A Pilot Study Using Chaos Theory to Predict Temperature Time Series in Malaysian
Semi Urban Area
Nor Zila Abd Hamid1*, Nur Hamiza Adenan2, Nurul Bahiyah Abd Wahid3, Biliana Bidin4,
Siti Hidayah Muhad Saleh5
1,2,3Faculty of Science and Mathematics, Sultan Idris Education University, 35900Tanjong Malim,
Perak, Malaysia
4Institute of Engineering Mathematics, University Malaysia Perlis,026000 Arau, Perlis, Malaysia 5Faculty of Computer and Mathematical Sciences, UiTM Negeri Sembilan Branch,
70300 Seremban, Negeri Sembilan nor.zila@fsmt.upsi.edu.my1*
Article History: Received: 10 November 2020; Revised: 12 January 2021; Accepted: 27January 2021;
Published online: 05April 2021
Abstract: Chaos theory draws more attention since it has been widely used in modeling various time series. Changes in temperature can cause bad effect on health and lead to death. Therefore, in this study, chaos theory was applied to the temperature time series. The temperature time series is observed hourly in one of Malaysiansemi urban area namely Tanjong Malim,located in the state of Perak. This pilot study begins by detecting the chaos nature in time series through phase space approach and Cao method. Next, the time series was predicted through the local approximation method, a method based on chaos theory. This study resulted that the nature of the observed temperature time series was chaos. Prediction through the local approximation method was success with correlation coefficient value 0.9138. This shows that there exist a strong relationship between the predicted and observed temperature time series. Therefore, chaos theorywas a good approach that can be used to determine the nature and predict temperature time series in the semi urban area. In implication, this findingwas expected to serve stakeholders such as Ministry of Higher Education, Meteorological Department as well as Department of Environment in temperature time series management.
Keywords: Chaos theory, phase space approach, Cao method, local approximation method, temperature time series
1. Introduction
The changes in temperature can cause bad effect on health and lead to death. In Malaysian communities, this phenomenon contributed to many negative impacts [1]. Global development and population growth have changed the physical character of the earth. Study by [2] determined that there exists a significant increase in Malaysian temperature time series. Pollution from various human activities such as vehicle emission and manufacturing operations have increased heat production and furthermore contribute to many negatives impact that lead in increasing of temperature. Thus, it was important to analyze the temperature time series. This will facilitate the stakeholder for making preparations in facing the uncertain climate. In this study, the analysis will be done through the chaos theory.
Research by [3] found that there were two types of time series’nature which were randomand deterministic. Time series with random nature cannot be predicted. Vice versa, time series with deterministic naturecan be predicted. According to [4], there exist one more type of times series’ nature namely chaos. Time series with chaos nature was in between the random and deterministic nature. Time series with chaos nature can be predicted. However, only short-term prediction can be done due to its character which was sensitive dependence upon initial condition[3].
Chaos theory has been applied to various time series such as river flow[5], sea level [6], carbon monoxide [7] and ozone pollution [8]. Chaos theory has been found to successfully modelling the temperature time series in several countries such as Nigeria [9], Iran [10] and Denmark [11]. In Malaysia, the chaos theory was used to model ozone [12], PM10[13] and river flow [14]. However, chaos theorywas still new to Malaysian temperature
time series. Study by [15] has successfully applied chaos theory to temperature time series which observed in highland area while study by [16] has applied the theory to temperature time series which observed in rural area. Therefore, this study will be a pilot study in application of chaos theory to temperature time series in semi urban area.
In previous studies, temperature is often predicted through multivariate method such asmultiple linear regressions and neural network ([17][18], [19]). Most prediction process usingmultivariate method involve factors such as relative humidity,maximum and minimum temperature as well as pressure. In the case of some information of the factors were incomplete, a univariatemethod was suggested to do the prediction. Thus, local
approximation method, a chaos theory basedmethod wasapplied throughout this research.The advantageof local approximation method is that prediction is conductedsolely using temperature time series data.
