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Mar 12, 2009 - a Jordan Meteorological Department, 11134 Amman, Jordan .... ANN and generalized regression (GR) ANN models as well as the linear ...
METEOROLOGICAL APPLICATIONS Meteorol. Appl. 16: 325–337 (2009) Published online 12 March 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/met.127

Artificial neural network models for forecasting intermittent monthly precipitation in arid regions Ahmad Dahamsheha and Hafzullah Aksoyb * a

b

Jordan Meteorological Department, 11134 Amman, Jordan Department of Civil Engineering, Istanbul Technical University, Hydraulics Division 34469 Maslak, Istanbul, Turkey

ABSTRACT: Forecasting monthly precipitation in arid regions is investigated by means of feed forward back propagation (FFBP) artificial neural networks (ANNs) and compared to the linear regression technique with multiple inputs (MLR). Four meteorological stations from different geographical regions in Jordan are selected. The ANNs and MLR processes are analysed based on the mean square error, relative/absolute error, determination coefficient as well as the central statistical moments such as mean, standard deviation, and minimum and maximum values. It is found that whilst on one hand the ANNs are slightly better than the MLR in forecasting the monthly total precipitation, on the other hand, both are found with to have limitations which should be improved by means of either changing the type and architecture of the ANNs or incorporating modelling tools such as Markov chains into the forecast model. Copyright  2009 Royal Meteorological Society KEY WORDS

artificial neural networks; linear regression; feed-forward back propagation; intermittent precipitation; Jordan; monthly precipitation

Received 30 April 2008; Revised 26 September 2008; Accepted 5 January 2009

1.

Introduction

Precipitation is one of the main sources of water without which humankind cannot survive, so understanding, modelling, predicting or forecasting of precipitation has always been important. In particular, precipitation becomes the unique source of water in arid regions where surface water courses are generally intermittent or ephemeral in nature. In such regions the main water supply is from groundwater storages that are again fed by precipitation. In such regions, therefore, precipitation becomes much more interesting than runoff to analyse as flow records are generally insufficient in length to obtain accurate estimations. Analysis of precipitation is evident not only because of this reason but also for agricultural and socio-economical activities, for increasing human and environmental demands as well as for the planning and management of water resources. Precipitation changes both in time and space and affects other components of the hydrological cycle: i.e. surface runoff, infiltration, groundwater, seepage, percolation, evaporation and transpiration (Chow et al., 1988). Time variation comes from seasonal climatological changes in the atmosphere, whereas spatial change is due to the topographical heterogeneity of the Earth’s surface. * Correspondence to: Hafzullah Aksoy, Department of Civil Engineering, Istanbul Technical University, Hydraulics Division 34469 Maslak, Istanbul, Turkey. E-mail: [email protected] Copyright  2009 Royal Meteorological Society

Precipitation is recorded at meteorological stations, each corresponding to a point on the Earth’s surface. A well-distributed network of stations is required to assess point-scale precipitation data spatially. The spatial distribution does not necessarily need to be homogeneous although the stations might cover all around a study area, and their heterogeneous scatter might result in significant differences than results obtained through their individual use. Structural characteristics (trend, discontinuity, probability distribution function and jump) of precipitation are important to detect. Trend and jumps (if any) should be removed from the data prior to modelling efforts being made to forecast future precipitation (Aksoy, 2007; Dahamsheh and Aksoy, 2007; Gedikli et al., 2008). It is also important to detect these characteristics as they could signal man-made or natural short- or long-term changes due, for example, to a wild-fire, a volcanic eruption, climate change or variability (Alexandrov et al., 2005; Aksoy et al., 2008b). Precipitation data have been analysed not only in semiarid to arid regions but also in semi-humid to humid areas. The following studies are only a few among those devoted to the analysis of precipitation from different countries and regions: for instance, China (Gemmer et al., 2004), the Balkan peninsula (Anagnostopoulou et al., 2003; Tosic, 2004; Tolika and Maheras, 2005), Turkey (Kadioglu et al., 1999), the Netherlands (Tu et al., 2004), Cuba (Naranjo-Diaz et al., 2007), and Saudi Arabia and the Middle East (Nasrallah and Balling, 1996; Subyani,

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2004; Zhang et al., 2005). Israel in the Middle East has been studied widely (Ben-Gai et al., 1993; 1994; 1998; 1999; Steinberger and Gazit-Yaari, 1996; Krichak et al., 2000) while studies devoted to the analysis of precipitation data in Jordan are rare. Variability in the precipitation data of Jordan was analysed by Shehadeh (1976) and Freiwan and Kadioglu (2007a), periodicity by Tarawneh and Kadioglu (2003), and the structural characteristics by Dahamsheh and Aksoy (2007) and Freiwan and Kadioglu (2007b). Modelling efforts for forecasting precipitation in a time step ahead is another issue on which studies were performed. In forecasting models an input vector is introduced into the model and then an output (precipitation in the time-step ahead) is determined by the mathematical rule developed in the model. The model can be based either on the physics of precipitation, in which each step of the model is clear to the user and the variables (such as temperature, wind, relative humidity, pressure) creating precipitation are introduced into the model, or on a conceptual or empirical (data-based) black-box model in which there is no interest in the formulation between the input and output vectors. Models using previously recorded precipitation data only as their input vector have commonly been developed, which are beneficial when the precipitation-related climatological variables are not available. For this aim, time series models such as the pioneering autoregressive integrated moving average (ARIMA) technology (Box and Jenkins, 1970) were popularly used in the 1970s (Delleur and Kavvas, 1978) and are still in common use (Toth et al., 2000; Soltani et al., 2007). Use of frequency analysis is also possible in which a probability distribution function is empirically fitted to the observed data and prediction is made from the population of the distribution fitted (Stidd, 1973). Precipitation can also be forecast by numerical weather prediction models together with radar information (Pedder et al., 2000; Ganguly and Bras, 2003; Kaufmann et al., 2003). The chaotic nature of rainfall time series is well documented in literature (e.g. Selvam et al., 1995; Sivakumar, 2001; Dix and Hunt, 2007; Chattopadhyay and Chattopadhyay, 2008a) and the suitability of an ANN in dealing with chaotic time series is well established (Prinicipe et al., 1992). In the present study, the ANN approach has been adopted to forecast precipitation in a time step ahead. Although it might be questioned (Aksoy et al., 2007; 2008a; Koutsoyiannis, 2007), there is no doubt that ANNs are useful tools in hydrological, meteorological or climatological practices (Jain et al., 1999; Coulibaly et al., 2000; Riad et al., 2004; Sudheer, 2005; Aytek et al., 2008; Zanetti et al., 2007). The ANN type of modelling is relatively easy and is able to reproduce statistical characteristics of observed precipitation sequences. Bodri and Cermak (2000) adopted a time-delay artificial neural network model in which precipitation in a given month was forecasted depending on the previous two months of the current year and on the precipitation of the given month in two previous years. A similar study was later extended to six Czech and four Hungarian meteorological Copyright  2009 Royal Meteorological Society

