Clean 2008, 36 (10 – 11), 905 – 912
Aytac Guven1 Ali Aytek1 Mehmet Ishak Yuce1 Hafzullah Aksoy2 1
Gaziantep University, Department of Civil Engineering, Hydraulics Division, Gaziantep, Turkey.
2
Istanbul Technical University, Department of Civil Engineering, Hydraulics Division, Maslak, Istanbul, Turkey.
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Research Article Genetic Programming-Based Empirical Model for Daily Reference Evapotranspiration Estimation Genetic programming (GP) is presented as a new tool for the estimation of reference evapotranspiration by using daily atmospheric variables obtained from the California Irrigation Management Information System (CIMIS) database. The variables employed in the model are daily solar radiation, daily mean temperature, average daily relative humidity and wind speed. The results obtained are compared to seven conventional reference evapotranspiration models including: (1) the Penman-Monteith equation modified by CIMIS, (2) the Penman-Monteith equation modified by the Food and Agricultural Organization (FAO 56), (3) the Hargreaves-Samani equation, (4) the solar radiation-based ET0 equation, (5) the Jensen-Haise equation, (6) the Jones-Ritchie equation, and (7) the Turc method. Statistical measures such as average, standard deviation, minimum and maximum values, as well as criteria such as mean square error and determination coefficient are used to measure the performance of the model developed by employing GP. Statistics and scatter plots indicate that the new equation produces quite satisfactorily results and can be used as an alternative to the conventional models. Keywords: Evapotranspiration; Genetic programming; Artificial intelligence; Gene expression programming; Received: January 11, 2008; revised: March 24, 2008; accepted: May 21, 2008 DOI: 10.1002/clen.200800009
1 Introduction Determining grass (or alfalfa) reference evapotranspiration (ET0) is particularly required for water resources management, irrigation system design, hydrological analysis, drainage studies, crop production, and environmental impact assessment. Since direct measurement of ET0 is difficult, and hence costly, the most common practice is to estimate ET0 from climatic variables such as solar radiation, air temperature, wind speed and relative humidity. ET0 is a vitally important component in the hydrological cycle. Therefore, it has been investigated by many researchers [1 – 9]. This study employs genetic programming (GP) which has been applied to a wide range of problems in artificial intelligence, artificial life, engineering and science, financial markets, industrial, chemical and biological processes, and mechanical models including symbolic regression, multi-agent strategies, time series prediction, circuit design and evolutionary neural networks. Research and application of evolutionary computing, over the years, have led to the independent development of five approaches, i. e., evolution
Correspondence: Dr. A. Aytek, Gaziantep University, Department of Civil Engineering, Hydraulics Division, 27310, Gaziantep, Turkey. E-mail:
[email protected] Abbreviations: CIMIS, California Irrigation Management Information System; ET0, Evapotranspiration; GA, Genetic algorithm; GEP, Gene expression programming; GP, Genetic programming; RH, Relative humidity
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strategies, evolutionary programming, classifier systems, genetic algorithms, and genetic programming. GP can be successively applied to areas where (i) the interrelationships among the relevant variables are poorly understood (or where it is suspected that the current understanding may well be wrong), (ii) finding the size and shape of the ultimate solution is difficult and a major part of the problem, (iii) conventional mathematical analysis does not, or cannot, provide analytical solutions, (iv) an approximate solution is acceptable (or is the only result that is ever likely to be obtained), (v) small improvements in performance are routinely measured (or easily measurable) and highly prized, (vi) there is a large amount of data in computer readable form, that requires examination, classification, and integration, e. g., molecular biology for protein and DNA sequences, astronomical data, satellite observation data, financial data, marketing transaction data, or data on the World Wide Web [10]. It has been observed that only a few studies exist in the literature related to the use of GP in the field of water resources engineering. GP has been applied to rainfall-runoff modeling [11 – 16], sedimentary particle settling velocity equations [17], velocity predictions in compound channels with vegetated floodplains [18], prediction and modeling of the rainfall-runoff transformation of a typical urban basin [19], Chezy resistance coefficients in corrugated channels [20], unit hydrographs of a typical urban basin [21], and suspended sediment modeling [22]. Neural network formulation that predicts ET0 as a function of climatic variables including solar radiation, mean air temperature, mean relative humidity, average vapor pressure, wind speed and
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soil temperature has recently been made available [23 – 26] and been widely discussed [27 – 29]. However, a GP-based explicit formulation for ET0 does not yet exist in the literature. Therefore, the purpose of this study is to develop a mathematical model for ET0 based on GP. Climatic variables considered here are solar radiation, Rs, air temperature, Ta, relative humidity, RH, and wind speed, u2. For this purpose, two climatic stations, one from northern California, and the other from southern California, USA, each with almost ten-year daily data were utilized in training and testing the developed model. The performance of the model was compared to seven conventional models available in the literature by using daily data from five more climatic stations located in different parts of California.