In order to predict particulate matter, local approximation method wasapplied by [20], while to predict carbon monoxide, the method was applied by[7]and next,[21]applied the method to predict nitrogen dioxide as well as sulfur dioxide. All results gain satisfied prediction. In Malaysia, the chaos theory was used by [12]to predict ozone, [13] to predict particulate matter and [14] to predict river flow. Researches by [6] and [8] have employed the method to predict temperature time series. Recently, [22] has applied chaos theory to predict carbon monoxide. All research obtained very satisfactory results. Thus, chaos theory based method is continuously applied.
The main objective of this study wasto apply local approximation method to predict the temperature time series in one of Malaysian semi urban area. Performance in prediction by local approximation method is determinedvia thecomputed correlation coefficient’s value and graphical plot of scatter diagram.
2. Temperature Time Series
The secondary data of temperature time series were collected by Department of Environment Malaysia. The temperature time series was observed hourly in units ℃at Tanjong Malim, one of Malaysiansemi urbanwhich located in Perakstate. In total, the area of Tanjong Malim is 950 km². Sultan Idris Education University, one of the leading education university is in Tanjong Malim. Department of Environment Malaysia categorized Tanjong Malim as semi urban area. Tanjong Malim is quite near to Klang Valley area. Thus, Tanjong Malim is also frequently visited by tourists and public. Hence, maintaining public health as well as prediction of temperature in Tanjong Malim is important.
This study started on year 2018. Therefore, the latest data is from year 2017. Since southwest monsoon contributes to high temperature, the chosen duration of observed data is from May 1st to August 31st, 2017.
Hence, overall, number of data is 𝑁 = 2952. From all the series, there were 174 missing data and were replaced with the time series at the same hour of previous day. Table 1 described the statistical information of the observed temperature time series.
Table 1.Statistical information of the temperature time series
Information Min Max Sum Mean Mode Median Variance
Value 21.6 35.6 80009.2 27.1 24.0 26.1 10.5
3. Methods
Chaos theory was applied to predict temperature time series that is described in previous section.
The overall process covered two parts namely reconstruction of phase space and prediction process. First parthelp to determine whether the observe time series is either random or chaos while the later part help in predictingfuture temperature time series.
By using the reconstructed phase space, through the method such as Lyapunov exponent, phase space approach and Poincare map research by [23],[9]as well as[24]resulted that the nature of their observed temperature time series is chaos. Phase space approach is quite simple while Cao method [25]can helpindifferentiatingthe time series’ nature. Hence, phase space approachas well as Cao method wereapplied. If the result reveal that the time series is chaos in nature, the prediction iscontinuedby applying the method of local approximation.
Reconstruction of Phase Space
The concept of phase space is simple yet powerful for characterizingthe nature of the time series. The nature of the time series can be represented geometrically by a phase space trajectory [3]. The obtained secondary temperature time series is recorded mathematically in the form of one-dimensional scalar data:
𝑋 = {𝑥1, 𝑥2, … , 𝑥𝑁} (1)
where𝑥 is the temperature time series at 𝑛-th hour and 𝑁 is the total hour of observation. 𝑋is then divided into two parts; 𝑋𝑡𝑟𝑎𝑖𝑛and 𝑋𝑡𝑒𝑠𝑡 . 𝑋𝑡𝑟𝑎𝑖𝑛wereused as a training data to calculate the unknown parameters, while
𝑋𝑡𝑒𝑠𝑡will bekept and later use to test the performance of the prediction model. In this study, the first three
months’ time series wereused as 𝑋𝑡𝑟𝑎𝑖𝑛 and the remaining one month time serieswere used as 𝑋𝑡𝑒𝑠𝑡. Thus, the
prediction is done for a period of one month data started from 1amAugust 1stand ended at 11pm August 31st
2017.
