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

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

Diserahkan: 30 September 2011 / Diterima: 21 Mei 2012

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

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*Pengarang untuk surat-menyurat; email: muzirah@fsmt.upsi.edu.my