2 Overview of Genetic Programming In this section, a brief overview of the GP and Gene expression programming (GEP) is given for motivation. Detailed explanations of GP and GEP are provided by Koza [30] and Ferreira [31], respectively. GP was first proposed by Koza [30]. It is a generalization of genetic algorithms (GAs) [32]. The fundamental difference between GA, GP, and GEP is due to the nature of the individuals. In the GA, the individuals are linear strings of fixed length (chromosomes). In the GP, the individuals are nonlinear entities of different sizes and shapes (parse trees), and in GEP the individuals are encoded as linear strings of fixed length (the genome or chromosomes), which are afterwards expressed as nonlinear entities of different sizes and shapes [33, 34]. GP is a search technique that allows the solution of problems by automatically generating algorithms and expressions. These expressions are coded or represented as a tree structure with its terminals (leaves) and nodes (functions). GP applies GAs to a “population” of programs, i. e., typically encoded as tree-structures. Trial programs are evaluated against a “fitness function” and the best solutions selected for modification and re-evaluation. This modification-evaluation cycle is repeated until a “correct” program is produced. There are five major preliminary steps for solving a problem by using GP. These are the determination of (i) the set of terminals, (ii) the set of functions, (iii) the fitness measure, (iv) the values of the numerical parameters and qualitative variables for controlling the run, and (v) the criterion for designating a result and terminating a run [30]. The first major step in preparing to employ the GP paradigm is to identify the set of terminals to be used in the individual computer programs in the population. The major types of terminal sets contain the independent variables of the problem, the state variables of the system and the functions with no arguments. These types of terminal sets are given in a table by Koza [30]. The second major step is to determine the set of functions, arithmetic operations, testing functions, (such as IF and CASE statements) and Boolean functions (AND, OR, NOT). The third major step is fitness measure, which identifies the way of evaluating how good a given program solves a particular problem. The terminals and the functions are the components of the programs that form the junctions in the tree. The choice of components of the terminals and functions (the program) and the fitness function establish the space that the GP searches for. The fourth major step is the selection of certain parameters to control the runs. The control parameters contain the size of the population, the rate of crossover, etc. The last step is determination of the
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Figure 1. Genetic programming flowchart.
criteria to terminate the run. In general, if the sum of the absolute differences between the model results and the data at hand becomes zero (or reasonably close to zero), then the model is considered acceptable. A GEP flowchart improved by Ferreira [34] is presented in Fig. 1. Once the terminal and non-terminal operators are specified, it is possible to establish the types. Each node will have a type, and the construction of child expressions with crossover and mutation operations needs to follow the rules of the nodal type [35], i. e., by respecting the grammatical rules specified by the user or investigator. Moreover, both specified operator sets must fulfill two requirements: closure and sufficiency, i. e., it must be possible to build correct trees with the specified operators, and the solution to the problem (the expression desired) must be able to be expressed by means of those operators. The automatic program generation is carried out by means of a process derived from Darwin's evolution theory, in which, after subsequent generations, new trees (individuals) are produced from old ones via crossover, copy, and mutation [36, 37]. Based on natural selection, the best trees will have more chances of being chosen to become part of the next generation. Thus, a stochastic process is established where, after successive generations, a well-adapted tree is obtained.
