII: Epilogue......Page 412
18.1.1 UNDERSTANDING MORE ABOUT THE COMPLEXITY OF THE NATURAL WORLD......Page 413
18.1.2 INVESTIGATION OF UNCERTAINTIES IN THE COGNITION OF GEOGRAPHIC OBJECTS......Page 414
18.1.5 DATA QUALITY CONTROL FOR GIS WITH COMPLEX UNCERTAINTIES......Page 415
18.2.1 GEOSTATISTICS......Page 416
18.3.1 ERROR AWARENESS......Page 417
18.3.3 QUALITY MARKS ON GIS DATA PRODUCTS......Page 418
18.4 INTERACTION WITH OTHER DISCIPLINES......Page 419
ACKNOWLEDGMENT......Page 420