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Vanmarcke Erik — Random Fields : Analysis and Synthesis
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Íàçâàíèå: Random Fields : Analysis and SynthesisÀâòîð: Vanmarcke Erik Àííîòàöèÿ: Random variation over space and time is one of the few attributes that might safely be predicted as characterizing almost any given complex system. Random fields or "distributed disorder systems" confront astronomers, physicists, geologists, meteorologists, biologists, and other natural scientists. They appear in the artifacts developed by electrical, mechanical, civil, and other engineers. They even underlie the processes of social and economic change. The purpose of this book is to bring together existing and new methodologies of random field theory and indicate how they can be applied to these diverse areas where a "deterministic treatment is inefficient and conventional statistics insufficient." Many new results and methods are included.
ßçûê: Ðóáðèêà: Ìàòåìàòèêà /Âåðîÿòíîñòü /Ñòîõàñòè÷åñêèå ïðîöåññû /Ñòàòóñ ïðåäìåòíîãî óêàçàòåëÿ: Ãîòîâ óêàçàòåëü ñ íîìåðàìè ñòðàíèö ed2k: ed2k stats Ãîä èçäàíèÿ: 1983Êîëè÷åñòâî ñòðàíèö: 393Äîáàâëåíà â êàòàëîã: 15.06.2005Îïåðàöèè: Ïîëîæèòü íà ïîëêó |
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Ïðåäìåòíûé óêàçàòåëü
Isotropic behavior of the variance function 243 Isotropic bounds on spectral bandwidth measures 13 145—147 155 Isotropic condition of moments of correlation function 242 287—288 Isotropic condition on 3-D spectral density function 105 Isotropic corner frequencies 274 299 Isotropic correlation measures 242—243 287 289 Isotropic correlation structure 81 242 287 Isotropic envelope statistics 155 Isotropic joint spectral moments 143 146—147 Isotropic lower bound on correlation function 81 Isotropic moving averages 126—127 Isotropic multidimensional spectra 103—105 Isotropic random fields 11 31 242 Isotropic two-dimensional spectra 98—99 ]04 122 243 244 Isserlis, L. 51 Jenkins, G.M. 7 341 Joint event 25 Joint spectral moments, definition and notation 11 96 141—145 Joint spectral moments, interpretation in partial derivatives 13 148—149 Joint spectral moments, second-order relation to quadrant symmetry 149 256—257 310 312—313 Kafritsas, J.C. 198 Kalman, R. 6 Khinchine, A.I. 5 see Kolmogorov, A.N. 6 7 56 72 Krige, D.G. 6 57 Kriging 57 Lag vector, notation 79 Langevin equation see "Brownian motion" Laplace operator 122—123 Lattice process, definition of 21 Lattice process, purely random 62—66 Lattice process, second-order input-output relations 109—110 Laws of disorder 17 218—219 262—264 321—322 Leadbetter, M.R. 12 151 Lee, Y.W. 7 Level crossings see "Threshold crossings" Level curves and surfaces 165 170 Level-crossing statistics if mean by local averages 213—217 258—261 313—315 Level-crossing statistics, mean size of excursion regions 165—169 260—261 313—315 Level-crossing statistics, square derivative exists 155—163 166 Lindenberg, K. 73 Linear estimation see "Optimal linear estimation" linear oscillator 124—125 Linear oscillator, spectral parameters of response to white noise 179—180 Linear systems, response analysis 105—110 217—219 262—264 321—322 Lm, Y.K. 7 82 183 Local averages of integrals, multidimensional 285—286 Local averages of integrals, one-dimensional 185—190 Local averages of integrals, scale of fluctuation and correlation measures 218 263 320 Local averages of integrals, two-dimensional 235 253—255 Local averages of integrals, variance for homogeneous-random fields 115—119 Local maxima of Gaussian processes 156—157 Local maxima, mean frequency or above high levels 157—159 215 262 315—316 Local maxima, tendency to cluster 161—163 Localized linear transformations of a random field 253—255 Lognormal probability distribution 53 Longuet-Higgins, M.C. 140 157 Low-pass white noise see "White noise" Lower bounds see "Bounds" Lumley, J.L. 7 84 195 Lyon, R.H. 161 Macroscale 3 9 225—226 Mandelbrot, B.B. 7 225 Marginal see "Bivariate" "Conditional" Mark, W.D. 183 Markov chains 70—71 Markov correlation structure 120—124 186 193—194 Markov models 178—179 274—275 383—385 Markov processes 69—73 Markov property 69—70 121 Markov, frequency-dependent correlation measures 274—275 303—305 Markov, historical review 3—4 Markov, memory length 178 Markov, nonexistence of mean square derivative 175 214 Markov, spectral parameters 178—179 Markov, threshold-crossing process, one-dimensional 164 see Marshal, T.A. 183 Matern, B. 81 Matheron, G. 6 57 Maximum values of independent random variables 35—36 Maximum values, probability distribution for local averages 15 215—216 317—319 Maximum values, probability distribution if mean square derivative exists 163—165 172—174 Mean excursion length for local averages 216—217 259—260 Mean excursion length if mean square derivative exists 159—161 166—170 see Mean of a random variable, definition 37 Mean of frequency 140 Mean recurrence region 314—318 Mean square derivatives 15 111 156 Mean square derivatives of composite models 224 Mean square derivatives of local averages 208—212 257—259 309—312 312—313 Mean square derivatives, existence 113—114 175 192 208—209 213—214 Mean square derivatives, sensitivity to details of correlation structure 113 175 213 Mean square derivatives, use in parameter estimation 328 Mean square differentiability conditions 113—114 175 208—209 Mean square error in optimal linear estimation 57—59 see Mean square of frequency 139—140 147 see Mean square of random variable, definition 37 Mean threshold-crossing rates if mean square derivative exists 155—163 Mean threshold-crossing rates of linear averages 215 230—360 Mean threshold-crossing rates, use in parameter estimation 328 Measurement errors 18 225 330—331 349 351 353 Measurement in linear estimation 58 61 Measurement of random fields 2 350 353 361 Measurement, limited resolution 2 9 213 219 225 330—331 Median, definition of 38 Mejia, J.M. 82 Memory length of Markovian models 178 Meteorology, applications in 1 82 353 Microscale 2 9 13—14 113 213 225 354 Middleton, D. 7 Mitsuyasu, H. 83 Moan, T. 7 Moments of correlation function 12 188 192 205—206 238 241—242 287 Moments of random variable 37 44 Moments of spectral density function 11 139—147 175—180 256 see Monin, A.S. 7 145 Monochromatic wave 135 Monte Carlo simulation 36 349 Moving average, correlation models 119 125—128 Moving average, nonexistence of mean square derivative 175 Moving average, random processes 185 235 285 Moving average, scale of fluctuation and correlation measures 218 263 320 Multicomponent modeling 18 183 221—224 265 296 330—331 349 Narrow-band processes 157 206—208 224 Nearest neighbor in Poisson process 75 76 Negative binomial distribution 67 Nichols, R. 141 Nonhomogeneous random fields 182—183 Normal distribution 47 see Nosko, V.P. 170 316 Ocean engineering 83 136 Ocean engineering, local maxima above heights 157 On-off process 90—92 One-sided spectral density function 86 133 see Optimal linear estimation 6 56—61 349—351 351—353 Ornstein, L.S. 4 72 73 Page, C.H. 183 Parameter estimation for composite processes 330—331 Parameter estimation from digitized records 335—336 345—347 Parameter estimation in Bayasian framework 327 346 Parameter estimation in frequency domain 339—347 Parameter estimation, based on correlation function 331—336 Parameter estimation, based on variance function 336—339 Parameter estimation, simple procedures 327—330 Parseval's theorem 343 Partial derivatives and spectral moments 148—149 Partial derivatives of envelope statistics 12 153—155 Partial derivatives of local average field 309—312 Partial derivatives, statistics 110—113 Parzen, E. 4 70 77 331 339 Peak factor 163 Perrin, J. 69 Petocz, P. 183 Physics, applications in 1 4 5 6 23 68—69 72—73 359—360 see Piersol, A.G. 7 Point process see "Impulse or point process" Poisson process 4 15 47 73—77 Poisson process, linear transformation 221 Poisson process, local maxima above high levels 163 166 170 174 358 see Poisson process, scale of fluctuation 221 232—233 Poisson process, spectral representation 91—92 Poisson variance in elementary probability 27—52 see Poisson variance in linear estimation 57—69 349—351 Positive definite, covariance function 41 Positive definite, covariance matrix 40 Positive definite, ellipsoidal correlation function, matrix 82 Power law approximation to the variance function 205—206 265—266 Prediction 18 349—351 361 see Priestley, M.B. 183 Prior probability 27 52 see Probabilistic independence see "Independence" Probability density function, notation 28 Probability mass function, notation 28 29 30 Pugachev, V.S. 56 118 Purely random fields 62—65 127 Purely random fields, spectral representation 90—91 Quadrant symmetry 11 80 81—82 Quadrant symmetry, input-output in frequency domain 108 Quadrant symmetry, joint spectral moment of second order 149 256—257 310 312—313 Quadrant symmetry, multidimensional homogeneous field 101—102 Quadrant symmetry, space-time correlation structure 133—134 266 Quadrant symmetry, two-dimensional homogeneous field 94—99 Quadrature spectrum 133 Qualified R-crossings 162 317 R-crossings 161—162 see "Envelope" Radial correlation function 81 103 242 274 287 299 Radial spectral density function 98 104 244 Radial spectrum 98 104 122 243 Rain-on-the-roof process 65 69 Random impulse process see "Impulse process" Random measurement errors see "Measurement errors" Random media, nature of uncertainty 2 219 285 347 360—362 Random partition of space, definition 21 219 Random series, definition 21 Random series, local integration 118—119 Random series, purely random 62—63 Random series, scale of fluctuation 230—231 Random series, spectral representation 90—91 Random series, sums or averages 65—66 229 Random vibration 7 18 183 343—344 348 Random walk, historical review 3 Random walk, historical review, analysis 71—73 see "Brownian Ratoosh, P. 24 Rayleigh distribution 54 Rayleigh distribution, envelope 