stations by Bodri and Cermak (2001). Hung et al. (2008) used meteorological parameters in developing an ANN rainfall forecast models. Chattopadhyay (2007) applied an ANN in the form of multilayer perceptron to forecast monsoon rainfall over India using some meteorological predictors and Chattopadhyay and Chattopadhyay (2008a) compared the performances of various backpropagation learning algorithms in forecasting monsoon rainfall time series using the monthly mean rainfall amounts as predictors. Chattopadhyay and Chattopadhyay (2008b) compared the performances of different ANNs with variable hidden layer size in forecasting rainfall time series. With the exception of Freiwan and Cigizoglu (2005), who used FFBP ANNs to predict monthly precipitation in the Amman meteorological station but did not succeed in obtaining a good forecast, no study has been performed for the purpose of forecasting precipitation in Jordan. In the meantime, while this study was under review, a comprehensive study was made available by Aksoy and Dahamsheh (2009) who constructed feed forward back propagation (FFBP) ANN, radial basis function (RBF) ANN and generalized regression (GR) ANN models as well as the linear regression with multiple inputs (MLR) for Baqura, Amman and Safawi stations. The present study focused on forecasting precipitation one-month ahead by using previously observed precipitation data. Input variables in the FFBP ANN were taken differently to those used by Freiwan and Cigizoglu (2005) and Aksoy and Dahamsheh (2009). For comparison purposes, the MLR was also used for forecasting precipitation at the selected meteorological stations. Results of the MLR and the ANN models were first analysed, then a comparison was made, and finally conclusions were drawn on the superiorities and drawbacks of the MLR and ANN models in forecasting intermittent monthly precipitation. Future possible improvements are also considered.

2.

Study area and data

Extending over an area of approximately 90 000 km2 in a semi-arid to arid climatic region in the Middle East, Jordan (Figure 1(a)) suffers from water scarcity and its uneven distribution throughout the country. High population growth, socio-economical development and continuous degradation in water quality exacerbate water shortage in the country where more than 90% of the country receives an annual total precipitation less than 200 mm on average and the amount of water that evaporates back to the atmosphere exceeds 90% of precipitation. Water demand in the country is shared by irrigation (73%), municipalities (22%), and industry (5%). The total water demand in the country is almost twice the conventional water supply that includes the safe yield of all available groundwater and surface water resources (Al-Weshah, 2005). Because of this, important basins such as Azraq supplying Jordanian major cities with drinking water started to suffer from depletion Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

ARTIFICIAL NEURAL NETWORK MODELS (a)

(b)

33 Baqura 1 32

Safawi

Amman

2 31

3

30 Aqaba

4

29 35

36

37

38

39

Figure 1. (a) Jordan topographical map (http://www.kinghussein.gov. jo/maps 1.htm), (b) the climatological regions of Jordan and the location of the Baqura, Amman, Safawi and Aqaba meteorological stations. This figure is available in colour online at www.interscience.wiley. com/ma

of groundwater levels, and became completely dry (Al-Kharabsheh, 2000). Jordan is divided between four climatologically different regions (Figure 1(b)): (1) The Ghor, Jordan Valley, (2) mountains, the hilly region, (3) Badia, the desert, and (4) Aqaba, the gulf. The Ghor, in the western part of the country, is the world’s lowest valley, extending from north to south between two mountain ranges with a length of about 400 km and a width varying from 10 km in the north to 30 km in the south. The elevation in this region changes from 400 m below mean sea level up to 170 m below mean sea level. Climate in the Ghor is classified as tropical. It is very hot in summer and warm in winter with an annual precipitation of 150–250 mm. Lying to the east of the Ghor and extending from north to south, the mountainous region, with elevation reaching up to Copyright  2009 Royal Meteorological Society

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1150 m, has a climate rather mild in summer and cold in winter. The amount of precipitation ranges from 300 to 600 mm per year. To the east of the mountainous region is the Badia region, a semi-desert plateau, with a climate characterized by dry hot summer and relatively cold dry winter with a precipitation 40–100 mm in the complete year. The Aqaba region, the most southern part in the country, is characterized by a very hot summer and a warm winter with an annual precipitation not exceeding 50 mm. The Jordan Meteorological Department (JMD) historical weather observation network has moderate-quality data sets of temperature, total precipitation and other climatic variables averaged at hourly, daily and monthly time scales in order to develop assistance in weather forecast and climate research in Jordan. The JMD network is comprised of 46 surface weather stations and one upper air station. The record period for each station varies but generally includes the period starting by 1970. There are two kinds of rain gauges used at JMD weather observation network: manual gauges and recording rain gauges. Rainfall is collected from manual gauges and recorders daily at 0800 Jordan local time. At most of the sites, the manual gauge had several more years of available records than did the recorders. Some differences in average annual precipitation are attributable to the slightly different record lengths. For the most part, however, the differences indicate some missing periods of data at the recorders during most years of data collection. The percentage of missing data for the synoptic and agro-climate stations is less than 13% for temperature and less than 7.5% for precipitation. Procedures for estimating missing data are not applied in the JMD, data from these stations are not adjusted for homogeneities. The JMD started to implement a new project for automatic weather stations, two stations were installed in Baqura and Amman, and recently two new stations with temperature, wind and rainfall sensors were installed in Deir Alla and Dhulail. Four meteorological stations operated by JMD were selected to be used in this study. The stations were chosen such that they cover the four climatological regions in Jordan and that they have long record periods. Stations and the climatological regions are seen in Figure 1(b), their statistical characteristics given in Table I. As a pre-treatment, the data sets were checked for the presence of homogeneity and trend. The standard normal homogeneity tests, the Pettitt test and the von Neuman ratio test were used for identifying homogeneity, while the t test and the Spearman rank order correlation coefficient test were employed for trend. The four data sets pass these tests at the 95% significance level, i.e. they are homogeneous and have no trend. There are no missing data in the four time series and, hence, no pre-treatment is required for missing data. The record period ranges from 38 to 83 years. The stations are located at elevations from 170 m below mean sea level for Baqura in the Ghor to 772 m above mean sea level for Amman in the mountainous region. Characteristics of the meteorological stations show that the highest Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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Table I. Characteristics of the Baqura, Amman, Safawi and Aqaba meteorological stations and statistics calculated from the annual total precipitation data. Station

