Evaluation of Hurst exponent for precipitation time series

Transcription

1 LATEST TREDS on COMPUTERS (Volume II) Evaluation of Hurst exponent for precipitation time series ALIA BĂRBULESCU Faculty of Mathematics an Computers Science Oviius University of Constantza 4, Mamaia Boulevar, Constantza ROMAIA CRISTIA SERBA (GHERGHIA) Faculty of Civil Engineereing Oviius University of Constantza 4, Mamaia Boulevar, Constantza ROMAIA CARME MAFTEI Faculty of Civil Engineereing Oviius University of Constantza 4, Mamaia Boulevar, Constantza ROMAIA Abstract: - A major issue in time series analysis an particularly in the stuy of meteorological time series behaviour is the long range epenence (LRD) Various estimators of LRD have been propose Their accuracy have been generally teste by using simulate time series since sometimes only their asymptotic property are nown, or worse, no asymptotic property have been prove It is well nown that the Hurst exponent (H) is a statistical measure use to classify time series In this article we etermine the Hurst exponent for precipitation time series collecte in Dobruja region, for 4 years an we compare the results Key-Wors: - Long range epenence, Hurst coefficient, precipitation, etrene fluctuation Introuction The stuy of time series with long-range epenence have been extensively evelope for applications in nature sciences, as well as in DA sequences, cariac ynamics, internet traffic [] an finance [] The Hurst [3] exponent provies a measure for long term memory an fractality [4] of a time series For etails on the topics, see the volumes of Beran [5], Embrechts an Maejima [6], an Palma [7] an the collections of Douhan et al [8] an Robinson [9] The values of the Hurst exponent range between 0 an Base on the Hurst exponent value H, the following classifications of time series can be realize: H = 05 inicates a ranom series; 0 < H < 0 5 inicates an anti - persistent series, which means an up value is more liely followe by a own value, an vice versa; 0 5< H < inicates a persistent series, which means the irection of the next value is more liely the same as current value Measuring Hurst exponent Let ( X t ) t be a time series, shortly enote by ( X t ) We say that ( X t ) it is wealy stationary if it has a finite mean an the covariance epens only on the lag between two points in the series Let ρ () be the autocorrelation function (ACF) of ( X t ) If the time series is wealy stationary, ACF is given by: E[( X µ )( µ )] ρ( ) = t X t+, where E( X t ) is the expectation of X t, µ is the mean an σ the variance σ ISS: ISB:

2 LATEST TREDS on COMPUTERS (Volume II) It is sai that the time series ( X t ) has the long range epenence (LRD) property if ρ( ) iverges = LRD can be thought of in two ways: [0] In the time omain it manifests as a high egree of correlation between istantly separate ata points In the frequency omain it manifests as a significant level of power at frequencies near zero The following methos are use for LRD analysis: R/S an Lo s moifie R/S statistic; Aggregate Variance; Absolute Moments; Detrene fluctuation analysis (variance of resiuals); Ratio of variance of resiuals; Perioogram; Whittle's approximate MLE an Local Whittle estimators; Wavelets We shall iscuss here the results obtaine using some of the methos escribe in the following R/S analysis The Hurst exponent can be calculate by rescale range analysis (R/S analysis) [3] To stuy the LRD in a time series, the following algorithm is use A time series ( X ) is ivie into sub-series, of length m For each sub-series n =,, : - Fin the mean, En an the stanar eviation, S n ; - ormalize the ata ( X in ) by subtracting the subseries mean: Zin = X in En, i=,, m; - Create a cumulative time series: - Fin the range i = Yin Z jn, i=,, m; R n = maxy, m jn min Y, m - Rescale the range R n / Sn; - Calculate the mean value of the rescale range for all sub-series of length m: n= jn ( R / S) m = Rn / Sn Hurst foun that (R/S) scales by power - law as time increases, which inicates ; t H ( R / S) = c t In practice, in classical R/S analysis, H can be estimate as the slope of log/log plot of ( R / S) t versus t Although Manelbrot [] gave a formal justification for the use of this test, Lo [] showe that this statistic was not robust to short memory epenence an moifie this statistic Lo efine moifie R/S statistic by: instea of consiering multiple lags, only focus on lag, the length of the series: After fining the overall mean X = X j create the cumulative time series: an fin the range = Y ( X X ), =,, ; R( ) = maxy =,, min Y, instea of using the stanar eviation to normalize R (), he uses the following sum: q = + ω Sq ( ) ( X j X ) j ( q) ( X i= j+ where j ω i ( q) =, q< q+ ; j X )( X The Lo s moifie R/S statistic is efine by: Vq ( ) = R( ) / Sq ( ) i j X ) Lo uses the interval [0809, 86] as the 95% asymptotic acceptance region for testing the null hypothesis: H 0 : the absence of LRD, against the alternative: H : the presence of LRD As iscusse in [3], the right choice of q in Lo s metho is essential For stuy, the following values are use: an ˆ ρ q =, ˆ [4] (*) ρ 4 3 ˆ ρ q =, 0 ˆ [5] (**) ρ ISS: ISB:

