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sistema de inferencia de redes neuronales artificiales para predicción de clima
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ORIGINAL PAPER
Adaptive Neuro-Fuzzy Inference System for drought forecasting
Ulker Guner Bacanli Mahmut Firat Fatih Dikbas
Published online: 24 October 2008
Springer-Verlag 2008
Abstract Drought causes huge losses in agriculture and
has many negative influences on natural ecosystems. In this
study, the applicability of Adaptive Neuro-Fuzzy Inference
System (ANFIS) for drought forecasting and quantitative
value of drought indices, the Standardized Precipitation
Index (SPI), is investigated. For this aim, 10 rainfall
gauging stations located in Central Anatolia, Turkey are
selected as study area. Monthly mean rainfall and SPI
values are used for constructing the ANFIS forecasting
models. For all stations, data sets include a total of 516 data
records measured between in 1964 and 2006 years and data
sets are divided into two subsets, training and testing.
Different ANFIS forecasting models for SPI at time scales
112 months were trained and tested. The results of ANFIS
forecasting models and observed values are compared and
performances of models were evaluated. Moreover, the
best fit models have been also trained and tested by Feed
Forward Neural Networks (FFNN). The results demon-
strate that ANFIS can be successfully applied and provide
high accuracy and reliability for drought forecasting.
Keywords Drought forecasting ANFIS Droughtindices Central Anatolia Turkey
1 Introduction
Drought is a threatening global and local problem that has
many damages in various ways. It causes huge losses in
agriculture and has many negative influences on natural
ecosystems. Drought causes degradation of soils and
desertification (Nicholson et al. 1990; Pickup 1998), social
alarm and famine and impoverishment. Studies on climate
change also drew attention to drought in recent years (Byun
and Wilhite 1999) and many studies were made to analyze
the spatial patterns of drought risk in order to assist agri-
cultural or environmental management (Dracup et al.
1980). The development of drought monitoring plans is
a priority in many of these studies (Wilhite 1997; Hayes
et al. 1999) and drought prediction is the subject of some
other studies that investigate the atmospheric causes of
droughts. Drought risk analysis aiming at improving tech-
niques for drought prediction and management are based
on the spatial variation of drought and are mainly focused
on the magnitude, duration, intensity and spatial extent of
droughts. Currently, indirect characteristic features of soil
moisture time series namely drought indices are widely
used. Spatial and temporal extent and severity of drought
can be determined by the help of these indices (Palmer
1995; McKee et al. 1993; Edwards and Mckee 1997; Hayes
1997; Guttmann 1998; Hayes 2000). The Standardized
Precipitation Index (SPI), developed by McKee et al.
(1993), is an effective drought index which has several
advantages over the others. Calculation of the SPI is easier
than the more complex indices such as the Palmer Drought
Severity Index (PDSI; Palmer 1965), because the SPI
requires only precipitation data, whereas the PDSI uses
several parameters. The SPI is comparable in both time and
space and it can be calculated for several time scales
(Srdas and Sen 2003; McKee et al. 1995) and it allows thedetermination of duration, magnitude and intensity of
droughts. The SPI identifies various drought types as
hydrological, agricultural or environmental and it has been
extensively used for drought analysis of many areas of the
world. Several studies focused on the SPIs calculation
U. G. Bacanli M. Firat (&) F. DikbasCivil Engineering Department, Faculty of Engineering,
Pamukkale University, 20017 Denizli, Turkey
e-mail: [email protected]
123
Stoch Environ Res Risk Assess (2009) 23:11431154
DOI 10.1007/s00477-008-0288-5
procedures, which identify the most appropriate frequency
distributions (Guttmann 1998), the effect of time scales on
the parameters (Ntale and Gan 2003), and spatial and
temporal comparability (Keyantash and Dracup 2002).
However, the SPIs spatial stability and coherence in
relation to time scales have not been analysed. Mishra et al.
(2008) investigated the distribution of drought interval
time, mean drought interarrival time, joint probability
density function and transition probabilities of drought
events using the alternative renewable process and run
theory in the Kansabati River basin in India. For this aim,
the Standardized Precipitation Index (SPI) series were
employed and the time interval of SPI was found to have a
significant effect of the probabilistic characteristics of
drought. Mishra and Desai (2005) used the linear stochastic
models known as ARIMA and multiplicative Seasonal
Autoregressive Integrated Moving Average (SARIMA)
models to forecast droughts based on the procedure of
model development. The models were applied to forecast
droughts using standardized precipitation index (SPI) series
in the Kansabati river basin in India. Cancelliere et al.
(2007) proposed two methodologies for the seasonal fore-
casting of SPI, under the hypothesis of uncorrelated and
normally distributed monthly precipitation aggregated at
various time scales. In the first methodology, the auto-
covariance matrix of SPI values was analytically derived,
as a function of the statistics of the underlying monthly
precipitation process. In the second methodology, SPI
forecasts at a generic time horizon M were analytically
determined, in terms of conditional expectation, as a
function of past values of monthly precipitation. The
results showed that the proposed methodologies can be
applied for drought monitoring system. Hughes and
Saunders (2002) used monthly SPIs at time scales of 3, 6,
9, 12, 18, and 24 months for characterizing the drought
climatology of Europe. Bonaccorso et al. (2003) used the
SPI for drought analysis in Italy and Loukas et al. (2004)
applied the SPI for drought forecasting in Greece. Vicente-
Serrano and Lopez-Moreno (2005) analyzed the usefulness
of different SPI time scales to monitor droughts in river
discharges and reservoir storages. The objective was to
determine the most adequate time scales of SPI to monitor
droughts in two basic water usable sources: river dis-
charges and reservoir storages. They found that Time
scales of SPI longer than 12 months do not seem useful to
monitor any drought type in their study areas. Moreira et al.