3 Derivation of ET0 based on GEP In general, the selection of input parameters does not completely define the environment from which the system will learn. The researcher must also choose specific past examples from the learning domain. Each example should contain data that represents one www.clean-journal.com
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instance of the relationship between the chosen inputs and the outputs. These examples are often referred to as “training cases” or “training instances” while they are called “fitness cases” in the case of GP. Collectively, all of the training instances are referred to as the “training set”. Once the training set is selected, one can say that, the learning environment of the system is defined. There are five major steps in preparing to use GEP of which the first is to choose the fitness function. The fitness of an individual program i for fitness case j is evaluated by Ferreira [31] using:
If E (ij) f p, then f(ij) = 1; else f(ij) = 0
(1)
where p is the precision and E(ij) is the error of an individual program i for fitness case j. For the absolute error, this is expressed by:
EðijÞ ¼ PðijÞ Tj
ð2Þ
Again for the absolute error, the fitness fi of an individual program i is expressed by:
fi ¼
n X R PðijÞ Tj
ð3Þ
j¼1
where R is the selection range, P(ij) is the value predicted by the individual program i for fitness case j (out of n fitness cases) and Tj is the target value for fitness case j. Therefore, for a perfect fit, P(ij) = Tj for all fitness cases and for maximum fitness, fmax = Rn, where n is the number of fitness cases. The second major step consists of choosing the set of terminals T and the set of functions F to create the chromosomes. In this problem, the terminal set obviously consists of the independent variables, i. e., T = {Rs, Ta, u2, RH}. The choice of the appropriate function set is not so obvious. However, a good guess can always be helpful in order to include all of the necessary functions. In this study, four basic arithmetic operators, i. e., (+, – , *, /), and some basic mathematpffi ical functions, i. e., ( , ln (x), log (x), ex, 10x, power) were utilized. The third major step is to choose the chromosomal architecture, i. e., the length of the head and the number of genes. Values of the length of the head, h = 8, and three genes per chromosome were employed. The fourth major step is to choose the linking function. In this study, the sub-ETs were linked by addition. Finally, the fifth major step is to choose the set of genetic operators that cause variation and their rates. A combination of all genetic operators, i. e., mutation, transposition and recombination, was used for this purpose. The parameters of the training of the GEP model are given in Tab. 1. The best of generation individual, chromosome 12, has a fitness of 797. An explicit formulation based on GEP for reference evapotranspiration was obtained as a function of climatic parameters as follows:
ET0 = f(Rs, Ta, u2, RH)
(4)
Fig. 2 shows the expression tree of the formulation, which is actually given by:
Figure 2. Expression Tree. Table 1. Parameters of the GEP model.
p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12
pffi +, – , *, /, , ln (x), log (x), ex, 10x, power Chromosomes 50 Head size 8 Number of genes 3 Linking function Addition Fitness function error type MAE (Mean absolute error), Custom fitness error function Mutation rate 0.044 Inversion rate 0.1 One-point recombination rate 0.3 Two-point recombination rate 0.3 Gene recombination rate 0.1 Gene transposition rate 0.1 Function set
ð=:d1::Sqrt:X2:=:G1C1:G1C2:d3:d3:G1C1:d0:d1:G1C2:G1C2:d1:d3Þ þð=:G2C1:Pow:Pow10:Ln:Log:Log:
where the constants in the formulation are:
þ:d2:d3:d2:G2C1:d2:G2C1:G2C1:d1:d2Þ þ ð=:d0: þ :d2:=:d0: þ :
G1C1 = – 3.44, G1C2 = 1.94, G2C1 = – 9.39, G3C1
þ:d3:d2:G3C1:G3C2:G3C2:G3C2:G3C1:G3C1:d3Þ
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ð5Þ
= – 9.39, G3C2 = – 8.43
(6)
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Table 2. Minimum and maximum values of input and output variables.
