Meta-heuristic ant colony optimization technique to forecast the amount of summer monsoon rainfall: skill comparison with Markov chain model

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1 Chapter -7 Meta-heuristic ant colony optimization technique to forecast the amount of summer monsoon rainfall: skill comparison with Markov chain model 7.1 Summary Forecasting summer monsoon rainfall with precision becomes crucial for the farmers to plan for harvesting in a country like India where the national economy is mostly based on regional agriculture. In the present study the meta-heuristic method of ant colony optimization (ACO) technique is proposed to forecast the amount of summer monsoon rainfall over an urban station Kolkata. ACO technique takes inspiration from the foraging behavior of some ant species. The ants deposit pheromone on the ground in order to mark a favorable path that should be followed by other members of the colony. A range of rainfall amount replicating the pheromone concentration is evaluated during the summer monsoon season. The maximum amount of rainfall during summer monsoon season (June September) is observed to be within the range of 7.5 to 35 mm, the Range 4 category set by the India Meteorological Department (IMD) during 1998 to The transitional probabilities of rainfall for consecutive two days during the summer monsoon season are computed using ACO technique and compared with Markov Chain Model (MCM). The result reveals that the accuracy in forecasting the amount of rainfall for two successive days using ACO technique and MCM are 95% and 83% respectively. The accuracy of the forecast is validated with the IMD observations from 2008 and Chaudhuri S., S. Goswami and A. Middey (2013) Meta-heuristic ant colony optimization technique to forecast the amount of summer monsoon rainfall: skill comparison with Markov chain model Theoretical and Applied Climatology, DOI /s y, (online first: ) 144

2 7.2 Materials and methods Meteorological Data The record and data of summer monsoon rainfall are collected from regional meteorological centre (RMC), Kolkata, India during the period from 1998 to 2007 on daily temporal scale for the months of June, July, August and September (JJAS) for the present study. The record and data from 2008 to 2012 are utilised for validation of the result Methodology Mainly two methods are adopted in this study. One is the ACO technique and the other is the MCM. The ant colony optimization (ACO) technique is implemented to forecast the amount of rainfall over Kolkata during the summer monsoon season with the optimum ranges of the different categories of the amount of rainfall. The statistical Markov chain model (MCM) is utilised to find out the amount summer monsoon rainfall for the very next day for comparing the performance of the ACO technique Ant Colony Optimization Swarm intelligence is a relatively new approach to problem solving that takes inspiration from the social behaviours of insects and of other animals. In particular, ants have inspired a number of methods and techniques among which the most studied and the most successful is the ant colony optimization ACO technique. The ACO takes inspiration from the foraging behaviour of some ant 145

3 species. These ants deposit pheromone on the ground in order to mark some favourable path that should be followed by other members of the colony. Ant colony optimization exploits a similar mechanism for solving optimization problems. ACO technique attracted the attention of many researchers and a relatively large amount of successful applications are now available since the early nineties, when the first ant colony optimization algorithm was proposed (Dorigo 1992). Moreover, a substantial corpus of theoretical results is becoming available that provides useful guidelines to researchers and practitioners in further applications of ACO. The purpose of this research is to survey the most notable applications and to introduce ant colony optimization technique in the study of summer monsoon rainfall over Kolkata, India. In the forties and fifties of the twentieth century, the French entomologist, Grass e (1946) observed that some species of termites react to what he called significant stimuli. He observed that the effects of these reactions can act as new significant stimuli for both the insect that produced them and for the other insects in the colony. Grass e used the term stigmergy (Grass e 1959) to describe this particular type of communication in which the workers are stimulated by the performance they have achieved. The two main characteristics of stigmergy that differentiate it from other forms of communication are: Stigmergy as an indirect, non-symbolic form of communication mediated by the environment: insects exchange information by modifying their environment; and 146

4 Stigmergy information as local which can only be accessed by those insects that visit the locus in which it was released (or its immediate neighbourhood). Examples of stigmergy can be observed in colonies of ants. In many ant species, ants walking to and from a food source deposit on the ground a substance called pheromone. Other ants perceive the presence of pheromone and tend to follow paths where pheromone concentration is higher. The mechanism enables the ants to transport food to their nest in a remarkably effective way. Deneubourg et al. (1990) investigated the pheromone laying and following behaviour of ants. In an experiment known as the double bridge experiment, the nest of a colony of ants was connected to a food source by two bridges of equal lengths. In such settings, ants start to explore the surroundings of the nest and eventually reach the food source. Along their path between food source and nest, the ants deposit pheromone. Initially, each ant randomly chooses one of the two bridges. However, due to random fluctuations, after some time one of the two bridges presents a higher concentration of pheromone than the other and, therefore, attracts more ants. This brings a further amount of pheromone on that bridge making it more attractive with the result that after some time the whole colony converges toward the same bridge. This colony-level behaviour, based on autocatalysis, that is, on the exploitation of positive feedback, can be exploited by ants to find the shortest path between a food source and their nest. Goss et al. (1989) considered a variant of the double bridge experiment in which one bridge 147

5 is significantly longer than the other. In this case, the stochastic fluctuations in the initial choice of a bridge are much reduced and a second mechanism plays an important role: the ants choose by chance the short bridge is the first to reach the nest. The short bridge receives, therefore, pheromone earlier than the long one and this fact increases the probability that further ants select it rather than the long one. It has been established from the double - bridge experiment that the concentration of pheromone deposition is inversely proportional to the path length between the nest and the food-source, which is mathematically represented as: k 1 ij (7.1) L k The amount of pheromone deposition if ant k, travel on edge i, j. L k = the path length = (ant hill value - critical value). The smallest path can be identified by noting high pheromone concentration on the path (Bonabeau et al. 2000) Goss et al. (1989) developed a model of the observed behaviour assuming that at a given moment in time m 1 ants have used the first bridge and m 2 the second one, the probability p 1 for an ant to choose the first bridge is: p 1 = (m 1 +k )h (m 1 +k )h+(m 2 +k )h (7.2) Where parameters k and h are to be fitted to the experimental data: P2 = 1 P 1 (7.3) 148

6 Monte Carlo simulations showed a very good fit for k 20 and h 2 (Pasteels et al. 1987) Markov Chain Model The simplest kind of discrete random variable pertaining to dichotomous (yes / no) forecast is made through the Markov chain model (MCM). A two state MCM is a statistical model for the persistence of binary events. The occurrence or non occurrence of summer monsoon rainfall on a given day is a simple meteorological example of a binary random event. The two states, first order MCM illustrated in terms of daily rainfall occurrences or non occurrences (Fig 1). The two states are labelled 0 for no rainfall in Day 1 and 1 for rainfall occurrences in Day 2. There are four transition probabilities controlling the state of a system in the next time period in case of first order MCM. These four probabilities are pairs of conditional probabilities: P 00 +P 01 = 1 P 10 +P 11 = 1 (7.4) (7.5) In particular, it is sufficient to estimate only two parameters for two state first order MCM, since the two pairs of conditional probabilities must sum up to 1. The parameter estimate procedure usually consists simply of computing the conditional relative frequencies, which yield the maximum likelihood estimators: 149

7 P 01 n n 01 o n oo n o 1 n 01 (7.6) P 11 n n 11 1 n 10 n11 n 11 (7.7) Here n 0 is the number of transitions from state 0 to state 1, n 11 is the number of pairs of time steps in which there are two consecutive 1 s in the series, n o is the number of 0 s in the series followed by another data point, and n 1. is the number of 1 s in the series followed by another data point. That is the subscript (.) indicates the total overall values of index replaced by this symbol, so that n n 0 n00 n01 1 n10 n11 (7.8) (7.9) 7.3 Implementation procedure Ant Colony Optimization (ACO) is a paradigm for designing metaheuristic algorithms for combinatorial optimization problems. The first algorithm which can be classified within this framework was presented in early 1990 s (Colorni et al. 1991, Dorigo 1991) and, since then, many diverse variants of the basic principle have been reported in the literature. The essential trait of ACO algorithms is the combination of a priori information about the structure of a promising solution with a posteriori information about the structure of previously obtained good solutions Maniezzo (2004). The method is implemented as follows; 150

8 The records and data of rainfall days with rainfall amounts are taken for the period from 1998 to 2011 throughout the monsoon season (JJAS). The range of monsoon rainfall as per the IMD norm is considered in this study. The values of each range of monsoon rainfall are computed using statistical method. Ranges of monsoon rainfall using ACO technique in terms of pheromone are evaluated. A critical value which signifies the threshold values for each range has been estimated. Thereafter an ant-hill value which indicates the maximum values of the respective ranges has been computed for the monsoonal rainfall. The pheromone deposition for the path has been computed using equation 1 (see section 2.2.1). A range of values of pheromone concentration is obtained applying the optimization procedure. The values of the ranges are computed for successive two days of rainfall using ant colony optimization (ACO) technique and Markov chain model (MCM). 151

9 The transitional probability is computed using ACO technique and MCM for successive two days to identify the most pertinent combination of the monsoonal rainfall. The possibility of rainfall with different combination of ranges for successive two days is estimated using Histogram plot. The results are validated with the observation of 2008, 2009, 2010 and 2011 monsoon rainfall. The quality of prediction is estimated from the performance of the test set of data. The percent errors of prediction (PE) is computed (Perez and Reyes 2001, Chaudhuri and Middey 2011) (chapter 4). The skill of ACO technique is compared with other models in forecasting the amount of rainfall for the next day over Kolkata during the summer monsoon season (Table 7.2). 7.4 Results and discussions The records of the amount of summer monsoon rainfall during the period from 1998 to 2011 over Kolkata are collected from Regional Meteorological Centre (RMC), Kolkata for the present research (Table 7.1). The amount of summer monsoon rainfall is categorized into nine ranges by the India Meteorological Department (IMD). The Range - 1 signifies no rainfall. Range - 2 is the category of very light rain between mm. The rainfall amount between mm is termed as light rain and falls under the category of Range - 3. Range 4 is 152

10 the category of moderate rainfall between mm. The heavy rainfall category begins from Range 5 with the amount of rainfall within mm. Range 6 is also considered as heavy rainfall category with rainfall amount remaining within mm. The Range - 7 falls under very heavy rainfall category when the amount of rainfall is within mm. The Range 8 designates extremely heavy rainfall category when the amount of rainfall exceeds mm rainfall. However, when the amount of rainfall is more than 12 cm it falls under exceptionally heavy rainfall category of Range - 9. The data set considered in this study show that the amount of rainfall remains within the category of Range 7 during the monsoon season (Fig 7.2). The result shows maximum variability in Range 7 category of monsoon rainfall. The Ant colony optimization technique is implemented to estimate the ranges of rainfall amount during summer monsoon season in terms of pheromone deposition (Fig 7.3). The result reveals that the maximum pheromone deposition and the highest amount of rainfall is observed for Range - 4 (R4) followed by Range - 2 (R2) (Fig 7.4). The result further shows that a sudden drop in pheromone concentration as well as amount of rainfall is recorded for Range - 3 category. It is also revealed that in Ranges 5, 6 and 7, the value of pheromone deposition and amount of summer monsoon rainfall are quite low. The result thus shows that the pheromone deposition of ACO technique is capable of simulating summer monsoon rainfall over Kolkata. The MCM is implemented to estimate the ranges of different 153

11 categories of summer monsoon rainfall amount for comparison with ACO technique (Fig 7.4). The transitional probabilities of sequential frequency of rainfall during summer monsoon over Kolkata is computed with ant colony optimization technique and Markov chain model (Fig 7.5). The transitional probabilities for maximum summer monsoon rainfall is observed to be P (1, 1) followed by P (0, 1), P (0, 0) and P (1, 0). It is further revealed that the probability of rainfall for two successive days is maximum and is within the categories of Range - 4 and Range - 2 (Fig 7.6). Different possible combinations have also been evaluated with the amount of rainfall for successive two days. It is observed from the pie chart that, maximum potential is observed for the combination of (R4, R4) and the second highest combination is (R2, R2). The combination of (R2, R4) is less significant followed by the combination (R4, R2). 7.5 Validation The monsoon rainfall of 2008, 2009, 2010 and 2011 are considered for validation of ACO technique and MC model. The probabilities of the monsoon rainfall for consecutive two days with most dominating range for 2008, 2009, 2010, 2011 and 2012 are estimated to be maximum for P (1, 1) using ant colony optimization technique and Markov chain model during validation. The most preferred combination of rainfall during 2008, 2009, 2010, 2011 and 2012 are observed to be (R4, R4) which is well validated (Fig 7.7). Fig 7.8 depicts that for all the 154

12 validating years the most significant dominating amount of rainfall remains within Range 4. It also indicates that if the amount of rainfall on the first day is within the category of Range - 4, then the following day will have the high probability of rainfall of category Range - 4. This figure shows the dominating combination is (R4, R4) validating the observations. It is further observed that the values of root mean square (RMSE) and mean absolute error (MAE) in validating the most preferred combination of (R4, R4) for successive two days rainfall during summer monsoon in the year 2008 are and respectively (Fig 7.9 ). The said errors for the year 2009 are and respectively. The values of RMSE and MAE for the year 2010 are and respectively. The error estimation of monsoon rainfall during 2011 are 1.5 and 1.2 respectively. The values of RMSE and MAE for the year 2012 are and 0.96 respectively. It can thus, be stated that to mm is the most dominating range of rainfall during summer monsoon over Kolkata. The pheromone concentration, on the other hand, remains within the range of 0.42 to 0.58, indicating that the maximum amount of pheromone also signifies the maximum amount of rainfall during the summer monsoon season. False Alarm Rate (FAR) is computed using the following formula (Chaudhuri et al. 2013); FAR = b a+b (7.10) 155

13 where a = number of Hit + number of false forecast and b = number of false forecast The validation data are checked with percentage false alarm rate (%FAR) and presented in figure The range R4, R4 for 2 consecutive days shows zero FAR in 2009 while maximum FAR (30%) has been observed in This result also supports the analysis in figure 7.9 where 2011 shows maximum amount of error and 2009 shows minimum. 7.6 Comparison with other methods Little et al. (2009) have investigated the spatiotemporal correlations in U.K. daily rainfall amounts over the Thames Valley and constructed statistical Markov chain generalized linear models (Markov GLM) of rainfall. Markov GLMs can be configured to produce good 1- day -ahead forecasts and reasonably skilful shortterm forecasts up to a couple of weeks ahead. Mandal et al. (2007) worked on the verification of quantitative precipitation forecasts with the Global Spectral Model, running at the National Center for Medium Range Weather Forecasting (NCMRWF), Noida, India and they found that the model is most efficient in predicting precipitation in the mm range but the efficiency decreases rapidly for higher thresholds. The root mean square error (RMSE) of model predicted monthly rainfall varies between 14 mm to 32 mm over Gangetic West Bengal. Sanso and Guenni (1999) used a stochastic model for tropical rainfall at a single location. They applied the model to estimate and predict both the amount of 156

14 rainfall and the probability of a dry period. The mean absolute deviation (MAD) of their precipitation forecast varies between and mm. Singh and Borah (2013) have used feed-forward back propagation neural network algorithm for Indian Summer Monsoon Rainfall forecasting. The ability of this model to predict rainfall amount in advance can be evaluated by the future years observations, which will draw more interesting findings and conclusions. A study of forecasting the summer monsoon rainfall was also carried out by Sahai et al. (2009). Current status of multimodel superensemble and operational NWP forecast of the Indian summer monsoon has been done by Mishra and Krishnamurti in 2007 (Table 7.2). 7.7 Conclusion The ant colony optimization (ACO) and Markov chain model (MCM) are implemented to forecast the amount of summer monsoon rainfall over Kolkata, India. The study is aimed to predict the transitional probabilities of rainfall for consequently two days during the summer monsoon months (JJAS). The most dominant combination is observed to be P (1, 1). The probability, P (1, 1) is 0.95 and 0.83 using ant colony optimization technique and Markov chain model respectively. The most significant combination of rainfall over Kolkata during summer monsoon is also evaluated. The maximum occurrences of rainfall are found to be within the categories of (R4, R4). The present study shows that the ant colony optimization technique is a reliable method in predicting the amount of 157

15 rainfall for the next day with the observation of the present day and is better than Markov chain model. The values of the error matrices in estimating the amount of rainfall during validation is also observed to be minimum with ACO technique. The ACO technique while compared with other existing models in forecasting the amount of rainfall is found to perform better. ACO technique, therefore, may be used as an operational model for forecasting the amount of rainfall as well as its ranges over Kolkata during the summer monsoon season. 158

