Archives Des Sciences
Vol 65, No. 12;Dec 2012
TEMPORAL CORRELATION BEHAVIOUR OF DAILY TEMPERATURE TIME SERIES ACROSS MALAYSIA
Fadhilah Yusof Department of Mathematical Sciences, Faculty of Science University of Technology Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia Tel: + 60 127578908
Email:
[email protected]
Ibrahim Lawal Kane Department of Mathematical Sciences, Faculty of Science University of Technology Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia Tel: + 60 169669415
Email:
[email protected]
Abstract Stochastic processes that are invariant in distribution under judicious scaling of time and space, called self-similar process, can parsimoniously model the long-run properties of phenomena exhibiting long memory. If a series exhibits long memory, there is a persistent temporal dependence even between distant observations; such series is characterized by distinct but non-periodic cyclical patterns. The temporal correlations of the daily temperature series of eight weather stations across Malaysia for the period 01/01/1968 to 31/12/2003 were studied by using the Modified rescaled range analysis (R/S), Whittle estimator and Detrended fluctuation analysis (DFA) Methods. The results show that the scaling exponents H are larger than 0.5 for all data sets. This finding indicates that the daily temperature series for all locations exhibit long memory. Hence the observations have a predictable component and therefore, past observations could be used to predict the future. Keywords: Long Memory, Modified Rescaled Range(R/S), Whittle estimator, Detrended Fluctuation Analysis 1.
Introduction
Long memory processes where positive correlations between observations far apart in time and space decay very slowly to zero with increasing time lag, occur quite frequently in fields such as meteorology, hydrology and economics. A series of observations shows long memory when the values observed in distant lags are correlated among themselves or if the effect of an event which occurs in an instant can be detected
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many lags afterwards. If a series exhibit long memory, then the observations have a predictable component and therefore, past observations could be used to predict the future. In empirical modeling of long memory processes, the autoregressive fractionally integrated moving average (ARFIMA) model that was proposed by Granger and Joyeux (1980) and Hosking (1981) is used. Many of the empirical studies of long range dependence are based on the estimation method by Geweke and Porter-Hudak (1983) (GPH). However, developments in optimal estimation of the long memory parameter have brought the statistical properties of different estimators in sharper focus (Hurvich and Deo, 1998). Applications of long memory models in meteorological time series data date back to the work of (Bloomfield, 1992; Koscielny-Bunde et al., 1998; Maraun et al., 2004; Gil-Alana, 2003; 2005; and 2009). Natural temperature variability has been frequently described as a stochastic process with long-range dependence (LRD) or long-memory (Cohn and Lins, 2005). Rao and Bhattacharya (1999) explored some monthly and annual hydrologic time series, including average monthly stream flow, maximum monthly stream flow, average monthly temperature and monthly precipitation, at various stations in the mid-western United States. They stated that there is little evidence of long-term memory in monthly hydrologic series, and for annual series, the evidence for lack of long-term memory is inconclusive. Some of the work on long memory has been found to be extremely relevant for the theoretical modeling of meteorological data (Gil-Alana, 2004). Various methods were used to characterize quantitatively the fluctuations and correlations of high frequency meteorological data. Power density spectra (Marple, 1987; Percival and Walden, 1993) have been routinely computed for decades. Pelletier (1997) determined the temperature spectra from hundreds of stations and ice core records and identified power-law behavior for continental and maritime locations. Koscielny-Bunde et al. (1998) reported to observe a near universal exponent value in the fluctuations of daily temperature; however Weber and Talkner (2001) found differences depending on the altitude of the meteorological station. A largest related study is presented by Eichner et al. (2003) on temperature records of 95 stations all over the globe. They obtained correlation exponent values in the range 0.55–0.9 centered strongly at around 0.65 over continents, but systematically higher values for islands. The last observation is supported by Monetti et al. (2003), where they detected much stronger persistence for sea-surface temperature than values for air over the ground. Kiraly et al. (2004) analyzed daily temperature anomaly records (61 for Australia, 18 for Hungary) by means of Detrended fluctuation analysis. Positive long range asymptotic correlations extending up to 5– 10 years are detected for each case. Contrary to earlier claims, the correlation exponent is not universal for continental stations. Interestingly, the dominant factor is the geographic latitude over Australia: the general tendency is a decrease of correlation exponent with increasing distance from the equator. This tendency is in a complete agreement with the results found by Tsonis et al. (1999). The Hurst exponent is a parameter to measure long-range dependence (LRD) a characteristic frequently used to describe the natural variability of temperature records. Persistence and trend in temperature and its geographical patterns have been subject to detection studies. However in Malaysia little or not much has been done in this area. The aim of this research is to analyze eight temperature records of different locations across Malaysia to exploit the extent of long range correlations in the time series using a battery of statistical
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techniques which has proven useful in revealing the correlation structures of a time series. The rest of the paper is organized as follows: Section 2 gives the data used, Methodology and results. Section 3 gives the summary and conclusion.
2. Material and Methods The time series data analyzed in this section correspond to the daily average observations of the temperature datasets for the time period 1 January 1968 – 31 December 2003, for Kuantan, Alorsetar, Ipoh, Mersing, Sitiawan Malacca, Bayen Lepas and Kota Bahru obtained from the Malaysian Meteorological Department (MMS) Malaysia. Any modeling effort on this data set will have to be based on an understanding of the variability of past data. Detecting long memory can only be considered interesting if it reveals something about the internal behavior of the process. Processes which induce nonstationarity in the mean are particularly problematic, since they are most likely to lead to spurious detection of long memory where it is not present. Thus, before applying the data to any test, the time series must be free from such processes. 2.1 Detection of Long Memory: Heuristic Methods To model observed persistence in temperature time series, the long memory stochastic models were introduced independently. A stationary stochastic process { exist a real number H and a finite constant C such that the
yt } is called a long memory process if there autocorrelation function ρ(τ) has the following
rate of decay:
( ) C2 H 2 ,
as τ→∞
(1)
The parameter H (Hurst Exponent), display the long memory property of the time series. A long memory time series is said fractionally integrated, where the fractional degree of integration d is related to the parameter H as follows: d = H – 0.5 The Hurst exponent takes values from 0 to 1 (0 ≤ H ≤ 1). If H = 0.5, the series is a random walk (Brownian time series). In a random walk there is no correlation between any element and a H < 1, the series indicates persistent behavior or long memory.
future element. If 0.5