Variable (unit)
Minimum
Maximum
Rs (MJ/m2 d) Ta (oC) RH (%) u2 (m/s) ETo (mm/d)
– 23.00 – 14.50 18.00 0.40 0.00
443.00 39.10 100.00 9.20 10.70
and the actual parameters are:
d0 = Rs, d1 = Ta, d2 = RH, d3 = u2
(7)
After replacing the corresponding values, the final equation becomes: lnðlogðRHÞÞ ETo ¼ 0:06 Ta u0:5 2 9:39ðu2 þ RHÞ
þRs RHm þ
Rs u2 þ RHm 9:39
1 ð8Þ
It should be noted that the proposed GEP formulation in Eq. (8) is valid for parameters ranging between the minimum and maximum values given in Tab. 2, and thus, it covers almost all possible cases that can be expected in practice.
4 Study Area and Data The data set used in this study was obtained from the California Irrigation Management Information System (CIMIS) database. 41-Calipatria station in Southern California (33o0293799 N, 115o2495699 W) and 43-McArthur station in Northern California (41o0395399 N, 121o2791699 W) were selected for model development. For comparison purposes, in addition, the following five stations from different parts of California were selected: 06-Davis (38o3290999 N, 121o4693299 W), 121-Dixon (38o2495599 N, 121o4791299 W), 122-Hasting (38o1695799 N, 121o4792499 W), 123-Suisun (38o1490299 N, 122o0790099 W), and 77Oakville (38o2690299 N, 122o2492599 W). Information on the CIMIS database can be acquired free of charge, from www.cimis.water.ca.gov. Data are available in the database at hourly and daily time scales, and includes solar radiation, air temperature, soil temperature, relative humidity, vapor pressure, wind speed, wind direction and precipitation. The total incoming solar radiation is measured by employing pyranometers at a height of 2.0 m above the ground. Air temperature is measured at a height of 1.5 m above the ground by using a thermistor. Soil temperature is measured at 15 cm (6 inches) below the soil surface. The relative humidity sensor is sheltered in the same enclosure with the air temperature sensor at 1.5 m above the ground. Wind speed is measured by utilizing three-cup anemometers at 2.0 m above the ground. Wind direction is measured by using a wind vane at 2.0 m above the ground. Wind direction values range from 0 to 3608 (both being true north) and are adjusted for declination of the Earth's axis. Rainfall is measured by employing tipping bucket rain gauges. Daily data for almost 10 years covering the period of October 1, 1997 to September 30, 2006 were analyzed for the purposes of developing an explicit formulation based on GP for ET0 and comparing its results with conventional ET0 models. Simple statistical analysis shows that, on average, ET0 increases from north to south together
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Figure 3. GEP estimations and CIMIS ETo for (a) training, and (b) testing sets.
with a higher variation in the north and a lower variation in the south. The maximum ET0 was recorded in the Central stations, i. e., Davis, Hasting, and Dixon. Data (6490 days in total) taken from the two stations were used as training and testing sets for the GP architecture. Among these, 1298 (20% of the total) were reserved for the test and the remaining data were used in the training. The overall performances of both sets were evaluated by the determination coefficient, R2, and mean standard error (MSE). In Fig. 3, the GEP estimates are compared to the CIMIS data via scatter plots for training, see Fig. 3a and testing sets, see Fig. 3b. It is clear from Fig. 3 that the proposed GEP model has accurately learned the nonlinear relationship between the input and the output variables with R2 = 0.984 and MSE = 0.084 (a = 0.965, b = 0.211). A comparison of the GEP predictions with the CIMIS data for the test stage demonstrates a high generalization capacity of the proposed model with relatively low error and high correlation (R2 = 0.967, MSE = 0.320, a = 0.874, b = – 0.123), which shows a successful perwww.clean-journal.com
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Figure 4. Comparison of GEP to the CIMIS model for (a) Davis, (b) Dixon, (c) Hasting and (d) Suissun stations.