16 Table 7.1 Ranges of the amount of monsoon rainfall as per IMD norm Range Descriptive Term used Rainfall amount in mm minimum maximum R1 No rain 0 0 R2 Very light rain R3 Light rain R4 Moderate rain R5 Rather heavy rain R6 Heavy rain R7 Very Heavy rain R8 Extremely Heavy rain R9 Exceptionally Heavy rain When the amount is near the highest recorded rainfall at or near the station for the month or the season. However, this term will be used only when the actual amount of rainfall exceeds 12 cm. 159

17 Table 7.2 Comparison of present study with other existing models / methods Models Markov Chain Model (MCM) forecast (Little et al. 2009). Global Spectral Model (GSM) forecast (Mandal et al., 2007). Stochastic Model (SM) forecast (Sanso and Guenni, 1999). Artificial neural network (ANN), a five neural network architectures Model forecast (Singh and Borah 2013) Artificial neural network (ANN) Model forecast (Sahai et al. 2000) Numerical Weather Prediction model (Multimodel superensemble and operational NWP) forecast (Mishra and Krishnamurti 2007). Ant Colony Optimization (ACO) Model forecast (present study) MAE RMSE - 14 to 32 Forecast error MAD to PE-7.67 Absolute % error- 0.5 to 15 RMSE 1.25 to 2.60 RMSE to 1.5 MAE to

18 P 01 State 0 State 1 P 00 P 11 (No Rainfall) (With rain) P 10 Figure 7.1 Two states, first order Markov Chain Model 161

19 Amonut of monsoon rainfall in mm Range 1 Range 2 Range 3 Range 4 Range 5 Range 6 Range 7 Different ranges of monsoon rainfall Figure 7.2 Blot plot showing the different ranges of monsoonal rainfall amount over Kolkata using statistical method for 1998 to

20 Amount of pheromone deposition Range 1 Range 2 Range 3 Range 4 Range 5 Range 6 Range 7 Different ranges of monsoon Figure 7.3 Box plot showing the different ranges of monsoonal rainfall amount in terms of pheromone deposition over Kolkata using ant colony optimization technique during 1998 to

21 Pheromone deposition Range 1 Range 2 Range 3 Range 4 Range 5 Range 6 Range 7 Different range of rainfall during monsoon Maximum Pheromone Deposition Percentage of Occurrence R ainfall occurrence (% ) Figure 7.4 The maximum pheromone deposition alongwith percentage of rainfall occurrences with different ranges of monsoon rainfall over Kolkata during 1998 to

22 Probability ofoccurrences P(00) P(01) P(10) P(11) Tranisitional probabilities Ant colony optimization Statistical Markov chain Figure 7.5 Transitional probability using for the occurrences of rainfall for successive two days during monsoon season using ant colony optimization technique and statistical Markov chain method for 1998 to 2005 over Kolkata 165

23 (a) (b) (R4,R2) 19% (R4,R4) 36% (R2,R4) 20% (c)(c ) (R2,R2) 25% Figure 7.6 The diagram showing the,maximum occurrences of rainfall in (a) Day 1 and (b) Day 2 in histogram plot and (c) maximum occurrences of rainfall with different combinations using ranges 4 and 2 throughout the summer monsoon season over Kolkata during the period from 1998 to

24 12 N o of occurrences R4 R R2 R R2 R R4 R Combination of rainfall ranges Figure 7.7 The diagram validating the most dominating combination of the amount of rainfall for consecutive two days (Day 1 and Day 2) is (R4, R4) in 2008, 2009, and 2012 throughout the monsoon season (JJAS) over Kolkata 167

25 R4,R4 34% R2,R2 31% R4,R2 22% R2,R4 13% R2,R4 R4,R4 19% 39% R2,R2 R4,R2 16% 26% a b R4,R4 35% R4,R2 24% R2,R4 31% R2,R2 10% R4,R4 29% R4,R2 21% R2,R2 25% R2,R4 25% c d R4,R4 39% R4,R2 19% R2,R4 25% R2,R2 17% 2012 e Figure 7.8 The diagram showing the most significant combination of the amount of rainfall for consecutive two days during validation with observation of 2008, 2010, 2011 and 2012 throughout the summer monsoon season(jjas) over Kolkata 168

26 2.500 Forecast error RMSE MAE Figure 7.9 Different error matrices in predicting the maximum occurrences of the most dominating combination of rainfall amount (R4,R4) of monsoon rainfall over Kolkata during 2008 to