# Modeling daily maximum temperature for thermal comfort in northern Malaysia.

INTRODUCTION

In the field of climatology, modeling is essential in investigating the pattern of the atmospheric condition. Researchers have been developing ways to come up with a model to forecast the weather, as if to imitate the real-life situation. However, it is hard to develop it since the state of the atmosphere is not a constant variable in an equation. Meteorological modelling and prediction practices are vital aspects, as they are the key to the development and sustainability of agriculture, urbanisation and health prospects.

In order to understand the irregularity of climate change, stochastic process is one of the used methods. Semenov and Barrow [1], Keith et al. [2] and Wilks [3] are examples of popular papers that use stochastic processes in their studies of climate change. In addition, Florescu & Levin [4] used stochastic approach in three different areas, which are finance, climate data and earthquake analysis using hidden Markov chain model. Among branches of stochastic processes, Markov chain is one of the popular methods. Both discrete and continuous approaches can be used to build a proper model. However, both approaches may yield different results because of the type of data and the variables involved in the process. The formulation of Markov chain model includes the order of the model and the number of classified states in the process.

In this paper, we would like to see the climate temperature in the future as it will affect the environment. In the study, the physiological equivalent temperature (PET) was used as the standard temperature. Matzarakis et al. [5] studied this subject to evaluate thermal comfort (favourable temperature) in different countries. By definition, thermal comfort is the condition of mind that expresses satisfaction with the thermal environment and is assessed by subjective evaluations [6]. It is known that people in different countries experience different thermal comfort and the definition of warmness itself is completely different. The importance of this study is the use of stochastic approach and cluster analysis in understanding the maximum temperature in Malaysia and its behaviour.

Data Descriptions:

This study covers the northern region of Malaysia, which comprises five meteorological observatory stations from five states. Table 1 shows the geographic background of the selected stations. A 10-year data of daily maximum temperature (2004-2013) was obtained from the Malaysian Meteorological Department (MMD) and used throughout this study. The geographical factors of the location of the stations were considered since the northwest and east regions are separated by the Titiwangsa Mountains.

Methodology:

The methods adopted in this study are divided into two parts. The first part is to model the daily maximum temperature data using Markov Chain. The Markov Chain model was by classifying the data into states of transition, performing test of independence, developing the count matrix into transition probability matrix and iterating of the matrix until equilibrium. The second part is to classify the meteorological stations based on annual maximum temperature data using cluster analysis.

Data Classification and Test of Independence:

The state of transition must be known before sorting the data. This is the most crucial part as it will decide the future result in the long term run. The PET index was used as a scale to determine the states of transition. Lin and Matzarakis [7] investigated thermal comfort analysis based on thermal indices in Sun Moon Lake, Taiwan. Although the scope of study only covers on the thermal comfort (neutral temperature), the range of temperature proposed, was used by Makaremi et al. [8] as the thermal perception classification (TPC) for (sub)tropical region when they studied on the thermal comfort condition in Malaysia.

As the values of the minimum and maximum temperature of the data of this study are 23.7[degrees]C and 38.6[degrees]C respectively, this TPC was fixed accordingly, so that the transition count matrix can be obtained. Thus, this study uses only four states of transitions--cool, neutral, slightly warm and warm as listed in Table 2. It is important to figure out whether the behaviour of the states of the transition is dependent on the stations. For this reason, the chi-square test for independence was used to determine whether there is a significant difference of the probabilities between the states of transition among the stations.

Development of Transition Probability Matrix:

The data are sorted into the transition count matrix, [M.sub.nxn] = [[M.sub.ij]] where {i = 1,2, ..., n; j = 1,2, ..., n} where n depends on the number of states present.

The placement of value into matrix M is somehow simple but may be confusing. The term i represents the current state of transition and the term j represents the next state of transition. Since this study is focusing on the first order Markov Chain, both previous (yesterday) and current (today) states were investigated. If the previous and current states are i and j, respectively, then the counted value will be placed at [M.sub.ij]. As four states of transition were used in this study, the dimension of the transition count matrix is 4 x 4 as shown below.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A count matrix M will be transformed into a transition probability matrix P. The matrix P represents the probability of the state's transition behaviour. In our case,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [P.sub.ij] = [M.sub.ij]/[summation over (j)] [M.sub.ij]; [4.summation over (j)] [P.sub.ij] = 1

Matrix Multiplication and Iteration:

This study aims to find the long-run proportion of time that the process will be in each transition state. In other words, the behaviour of the state in the long-run will be known. This method is known as limiting probability. In the limiting state probability, the n-step of transition probability matrix P converges independently as n is approaching infinity. After a sufficiently long time, the matrix P will reach a unique steady-state probability.

The probability can be obtained by multiplying the matrix P by itself, resulting transition probability matrix [P.sup.2]. Then, the matrix is iterated until all the transition probability in the matrix reach equilibrium. The resulting matrix [rho] contains the values of the probability of the state in the long-run. The computation of the matrix multiplication was also done using a software.

Cluster Analysis:

Cluster analysis classifies data into groups that are similar to each other but different from other groups. The clustering method can be conducted by hierarchical clustering technique which combines observations into a hierarchy [9]. The technique proceeds by either using agglomerative or divisive approaches. As described by Hasan et al. [10], the agglomerative approach begins with all observations as their own cluster. Next, using a similarity measure, two similar clusters are combined into a new cluster. The combination process is repeated n - 1 times (n is the number of observations) until all observations are classified into a single cluster.

One of the most commonly recognized distance measures is Euclidean distance or straight-line distance:

[D.sub.ij] = [square root of [n.summation over (k=1)] [([x.sub.ik] - [x.sub.jk]).sup.2]

where [D.sub.ij] is the distance between [x.sub.i] and [x.sub.j]. and n is the number of observations.

In this study, an agglomerative hierarchical clustering using Ward's method and an Euclidean distance matrix was used to form a cluster of meteorological stations as applied by Hasan et al. [10], Piticar and Ristoiu [11] and Rebetez and Reinhard [12]. The clustering result can be displayed in a dendrogram, a tree diagram which is frequently used to present the cluster arrangement by hierarchical clustering.

RESULTS AND DISCUSSIONS

Data Classification & Independence Test:

Table 3 shows the count data of 10 years. The result shows that most of the maximum daily temperature in Malaysia is between 30[degrees]C-34[degrees]C. The figure is almost close to the latest actual observation (29.9[degrees]C-33.2[degrees]C), reported by the Malaysian Meteorological Department in their November 2014 report. The transition count matrix is as shown below.

Comparatively, it seems that Chuping and Alor Setar experienced warmer temperature (above 30[degrees]C) compared to other stations. This is perhaps due to the location of the stations since they are close to the equator line [13]. To see whether the states of transition are independent of the stations, the chi-square test was performed. The chi-square statistics obtained was 2859.5417 with p-value of 0.00001 (<0.5). The null hypothesis which stated that the states of transition and the stations are independent is rejected. It is statistically proven that the states of transition are dependent of the observatory stations.

Transition Probability Matrix:

The behaviour of daily maximum temperature at each station can be observed from the individual transition count matrix as shown in the Table 4. From the obtained count matrix M, the transition probability matrices were computed and the values are summarized in Table 5. The result shows that there are no records of the transition from the state of slightly cool (C) moving to the state of warm (W). In addition, all stations showed no evidence of the warm state (W) moving to slightly cool (C) state, except for station in Chuping with a very small probability.

Common trends can be spotted with all of the stations tend to experience two consecutive days of the state of slightly warm (S). If today is slightly cool, the next day will be neutral (N) for all stations except Alor Setar station where it tends to be slightly cool too on the next day. If today is neutral, the chance to be slightly warm is higher for the stations in the West (Chuping, Alor Setar and Bayan Lepas) but not for the stations in the East (Kota Bharu and Kuala Terengganu) where it tends to experience the same neutral condition. If it is warm today, then it will be warm on the next day for Chuping and Alor Setar stations but slightly warm for Bayan Lepas, Kota Bharu and Kuala Terengganu.

Limiting State Probabilities:

Table 6 shows the final matrix after a number of iteration until the matrix reaches equilibrium. After a sufficiently long time, there is a high probability for the states in the northern of Malaysia to be experiencing the range of temperature from 30[degrees]C to 34[degrees]C (slightly warm). It can be observed that the Bayan Lepas station has the lowest long-run proportion of time to experience slightly cool temperature but has the highest proportion to experience slightly warm temperature, compared to the other stations. This situation may occur due to the location of the station at the south of Penang Island. The circulation of air temperature and wind formation will be affected due to the island being surrounded by the sea.