Table 3. Statistical measures.
Station
Model
Average (mm)
Standard Deviation (mm)
Minimum (mm)
Maximum (mm)
MSE
R2
a
b
77-Oakville
CIMIS GP CIMIS GP CIMIS GP CIMIS GP CIMIS GP
3.17 3.03 3.85 3.85 3.76 3.81 3.93 4.08 3.51 3.60
2.02 1.74 2.49 2.23 2.43 2.20 2.60 2.45 2.17 2.05
0.00 – 1.31 0.00 – 0.06 0.00 – 0.92 0.00 – 0.17 0.00 – 1.17
8.52 8.10 12.85 11.18 11.12 15.01 12.40 12.54 8.68 9.71
0.224
0.965
0.884
0.351
0.234
0.971
0.881
0.465
0.347
0.946
0.880
0.506
0.299
0.961
0.924
0.466
0.232
0.954
0.924
0.365
06-Davis 121-Dixon 122-Hasting 123-Suisun
formance of the GEP model for estimating ET0 both at the training and testing stages.
5 Further Application of the Model Once the GEP model that best approaches the CIMIS data was constructed, the applicability of the developed GEP model was then shown for any other nearby stations, for other regions in California and for any other geographical region with different climatic conditions from that of California. For this purpose, 77-Oakville, 06-Davis, 121-Dixon, 122-Hasting, and 123-Suissun stations detailed above were selected to reach a more generalized conclusion that the GEP
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model can be used for any location chosen within California. ET0 forecasts were found to be as satisfactory as those for the two stations used in the development of the model. The performance criteria for these five stations were also perceived to be satisfactory, see Tab. 3 and Fig. 4. It should be mentioned that climatic conditions, which obviously have an effect on ET0, in Northern California are completely different from Southern California and even from Central California. Fig. 4 shows the performance of the GEP for stations selected from Central California. Taking the results achieved in Fig. 4 into account, it could be said that the GEP model developed in this work for the prediction of ET0 is a generalized and impressively satisfactory model. The GEP model in Eq. (8) can be used as an alternative to the CIMIS www.clean-journal.com
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Table 4. Statistics and performance criteria of models compared to the CIMIS model for Davis station.
Model
1 2 3 4 5 6 7
CIMIS GP-based FAO 56 Hargreaves-Samani Rs-based Jensen-Haise Jones-Ritchie Turc
Average (mm)
Standard deviation Minimum (mm2) (mm)
Maximum (mm)
MSE (mm2)
R2
3.85 3.85 3.84 3.79 3.27 3.40 4.20 3.72
2.49 2.23 2.22 2.23 1.12 2.88 2.49 2.22
12.85 11.18 12.90 9.28 5.44 11.77 10.37 9.59
– 0.234 0.356 1.005 2.769 1.506 0.630 0.528
– 0.971 0.949 0.837 0.799 0.842 0.920 0.854
0.00 – 0.06 0.28 0.36 0.82 0.01 0.14 – 0.39
Figure 5. Comparison of the CIMIS model to the conventional models (Numbers on the horizontal axis refer to models in Tab. 4 with the exception that (1) CIMIS was replaced by GEP).
model, and therefore, can be used in Central California as well as northern and southern parts of the state. Although further studies may need to be conducted for a more solid conclusion, it is worth stressing that the GEP model could be used for any location within California that presents humid, semi-arid, and arid weather conditions [38]. Again, the finding, very simply, is that the GEP model in Eq. (8) is an alternative to the CIMIS model, and therefore, can be used in Central California as well as northern and southern parts of the state.
