Stochastic modelling of skewed data exhibiting long range dependence

Transcription

1 IUGG XXIV General Assembly 27 Perugia, Italy, 2 3 July 27 International Association of Hydrological Sciences, Session HW23 Analysis of Variability in Hydrological Data Series Stochastic modelling of skewed data exhibiting long range deendence S.M. Paalexiou Deartment of Water Resources, National Technical University of Athens

2 . Abstract Time series with long-range deendence aear in many fields including hydrology and there are several studies that have rovided evidence of long autocorrelation tails. Provided that the intensity of the long-range deendence in time series of a certain rocess, quantified by the self-similarity arameter, also known as the Hurst exonent H, could not be falsified, it is then essential that the variable of interest is modelled by a model reroducing long-range deendence. Common models of this category that have been widely used are the fractional Gaussian noise (FGN) and the fractional ARIMA (FARIMA). In case of a variable exhibiting skewness, the revious models can not be imlemented in a direct manner. In order to reserve skewness in the simulated series, a normalizing transformation is tyically alied in the real-life data at first. The models are then fitted to the normalized data and the roduced synthetic series are finally denormalized. In this aer, a different method is roosed, consisting of two arts. The first one regards the aroximation of the long-range deendence by an autoregressive model of high order AR(), while the second one regards the direct calculation of the main statistical roerties of the random comonent, that is mean, variance and skewness coefficient. The skewness coefficient calculation of the random comonent is done using joint samle moments. The advantage of the method is its efficiency and simlicity and the analytical solution.

3 2. Motivation Since Hurst (95) observed the long-term ersistence henomenon in the annual average streamflows of Nile, the same behaviour has been identified in numerous natural rocesses while, its imortance has been underlined by scientists in many controversial discilines. It seems that the Hurst henomenon is ubiquitous in nature and this makes it necessary to find adequate ways to model it. Many models have been roosed in the literature that reserve the Hurst behaviour, such as Fractional Gaussian Noise (FGN) (i.e. Mandelbrot, 969; Mandelbrot and Wallis 969), fast FGN (Mandelbrot, 97), broken line models (i.e. Ditlevsen, 97), fractional ARIMA (Hosking, 98), and recently symmetric moving average models (SMA) (Koutsoyiannis, 2; 22). If the Hurst behaviour aears in a rocess, it needs to modeled as it affects dramatically the time series structure. Another distinguished characteristic of hydrological rocesses, that needs to be modeled, is asymmetry. In this direction have been made many attemts to adat standard models to reserve the skewness (i.e. Matalas and Wallis, 976). Some of the revious models are not easy to aly as the arameters are not easy to estimate. while other can reserve the skewness but not the Hurst behaviour and vice versa. Other roblems are the narrow tye of autocorrelation functions that those model can simulate (excetion is the SMA model). In this study is roosed a general methodology to reserve both the Hurst behaviour and skewness. The framework of the methodology is simle: the Hurst henomenon is modeled from an autoregressive model of high order, AR(), while the skewness is reserved by evaluating the skewness coefficient of the random comonent of the model. The model should be easy to aly and suitable for any ractical uroses such as hydrologic design or water resources management.

4 3. Modelling Aroach In order to reserve the long-range deendence or the Hurst henomenon in the simulated time series, a high order autoregressive model is imlemented. The long-range deendence behaviour, is essentially the slow decay of the autocorrelation function with time. On the contrary, the AR() models are considered to be short-range deendence models. Nevertheless, as this study reveals, AR() models of high order can reroduce the Hurst henomenon sufficiently enough for any ractical modelling uroses. In the general case of order, the AR() model takes the following form: X t t + i= where ε t is the innovation or the random comonent and α i are coefficients. In order to fit the model to a dataset, the α i coefficients and the basic statistics (mean, standard deviation) of the ε t have to be estimated. The auto-covariance function γ k of the AR() model for lag k and for k> is given by g k a i g i-k i= X t-i a i The relacement of γ k with the samles estimates and the imlementation of the last equation times gives a linear system of equations that can be solved straightforwardly, evaluating therefore the α i coefficients. Finally, the mean and the variance of the ε t can be estimated using the following two equations. m t m Xt i j - y a i g i z k i= { s t 2 g - i= a i g i