3 LATEST TREDS on COMPUTERS (Volume II) where: - is the length of the series, - ρˆ is the estimate first orer correlation coefficient, - [ ] is the greatest integer function Aggregate variance metho [3] A series of length is ivie into sub-series of length m For each subseries, the aggregate series, forme by the means X m m ( ) = X i, =,,, m i= ( ) m+ is calculate, as well as, its sample variance: VarX = = ( X ( ) X ) For successive values of m, the sample variance is plotte against m on a log-log plot Fitting a least squares line to the points of the plot, the Hurst coefficient is calculate, nowing that the straight line slope is H- In orer to istinguish between the nonstationarity an LRD, Teverovsi an Taqqu [0] propose to use this metho, together with the stuy of successive ifferences of variances or fitting a function H (m) C + C m to the VarX 3 Absolute Moments Metho [3] (m) This metho is analogous to 3, but instea of VarX, the n th absolute moments are calculate for the aggregate series, n AM = X X = For successive values of m, the sample absolute moment is plotte versus m on a log-log plot Fitting a least squares line to the points of the plot, the Hurst coefficient is calculate, nowing that the straight line slope is n(h-) 4 Detrene fluctuation analysis (DFA) Detrene fluctuation analysis was originally propose as a technique for quantifying the nature of long-range correlations by Peng et al [6] It was introuce in orer to permit the etection an quantification of longrange correlations in DA sequences In recent years the DFA metho has been applie in analysis of ifferent time series that appear in ifferent fiels as DA sequences [7], heart rate stuy [8], human gait, meteorology, economics, an physics DFA metho, involves the following steps [9]: n Starting with a time series, ( X t ) with the length, it is integrate, obtaining: Y = i= ( X X ) The integrate series in ivie into sub-series of equal length m In each sub-series, fit X t, using a polynomial function of orer l which represents the tren of that sub-series The orinate of the fit line in each box is enote by y m () The integrate series is etrene by subtracting the local tren y m () in each sub-series of length m For a given sub-series length, m, the root meansquare fluctuation for the integrate an etrene series is calculate: F = [ Y ym( ) ] = The above computation is repeate for a broa range of scales (m) to provie a relationship between F(m) an the box size m A power-law relation between the average rootmean-square fluctuation function F(m) an the box size m inicates the presence of scaling: F(m) H m Ratio of Variance of Resiuals This metho estimates the parameter alpha characterizing the intensity of heavy tails, instea of estimating the long-range epenence parameter H The metho is base on the Variance of Resiuals metho It calculates the Variance of Resiuals in two ways, an taes their ratio to obtain a statistic, an then fits a leastsquares line to the logarithm of that statistic This enables one to estimate alpha[3] 5 Perioogram Gewee an Porter-Hua [0] propose a semiparametric approach to test for long-memory memory of a fractionally integrate process The fractional ifference parameter can be estimate by regression equations Let ( X ) be a time series, an, ijλ I ( λ) = π X j e, where λ is a frequency It was shown that a series with LRD shoul have a perioogram proportional to H λ in the origin neighbourhoo, so a log-log plot of a perioogram ISS: ISB:

4 LATEST TREDS on COMPUTERS (Volume II) against the frequency shoul give the coefficient H = Moifie perioogram, as well cumulative perioogram have also been use [3] 3 Results an iscussions In the following we present the results of the calculus of Hurst coefficient, for ten annual precipitation time series (Fig), collecte between 965 an 005, in Dobruja, Romania an we iscuss the results Fig 3 The V - statistic associate to Aamclisi series Precipitation (mm) year Fig 4 The R/S chart of Megiia series Aamclisi Cernavoa Megiia Harsova Corugea Tulcea Sulina Jurilovca Constanta Mangalia Fig The series of annual mean precipitation in Dobruja region (965-00) For R/S analysis, a part of results is given in [] an those obtaine by using KaotiXL [], in Table The R/S charts an the evolution of V - statistics associate are exemplifie in Figs 7, where R is the etermination coefficient Fig 5 The V - statistic associate to Megiia series Table Hurst coefficients for annual series, calculate by KaotiXL (rescale metho) Fig 6 The R/S chart of Sulina series Fig The R/S chart of Aamclisi series Fig 7 The V - statistic associate to Sulina series ISS: ISB:

5 LATEST TREDS on COMPUTERS (Volume II) For Lo s moifie statistic, the values use for q were calculate by (*) an (**) As result of formula (*), q was respectively: 3 for Sulina an Jurilovca series an for the rest Using formula (**), the coefficients were not very ifferent The results are presente in Table an those obtaine by perioogram metho, in Table 3 Table Hurst coefficients for annual series, calculate by Lo s metho Table 3 Hurst coefficients for annual series, calculate by perioogram metho It can be remare that the results for each series are very ifferent an a correct conclusion on the long range epenence property of these time series coul not be extract applying only one metho Discussing, for example, about Aamclisi, Cernavoă or Megiia series, after the application of normality, homosceasticity an correlation tests, it was conclue that they are Gaussian white noises [] But not all the values from the previous tables are in concorance with these conclusions In [] a FARIMA moel was foun for Sulina series But the values of H, in Tables an 3 are iscorance to the LRD property of Sulina series The reasons for which these results are so ifferent coul be: q is too small in the case of Lo s statistic (an it oes not account for the autocorrelation of the process) an the small number of ata The results of DFA are closer to those of Lo s metho (for example, for Aamclisi series), other being closer to those of perioogram metho 4 Conclusion In this article we presente the results of LRD analysis for mean annual precipitation time series They were compare to the results of statistic tests We remar that they are not concorant As consequence, it is inicate to use with circumspection ifferent LRD analysis methos, since sometimes the statistics attache to them have properties that have been prove only for some well establishe time series or their properties are not nown References: [] W Willinger et all, Self-similarity in high-spee pacet traffic: analysis an moeling of Ethernet traffic measurements, Statistical Science, 0, 995, pp67-85 [] A Lo, Fat tails, long memory, an the stoc maret since the 960s, Economic otes, Banca Monte ei Paschi i Siena, 00 [3] H E Hurst, Long-term storage of reservoirs: an experimental stuy, Transactions of the American society of civil engineers, 6, 95, pp [4] BB Manelbrot, J R Wallis, Robustness of the rescale range R/S in the measurement of noncyclic long-run statistical epenence, Water Resources, 5, 969, pp [5] J Beran, Statistics for Long - Memory Processes, Chapman an Hall, ew Yor, 994 [6] M Embrechts, P Maejima, Self - similar processes Princeton, University Press, Princeton an Oxfor, 00 [7] W Palma, Long-Memory Time Series Theory an Methos, Wiley - Interscience, 007 [8] P Douhan, G Oppenheim, an M Taqqu, Theory an Applications of Long-Range Depenence, Birhauser, 003 [9] P M Robinson, Time Series with Long Memory, Oxfor University Press, 003 [0] V Teverovsy, M S Taqqu, W Willinger, On Lo s moifie R/S statistic, Preprint, 997 [] B Manelbrot, Statistical methoology for non - perioic cycles: from the covariance to R/S analysis, Annals of Economic an Social Measurement, vol, 97 [] A W Lo, Long term memory in stoc maret prices, Econometrica, vol 59, 99, pp79-33 [3] 3 M S Taqqu, V Teverovsy, W Willinger, Estimators for long range epenence: an empirical stuy, Fractals, vol3, no4, 995, pp [4] D W K Anrews, Heteroseasticity an autocorrelation consistent covariance matrix estimation, Econometrica, 59, 99, pp [5] W Wang et all, Detecting long-memory: Monte Carlo simulations an application to aily streamflow processes, wwwhyrol-earth-syst-sci-iscussnet/3/603/006/ [6] C- K Peng et all, Mosaic organisation of DA nucleoties, Physical Review E, vol 49, no, 994, pp [7] S V Bulyrev et all Long-Range Correlation Properties of Coing an oncoing DA Sequences: GenBan Analysis, Physical Reviews E, vol 5, 995, pp [8] Y Ashenazy et all, Magnitue an Sign Correlations in Heartbeat Fluctuations, Physical Review Letters, vol 86, 00, pp ISS: ISB:

6 LATEST TREDS on COMPUTERS (Volume II) [9] etic/tns/paper/noehtml [0] J Gewee, S Porter-Hua, The estimation an application of long memory time series moels, Journal of Time Series Analysis, 4, 983, pp 38 [] A Bărbulescu, E Pelican, ARIMA moels for the analysis of the precipitation evolution, Recent Avances in Computers, Proceeings of the 3 th WEAS International Conference on Computers, 009, pp 6 [] XL/ html Acnowlegements: This article was supporte by CCSIS UEFISCSU, project number PII IDEI 6/007 ISS: ISB: