Stochastic simulation for the precipitation frequency over some Brazilian cities through the Metropolis Hastings algorithm

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METEOROLOGICAL APPLICATIONS Meteorol. Appl. 23: 420 424 (2016) Published online 20 April 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/met.

1 METEOROLOGICAL APPLICATIONS Meteorol. Appl. 23: (2016) Published online 20 April 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: /met.1566 Stochastic simulation for the precipitation frequency over some Brazilian cities through the Metropolis Hastings algorithm Marconio Santos, a, * Paulo Sérgio Lucio a and Ana Carla Gomes b a Federal University of Rio Grande do Norte (UFRN) Campus Central, Natal, Brazil b Federal University of Western Pará (UFOPA), Belém, Brazil ABSTRACT: This work analyses precipitation patterns in Brazilian cities with different climatic classifications, according to Köppen Geiger, in order to estimate the probability of precipitation on any day of a particular month. It is focused on the frequency of precipitation and its probability. For this aim, the R software was used to perform stochastic simulations by the Metropolis Hastings algorithm, assuming that the days of a particular month are distributed identically regarding the frequency of rainfall. It is also assumed that this frequency can be modelled by the Beta distribution, with adjusted parameters. The distribution proposed in this work is suitable for adequately estimating the frequency of precipitation when the average chosen rates are higher than 50%. However, this frequency could not be approximated by the proposed method, because the average chosen rates were <50%. The results can contribute to numerical prediction models for cities with a shortage of or missing rainfall data, providing information that contributes to urban planning in these cities. KEY WORDS stochastic simulation; Metropolis Hastings; precipitation Received 12 November 2014; Revised 30 December 2015; Accepted 23 January Introduction Precipitation is a variable related to weather and to climate that affects the economy in a region, whether rural or urban (Mello and Silva, 2009). The impacts of weather conditions can either decrease or increase poverty and vulnerability, in cyclical fashion (Suliano et al., 2009). The differences in climate conditions among Brazilian regions are considered factors explaining the levels of vulnerability of each region, and will vary differently with climate changes. Differences in surface warming cause changes in atmospheric conditions, such as average temperature. Changes in heating affect also cause changes in humidity, prevailing winds and rainfall (Varejão-Silva, 2000). To estimate the probability of precipitation, several large Brazilian cities were selected (Belém, Curitiba, Manaus, Natal, Petrolina, Porto Alegre, São Paulo and Uberaba; see Figure 1), which have different rainfall regimes and climatic classifications. In regions close to the Equator, the total precipitation is maximum in summer and minimum in winter (Reboita et al., 2012). In the Southern Hemisphere summer, the weather systems are displaced to the south. The Intertropical Convergence Zone (ITCZ) favours the occurrence of rainfall in the north and northeast of Brazil. The northeast trade winds are more intense in this season. In summer and autumn, the ITCZ migrates to the south, contributing to the rainy season in this region (Reboita et al., 2010). However, the diurnal rainfall can be influenced by the sea breeze at any time of year in northeast of Brazil. The land breeze and trade winds from the southeast influence nocturnal rainfall in any period (Kousky, 1980). * Correspondence: M. Santos, Programa de Pós-graduação em Ciências Climáticas - PPGCC. Universidade Federal do Rio Grande do Norte - UFRN. Centro de Ciências Exatas e da Terra - CCET. Campus Universitário - Lagoa Nova Natal - RN - Brazil. marconio@ccet.ufrn.br In the north region of Brazil, where the cities of Belém and Manaus are located, three sub-regions of abundant precipitation are found: 1. the northwestern region of Amazonas state, with precipitation above 3000 mm year 1, associated with the condensation of humid air brought by easterly winds of the ITCZ; 2. the central part of the Amazon basin, around 5 S, with average rainfall of 2500 mm year 1 ;and 3. the eastern part of the Amazon basin, near Belém, with precipitation of 2800 mm year 1 (Nimer, 1979; Nobre, 1983). In the northeast of Brazil, where the cities of Petrolina and Natal are located, the ITCZ is the main atmospheric system causing precipitation. Other systems that affect rainfall in this region are cold fronts, easterly waves, upper tropospheric cyclonic vortices, sea and land breezes systems, and the movement of large-scale direct circulatory tropical convective cells associated with the day Madden and Julian Oscillation (Nogués- Peagle and Mo, 1997; Molion and Bernardo, 2002; Ferraz, 2004). The cities of São Paulo and Uberaba are located in the southeast of Brazil. This region is affected by synoptic systems that reach the south of the country, with some differences of intensity and seasonality. The troughs are active mostly during the winter, causing moderate weather conditions (Fernandes and Satyamurty, 1994). Upper tropospheric cyclonic vortices, originating from the Pacific, are organized with intense convection associated with the instability caused by the subtropical jet stream (Cavalcanti et al., 1982). Squall lines, generated from the association of dynamic factors, favour intense rainfall during the months with greatest convective activity. The South Atlantic Convergence Zone (SACZ) is one of the main systems influencing rainfall regimes in this region (Carvalho et al., 2004). Finally, the cities of Curitiba and Porto Alegre are located in the south of Brazil, where the annual average rainfall varies from 1250 to 2000 mm. Some weather systems in this region are 2016 Royal Meteorological Society

2 Stochastic simulation for the precipitation frequency 421 Figure 1. Map of Brazil with the cities chosen for the study. Source IBGE (adapted). essential to precipitation levels. Frontal systems are among the most important (the trajectory of these systems is associated with the location and intensity of the subtropical jet stream in South America). The troughs influence the development of severe climate conditions over this region, and mesoscale convective systems (SC) are also responsible for large precipitation totals (Custódio and Herdies, 1994; Fernandes and Satyamurty, 1994). Rainfall can be predicted, in probabilistic terms, by theoretical distributions adjusted to a series of data (Moreira et al., 2010). The generated models, after adjustment to theoretical distribution, can provide useful information for the planning of several activities (Fietz et al., 2008). This work analysed precipitation patterns in Brazilian cities with different climatic classifications, according to Köppen Geiger, in order to estimate the probability of precipitation on any day of a particular month. It is a work focused on the frequency of precipitation and its probability. The free statistical software R was used to perform stochastic simulations by the Metropolis Hastings algorithm, assuming that the days of a particular month are distributed identically regarding the frequency of precipitation. It is also assumed that this frequency can be modelled by the Beta probability distribution, with adjusted parameters. 2. Materials and methods The data were collected from the National Institute of Meteorology (INMET, 2014). These data were analysed by the Markov chain Monte Carlo method (MCMC) through the Metropolis Hastings algorithm. Each city in this study has a different type of climate, among the most prevalent in Brazil, by Köppen Geiger climate classification. This classification was first proposed in 1900 by Wladimir Köppen, and improved in 1936 in collaboration with Rudolf Geiger. To choose the cities, an updated version of the map of South America for the Köppen Geiger climate classification (Peel et al., 2007), and a recent update to the map of Brazil for this classification (Alvares et al., 2013) were considered. Some concepts and definitions concerning the methods used in this work will be introduced, such as the Markov chain Monte Carlo method, Bernoulli distribution and Beta distribution. The Markov chain Monte Carlo method (MCMC) is a numerical random sampling method that consists of building an ergodic Markov chain such that its limit distribution numerically approximates the desired probability distribution, regarding large samples (Kruschke, 2010). In this work, it was assumed that the days of a particular month are identically distributed regarding the frequency of precipitation, and that the distribution is associated with that particular month. The probability of precipitation was considered the parameter of a Bernoulli distribution, which can be generated by a Beta distribution. A random variable X has Bernoulli distribution when it only takes the values 0 or 1. Its probability function is given by: p (x) = p x (1 p) 1 x, x {0, 1}. (1)

3 422 M. Santos et al. In this distribution, the value 1 represents the success of the experiment (Magalhães, 2006). In this work, the occurrence of precipitation was considered a successful result. The success probability p is the parameter of the Bernoulli distribution. Because this parameter varies between 0 and 1, it is assumed that it may be obtained by a Beta distribution defined on the interval (0, 1). If a random variable X has Beta distribution with parameters a > 0andb > 0 (notation: X Beta (a, b)), its density function is given by: in which: f (x; a, b) = 1 β (a, b) xa 1 (1 x) b 1, x (0, 1), (2) 1 β (a, b) = t a 1 (1 t) b 1 dt. (3) 0 A random variable X Beta (a, b) has expectation and variance given, respectively, by: E [X] = a a + b and V [X] = ab (4) (a + b + 1)(a + b) 2 A known method to estimate the parameters a and b of the Beta distribution is the method of moments, which consists of replacing the expectation and the variance in Equation (4) by the respective first two sample moments (mean and variance). The data sample is not sufficiently large in the present work to use such a method. In order to obtain the precipitation frequency in a particular month, the data of the month were isolated to construct a Bernoulli sample as follows: each day with an observed value of precipitation >0 was considered a success (1), otherwise it was considered a failure (0). Thus, to obtain the frequency θ, the total number of successes was divided by the number of observed data points. This frequency was considered to be a sample of length n = 1 from the Beta distribution. In this way, the frequency also coincides with the sample average. The expectation in Equation (4) was then replaced by the frequency value. The variance was not used because in the sample there was only one value, and this is not adequate for the method of moments. Thus, the value of b was established, so that the Beta density function is in agreement with the frequency obtained by observed data. For a fixed value of b: θ 0 = a a + b a = bθ 0, (5) 1 θ 0 that is, the value of a is selected so that the expectation is θ 0. This value is denoted by a 0. Therefore, the Metropolis Hastings algorithm can be used to generate a Markov chain of values θ 0, θ 1, obtained from the Beta distribution with parameters a and b, inwhicha is a sequence a 0, a 1, with terms given by Equation (5) and b is fixed. It is noteworthy that only the parameter a is considered to be variable (random variable) in this work, although there are two parameters in the Beta distribution. Both the a priori Beta (a 0, b) and auxiliary Beta distributions were used to construct the algorithm. Thus, the choice (acceptance) is given by: α ( { ) θ t,θ t+1 = min 1, f ( θ t+1 ; a 0, b ) f ( θ t ; a 0, b ) f ( θt ; a t+1, b ) } f ( θ t+1 ; a t, b ) (6) with f given by Equation (2) (Robert and Casella, 1999). The data used in this study were collected from the database of the National Institute of Meteorology of Brazil (INMET). These City Belém Manaus Natal Petrolina Porto Alegre Curtiba São Paulo Uberaba Table 1. Brazilian cities chosen for the study. Climate type Af Tropical without dry season Am Tropical monsoon Aw Tropical with dry winter BSh Dry Semi-arid low latitude and altitude Cfa Humid subtropical without dry season with hot summer Cfb Humid subtropical without dry season with temperate summer Cwa Humid subtropical with dry winter and hot summer Cwb Humid subtropical with dry winter and temperate summer Sources: Peel et al. (2007) and Alvares et al. (2013). Table 2. Highest and lowest precipitation frequencies. City Highest frequency Lowest frequency Value Month Value Month Belém February October Manaus February August Natal June December Petrolina March 0.03 September Porto Alegre June April Curitiba January August São Paulo January August Uberaba January July data include values of precipitation from January 1961 to December 2012, for the cities in Table 1. The simulations were performed by the R software, v3.0.1, for the months having highest and lowest frequencies of precipitation, in each city, according to the data of INMET. The R software is a language and environment for statistical computing and graphics (R Development Core Team, 2013). 3. Results and discussion Initially, the parameter value for b of the Beta (a, b) distribution was set so that the density function is consistent with the observed data, as previously mentioned. These simulations were carried out for each chosen city, for the months having highest and lowest frequencies of precipitation, as shown in Table 2, according to the collected data. Taking into account the monthly number of rainy days, it was decided to fix the value b = 5, considering that the Beta density function with this b value does not exclude the possibility of a few rainy days in the months with lowest frequency, nor does it exclude the possibility of rainfall every day in the months with highest frequency. All datasets for each city have some missing values. The missing values do not invalidate the use of the proposed method because it is assumed that the data from the same month are an independent and identically distributed random sample, not a time series. The choice in Equation (6) was adopted within the execution of the Metropolis Hastings algorithm, generating one thousand values in the Markov chain. The chosen values were then counted. These chosen values were considered a Beta distribution sample with parameters of the a priori distribution. One hundred iterations of the Metropolis Hastings algorithm were computed

4 Stochastic simulation for the precipitation frequency 423 (a) Beta(1.22, 5) probability density function (b) Beta(59.09, 5) probability density function Figure 2. Beta density functions with respect to the data with highest and lowest chosen rates in the simulations. (a) August in São Paulo: observed frequency 19.6%, chosen rate 54.9%. (b) February in Belém: observed frequency 92.2%, chosen rate 68.1%. (a) Beta(0.15, 5) probability density function (b) Beta(0.29, 5) probability density function Figure 3. Beta density functions with respect to the data with low average chosen rates in the simulations. (a) September in Petrolina: observed frequency 3%, chosen rate 25.2%. (b) July in Uberaba: observed frequency 5.6%, chosen rate 30.8%. for the months with lowest and highest frequencies. There was a percentage of chosen values for each iteration, and its average was defined as the average chosen rate. The average chosen rate in most datasets in these simulations varied from to 0.681, for August in São Paulo and February in Belém, respectively. The a priori density functions for these data are shown in Figure 2, according to the model adopted in this work. However, the proposed model, regarding the months with lower frequency in Petrolina and Uberaba, could not fit two datasets. For these datasets, the average chosen rates were and 0.308, respectively. These low values were considered inappropriate for data adjustments according to the model. In Table 2, the very low frequencies in these cities can be noted, and the a priori density functions are similar to declining exponential curves (as shown in Figure 3). The results of these simulations indicate that the proposed method can estimate the probability of precipitation in conformity with the observed data. This work can contribute to numerical prediction models for cities with scant rainfall data or large amounts of missing data, providing information that contributes to urban planning in these cities. 4. Conclusion Based on the results obtained for the average chosen rate, previously defined, the proposed Beta density function was considered adequate to describe the probability of precipitation on any day of a particular month, where rainy days are not rare events. It is noteworthy that the value of the parameter b was fixed and the parameter a t was controlled by Equation (5). Thus, the value of θ t is expected in the Beta (a t, b) distribution. Therefore, the fixed value of b was considered appropriate to control the variance of the generated sample by the method used, because the frequency, concerning the observed data, was calculated for all the years, and there a random sample of frequencies did not exist.