5 4. Preserving the Skewness in an AR() Model To reserve asymmetry in the simulated time series, it is necessary to evaluate the skewness coefficient of the innovation, Csk ε. It can be shown that the third central moment of the innovation of the AR() model is Defining as multi-auto-covariance of order (m, m 2,,m n ) and lag (l, l 2,, l 3 ), E i j k i= it can be roven that the following equation is valid, y a i X t-i z { 3 i= (, ) - HL a3 i + 3 i= m H3L m 3 t m 3 X t - E i j k i= m H2,L j=i+ - H, j-il a 2 i a j + 3 i= m H,2L j=i+ -2 H, j-il a i a2 j + 6 i= Relacing the multi-auto-covariance terms in the revious equation with the samle estimates, given by y a i X t-i z { ( m ) ( l ) 2, l2 l m m n = E ( X m m m t l X) ( Xt l X) ( Xt l X) μ μ μ μ 2 n n - j=i+ k= j+ H, m j-i,k-il H,,L a i a j a k k max ( ) ( l, l2 l ), n + l l2 l m m2 m n n μ ( m, m2 m ) = ( ) ( x n i l μ X ) ( xi l μ 2 X ) ( xi l μ X ) k max l, l2 ln + i= it is then straightforward to estimate the μ 3ε in () and thus the μ 3ε C sk = ε 3 σ ε ( )

6 5. The Generalized AutoCorrelation Function (GACF) The major criticism of a high order AR() model would focus on the lack of arsimony, as estimation of the autocorrelation function u to lag is required to fit the model. Moreover, it is well known that the estimator of the ACF is highly variable and that it increases its variability with increasing lag (Bras and Rodriquez, 985). Consequently, the uncertainty in the estimation of the ACF would lead to uncertain validation of the model arameters. To overcome this disadvantage, it is roosed to fit a generalized ACF, ρ (G), to the first few emirical ACF values (where α, β, δ are ositive arameters and j is the lag). Subsequently, the fitted GACF can be used to extraolate ACF values for Emirical ACF high j. The figure deicts the emirical autocorrelation function of the Nilometer dataset (analysis follows) and the fitted GACF. The GACF has been fitted to the first emirical values of the ACF by minimizing the square error. ρ ( ) ( ) G = α β + j j δ β ACF Fitted GACF Lag

7 6. The Generalized Lambda Distribution (GLD) In order to reserve the skewness in the simulated series, the innovation ε t must be samled from a distribution with variable skewness. Such a flexible distribution is the GLD family. The GLD(λ, λ 2, λ 3, λ 4 ) family distributions originated from the one-arameter lambda distribution roosed by John Tukey (96) and was generalized for Monte Carlo simulation uroses by John Ramberg and Bruce Schmeiser (974). Although the GLD has been alied in many fields since the early 97s (Karian and Dudewicz, 2), it has never been used in hydrology. The GLD family with arameters λ, λ 2, λ 3 and λ 4, is defined in terms of its ercentile function, QH yl yl 3 - H - yl l 4 +l l 2 where < y <. The arameters λ and λ 2 are, resectively, location and scale arameters, while λ 3 and λ 4 determine the skewness and kurtosis of the distribution. The GLD robability density function is f HxL l 2 l 3 y l H - yl l 4 - l 4 The restrictions on λ, λ 2, λ 3 and λ 4, that yield a valid GLD distribution, the arameter sace and the skewness kurstosis sace are discussed in detail by Karian and Dudewicz (2). In the next figure GLD dfs are lotted with mean =, variance = and skewness coefficient Csk ranging from 4.5 at x QH yl PDF Csk Random Variable