Observation Latitude Longitude Elevation Mean period (m) (mm)

Baqura Amman Safawi Aqaba

1968–2005 1923–2005 1943–2005 1946–2005

32.63 31.98 32.20 29.55

35.62 35.98 37.13 35.00

−170 772 672 51

St. Median Dev (mm) (mm)

397.6 125.9 270.4 90.0 71.9 39.3 29.9 22.5

377.0 263.3 64.7 28.4

Cv

0.32 0.33 0.55 0.75

Cs

r1

Max. Year Min. Year (mm) (mm)

1.13 −0.110 822.7 1992 168.2 1999 0.30 −0.133 476.5 1938 98.0 1995 1.37 −0.083 213.5 1988 7.5 1958 0.91 0.058 96.9 1954 2.0 1970

St. Dev, standard deviation; Cv , coefficient of variation; Cs , Skewness coefficient; r1 , lag-one correlation coefficient; Max, maximum; Min, minimum.

amount of precipitation is observed in Baqura and the second highest in Amman. Safawi, due to its location in the desert region, and Aqaba, due to its location in lower latitudes, receive the least annual total precipitation of 71 and 30 mm, respectively. The median value and the standard deviation have the same ranking at the annual scale. Where the variability is concerned, the desert station tends to be more variable than the other two stations. Frequency distributions of the recorded precipitation are far from the normal distribution and the interdependency between the subsequent years has a negative correlation. The range between the minimum and the maximum precipitation is highest in Baqura and smallest in Aqaba. However, when the ratio of maximum precipitation to minimum precipitation is considered, Safawi and Aqaba are found to have maximum precipitation 28.5 and 48 times higher than their minima due to the high variability mentioned above. Maximum precipitation observed in Baqura and Amman is about five times greater than their minima. A monthly analysis was also performed. When Table II is considered, discrepancies were observed between the stations although the four stations show similar patterns. For instance, in the north (Baqura and Amman), on average, the highest precipitation falls in January while the highest precipitation is shifted back to December in Safawi and Aqaba. Monthly total precipitation in the 7-month period April–October (both inclusive) remain under the long-term average in Baqura, Amman and Safawi. This period is from May to October in Aqaba, a month shorter than at the other three stations. It is, therefore, considered a dry period. Four months (June–September) in the dry period are completely dry with no precipitation at all when monthly total precipitation less than 1 mm is ignored. In Aqaba, the total precipitation in May is 0.9 mm and therefore the total number of dry months is five in this particular station. Five months (November–March in northern stations) cover the wet period. The standard deviation has similar patterns throughout the year while the coefficient of variation shows an inverse pattern in which variability increases in the dry period. It should be noted that, by definition, coefficients of variation and skewness cannot be calculated in months with no precipitation (August in Amman, July and August in Baqura and Safawi and June–August Copyright  2009 Royal Meteorological Society

in Aqaba). When these months are excluded, it is seen that the dry period has more variable and more skewed precipitation in the three stations as in other stations in the country (Dahamsheh and Aksoy, 2007). The maximum monthly precipitation was recorded in January and December (months with the highest monthly average) for Amman and Safawi, respectively. In Baqura and Aqaba, however, the maximum was observed in February and not in the months with the highest average, e.g. January for Baqura and December for Aqaba. In the case of Baqura station, the maximum precipitation observed in January was ranked the third after the maximum observed in December. The highest minima were recorded in January (Baqura) and February (Amman). In Safawi and Aqaba, each month has dried out at least once during the 63year observation period. The autocorrelation coefficient calculated between the subsequent months was found to be low (close to zero) when a few summer months where the autocorrelation cannot be calculated by definition are excluded. Finally, the zero-precipitation ratio was found to be similar in the four stations. This ratio is calculated, for each month, by dividing the number of years with no precipitation in the particular month by the total number of years of observation. It was seen that four months (June–September) in Amman, five months (May–September) in Baqura, six months (May–October) in Safawi and seven months (April–October) in Aqaba have higher number of years with no precipitation than the average number. The four stations are ranked from the wettest to the driest as Baqura (zero precipitation 37.9% of the time), Amman (38.2%), Safawi (46.8%) and Aqaba (63.2%). 3. Artificial neural networks and the linear regression 3.1. Feed forward back propagation (FFBP) artificial neural network (ANN) The FFBP can be made of one or more hidden layers where there is a function intervening between the external input and the network output. Each layer consists of a number of nodes, also called neurons. The neurons in the hidden layer(s) are able to extract higher order statistics, particularly when large amounts of input data Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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Table II. Statistical characteristics of the monthly total precipitation data recorded in the Baqura, Amman, Safawi and Aqaba meteorological stations. Month Baqura