The Kota Bharu and Kuala Terengganu stations have the highest long-run proportion of time to experience neutral state (26[degrees]C-30[degrees]C), but have the lowest proportion to experience warm temperature. This is due to geographical factors such as Titiwangsa Mountain as the backbone of Peninsular Malaysia which may restrict the movement of the wind and cloud. The altitude of the stations may also contribute to the changes in air temperature since these two have the highest altitude compared to the other stations. Stations in Chuping and Alor Setar which are located closer to the equator also contributed to the surrounding temperature. From the result, it is likely for both stations to experience slightly warm temperature, followed by warm temperature.

Hierarchical Cluster Analysis:

The hierarchical cluster analysis was conducted on annual maximum temperature data of five meteorological stations for the period of 2004 until 2013. As described in methodology section, the agglomerative hierarchical clustering using Ward's method and the Euclidean distance has been selected as a method for this cluster analysis. Figure 1 shows a dendrogram plot of the hierarchical cluster.

From the plot, the result reveals that the stations can be clustered into two groups with major difference between the groups given by the rescaled distance of 25. The first group contains two stations (Alor Setar and Chuping) while the other group consists of three stations (Bayan Lepas, Kota Bharu and Kuala Terengganu). The distance among the members of the group are less than 5 which make them connected as one major group. Thus, the hierarchical cluster result indicates that Alor Setar has the same behaviour with Chuping station while Bayan Lepas, Kota Bharu and Kuala Terengganu stations share the similar behaviour among them. This result corresponds to the final result of long-time proportion for each station.

Conclusion:

Based on the results from the 10-year data, it is found that Malaysia is currently experiencing slightly warm temperature (based on PET index) with the range of 30[degrees]C to 34[degrees]C. The result is consistent with the average range of temperature observed, 29.9[degrees]C to 33.2[degrees]C. This shows that the use of PET index to measure thermal comfort is the best to represent the climate temperature activity in Malaysia. Analysis from the chi-square test of independence has statistically proven that the behaviours of the states of transition are dependent of the observatory stations.

After a sufficiently long time, Chuping and Alor Setar have a high long-run proportion of time in experiencing slightly warm temperature, followed by warm temperature. On the other hand, Kota Bharu, Kuala Terengganu and Bayan Lepas are likely to experience slightly warm temperature followed by neutral. This result coincides with the result of cluster analysis which classify Chuping and Alor Setar in the same behavioural group while Kota Bharu, Kuala Terengganu and Bayan Lepas in another group.

Although Bayan Lepas have the highest long-run proportion of time of experiencing slightly warm temperature compared to the rest of the states, the chance for the next day to be neutral and warm does not differ very much. The behaviour of the transition state shows that the state of slightly warm temperature has the highest long-run proportion of time among five stations in the northern of Malaysia. This common trend is obvious when the data revealed that Malaysia is experiencing two consecutive days of slightly warm temperature. Since Malaysia is grouped as a tropical region, the range of temperature is considered normal.

ARTICLE INFO

Article history:

Received 3 October 2015

Accepted 10 October 2015

Published Online 13 November 2015

REFERENCES

[1] Semenov, M.A. and E.M. Barrow, 1997. Use of a stochastic weather generator in the development of climate change scenarios. Climatic Change, 35(4): 397-414.

[2] Keith, D.A., H.R. Akqakaya, W. Thuiller, G.F. Midgley, R.G. Pearson, S.J. Phillips, H.M. Regan, M.B. Araujo, and T.G. Rebelo, 2008. Predicting extinction risks under climate change: coupling stochastic population models with dynamic bioclimatic habitat models. Biology Letters, 4(5): 560-563.

[3] Wilks, D., 1992. Adapting stochastic weather generation algorithms for climate change studies. Climatic Change, 22: 67-84.

[4] Florescu, I.I. and F. Levin, 2008. Estimation procedure for a hidden Markov chain model with applications to finance, climate data and earthquake analysis. Retrieved July 28, 2015, from http://www.math.ubordeaux1.fr/~pdelmora/hmm-finance.pdf

[5] Matzarakis, A., H. Mayer and M.G. Iziomon, 1999. Applications of a universal thermal index: physiological equivalent temperature. International Journal of Biometeorology, 43(2): 76-84.

