Sains Malaysiana 41(11)(2012): 1367–1376

 

The Existence of Long Memory in Ozone Time Series

(Kewujudan Ingatan-Panjang dalam Siri Masa Ozon)

 

Muzirah Musa*

Department of Mathematics, Faculty of Science and Mathematics

Universiti Pendidikan Sultan Idris, 35900 Tanjong Malim, Perak, Malaysia

 

Kamarulzaman Ibrahim

School of Mathematical Sciences, Faculty of Science and Technology

Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor, Malaysia

 

Received: 30 September 2011 / Accepted: 21 May 2012

 

ABSTRACT

 

ABSTRACT

 

Long-memory is often observed in time series data. The existence of long-memory in a data set implies that the successive data points are strongly correlated i.e. they remain persistent for quite some time. A commonly used approach in modelling the time series data such as the Box and Jenkins models are no longer appropriate since the assumption of stationary is not satisfied. Thus, the scaling analysis is particularly suitable to be used for identifying the existence of long-memory as well as the extent of persistent data. In this study, an analysis was carried out on the observed daily mean per hour of ozone concentration that were available at six monitoring stations located in the urban areas of Peninsular Malaysia from 1998 to 2006. In order to investigate the existence of long-memory, a preliminary analysis was done based on plots of autocorrelation function (ACF) of the observed data. Scaling analysis involving five methods which included rescaled range, rescaled variance, dispersional, linear and bridge detrending techniques of scaled windowed variance were applied to estimate the Hurst coefficient (H) at each station. The results revealed that the ACF plots indicated a slow decay as the number lag increased. Based on the scaling analysis, the estimated H values lay within 0.7 and 0.9, indicating the existence of long-memory in the ozone time series data. In addition, it was also found that the data were persistent for the period of up to 150 days.

 

Keywords: Hurst coefficient; long-memory; ozone; scaling analysis

 

ABSTRAK

Ingatan-panjang sering diperhatikan dalam data siri masa. Kewujudan ingatan-panjang dalam set data menunjukkan bahawa titik-titik data yang berturut adalah amat berkait rapat dan berterusan dalam suatu tempoh. Satu pendekatan yang biasa digunakan dalam pemodelan data siri masa seperti model Box dan Jenkins tidak lagi sesuai berikutan andaian kepegunan tidak dipenuhi. Oleh itu, analisis penskalaan sangat sesuai untuk digunakan bagi mengenal pasti kewujudan ingatan-panjang serta sifat keberterusan data. Dalam kajian ini, analisis dijalankan ke atas data cerapan purata harian per jam siri masa ozon yang diperoleh daripada enam stesen pemantauan yang terletak di kawasan Bandar di Semenanjung Malaysia dari tahun 1998-2006. Bagi tujuan penyiasatan kewujudan memori panjang, analisis awal dilakukan berdasarkan kepada plot fungsi autokorelasi (ACF) bagi data yang dicerap. Analisis penskalaan yang terdiri daripada lima kaedah penskalaan yang berbeza yang merangkupi julat penskalaan semula, varians penskalaan semula, penyerakan, teknik penyahan tren linear dan jambatan dalam penskalaan varians tertingkap (SWV) digunakan untuk menganggar pekali Hurst (H) bagi setiap stesen. Keputusan menunjukkan bahawa plot ACF siri data menyusut dengan lambat selaras peningkatan lag. Berdasarkan analisis penskalaan, purata anggaran H terletak di antara 0.7 dan 0.9 menunjukkan kewujudan memori-panjang dalam siri masa ozon. Selain itu, didapati bahawa data tersebut berterusan sehingga ke 150 hari.

 

Kata kunci: Analisis penskalaan; ingatan-panjang; ozon; pekali Hurst

REFERENCES

 

Afroz, R., Hassan, M.N. & Ibrahim, N.A. 2003. Review of air pollution and health impacts in Malaysia. Environmental Research 92(2): 71-77.

Azmi, S., Latif, M., Ismail, A., Juneng, L. & Jemain, A. 2010. Trend and status of air quality at three different monitoring stations in the Klang Valley, Malaysia. Air Quality, Atmosphere & Health 3(1): 53-64.

Beran, J. 1994. Statistics for Long-Memory Processes. New York: Chapman & Hall.

Cajueiro, D.O. & Tabak, B.M. 2005. The rescaled variance statistic and the determination of the Hurst exponent. Mathematics and Computers in Simulation 70(3): 172-179.

Chelani, A.B. 2009. Statistical persistence analysis of hourly ground level ozone concentrations in Delhi. Atmospheric Research 92(2): 244-250.

Delignieres, D., Ramdani, S., Lemoine, L., Torre, K., Fortes, M. & Ninot, G. 2006. Fractal analyses for ‘short’ time series: Are-assessment of classical methods. Journal of Mathematical Psychology 50(6): 525-544.

Hurst, H.E., Black, R.P. & Simaika, Y.M. 1965. Long-term storage : An Experimental Study. London: Constable & Co.Ltd.

Juneng, L., Latif, M.T., Tangang, F.T. & Mansor, H. 2009. Spatio-temporal characteristics of PM10 concentration across Malaysia. Atmospheric Environment 43(30): 4584-4594.

Junninen, H., Niska, H., Tuppurainen, K., Ruuskanen, J. & Kolehmainen, M. 2004. Methods for imputation of missing values in air quality data sets. Atmospheric Environment 38(18): 2895-2907.

Kai, S., Chun-Qiong, L., Nan-Shan, A. & Xiao-Hong, Z. 2008. Using three methods to investigate time-scaling properties in air pollution indexes time series. Nonlinear Analysis: Real World Applications 9(2): 693-707.

Koutsoyiannis, D. 2003. Climate change, the Hurst phenomenon and hydrological statistics. Hydrological Sciences Journal 48(1): 3-24.

Li, Z. & Zhang, Y-K. 2007. Quantifying fractal dynamics of groundwater systems with detrended fluctuation analysis. Journal of Hydrology 336(1-2): 139-146.

Mandelbrot, B.B. & Ness, J.W.V. 1968. Fractional brownian motions, fractional noises and applications. SIAM Review 10(4): 422-437.

Peter, E.E. 1994. Fractal Market Analysis: Applying Chaos Theory to Investment and Economic. New York: John Wiley & Son, Inc.

Samorodnitsky, G. 2006. Long Range Dependence. New York, USA: Now Publishers Inc.

Varotsos, C., Ondov, J. & Efstathiou, M. 2005. Scaling properties of air pollution in Athens, Greece and Baltimore, Maryland. Atmospheric Environment 39(22): 4041-4047.

Weng, Y-C., Chang, N-B. & Lee, T.Y. 2008. Nonlinear time series analysis of ground-level ozone dynamics in Southern Taiwan. Journal of Environmental Management 87(3): 405-414.

Windsor, H.L. & Toumi, R. 2001. Scaling and persistence of UK pollution. Atmospheric Environment 35(27): 4545-4556.

Zhu, J. & Liu, Z. 2003. Long-range persistence of acid deposition. Atmospheric Environment 37(19): 2605-2613.

 

*Corresponding author; email: Muzirah@Fsmt.Upsi.Edu.My

 

 

previous