1

2

3

4

5

Mean (mm) 96.6 75.1 58.0 20.8 St. Dev (mm) 52.5 63.1 32.8 24.0 0.54 0.84 0.57 1.16 Cv 0.85 2.44 0.40 2.56 Cs Max. (mm) 235.5 333.7 124.8 126.1 Min. (mm) 25.8 9.1 9.0 0.0 −0.23 −0.19 −0.11 0.15 r1 Zero (%) 0.0 0.0 0.0 5.3

6

4.6 10.7 2.33 3.11 49.0 0.0 −0.10 50.0

7

8

0.3 0.0 1.5 0.0 a 5.58 a 6.06 9.2 0.0 0.0 0.0 a −0.03 94.7 100.0

Amman Mean (mm) 64.1 62.3 42.8 13.7 3.4 St. Dev (mm) 38.2 44.5 35.6 20.7 6.4 0.60 0.71 0.83 1.51 1.88 Cv 1.18 0.96 1.45 4.12 2.50 Cs Max. (mm) 235.2 199.8 168.7 151.3 30.6 Min. (mm) 1.7 5.2 0.0 0.0 0.0 0.07 −0.05 0.12 0.06 0.11 r1 Zero (%) 0.0 0.0 1.2 7.2 37.3

0.0 0.2 4.70 4.94 1.2 0.0 −0.05 95.2

Safawi

Mean (mm) St. Dev (mm) Cv Cs Max. (mm) Min. (mm) r1 Zero (%)

12.0 11.5 0.96 1.31 44.5 0.0 0.07 3.2

13.3 14.2 1.07 2.04 73.5 0.0 −0.01 3.2

10.7 9.8 0.92 1.64 51.6 0.0 −0.11 7.9

5.2 2.6 8.9 5.4 1.72 2.09 2.31 2.57 39.3 24.5 0.0 0.0 −0.01 0.20 28.6 50.8

0.0 0.0 0.2 0.0 a 7.94 a 7.94 1.2 0.0 0.0 0.0 a −0.02 98.4 100.0

Aqaba

Mean (mm) St. Dev (mm) Cv Cs Max. (mm) Min. (mm) r1 Zero (%)

4.6 7.9 1.72 2.77 36.8 0.0 0.09 21.7

4.9 10.4 2.12 4.19 67.2 0.0 0.05 31.7

4.2 6.4 1.53 2.58 30.6 0.0 0.06 26.7

3.4 0.9 0.0 8.0 2.8 0.0 a 2.37 2.92 a 3.16 3.42 39.9 14.1 0.0 0.0 0.0 0.0 a 0.05 −0.05 55.0 80.0 100.0

a By

9

0.0 0.0

10

0.4 1.8 a 3.99 a 4.80 0.0 10.0 0.0 0.0 a −0.07 100.0 89.5

11

12

13.2 44.7 84.1 16.6 43.6 58.4 1.26 0.98 0.70 1.78 1.70 1.06 69.8 176.0 238.1 0.0 0.0 8.6 −0.08 −0.19 0.07 10.5 5.3 0.0

0.0 0.0 0.3 6.3 28.0 49.5 0.0 0.0 1.8 9.9 31.2 43.6 a 9.11 6.02 1.58 1.11 0.88 a 9.11 7.40 2.44 1.88 1.23 0.2 0.0 15.4 54.6 136.8 179.8 0.0 0.0 0.0 0.0 0.0 0.4 a −0.01 −0.03 0.06 −0.06 −0.20 98.8 100.0 91.6 22.9 3.6 0.0

0.0 0.0 a a

0.0 0.0 a

100.0

0.0 0.0

0.8 4.1 5.7 9.4 a 7.50 2.30 a 7.92 4.13 0.0 45.0 60.8 0.0 0.0 0.0 a −0.02 −0.02 100.0 95.2 52.4

9.3 11.3 1.22 1.48 45.0 0.0 0.07 17.5

13.9 15.9 1.14 2.00 79.0 0.0 −0.12 4.8

0.0 0.0

2.8 5.7 2.03 2.67 27.2 0.0 0.05 46.7

7.3 10.4 1.43 1.56 43.3 0.0 −0.12 30.0

0.0 1.8 0.1 5.1 a 6.36 2.75 a 7.16 3.71 0.0 0.8 28.0 0.0 0.0 0.0 a −0.03 −0.08 100.0 96.7 70.0

definition the statistical characteristics cannot be calculated

are available. The output of any layer is used as input to the next layer, i.e. the neurons in each layer of the network only have the outputs of the preceding layer as their inputs. One drawback to be mentioned about the FFBP is that it requires a long calibration duration with a high number of iterations to fix the weights for the optimal solution. The inputs xk , k = 1, . . . , K to the neurons in the input layer are multiplied by initial weights iwkl and summed up together with the constant initial bias ibl to result in al as:

al =

K 

iwkl xk + ibl ,

l = 1, . . . , L

(1)

k=1

which is the input to the hidden layer activation function f for which the mathematically convenient hyperbolic tangent or sigmoid function are most commonly used (Bodri and Cermak, 2001). In this study, the hidden layer Copyright  2009 Royal Meteorological Society

activation function (sigmoid) is defined as: f (a) =

1 1 + exp(−a)

(2)

The output of node l in the hidden layer then becomes: K  iwkl xk + ibl ), yl = f (

l = 1, . . . , L

(3)

k=1

The outputs of the hidden layer nodes are multiplied by layer weights lwlm and summed up together with constant layer bias lbm as: bm =

L 

lwlm yl + lbm ,

m = 1, . . . , M

(4)

l=1

The activation function of the output layer was selected the same as the input layer: f (b) =

1 1 + exp(−b)

(5)

Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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The final output of the model is then obtained by: ym = f

 L 



lwlm yl + lbm ,

m = 1, . . . , M

Table III. Correlation coefficients between input variables and the output variable. Station

xt−1

xt−12

x

(6) The batch rule was used in this study for the weight update. The FFBP ANN was trained using the Levenberg–Marquardt training algorithm.