5 424 M. Santos et al. In conclusion, it follows that the a priori distribution is suitable for estimating adequately the frequency of precipitation when the average chosen rates are higher than 50%. However, in the months that had very low precipitation frequency values, this frequency could not be approximated by the proposed method because the average chosen rates were <50%. The utility of the proposed model is that just one parameter of the Beta distribution needs to be adjusted to estimate the probability of precipitation, contributing to numerical predictions of precipitation for months with high frequency of rainfall, in cities for which there are few data. References Alvares CA, Stape JL, Sentelhas PC, de Moraes Gonçalves JL, Sparovek G Köppen s climate classification map for Brazil. Meteorol. Z. 22(6): Carvalho LMV, Jones C, Liebmann B The South Atlantic convergence zone: intensity, form, persistence, and relationships with intraseasonal to interannual activity and extreme rainfall. J. Clim. 17: Cavalcanti IFA, Ferreira NJ, Kousky VE Análise de um caso de atividade convectiva associado a linhas de instabilidade na Região Sul e Sudeste do Brasil. In Congresso Brasileiro de Meteorologia, Sociedade Brasileira de Meteorologia: Pelotas, RS. Custódio MAM, Herdies DL O jato de baixos níveis e leste da Cordilheira do Andes: um estudo de caso. In Congresso Brasileiro de Meteorologia, Sociedade Brasileira de Meteorologia: Belo Horizonte, MG. (accessed 10 March 2015). Fernandes KA, Satyamurty P Cavados invertidos na região central da América do Sul. In Congresso Brasileiro de Meteorologia, Sociedade Brasileira de Meteorologia: Belo Horizonte, MG, Vol. 8; (Anais II). Ferraz SET Variabilidade Intrasazonal no Brasil e Sul da América do Sul., Doctoral theses in Meteorology, Instituto de Astronomia, Geofisica e Ciências Atmosféricas, IAG-USP. Fietz CR, Comunello E, Cremon C, Dallacort R, Pereira SB Estimativa da precipitação provável para o Estado de Mato Grosso. Embrapa Agropecuária Oeste: Dourados. embrapa.br/publicacoes/online/zip/doc zip (accessed 15 March 2015). Instituto Nacional de Meteorologia Banco de Dados Meteorológicos para Ensino e Pesquisa (BDMEP). br/portal/index.php?r=bdmep/bdmep (accessed 20 October 2014). Kousky VE Diurnal rainfall variation in Northeast Brazil. Mon. Weather Rev. 108: Kruschke JK Doing Bayesian Data Analysis: A Tutorial with R and BUGS. Academic Press. Digital version available at Google Books. (accessed 25 October 2014). Magalhães MN Probabilidade e Variáveis Aleatórias. EDUSP : São Paulo, SP. Mello CR, Silva AM Modelagem estatística da precipitação mensal e anual e no período seco para o estado de Minas Gerais. Rev. Brasil. Eng. Agríc. Ambient. 13(1): Molion LCB, Bernardo SO Uma Revisão da Dinâmica das Chuvas no Nordeste Brasileiro. Rev. Brasil. Meteorol. 17: Moreira PSP, Dallacort R, Magalhães RA, Inoue MH, Stieler MC, Silva DJ, et al Distribuição e probabilidade de ocorrência de chuvas no município de Nova Maringá-MT. Rev. Ciênc. Agro-Ambient. 8: Nimer E Climatologia do Brasil. Superintendência de Recursos Naturais e Meio. Ambiente/IBGE: Rio de Janeiro, RJ. Nobre C The Amazon and climate. In Proceedings of Climate Conference for Latin America and the Caribbean. World Meteorological Organization: Geneva. Nogués-Peagle J, Mo KC Alternating wet and dry conditions over South America during summer. Mon. Weather Rev. 125: Peel MC, Finlayson BL, McMahon TA Updated world map of the Köppen-Geiger climate classification. Hydrol. Earth Syst. Sci. 11: R Development Core Team R: a language and environment for statistical computing. R Foundation for Statistical Computing: Vienna, Austria. (accessed 15 December 2013). Reboita MS, Gan MA, Da Rocha RP, Ambrizzi T Regimes de precipitaçãoo na América do Sul: uma revisão bibliográfica. Rev. Brasil. Meteorol. 25(2): Reboita MS, Krusche N, Ambrizzi T, Da Rocha RP Entendendo o Tempo e o Clima na América do Sul. Terra e Didatica. 8: Robert CP, Casella G Monte Carlo Statistical Methods. Springer: New York, NY. Suliano DC, Magalhães KA, Soares RB A influência do clima no desempenho da economia cearense. Instituto de Pesquisa e Estratégia Econômica do Ceará (IPECE): Fortaleza, CE. Varejão-Silva MA Meteorologia e climatologia. INMET: Brasília, DF.

Journal of Environmental Analysis and Progress

Journal of Environmental Analysis and Progress ISSN: 2525-815X Journal homepage: www.jeap.ufrpe.br/ 10.24221/jeap.2.4.2017.1467.526-535 Simulations of meteorological fields in the Amazon and Northeast