8 7. Fitting the GLD and Samling If X is GLD(λ, λ 2, λ 3, λ 4 ) with λ 3 >-/4 and λ 4 >-/4, then its first four moments (Ramberg et al., 979), μ, μ 2, μ 3, μ 4 (mean, variance, skewness coefficient, and kurtosis coefficient), are given by m A l 2 where +l m 2 B-A2 l2 C sk m 3 s3 2 A3-3 BA+C s3 l2 3 C k m 4 s4-3 A4 +6 BA 2-4 CA+D s4 l2 4 A l l 4 + B -2 ΒHl 3 +, l 4 + L + 2 l l 4 + C 3 ΒHl 3 +, 2 l 4 + L - 3 ΒH2 l 3 +, l 4 + L + 3 l l 4 + D -4 ΒHl 3 +, 3 l 4 + L + 6 ΒH2 l 3 +, 2 l 4 + L - 4 ΒH3 l 3 +, l 4 + L + 4 l l 4 + and B is the Beta function defined as ΒHa, bl = t a- H - tl b- dt If we consider the innovation ε t as a random variable with known estimation of the mean, the variance, the skewness and the kurtosis coefficient, a GLD distribution can be fitted by solving numerically the revious nonlinear system. The mean, the variance and the skewness coefficient of ε t can be analytically estimated as described in slide four. At the moment there is no analytical way to estimate the kurtosis coefficient of ε t, but heuristically for this study was taken the minimum so as λ 3 < and λ 4 <, which imlies that the fitted GLD ranges form - to (Karian and Dudewicz, 2). Once the arameters λ, λ 2, λ 3, and λ 4 of the GLD are estimated, the samling is very easy as the ercentile function has a simle and analytical formulae.

9 8. Simulation Organogram Recorded data Standardized data Emirical Autocorrelation Function Generalized Autocorrelation Function Innovation arameters μ σ α i arameters 2 ε, ε, sk C ε Generalized Lambda Distribution Simulation of series for samle size equal to the recorded data. Autoregressive model X = a X + ε t i t i t i= Statistical analysis of the synthetic series

10 9. Original Data I: Nilometer Index Standardized Nilometer series indicating the annual minimum water level of the Nile river for the years 622 to 284 A.D. (663 years; Beran, 994) A data with small ositive skewness but with a large Hurst exonent value that verifies the multiscale fluctuations. Standardized Nilometer Index Annual scale 3 year scale year scale Year A.D. Frequency Nilometer Index Samle size Average St. deviation Skewness Hurst Exonent log Σ k H.5 R H Standardized Nilometer Index log k

11 . Original Data II: Annual Temeratures Northern Hemishere temerature anomalies in C with reference to mean (Standardized in the figure on the right). (2 years, Moberg et al., 25) A highly variable Northern Hemishere temerature reconstruction that reveals the large natural variability of the climate in multile scales. The multiscale variation is verified by the large Hurst exonent value. Standardized Average Temerature Annual scale 3 year scale year scale Year A.D. Frequency Annual Temeratures Samle size 979 Average St. deviation Skewness -.3 Hurst Exonent.9 log Σ k H.5 R H Standardized Annual Average Temerature log k

12 . Original Data III: Daily Average Temeratures Standardized average daily temeratures in July recorded at Den Helder station in Netherlands, from 9 to25. (source: Royal Netherlands Meteorological Institute) The histogram below deicts the ositive asymmetry of the dataset, while the long range deendence is manifested from the high Hurst exonent value. Standardized Average Temerature Daily scale day scale Monthly scale Time day Frequency Daily Temeratures Samle size 369 Average St. deviation Skewness.93 Hurst Exonent.72 log Σ k R H H Standardized Daily Average Temerature log k