Baqura Amman Safawi Aqaba

0.424 0.385 0.252 0.111

0.453 0.449 0.227 0.133

0.720 0.686 0.497 0.357

3.2.

xt−1 : Current precipitation, xt−12 : previous year’s precipitation of the month to be forecast, x: mean value of the month to be forecast.

l=1

Multiple linear regressions (MLR)

Linear regression techniques still represent one of the most frequently used time series analysis methods. The advantage of the MLR is that it is a commonly used and well investigated technique that is simple to interpret. Its disadvantage is, however, that it assumes a linear relationship and the data set should fulfill a number of criteria. The mapping of multiple linear regressions has the form: K  y =c+ dk xk (7) k=1

where c is a constant and dk , k = 1, . . . , K are the regression coefficients. The regression coefficients were calculated by using the least square error method.

4. 4.1.

Table IV. Mean square error (MSE ) and determination coefficient (R 2 ) for different ANN model architectures. Neurons

1 2 3 4 5 6 7 8 9 10

Baqura

Amman 2

MSE

R

1209.7 1209.7 1196.4 1129.3 1113.8 929.4 1018.2 1012.0 977.3 882.3

0.51 0.51 0.51 0.54 0.54 0.62 0.58 0.59 0.60 0.64

Safawi 2

MSE

R

723.2 723.2 723.5 719.6 697.8 680.3 695.3 658.2 677.8 659.6

0.46 0.46 0.46 0.46 0.48 0.49 0.48 0.51 0.49 0.50

Aqaba 2

MSE

R

93.1 91.3 86.5 87.2 83.4 83.8 83.1 82.5 82.5 77.9

0.23 0.25 0.30 0.28 0.31 0.31 0.32 0.32 0.32 0.36

MSE

R2

40.6 40.6 40.6 39.6 38.3 37.9 37.7 35.5 36.9 37.9

0.15 0.15 0.15 0.17 0.19 0.20 0.21 0.25 0.22 0.22

Model Development Model Architecture

The variables to be used as input into the model were determined by calculating the autocorrelation coefficient between the observed and forecast total precipitation in each month. Once the input variables are determined it is then a trial-and-error effort to decide on the architecture of the model, i.e. selection of the appropriate number of layers in the model and the number of neurons in each layer. The number of input and output variables directly gives the number of neurons in the input and output layers, respectively. In order to forecast precipitation in a particular month, t, several model architectures were built. In these trials the observed precipitation of some previous months, the first- and second-order central moments (average and standard deviation) in those months as well as in the month to be forecast, and the periodic component, were used as input variables in different combinations. Here, no information was given on these models as the performance of none was found satisfactory. In Table III, it can be seen how the model output, the precipitation of the month to be forecast (xt ), is correlated to the selected model inputs, the previous year’s precipitation of the month to be forecast (xt−12 ); the current month’s precipitation (xt−1 ), and the mean value of precipitation in the month to be forecast (x t ). For the four stations, it was calculated that precipitation to be forecast was correlated to its long-term mean value and its previous year record as well as precipitation observed in the current month. It was also observed that the correlation Copyright  2009 Royal Meteorological Society

was reduced with decrease in the amount of precipitation: e.g. the higher the precipitation at the station, the higher the correlation between the observed and forecasted precipitation, or vice versa. Finally, an ANN architecture accepting these variables was developed and, hence, a three-input and one-output model was obtained. In order to decide on the number of neurons in the hidden layer of the developed model, the number of neurons was changed for each station from 1 to 10. The decision was made on the determination coefficient (R 2 ) and the mean square error (MSE ) calculated for each case from the observed and modelled precipitation as: 

N  (Po,i − P o )(Pf,i − P f )

   i=1 R2 =    N    (P

o,i

 N  − P o ) (Pf,i − P f )

i=1 N 

MSE =

     (8)   

i=1

(Po,i − Pf,i )2

i=1

N

(9)

where Po,i and Pf,i represent, respectively, the observed and the forecasted total precipitation in month i, which goes up to the total number of months, N , and P o and P f correspond, respectively, to the long-term averages of the observed and the forecasted precipitation (Table IV). The Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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the models over the last ten years (120 months) covering the common period of 1996–2005 in validating (testing in the ANN terminology) the models. Table V shows performance criteria of the calibration stage of the FFBP ANN model and the MLR technique for the four stations. Although none is perfect, an increase in the appropriateness is observed with increase in precipitation. R 2 increases and a, the slope of the fitted line tends to approach unity with increasing precipitation. During the calibration it was ensured that forecasts were not allowed to reach a value higher than a specified threshold. Therefore, maxima are not forecast well. For instance, the highest precipitation reached in the calibration stage of the ANN model was 220.2 mm in Baqura, 108.9 mm in Amman, 65.2 mm in Safawi and 27.3 mm in Aqaba while the maximum precipitation recorded during the calibration period in these stations was 333.7, 235.2, 79 and 67.2 mm, respectively. Similarly, the maxima were considerably underestimated in the calibration stage of MLR (100.4 mm in Baqura, 65.2 mm in Amman, 16 mm in Safawi and 8.6 mm in Aqaba). This is considered an important problem that was faced in this study and also happened in the study of Freiwan and Cigizoglu (2005) and Aksoy and Dahamsheh (2009) when intermittent precipitation records including zeros were modelled.

