Parameter estimation of random fields with long-range dependence. (English) Zbl 0829.76070
Summary: This paper considers random fields with spectral density having a pole of fractional order at the origin. This class of models offers a suitable framework for studying long-range spatial correlation. An approximation to the likelihood function is given for the estimation of model parameters. The convergence rate of the approximation is obtained, and an iterative method for maximising the approximate likelihood function is indicated. The method is applied to analyse a time series of maximum daily ozone levels and some texture data.
MSC:
| 76M30 | Variational methods applied to problems in fluid mechanics |
| 86A10 | Meteorology and atmospheric physics |
Keywords:
photochemical pollution; air quality modelling; fractal index; spectral density; likelihood function; convergence; iterative method; time series; maximum daily ozone levelsReferences:
| [1] | Melli, P.; Spirito, A., Atmospheric diffusion modelling by stochastic differential equations, (Proc. \(14^{th}\) NATO/CCMS International Technical Meeting on Air Pollution Modelling and Its Applications. Proc. \(14^{th}\) NATO/CCMS International Technical Meeting on Air Pollution Modelling and Its Applications, Copenhagen, September 1983 (1983)), 771-784 |
| [2] | Thomson, D. J., Criteria for the selection of stochastic models of particle trajectories in turbulent flows, J. Fluid Mech., 180, 529-556 (1987) · Zbl 0644.76065 |
| [3] | Sawford, B. L., Recent development in the Lagrange stochastic theory of turbulent dispersion, Boundary Layer Meteorology, 62, 197-215 (1993) |
| [4] | Selvam, A. Mary; Pethkar, J. S.; Kulkarni, M. K., Signatures of a universal spectrum for atmospheric interannual variability in rainfall time series over the Indian region, International J. Climatology, 12, 137-152 (1992) |
| [5] | Haslett, J.; Raftery, A. E., Space-time modelling with long-memory dependence: Assessing Ireland’s wind power resource, Appl. Statist., 38, 1-50 (1989) |
| [6] | Mann, H.; Wald, A., On the statistical treatment of linear difference equations, Econometrica, 11, 173-220 (1943) · Zbl 0063.03773 |
| [7] | Whittle, P., Estimation and information in time series analysis, Skand. Aktuar., 35, 48-60 (1952) · Zbl 0049.37301 |
| [8] | Whittle, P., On stationary processes in the plane, Biometrika, 41, 434-449 (1954) · Zbl 0058.35601 |
| [9] | Guyon, X., Parameter estimation for a stationary process on a \(d\)-dimensional lattice, Biometrika, 69, 95-105 (1982) · Zbl 0485.62107 |
| [10] | Sharma, G.; Chellappa, R., A model-based approach for estimation of two-dimensional maximum entropy power spectra, IEEE Trans. Info. Theory, 31, 90-99 (1985) · Zbl 0563.62069 |
| [11] | Künsch, H., Thermodynamics and statistical analysis of Gaussian random fields, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 58, 407-421 (1981) · Zbl 0458.60053 |
| [12] | Heyde, C. C.; Gay, R., On asymptotic quasi-likelihood estimation, (Stoch. Proc. Appl., 3 (1989)), 223-236 · Zbl 0684.62067 |
| [13] | Heyde, C. C.; Gay, R., Smoothed periodogram asymptotics and estimation for processes and fields with possible long-range dependence, (Stoch. Proc. Appl., 45 (1993)), 169-182 · Zbl 0771.60021 |
| [14] | Anh, V. V., Asymptotic maximum likelihood estimation of random fields, QUT Working Paper in Mathematics 7/93 (1993) |
| [15] | Chiang, T.-P., Entropy of the stationary random fields, Acta Scientiarum Naturalium Universitatis Pekinensis, 1, 25-35 (1987) · Zbl 0629.60059 |
| [16] | Dahlhaus, R., Efficient parameter estimation for self-similar processes, Ann. Statist., 17, 1749-1766 (1989) · Zbl 0703.62091 |
| [17] | Dzhaparidze, K., Parameter Estimation and Hypothesis Testing in Spectral Analysis of Stationary Time Series (1986), Springer-Verlag · Zbl 0584.62157 |
| [18] | Dahlhaus, R.; Künsch, H., Edge effects and efficient parameter estimation for stationary random fields, Biometrika, 74, 877-882 (1987) · Zbl 0633.62094 |
| [19] | Anh, V. V., Information divergence and maximum likelihood estimation of random fields with possible long-range dependence, QUT Working Paper in Mathematics 8/94 (1994) |
| [20] | Anh, V. V., Maximum entropy spectral estimation of random fields, QUT Working Paper in Mathematics 7/94 (1994) |
| [21] | V.V. Anh and K.E. Lunney, Asymptotic maximum likelihood estimation of random fields with long-range dependence (under preparation).; V.V. Anh and K.E. Lunney, Asymptotic maximum likelihood estimation of random fields with long-range dependence (under preparation). · Zbl 0829.76070 |
| [22] | Brodatz, P., Textures: A Photographic Album for Artists and Designers (1966), Dover: Dover New York |
| [23] | Anh, V. V.; Gras, F.; Tsui, H. T., A class of fractional random fields for image texture modelling, (Proc. 1994 International Symposium on Speech, Image Processing & Neural Networks. Proc. 1994 International Symposium on Speech, Image Processing & Neural Networks, Hong Kong (1994)) · Zbl 0871.60041 |
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