13 2. Original Data IV: Daily Average Dew Points Standardized average daily dew oints in January recorded at Den Helder station in Netherlands, from 9 to 25. (source: Royal Netherlands Meteorological Institute) This dataset is suitable for the uroses of this study as it shows high negative asymmetry and a high Hurst exonent value. Standardized Average Dew Points Daily scale day scale Monthly scale Time day Frequency Daily Dew Points Samle size Average St. deviation Skewness Hurst Exonent log Σ k.5..5 R H H Standardized Daily Average Dew Points log k

14 3. Simulated Data I: Nilometer Index An auto regressive model of order 3, AR(3), was fitted to the original dataset in order to reserve the Hurst behaviour. The skewness coefficient of the innovation was evaluated, according to the methodology analysed in this study, to Csk ε =.46. A simulated series with statistics in close roximity to the observed ones is resented here. Standardized Nilometer Index Annual scale 3 year scale year scale Time Year Frequency Nilometer Index Samle size Average St. deviation Skewness Hurst Exonent log Σ k R H H Standardized Nilometer Index log k

15 4. Simulated Data II: Global Temeratures An auto regressive model of order 4, AR(4), was fitted to the original dataset in order to reserve the Hurst behaviour. The skewness coefficient of the innovation was evaluated, according to the methodology analysed in this study, to Csk ε =.48. A simulated series with statistics in close roximity to the observed ones is resented here. Standardized Average Temerature Annual scale 3 year scale year scale Time Year Frequency Standardized Annual Average Temerature Annual Temeratures Samle size 979 Average.4 St. deviation.7 Skewness -.2 Hurst Exonent.9 log Σ k.5..5 H.5 R 2.99 H log k

16 5. Simulated Data III: Daily Average Temeratures An auto regressive model of order 2, AR(2), was fitted to the original dataset in order to reserve the Hurst behaviour. The skewness coefficient of the innovation was evaluated, according to the methodology analysed in this study, to Csk ε =.94. A simulated series with statistics in close roximity to the observed ones is resented here. Standardized Average Temerature Daily scale day scale Monthly scale Time Day Frequency Daily Temeratures Samle size 369 Average.2 St. deviation Skewness.94 Hurst Exonent.7 log Σ k H.5 R H Standardized Daily Average Temerature log k

17 6. Simulated Data IV: Daily Average Dew Points An auto regressive model of order 2, AR(2), was fitted to the original dataset in order to reserve the Hurst behaviour. The skewness coefficient of the innovation was evaluated, according to the methodology analysed in this study, to Csk ε = A simulated series with statistics in close roximity to the observed ones is resented here. Standardized Average Dew Points Daily scale day scale Monthly scale Time Day Frequency Daily Dew Points Samle size Average St. deviation Skewness Hurst Exonent log Σ k.5..5 R H H Standardized Daily Average Dew Points log k

18 7. Simulated Mean and Standard Deviation The box lots below deict the estimated mean values of the simulated series. Blue dots reresent the observed means of each dataset ( as the series were standardised). The effect of the Hurst behaviour is clearly manifested by larger variability of the estimator in the series with the larger Hurst exonent values. The box lots below deict the estimated standard deviation values of the simulated series. Blue dots reresent the standardized values. The variability of the classic standard deviation estimator, is larger in the series with larger values of Hurst exonent, and it is also negative biased Simulated Mean Values Simulated Standard Deviation Values Nilometer Index Annual Temeratures Daily Temeratures Daily Dew Points Nilometer Index Annual Temeratures Daily Temeratures Daily Dew Points

19 8. Simulated Skewness and Hurst Exonent The box lots below deict the estimated skewness coefficient values of the simulated series. Blue dots reresent the observed values of each dataset. Given that the estimation of the Csk is highly uncertain, as its value is sensitive to outliers, it is encouraging that the model reroduces Csk values in roximity with the observed ones and with low variability..5 The box lots below deict the estimated Hurst exonent values of the simulated series. Blue dots reresent the observed values. Again the model, as the lot reveals, manages to reroduce sufficiently the Hurst behaviour, with the mean Hurst exonent value of the simulated series in agreement with the observed ones. Simulated Skewness Coefficient Values Simulated Hurst Exonent Values Nilometer Index Annual Temeratures Daily Temeratures Daily Dew Points Nilometer Index Annual Temeratures Daily Temeratures Daily Dew Points