The time series of 𝑋𝑡𝑟𝑎𝑖𝑛 from (1) will be reconstructed into below𝑚-dimensional phase space:
𝒀𝑗𝑚= {𝑥𝑗,𝑥𝑗+𝜏, 𝑥𝑗+2𝜏, … , 𝑥𝑗+(𝑚−1)𝜏} (2)
with delay time 𝜏and embedding dimension 𝑚. Phase space (2) is used in order to determine the chaos nature of the time series and furthermore, to predict the future time series.
It can be observed that there exist two parameters in (2) namely the delay time 𝜏 and embedding dimension 𝑚. Parameter 𝜏 is important to reflect the structure of the phase space attractor. If 𝜏 is too small, the phase space is not independent, and result in a loss of attractor characteristics.
However, if 𝜏 is too large, the time series may be too independent and not correlate with each other. Some studies (e.g. [26], [27] and [28]) used 𝜏 = 1 and some more applied few methods such as autocorrelation function and average mutual information. Following the successful researches by [26], [27] and [28], 𝜏 = 1is selected.
Parameter 𝑚 is the embedding dimension of the phase space. In this research, 𝑚 is determined through Cao method. Cao method, introduced by[25]is chosen because this method is not dependent on the number of time series used, only involving parameter 𝜏and is also able to differentiate between random and chaos nature of the time series. Beside a parameter to phase space, 𝑚 is also the minimum number of affecting factors that influenced the observed temperature time series[29].
Phase Space Plot
In chaos theory, phase space is referred to the graph that is plotted in the plane of{𝑥𝑡, 𝑥𝑡+𝜏}using 𝑋𝑡𝑟𝑎𝑖𝑛. The
nature of the time series is chaos if there exist a well-defined trajectory [27]. On the other hand, if all points of the time series filled the entire plane or scattered, then the nature of the time series is categorized as random.
Cao Method
According to Cao[25], Cao method only depending on the value of 𝜏and does not containing any other parameters. Furthermore, Cao method does not depending on the number of observed time series. Hence, Cao method wasapplied in this research. Parameter 𝑚through Cao method is determined by
𝐸1(𝑚) = 𝐸(𝑚+1)
𝐸(𝑚) (3).
where‖∗‖ is the maximum norm and 𝒀𝑛𝑚 is the nearest neighbor to 𝒀𝑗𝑚,
𝐸(𝑚) = 1 𝑁−𝑚𝜏∑ 𝒀𝑗𝑚+1−𝒀𝑛𝑚+1 𝒀𝑗𝑚−𝒀𝑛𝑚 𝑁−𝑚𝜏 𝑗=1 (4).
If 𝐸1(𝑚)startssaturating when the value of 𝑚 is greater than the value of 𝑚0, then 𝑚 = 𝑚0+ 1 is the
minimum embedding dimension.Furthermore, the saturating 𝐸1(𝑚)also reflect that the nature of the time series
is chaos. Conversely, if 𝐸1(𝑚)does not saturating, then, the nature of the series is categorized as random.
Furthermore, Cao [25]introduced the second parameter of 𝐸2(𝑚) which computed through:
𝐸2(𝑚) = 𝐸∗(𝑚+1) 𝐸∗(𝑚) (5) Where 𝐸∗(𝑚) = 1 𝑈−𝑚𝜏∑ |𝑥𝑗+𝑚𝜏 𝑚 − 𝑥 𝑤+𝑚𝜏𝑚 | 𝑈−𝑚𝜏 𝑗=1 (6).
Following Cao [25], if𝐸2(𝑚) = 1for all 𝑚,the nature of the time series is concluded as random. Conversely, the
existence of 𝐸2(𝑚) ≠ 1 shows that the nature of the observed series is chaos. Local Approximation Method
The 𝑝-hour ahead prediction is doneusing equation
𝒀𝑗+𝑝𝑚 = 𝑓(𝒀𝑗𝑚) (7).
Since this is the pilot study, then, the shortest 𝑝 is chosen. Thus, 𝑝 = 1 and (7) turns to: 𝒀𝑗+1𝑚 = 𝑓(𝒀
𝑗𝑚) (8).