156—157 160 162 215 Regenerative property of variance functions 219—220 322 333 Regions of excursion, statistics 165—170 313—315 see Regions of excursion, summary 16 Reliability function 163—165 172—174 Reliability function for local averages 215—216 262 317—319 Rice, S.O. 4 6 150 155 157 Richards, P.G. 6 Richterdyn, N. 7 Rodriguez-Iturbe, I. 7 Root mean square (r.m.s.) 37 Root mean square (r.m.s.), spectral equivalent 139 Safe region see "Reliability function" Sample space 25 Sample Spectra as random processes 18 341—345 Sample statistics, correlation or covariance function 332—336 Sample statistics, invariant uncertainty density 332 340 Sample statistics, scale of fluctuation 332—335 336—340 346 Sample statistics, spectral 18 339—340 Sample statistics, variance 332 336 343—344 Sample statistics, variance function 336—339 Sampling intervals, relationship to scale of fluctuation 18—19 330 336 345—346 351—355 361 Savage, L.J. 27 Scale of fluctuation for wide-band random processes, sufficient descriptor of 224 349 355 Scale of fluctuation in Brownian motion 339 Scale of fluctuation of composite models 224—225 323 Scale of fluctuation of fractional noise 225 Scale of fluctuation of linear transformations see also "Invariant" Scale of fluctuation of local average processes 219 320 Scale of fluctuation of random series 230—231 Scale of fluctuation of sample spectra 341 Scale of fluctuation, conditional 16 247—250 291—295 Scale of fluctuation, direction-dependent 239—240 288 289—294 294—295 Scale of fluctuation, ergodicity 346 Scale of fluctuation, existence conditions 192 240 287 Scale of fluctuation, frequency-dependent 17 267—275 294 300 304 305 Scale of fluctuation, one-dimensional 14 185 187—190 220—221 233 236 Scale of fluctuation, piecewise linear approximation 248 293—294 Scale of fluctuation, relationship to optimal sampling intervals 18 330 351—353 Scale of fluctuation, statistical estimation 327—347 327 346 Scale of fluctuation, zero-scale narrow-band process 206—208 Scale of measurement 1—3 9 24 63 Scale of measurement, relationship to fractional noise 7 225 Schwartz's inequality, spectral moments of 140 Schweppe, F.C. 5 Self-similar in patterns of random variation 7 see Separable random field, correlation structure 16 82 97 241—242 245 290 Separable random field, frequency-dependent correlation measures 274 300 Seshadri, V. 73 Sherman, C. 136 Shmazuka, M. 7 136 Shot noise 4 see Simple events 26 Simulation 36 349 Sinusoid with random phase angle, spectral parameters of 176—177 Slutzky, E. 7 Smith, F. 206 Smoluchovski, M. 4 72 Space-time process, correlation structure 128—137 266—267 267—275 Space-time process, effect of spatial averaging on frequency content 275—282 Space-time process, frequency-dependent spatial scale of fluctuation 267—275 Space-time process, nature of the uncertainty 2 10 Space-time process, summary of results 16 18 Spectral bandwidth measures 12 139—140 145—147 Spectral bandwidth measures of local averages 212—213 217 314 Spectral bandwidth measures, bounds 12 Spectral bandwidth measures, examples 175—182 Spectral bandwidth measures, existence 174—175 213—214 Spectral bandwidth measures, relationship to excursion statistics 152—155 156—158 161 261 Spectral density function of composite models 222—223 225 323 Spectral density function of fractional noise 225 Spectral density function of random impulse process 91—92 Spectral density function of random series 90—91 Spectral density function under local averaging, limiting form 321 Spectral density function, behavior near zero frequency 191—193 238—239 242 287—288 352 Spectral density function, common models 174—182 193—194 Spectral density function, constraints on isotropic models 105 242 288 Spectral density function, derivative evaluated at zero 192 240 287 Spectral density function, effect of spatial averaging in space-time process 278—282 300—303 Spectral density function, estimation procedures 327 339—347 Spectral density function, historical review 5—7 10 Spectral density function, input-output for linear systems 102 110 Spectral density function, moments see "Spectral moments" Spectral density function, one-dimensional 84—92 Spectral density function, one-sided versus two-sided 85 86 133 Spectral density function, radial 98 104 243 Spectral density function, relationship to variance function 191 236—239 251 287 Spectral density function, sample spectra as random functions of frequency 18 341—345 347 Spectral density function, sensitivity to microscale variations 13—14 175 213 Spectral density function, two-dimensional 92—99 128—137 244 267 Spectral density function, unit-area, analogy with probability density function 139—140 Spectral distribution function 87 96—97 103 Spectral distribution function, one-dimensional bounds 140—141 Spectral moments of local averages 210—213 256 Spectral moments, definition 139—147 Spectral moments, examples 175—182
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