MSE was minimized and R 2 was maximized when 10 neurons were used for Baqura and Safawi, and 8 neurons for Amman and Aqaba. The final FFBP ANN models are then obtained as 3-10-1 (for Baqura and Safawi) and 3-8-1 (for Amman and Aqaba). For MLR, the following equations were fitted for Baqura, Amman, Safawi and Aqaba, respectively: y = 0.126 + 1.092x1 − 0.097x2 + 0.005x3

(10)

y = −0.005 + 1.024x1 − 0.025x2 + 0.001x3 (11) y = −0.043 + 0.969x1 − 0.015x2 + 0.056x3 (12) y = −0.017 + 1.013x1 − 0.007x2 + 0.012x3 (13) In Equations (10)–(13), x1 , x2 and x3 refer to the same input variables used in the ANN model, which are the long-term mean (x t ) of the month to be forecast, the observed value of the current month (xt−1 ) and the observed value of the month to be forecast in the previous year (xt−12 ), respectively. Encouraging results were obtained and are detailed below. 4.2. Model calibration (training) The calibration (in the classical modelling terminology) or training (in the recent common ANN terminology) of models was performed in order to minimize the MSE and maximize R 2 . At the same time, the unit slope and zero intercept were aimed in the best-fit line of the scatter diagram of the observed and forecasted precipitation. The ANN model and the MLR technique were applied to the monthly total precipitation of the four meteorological stations, Baqura, Amman, Safawi and Aqaba. Monthly data of the first 28, 73, 53 and 50 years (336, 876, 636 and 600 months) of each station were used for calibration of

4.3. Model validation (testing) Based on the calibration explained above, the models (ANN and MLR) were validated using the last 10 years’ data (Table VI). It is clear that the overall performance of both models is not perfect. It is worth, however, noting that results in Freiwan and Cigizoglu (2005) are worse even with models using the Amman precipitation data divided into subsets. As in the study by Freiwan

Table V. Performance criteria in the calibration (training) stage. Criteria

R2 MSE MAE a b

Baqura

Amman

Safawi

Aqaba

ANN

MLR

ANN

MLR

ANN

MLR

ANN

MLR

0.64 882.3 16.7 0.65 12.0

0.51 1206.5 19.0 0.51 17.0

0.51 658.2 14.4 0.50 11.0

0.46 720.3 15.0 0.46 12.0

0.36 77.9 5.1 0.36 3.8

0.24 92.6 5.5 0.24 4.7

0.25 35.5 3.0 0.25 2.1

0.15 40.6 3.3 0.15 2.4

Table VI. Performance criteria in the validation (testing) stage. Criteria

R2 MSE MAE a b

Baqura

Amman

Safawi

Aqaba

ANN

MLR

ANN

MLR

ANN

MLR

ANN

MLR

0.46 1318.3 20.4 0.59 15.0

0.55 1003.8 18.6 0.56 16.0

0.53 442.4 12.4 0.63 11.0

0.60 362.4 11.7 0.65 10.0

0.31 53.6 4.3 0.38 3.8

0.33 51.4 4.2 0.36 4.1

0.01 37.8 3.3 0.09 3.1

0.04 22.0 2.8 0.13 2.6

Copyright  2009 Royal Meteorological Society

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and Cigizoglu (2005), the main problem is the failure in the generation of the maximum precipitation. In the validation stage of both models (Figures 2 and 3), high precipitation values were not reached as it was the case in the calibration. Both models perform in a similar fashion and alternate each other. Again, similar to the calibration stage, the performance of the models is

reduced with increase in the number of months without precipitation. 4.4.

Possible reasons for not having good forecasts are numerous. Forecast performance is affected by the time interval 250

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Discussion and further analysis of results

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Figure 2. Time series and scatter diagrams of observed and forecasted precipitation in the validation stage of the feed forward back propagation artificial neural network model for (a) Baqura, (b) Amman, (c) Safawi and (d) Aqaba meteorological stations. Copyright  2009 Royal Meteorological Society

Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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Figure 3. Time series and scatter diagrams of observed and forecasted precipitation in the validation stage of the multiple-input linear regression technique for (a) Baqura, (b) Amman, (c) Safawi and (d) Aqaba meteorological stations.

of the process, statistical similarity of the calibration and validation segments of the data, the homogeneity and seasonality behaviour of the data, and existence of other structural characteristics such as jump or trend. The correlation structure of the process is also considered important in obtaining good forecasts. Other than all these possible reasons, based on results of this study, it can be stated that performance of the model is reduced with increasing Copyright  2009 Royal Meteorological Society

variability in the precipitation. This was observed not only for the ANN model but also for the MLR technique. Among the four stations, performance of the models is the best in Baqura, station with the lowest variability, and the worst in Aqaba with the highest variability. Another reason that could be specific to this study is the intermittent structure of the precipitation sequences. The zero-values in the time series prevented the models Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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producing maxima in the sequence of precipitation. When two data sets coming from two populations (zero and non-zero populations) with different characteristics from each other are put together one should keep in mind that perfect results cannot be expected. Indeed, this type of hydrometeorological variable should be divided into subsets and analysed or modelled separately (Wang and Singh, 1995). The following mathematical reasoning can help in understanding why the models do not reach values higher than a specified threshold. The ANN models basically depend on the non-linear relation between the input and output variables. Each input variable has its own weight factor determined from the data. In this study, the precipitation sequence is intermittent and therefore a number of zero-precipitation is recorded. Existence of the dry period in the precipitation sequence led to lower weight factors resulting in lower precipitation forecasts. Also, seasonality is considered important, because, even if the weight vector is high, the forecast cannot reach a specified value as long as the input variable vector has zeros which is the case in dry months. A further analysis at annual, seasonal and monthly scales was made and is presented in Table VII. When the mean value of annual total precipitation is concerned, the MLR technique performed better than the ANN model in Baqura, Amman and Aqaba while the ANN model

forecast better in Safawi. The relative error in Baqura was 5.4% for the MLR and 7.3% for the ANN. The relative error for Amman, Safawi and Aqaba changed from about 8 to 115%. This means the models can almost forecast the annual total precipitation well. Both the ANN models and the MLR gave much lower variability than its observed value for the four stations. The minimum annual total was overestimated and the maximum was underestimated, i.e. higher total annual minima and lower maxima were forecasted. Better forecasts for the mean value of the winter precipitation were obtained by MLR in Baqura, Safawi and Aqaba and by the ANN in Amman although the difference between the MLR and the ANN is not too big. Indeed, the performance of the models can be considered as good as each other. Here, it is considered important to state that the ANN architecture was built on how well it is in forecasting the maxima rather than obtaining the mean value (Sudheer et al., 2003). Therefore, as detailed below, even the performance of the ANN model looks weaker than MLR with respect to forecasting the mean value, the ANN model generally behaves better. The winter precipitation forecast can be considered very well in Amman (∼10% relative error) and even perfect in Safawi (∼3–5%), Baqura (0.4–0.7%), but it is not good in Aqaba, with relative errors higher than 100%. In the spring, the MLR model is better than