𝒀𝑗𝑚is the last reconstructed phase space and 𝒀𝑗+1𝑚 is a 𝑗 + 1 ahead phase space. There werevarious type of local
approximation method which were based on chaos theory. However, the common used method is local linear approximation method. With parameters 𝐴 and 𝐵were computed from the least squaremethod, prediction through the local linear approximation method is done through:
𝒀𝑗+1𝑚 = 𝐴𝒀𝑗𝑚+ 𝐵 (9).
The performance of the local linear approximation method wasevaluated through the value of correlation coefficient𝑟. Therange of parameter𝑟is between value -1 and +1. The closer 𝑟 to -1 or +1shows that there exist a strong relationship between the observed and predicted temperature time series.
4. Results and Discussion
Chaos Nature of Temperature Time Series
The overall process covered two parts namely reconstruction of phase space and prediction process. First part determine whether the observe time series is either random or chaos while the later part help in predicting future temperature time series.
The phase space for temperature time series is plotted on the plane{𝑥𝑡, 𝑥𝑡+𝜏}. As 𝜏 = 1, thus, plot of{𝑥𝑡, 𝑥𝑡+1}
is graphed as Figure 1. From Figure 1, it can be observed that most of the points were converging to the center of the plane. According to[27], this is an attractor. Attractor exists reflects that the nature of the observed temperature time series is chaos.
Figure 1.Results from phase space plot
Furthermore, Figure 2 is the Cao method’s result. Embedding dimension is𝑚 = 5 and this reflects that at least fivefactors influenced the temperature time seriesinsemi urbanarea of Perak. Beside determining the value of𝑚, 𝐸1(𝑚)in Figure 2starts to saturate at 𝑚 = 4. Therefore, following Cao [25], the nature of the observed
Figure 2.Results from Cao method
Next, 𝐸2(𝑚) ≠ 1exists at = 1. This reflects that the nature of temperature time series is chaos. Results from
both Cao methodas well asphase space approachwere consistent. Hence, it can be concluded that the nature of temperature time series in Tanjong Malim educational area is chaos.
Temperature Time Series’ Affecting Factors
Following 𝑚 = 5 from Cao method, this reflects that at least five factors influenced the temperature time series in semi urban area of Tanjong Malim. Studycarried out by[30]listed that temperature time series’ affecting factors were such as rainfall, humidity, wind speed, pressure and solar radiation. Besides, research by[1]determined that building and house development, urbanization, as well as industrialization also contributed to the changing in temperature. The affecting factors resulted from both research above concluded that there exist more than five affecting factors. Thus, this finding is compatible. Hence, 𝑚 = 5 from Cao method is consistent and reliable. Therefore, at least five factors influenced the temperature time series in semi urban area of Tanjong Malim.
Prediction Results through the Local Approximation Method
Prediction of the hourly temperature time series is done for one month started at 1am August 1stand ended at
11pm August 31st2017. By using 𝜏 = 1and = 5 , results found that the local linear equation to predict future
temperature is
𝑥𝑖+1= 0.9650𝑥𝑖+ 0.4655 (10).
With real value of 𝑥𝑖, prediction of 𝑥𝑖+1 is done through equation (10). The result of one-hour ahead
prediction is as shown in Figure 3. It can be observed that the trend of temperature time series (up and down) has been well predicted.
Figure 3. The one-hour ahead prediction
The computed 𝑟 between the observed and predicted temperature time series is 0.9138. The value shows that there exist a strong relationship between the predicted and observed temperature time series. Figure 4, the scatter diagram between the predicted and observed temperature agreed with the statement that there exist a strong relationship between both values. Thus, the results reflect that the local approximation method is good in predicting the observed temperature time series in semi urban area. Hence, chaos theory is a good theory that can be used to determine the nature and predict temperature time series in the semi urban area.