speed is taken into account only in the modified Penman-Monteith models while average vapor pressure is considered only in FAO-56. The CIMIS model was selected as the reference model in this study. Therefore, MSE and R2 were calculated to measure the deviation from, and approximation to the CIMIS model. For this purpose, climatic data from Oakville, Davis, Dixon, Hasting and Suisun stations in California were used as inputs into the above mentioned models including GP, and then ET0 was calculated. Based upon the MSE criterion, a comparison to the results of the CIMIS model is presented in Tab. 4 for Davis, in which the GEP model is seen as the closest approximation. This is expected since the GEP model was trained on the CIMIS model. However, keeping in mind that training of the GEP was performed with another data set, it can then be concluded that GEP is a generally valid model to be used in the calculation of ET0. The FAO-56 and Jones-Ritchie models had the second and third closest approximations, respectively, while the Turc model was ranked fourth. The least approaching models were noted to be the Rs-based model, Hargreaves-Samani and Jensen-Haise. The stationbased comparison of each model to the GEP model is presented in Fig. 5, where the MSE between CIMIS and the other models (including GEP) are also presented. The cumulative ET0 values for 77-Oakville station are given in Fig. 6. Again, it is easily seen that GEP performs well even under climatic conditions (wet northern California and dry southern California) different from those of Central California where data was obtained for training the proposed GEP model. This shows the applicability of the developed GEP model in the whole territory of California.
7 Conclusions 6 Comparison to Existing Models Seven models which predict ET0 were taken into consideration for comparison to the new model. These were: (1) the Penman-Monteith equation modified by CIMIS [9], (2) the Penman-Monteith equation modified by the Food and Agricultural Organization (FAO 56) [7], (3) the Hargreaves-Samani equation [5], (4) the solar radiation-based ET0 equation [8], (5) the Jensen-Haise equation [4], (6) the Jones-Ritchie equation [6], and (7) the Turc method [3]. Air temperature and solar radiation seem to be the two most widely employed climatic parameters in the ET0 models mentioned above. Even though minimum and maximum air temperatures are employed in some of these models, i. e., FAO 56, Hargreaves-Samani and Jones-Ritchie, in general, the average air temperature is utilized in the vast majority of them. Regardless of the fact whether ET0 is estimated for humid or arid climates, relative humidity is expected to be one of the indispensable parameters in ET0 models. Wind
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This study reports a new and efficient approach for the formulation of reference evapotranspiration, ET0, using GEP for the first time. The proposed GEP model is empirical and based on daily atmospheric variables, i. e., solar radiation, air temperature, relative humidity, and wind speed, obtained from the California Irrigation Management Information System (CIMIS) database. The GEP model is explicit and simple such that it can be used, by anyone not necessarily being familiar with GEP, in a spreadsheet on a very simple PC, even on a hand-held calculator. The results of the GEP model are compared to seven conventional models and are found to be in good agreement. The model gives a fast and practical method for the estimation of ET0 to obtain accurate results and encourages the use of GEP in other aspects of water engineering studies. Therefore, the GEP is considered as a new and efficient approach. It is worth noting that empirical evapotranspiration formulations developed to date are mostly based on predefined functions. However, in the www.clean-journal.com
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Figure 6. Cumulative evapotranspiration for 77-Oakville station (from October 1997 to September 2006).
GEP approach, there is no predefined function to be considered, i. e., GEP creates randomly formed functions and selects the one that best fits the experimental results. Moreover, there is no restriction in the complexity and structure of the randomly formed functions. Finally, it is important to stress that GEP may serve as a robust approach and may open a new area for the development of accurate and effective explicit formulation of many water resources engineering problems.
Acknowledgements The authors thank the Research Foundation of Gaziantep Univesity for the support provided for the study.
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