Table VII. Annual, seasonal and monthly analysis of observed and forecasted precipitation. Baqura

Amman

Safawi

Aqaba

Obs

ANN

MLR

Obs

ANN

MLR

Obs

ANN

MLR

Obs

ANN

MLR

Annual

Mean (mm) St. Dev (mm) Min. (mm) Max. (mm)

385.4 97.9 168.2 511.8

413.4 95.2 278.3 616.9

406.1 10.4 393.0 426.2

232.5 66.0 109.4 325.6

282.7 18.8 255.2 316.4

276.9 1.9 274.3 279.8

65.4 27.1 32.9 103.0

70.7 8.0 57.3 78.3

73.1 1.1 71.2 74.6

15.4 12.3 2.1 41.1

38.1 8.4 26.5 52.9

33.1 0.2 32.9 33.5

DJF

Mean (mm) St. Dev (mm) Min. (mm) Max. (mm)

255.2 99.3 120.8 490.8

253.3 60.8 144.4 331.2

254.3 9.9 231.3 267.3

161.7 47.4 90.7 267.5

177.0 20.2 147.7 211.0

178.3 1.5 175.3 180.8

38.9 14.1 9.0 55.9

36.9 8.2 28.1 51.2

40.0 1.4 38.1 42.1

8.5 8.6 1.2 29.0

22.3 7.5 16.0 41.4

18.6 0.1 18.5 18.8

MAM

Mean (mm) St. Dev (mm) Min. (mm) Max. (mm)

79.8 45.4 10.5 138.1

96.2 37.2 65.9 189.6

85.9 4.4 79.8 92.4

47.8 23.1 18.7 77.2

71.1 8.9 60.6 87.3

61.9 0.6 61.1 62.6

14.1 11.5 1.2 39.3

20.0 5.4 14.1 31.8

19.7 0.7 19.0 21.3

2.8 4.6 0.0 14.3

9.0 4.7 4.3 21.4

9.8 0.1 9.7 10.0

JJA

Mean (mm) St. Dev (mm) Min. (mm) Max. (mm)

0.0 0.0 0.0 0.0

2.4 0.0 2.3 2.4

0.8 0.1 0.8 1.0

0.0 0.0 0.0 0.0

1.9 0.1 1.8 2.0

0.0 0.0 0.0 0.1

0.0 0.0 0.0 0.0

0.7 0.0 0.7 0.7

−0.1 0.1 −0.1 0.0

0.0 0.0 0.0 0.0

0.3 0.3 0.2 1.2

0.0 0.0 −0.1 0.0

SON

Mean (mm) St. Dev (mm) Min. (mm) Max. (mm)

45.2 42.9 0.4 154.6

61.6 24.0 31.4 104.4

64.9 4.2 54.2 69.1

21.6 20.5 0.5 66.1

33.2 4.8 27.8 40.3

36.7 0.5 35.6 37.3

13.8 15.0 0.0 45.1

13.4 3.9 6.4 17.9

13.7 0.3 13.3 14.4

4.1 8.9 0.0 28.0

6.5 5.8 3.3 22.7

4.8 0.1 4.6 5.1

Monthly

Mean (mm) St. Dev (mm) Min. (mm) Max. (mm)

32.1 47.5 0.0 243.8

34.5 41.8 0.7 171.4

33.8 35.5 0.1 100.1

19.4 29.7 0.0 140.1

23.6 25.6 0.5 88.1

23.1 24.9 0.0 65.2

5.5 8.8 0.0 45.0

5.9 5.9 0.2 20.3

6.1 5.4 0.0 16.2

1.3 4.2 0.0 28.3

3.2 4.5 0.0 25.5

2.8 2.7 0.0 8.3

DJF, December-January-February; MAM, March-April-May; JJA, June-July-August; SON, September-October-November. Copyright  2009 Royal Meteorological Society

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the ANN model and the results are encouraging. For instance, the seasonal total was forecasted by MLR with only 7.6% relative error in Baqura. No precipitation was forecasted for the summer months, completely dry months, for which the MLR gave either zero or a negligible precipitation total less than 1 mm. The ANN model, however, never generated zero forecasts. It might be interesting to note that MLR produced a perfect seasonal total (0.7%) for rainfall in Safawi, and also that slightly better forecasts were obtained by the ANN model in Baqura and Amman with higher error. Minima and maxima were almost all better approximated by the ANN, although the variability was again lower than its observed counterpart, ending with higher minima and lower maxima in all seasons. In general, the ANN model worked better than the MLR technique when the above mentioned exceptions in the annual total and winter total are excluded. A monthly analysis was also made, in which it was seen that both models are as good as each other. Relative error in Baqura was calculated as 5.3% for MLR and 7.5% for ANN in Baqura; 7.3% for ANN and 10.9% for MLR in Safawi. The relative error in Amman was ∼20% and the models alternate each other with respect to better forecast of the mean value. Depending on the low precipitation in Aqaba, the relative error is found to be quiet high. The variability is again lower and it results in underestimated monthly maxima. It should, however, be noted that the minima at the monthly scale were set as zero in the MLR and almost zero in the ANN. This makes it clear that the models are able to generate summer precipitation correctly regarding the minimum value observed in that particular season. In general, both models produced standard deviations lower than their observed values in the three seasons, i.e. higher minima and lower maxima were generated. As a result of this, much higher minima and much lower maxima were obtained for the seasonal total precipitation forecast by both models in the three stations. The departure between the observed and forecasted precipitation becomes more evident with decreasing precipitation. Although none of the models is considered good in reproducing the variability in the seasonal precipitation, the ANN was seen to approach the observed standard deviation better than MLR. Indeed, as previously used in Unal et al. (2004), in cases where the precipitation is too low, the relative error is too high, which can then become meaningless as a comparison criterion and the absolute error can therefore be accepted instead. Although the relative errors calculated are too high, in general, it was observed that the ANN model produced lower minima than the MLR technique and, therefore, found to approach the observed minimum seasonal precipitation better. For maxima, the same is valid, i.e., the ANN gave higher maxima and approached better than the MLR the observed maximum seasonal precipitation, although both models underestimated. Again, Copyright  2009 Royal Meteorological Society

generally, better results were obtained in the wetter season. 5.