Figure 4.Scatter diagram between the predicted and observed temperature 5. Conclusion
In this paper, a pilot study to predict temperature time series in Malaysian semi urban area wascarried out using chaos theory. The nature of the observed temperature time series is concluded as chaos through the phase space plor and Cao method. Results shown that there exist a strong relationship between the predicted and observed temperature time series graphically, as well as the calculation of correlation coefficient. In implication, it is hoped that these findings can assist stakeholders such as Ministry of Higher Education, Meteorological Department and Department of Environment Malaysia in having a better temperature management.
In future, the approach can be expended to be applied on other time series data such as wind speed, humidity as well as sea level. Study by [15] has successfully applied chaos theory to temperature time series which observed in highland area while study by [16] has applied the theory to temperature time series which observed in rural area. Furthermore, this study applied chaos theory to temperature time series in semi urban area. In further study, chaos theory can applied to temperature time series in other area such as urban, industrial as well as port area.
6. Acknowledgment
The time series data were supplied by the Department of Environment, Malaysia and Malaysian Meteorological Department while the fund was sponsored by Research University Grant with code 2016-0188-102-01. Thank you to all contributions.
References
1. N. M. Hashim, “Analisis tren pemanasan global dan kesannya terhadap aspek dayahuni bandar di Malaysia,” Geogr. Online - Malaysian J. Soc. Sp., vol. 6, no. 2, pp. 72–88, 2010.
2. N. M. Wai, A. Camerlengo, and A. K. A. Wahab, “A study of global warming in Malaysia,” J. Teknol., vol. 42, pp. 1–10, 2005.
3. J. C. Sprott, Chaos and Time-Series Analysis. Oxford University Press, 2003.
4. H. D. I. Abarbanel, Analysis of Observed Chaotic Data. New York: Springer-Verlag, 1996.
5. M. A. Ghorbani, R. Daneshfaraz, H. Arvanagi, A. Pourzangbar, S. Mahdi, and S. K. Kar, “Local Prediction in River Discharge Time Series,” J. Civ. Eng. Urban., vol. 2, no. 2, pp. 51–55, 2012.
6. M. De Domenico, M. Ali, O. Makarynskyy, and D. Makarynska, “Chaos and reproduction in sea level,” Appl. Math. Model., vol. 37, no. 6, pp. 3687–3697, 2013.
7. A. B. Chelani and S. Devotta, “Prediction of ambient carbon monoxide concentration using nonlinear time series analysis technique,” Transp. Res. Part D, vol. 12, pp. 596–600, 2007.
8. P.Indira, S. S. R. Inbanathan, R. S. Selvaraj, and A. A. Suresh, “Chaotic analysis on surface ozone measurements at tropical urban coastal station Chennai, India,” IOSRD Int. J. Earth Sci., vol. 2, no. 1, pp. 1–8, 2016.
9. I. M. Echi, E. V Tikyaa, and B. C. Isikwue, “Dynamics of daily rainfall and temperature in Makurdi,” Int. J. Sci. Res., vol. 4, no. 7, pp. 493–499, 2015.
10. H. Millan, G. Ghanbarian-Alvijeh, and I. Garcia-Fornaris, “Nonlinear dynamics of mean daily temperature and dewpoint time series at,” Atmos. Res., vol. 98, no. 1, pp. 89–101, 2010.
11. R. Berndtsson, B. Sivakumar, J. Olsson, and L. P. Graham, “Dynamic Characteristics of Temperature, Precipitation and Runoff to the Baltic Sea,” Vatten, vol. 67, pp. 185–192, 2011.
12. N. Z. A. Hamid and M. S. M. Noorani, “An improved prediction model of ozone concentration time series based on chaotic approach,” Int. J. Math. Comput. Sci. Eng., vol. 7, no. 11, pp. 206–211, 2013. 13. N. Z. A. Hamid and M. S. M. Noorani, “A pilot study using chaotic approach to determine
characteristics and forecasting of PM10 concentration time series,” Sains Malaysiana, vol. 43, no. 3, pp. 475–481, 2014.