20 9. Conclusions While the autoregressive models are considered to be short range ersistence models, it is concluded in this study that a higher order AR model reserves adequately the Hurst behaviour, for Hurst exonent values as high as.9. It seems that the model can reserve even more intense long-term ersistence but this needs to be further examined. To reserve the asymmetry, an analytical exression for the estimation of the skewness coefficient of the innovation is given. Subsequently, the innovation sequence is samled from a flexible skewed distribution, the so-called Generalized Lambda Distribution. The model manages to reserve sufficiently the skewness as the mean skewness coefficient of the simulated series is in roximity with the observed ones. As the simulated series are in accordance with the observed ones, the model can be used for any ractical modeling uroses. Overall, the roosed methodology is simle and robust.

21 2. References Beran, J., Statistics for Long Memory rocesses, Volume 6 of Monograhs on Statistics and Alied Probability, New York: Chaman and Hall, 994. Bras, R. L., and I. Rodriguez-Iturbe, Random Functions and Hydrology, 559., Addison-Wesley-Longman, Reading, Mass., 985. Ditlevsen, O. D., Extremes and first assage times, Doctoral dissertation, Tech. Univ. of Denmark, Lyngby, Denmark, 97. Hosking, J. R. M.,Fractional Differencing, Biometrika, vol. 68, , 98. Hurst, H., Long-term storage caacity of reservoirs, Transactions of the American Society of Civil Engineers 6, 77 88, 95. Karian, Z., Dudewicz, E., Fitting Statistical Distributions: The Generalized Lambda Distribution and Generalized Bootstra Methods, Boca Raton: CRC Press, 2. Koutsoyiannis, D., A generalized mathematical framework for stochastic simulation and forecast of hydrologic time series, Water Resources Research, 36(6), , 2. Koutsoyiannis, D., The Hurst henomenon and fractional Gaussian noise made easy, Hydrological Sciences Journal, 47(4), , 22. Mandelbrot, B. B., Une class de rocessus stochastiques homothetiques a soi: Alication a la loi climatologique de H. E. Hurst, C. R. Hebd. Seances Acad. Sci., 26, , 965. Mandelbrot, B. B., A fast fractional Gaussian noise generator, WaterResour. Res., 7(3), , 97. Mandelbrot, B. B., and J. R. Wallis, Comuter exeriments with fractional Gaussian noises,, Averages and variances, Water Resour. Res., 5(), , 969a. Mandelbrot, B. B., and J. R. Wallis, Comuter exeriments with fractional Gaussian noises, 2, Rescaled ranges and sectra, Water Resour. Res., 5(), , 969b. Mandelbrot, B. B., and J. R. Wallis, Comuter exeriments with fractional Gaussian noises, 3, Mathematical aendix, Water Resour. Res., 5(), , 969c. Matalas, N. C., and J. R. Wallis, Generation of synthetic flow sequences, in Systems Aroach to Water Management, edited by A. K. Biswas, McGraw-Hill, New York, 976. Moberg, A., D. M. Sonechkin, K. Holmgren, N. M. Datsenko, and W. Karlen, Highly variable Northern Hemishere temeratures reconstructed from low- and high-resolution roxy data, Nature, 433(726), 63 67, 25. Ramberg, J., Schmeiser, B., An aroximate method for generating asymmetric random variables, Communications of the ACM 7(2):78 82, 974. Ramberg, J. S., Dudewich, E. J., Tadikamalla, P. R., Mykytka, E. F., A robability distribution and its uses in fitting data. Communications in Statistics Theory and Methods 2(2):2 24, 979. Tukey, J. W., The Practical Relationshi Between the Common Transformations of Percentages of Counts and of Amounts, Technical Reort 36, Statistical Techniques Research Grou, Princeton University,96.