Conclusions and ideas for future studies

1. It is reinforced that intermittent monthly precipitation data of semi-arid to arid regions should be analysed with more care than perennial precipitation time series of the humid regions. 2. Based on the experience obtained from this study and gained from the hydrometeorological use of ANN models, it should also be said that the FFBP ANN model has drawbacks in forecasting the intermittent monthly total precipitation sequence, although it was in general found to be slightly better than the classical linear regression with multiple inputs (MLR). Therefore, in order to improve the results, different types of ANN model, such as the radial basis function (RBF) and the generalized regression neural network (GRNN), should be tried, together with different calibration algorithms, different activation functions and spread coefficients (when available). 3. With a higher variability of precipitation the ANN model performed worse: i.e. increasing variability reduces the success of the model in approaching the observed precipitation sequence. 4. In order to improve the performance of the models, monthly analysis can be performed and monthly models can be proposed, with the cost of increase in the model parameters, weights and biases. 5. If a parsimonious model is desired and the monthly analysis is, therefore, found too costly with respect to model parameters, then the dry and wet periods in the year can be modelled separately. The model can, for instance, simply be separated by the period above and under the long-term average precipitation taken as a threshold under which one model can be constructed and above which another one. This model can be considered a quick solution to the problems faced in this study although more detailed studies can/should be foreseen. This will obviously end with much better forecasts as the effect of minima on maxima and that of maxima on minima will have been minimized, even be removed from the model. 6. The ANN model presented in this study uses the characteristics of the precipitation recorded in previous months. In other words, the constructed models are precipitation-precipitation models. Forecasting precipitation by using previously observed precipitation does not seem to be a well working way instead which forecasting based on other climatological variables, such as temperature, wind speed, relative humidity and pressure among other inputs, can be offered. 7. If the intermittent monthly precipitation record as a whole needs to be analysed, then some help tools should be incorporated into the forecasting models. For instance, in an ongoing study, Markov chains are used together with ANNs. Promising results are expected. Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met

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8. Finally, the local calibration procedure (shifting up the best-fit line as fitted to the scatter diagrams in Figures 2 and 3) can be used in order to arrive at higher maxima and lower minima than those obtained in this study. When the maxima are made higher by local calibration, then the minima will be lower than before, after which negative forecasts might be possible. Such negative forecasts should be eliminated from the generated precipitation time series by the use of Markov chains, which are being employed in an ongoing study for which promising results have already been seen for forecasting precipitation, particularly in the dry months of the year. Acknowledgements This study is based on the first author’s earlier work for his PhD thesis entitled ‘Forecasting monthly precipitation for arid regions using conditional artificial neural networks combined with Markov chain’ submitted to the Institute of Science and Technology, Istanbul Technical University, Istanbul, Turkey. References Aksoy H. 2007. Hydrological variability of the European part of Turkey. Iranian Journal of Science and Technology, Transaction B 31(B2): 225–236. Aksoy H, Dahamsheh A. 2009. Artificial neural network models for forecasting monthly precipitation in Jordan. Stochastic Environmental Research and Risk Assessment in press. DOI:10.1077/S00477008-0267-x. Aksoy H, Guven A, Aytek A, Yuce MI, Unal NE. 2007. Discussion of ‘Generalized regression neural networks for evapotranspiration modelling’. Hydrological Sciences Journal 52(4): 825–828. Aksoy H, Guven A, Aytek A, Yuce MI, Unal NE. 2008a. Comment on ‘Evapotranspiration modelling from climatic data using a neural network computing technique’. Hydrological Processes 22: 2715–2717. Aksoy H, Unal NE, Alexandrov V, Dakova S, Yoon J. 2008b. Hydrometeorological analysis of northwestern Turkey with links to climate change. International Journal of Climatology 28: 1047–1060. Alexandrov V, Genev M, Aksoy H. 2005. The Impact of Climate Variability and Change on Water Resources in the Western Coastal Zone of Black Sea, Vol. 295. IAHS Publications: Wallingford, UK; 62–71. Al-Kharabsheh A. 2000. Ground-water modeling and long-term management of the Azraq basin as an example of arid area conditions (Jordan). Journal of Arid Environments 44(2): 143–153. Al-Weshah RA. 2005. Jordan’s Water Resources: Technical Perspective, http://www.wsta-gcc.org/includes/GWC 4th/doc/w6.htm (last visited on March 16, 2005). Anagnostopoulou C, Maheras P, Karacostas T, Vafiadis M. 2003. Spatial and temporal analysis of dry spells in Greece. Theoretical and Applied Climatology 74(1–2): 77–91. Aytek A, Guven A, Yuce MI, Aksoy H. 2008. An explicit neural network formulation for evapotranspiration. Hydrological Sciences Journal 53(4): 893–904. Ben-Gai T, Bitan A, Manes A, Alpert P. 1993. Long-term changes in October rainfall patterns in southern Israel. Theoretical and Applied Climatology 46(4): 209–217. Ben-Gai T, Bitan A, Manes A, Alpert P. 1994. Long-term changes in annual rainfall patterns in southern Israel. Theoretical and Applied Climatology 49(2): 59–67. Ben-Gai T, Bitan A, Manes A, Alpert P, Rubin S. 1998. Spatial and temporal changes in rainfall frequency distribution patterns in Israel. Theoretical and Applied Climatology 61(3–4): 177–190. Ben-Gai T, Bitan A, Manes A, Alpand P, Rubin S. 1999. Temporal and spatial trends of temperature patterns in Israel. Theoretical and Applied Climatology 64(3–4): 163–177. Copyright  2009 Royal Meteorological Society

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Meteorol. Appl. 16: 325–337 (2009) DOI: 10.1002/met