14. N. H. Adenan and M. S. M. Noorani, “Nonlinear prediction of river flow in different watershed acreage,” KSCE J. Civ. Eng., vol. 18, no. 7, pp. 2268–2274, 2014.
15. N. Z. A. Hamid, “Application of chaotic approach in forecasting highland’s temperature time series,” IOP Conf. Ser. Earth Environ. Sci., vol. 169, no. 012107, pp. 1–10, 2018.
16. M. Bahari and N. Z. A. Hamid, “Analysis and Prediction of Temperature Time Series Using Chaotic Approach,” J. Qual. Meas. Anal., vol. 15, no. 1, pp. 43–52, 2019.
17. G. B. Sahoo, S. G. Schladow, and J. E. Reuter, “Forecasting stream water temperature using regression analysis, artificial neural network, and chaotic non-linear dynamic models,” J. Hydrol., vol. 378, pp. 325–342, Nov. 2009.
18. M. Zounemat-Kermani, “Hourly predictive Levenberg-Marquardt ANN and multi linear regression models for predicting of dew point temperature,” Meteorol. Atmos. Phys., vol. 117, no. 3–4, pp. 181– 192, 2012.
19. Paras and Sanjay Mathur, “A simple weather forecasting model using mathematical regression,” Indian Res. J. Ext. Educ. Spec. Issue, vol. 1, no. January, pp. 161–168, 2012.
20. A. B. Chelani and S. Devotta, “Nonlinear analysis and prediction of coarse particulate matter concentration in ambient air,” J. Air Waste Manag. Assoc., vol. 56, pp. 78–84, 2006.
21. V. N. Khokhlov, A. V. Glushkov, N. S. Loboda, and Y. Y. Bunyakova, “Short-range forecast of atmospheric pollutants using non-linear prediction method,” Atmos. Environ., vol. 42, no. 31, pp. 7284– 7292, Oct. 2008.
22. A. B. Ruslan and N. Z. A. Hamid, “Application of Improved Chaotic Method in Determining Number of k -Nearest Neighbor for CO Data Series,” Int. J. Eng. Adv. Technol., vol. 8, no. 6S3, pp. 10–14, 2019.
23. P. Indira, S. Stephen Rajkumar Inbanathan, R. Samuel Selvaraj, and A. Antony Suresh, “Forecasting daily maximum temperature of chennai using nonlinear prediction approach,” Indian J. Sci. Technol., vol. 9, no. 39, pp. 1–6, 2016.
24. A. Antony and R. S. Selvaraj, “A complete chaotic analysis on daily mean surface air temperature and humidity data of Chennai,” J. Ind. Geophys. Union, vol. 21, no. 4, pp. 277–284, 2017.
25. L. Cao, “Practical method for determining the minimum embedding dimension of a scalar time series,” Phys. D, vol. 110, pp. 43–50, 1997.
26. A. W. Jayawardena, “Runoff forecasting using a local approximation method,” in IAHS, 1997, no. 239, pp. 167–171.
27. B. Sivakumar, “A phase-space reconstruction approach to prediction of suspended sediment concentration in rivers,” J. Hydrol., vol. 258, pp. 149–162, 2002.
28. B. Sivakumar, “Forecasting monthly streamflow dynamics in the western United States : a nonlinear dynamical approach,” Environ. Model. Softw., vol. 18, pp. 721–728, 2003.
29. S. K. Regonda, B. Rajagopalan, U. Lall, M. Clark, and Y. Moon, “Local polynomial method for ensemble forecast of time series,” Nonlinear Process. Geophys., vol. 12, pp. 397–406, 2005.
30. Adiwijaya, U. N. Wisesty, and F. Nhita, “Study of line search techniques on the modified backpropagation for forecasting of weather data in Indonesia,” Far East J. Math. Sci., vol. 86, no. 2, pp. 139–148, 2014.