[6] American National Standard Institute (ANSI), 2013. ASHRAE Standard.

[7] Lin, T.-P. and A. Matzarakis, 2008. Tourism climate and thermal

comfort in Sun Moon Lake, Taiwan. International Journal of Biometeorology, 52(4): 281-290.

[8] Makaremi, N., E. Salleh, M.Z. Jaafar and A. GhaffarianHoseini, 2012. Thermal comfort conditions of shaded outdoor spaces in hot and humid climate of Malaysia. Building and Environment, 48: 7-14.

[9] Davids, D.L., 1998. Recovery effects in binary aluminum alloys, Ph.D. Thesis, Harvard University.

[10] Hasan, H., N.H. Mohd Salleh and S. Kassim, 2014. Stationary and non-stationary extreme value modeling of extreme temperature in Malaysia. AIP Conf. Proc, 1613: 355-367.

[11] Piticar, A. and D. Ristoiu, 2012. Analysis of air temperature evolution in northeastern Romania and evidence of warming trend. Carpathian Journal of Earth and Environmental Sciences, 7(4): 97-106.

[12] Rebetez. M. and M. Reinhard, 2007. Monthly air temperature trends in Switzerland 1901-2000 and 1975-2004. Theor. Appl. Climatol, 91: 27-34.

[13] Chung-hoi, Y., 2012. Why is the equator very hot and the poles very cold? Retrieved July 28, 2015, from http://www.hko.gov.hk/education/edu06nature/ele_srad_e.htm.

(1) Husna Hasan, (2) Muhamad Asyraf bin Che Nordin, (3) Nur Hanim Mohd Salleh

(1,2,3) School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Minden, Pulau Pinang, Malaysia

Corresponding Author: Husna Hasan, School of Mathematical Sciences, Universiti Sains Malaysia, 11800, Minden, Pulau Pinang, Malaysia.

E-mail: husna@cs.usm.my.
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Table 1: Geographic background of selected meteorological
observatory stations

Regions      Station             Latitude

Northwest    Chuping, Perlis     6[degrees]29' N

Alor Setar,         6[degrees]12' N
Kedah

Bayan Lepas,        5[degrees]18' N
Pulau Pinang

East         Kota Bharu,         6[degrees]10' N
Kelantan

Kuala Terengganu,   5[degrees]23' N
Terengganu

Regions      Station             Longitude          Altitude
(m)

Northwest    Chuping, Perlis     100[degrees]16'E   21.7

Alor Setar,         100[degrees]24'E   3.9
Kedah

Bayan Lepas,        100[degrees]16'E   2.8
Pulau Pinang

East         Kota Bharu,         102[degrees]17'E   4.6
Kelantan

Kuala Terengganu,   103[degrees]06'E   5.2
Terengganu

Table 2: Index to fit as transition state

State             Label   Range of Temperature

Slightly Cool     (C)     (23,26)

Neutral           (N)     [26,30)

Slightly Warm     (S)     [30,34)

Warm              (W)     [34, 39)

Table 3: Summary of the sorted data according
to state of transition

State of transition

Station        C    N     S      W

Chuping        19   277   2515   842
Alor Setar     34   212   2594   813
Bayan Lepas    5    309   3259   80
Kota Bharu     36   825   2729   63
Kuala          26   783   2822   22
Terengganu

Table 4: Transition count matrix of five stations

Station        Count matrix, M

Chuping        [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Alor Setar     [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Bayan Lepas    [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Kota Bharu     [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Kuala          [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Terengganu

Table 5: Transition probability matrix for five stations

Station                  Transition probability matrix, P

Chuping        [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Alor Setar     [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Bayan Lepas    [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Kota Bharu     [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Kuala          [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Terengganu

Table 6: The Final Result of Long-time Proportion for Each Station

State of transition

Station        C        N        S        W

Chuping        0.005    0.076    0.688    0.231
Alor Setar     0.009    0.058    0.710    0.223
Bayan Lepas    0.001    0.085    0.892    0.022
Kota Bharu     0.010    0.226    0.746    0.017
Kuala          0.007    0.214    0.773    0.006
Terengganu
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