(2006) analyzed the SPI with the 12-month time scale
through adjusting loglinear models to the probabilities of
transitions between the SPI drought classes.
The new techniques such as artificial neural networks
(ANN), Fuzzy Logic (FL) and ANFIS have been recently
accepted as an efficient alternative tool for modeling
of complex hydrologic systems and widely used for
forecasting. Some specific applications of ANN to
hydrology include modeling rainfall-runoff process (Jeong
and Kim 2005; Kumar et al. 2005; Rajurkar et al. 2004),
hydrologic time series modeling (Jain and Kumar 2007),
sediment concentration estimation (Nagy et al. 2002), esti-
mation of heterogeneous aquifer parameters (Mantoglou
2003), runoff and sediment yield modeling (Agarwal et al.
2006). Morid et al. (2007) examined the utility of ANN
approach for medium and long-term forecasting of both
the likelihood of drought events and their severity. Mishra
and Desai (2006) applied the feed-forward recursive
neural network and ARIMA models for drought fore-
casting using standardized precipitation index (SPI) series
as drought index. The results have demonstrated that
neural network method can be successfully applied for
drought forecasting. Wu et al (2008) applied the neural
network method to establish a risk evaluation model of
heavy snow disaster using back-propagation artificial
neural network (BP-ANN). According to results, BP-ANN
model showed an advantage in heavy snow risk evalua-
tion in Xilingol compared to the conventional method.
Moreover, ASCE Task Committee reports (2000) did a
comprehensive review of the applications of ANN in the
hydrological forecasting context. On the other hand,
several studies have also been carried out using FL in
hydrology and water resources planning (Mahabir et al.
2000; Liong et al. 2000; Nayak et al. 2005; Altunkaynak
et al. 2005). In recent years, Adaptive Neuro-Fuzzy
Inference System (ANFIS), which is integration of ANN
and FL methods, has been used in the modeling of non-
linear engineering and water resources problems (Chang
and Chang 2006; Nayak et al. 2004; Sen and Altunkaynak
2006; Firat 2007; Firat and Gungor 2007, 2008). More-
over, Chou and Chen (2007) have used the neuro fuzzy
computing technique for the development of drought early
warning index. For this aim, an approach has been pro-
posed to develop drought early warning index (DEWI) for
southern Taiwan to detect the drought in advance for
setting up proper plans to mitigate the water shortage
impact.
Drought forecasting plays an important role in the
mitigation of impacts of drought on water resources systems.
Because SPI is one of the most widely used methods
related to drought, accurate and reliable estimation of SPI
is very important. Traditional methods like regression
analysis and autoregressive moving average models
are commonly used in the estimation of hydrological
processes. Moreover FL and ANN methods offer real
advantages over conventional modeling especially when
the underlying physical relationships are not fully under-
stood. FL is employed to describe human thinking and
reasoning in a mathematical framework. The main problem
with FL is that there is no systematic procedure to define
1144 Stoch Environ Res Risk Assess (2009) 23:11431154
123
the MF parameters and to design of fuzzy rules. The con-
struction of the fuzzy rule necessitates the definition of
premises and consequences as fuzzy sets.
In this paper, Adaptive Neuro-Fuzzy Inference System
(ANFIS), which is an integration of ANN and FL methods,
is proposed as an alternative to the traditional methods for
drought forecasting using SPI for multiple time scales. The
main contribution of ANFIS method is that it eliminates the
basic problems in fuzzy modeling (defining the member-
ship function parameters and design of fuzzy ifthen rules)
by using the learning capability of ANN for automatic
fuzzy rule generation and parameter optimization. To
illustrate the applicability of ANFIS method in drought
forecasting, 10 rainfall gauging stations located in Central
Anatolia, Turkey are selected as study area. Monthly mean
precipitation and SPI values are used for constructing the
ANFIS forecasting models. The best fit forecasting model
structure was determined by comparing the forecasted and
observed values.
2 Standard Precipitation Index (SPI)
Standard Precipitation Index calculation is based on long-
term precipitation data. SPI is obtained by dividing the
difference between precipitation and mean to standard
deviation in a specific duration (McKee et al 1993). SPI is a
dimensionless index that takes negative values in drought
periods and positive values in wet periods. The magnitude,
length and duration of drought can be calculated with SPI.
The calculation of SPI is complex because the precipitation
does not fit normal distribution for the periods of
12 months and less and for this reason the precipitation
series are fitted to normal distribution
SPI xi xir
: 1
SPI permits to determine the rarity of a drought or an
anomalously wet event at a particular time scale for any
location that has a precipitation record. A drought event is
considered to occur at a time when the value of SPI is
continuously negative and end when SPI becomes positive
(Mishra et al. 2008). The classes according to the SPI index
are given in the Table 1.
The following steps are applied in the SPI method:
1. Monthly precipitation data sets are organized for a
period of at least 30 years. Different time steps are
determined like 3, 6, 9, 12, 24 or 48 months to monitor
the variations of the indices by considering the
influence of precipitation deficit on various resources.
The time steps may vary according to the condition of
water resources in the area. In the proposed study,
estimation models were constructed with ANFIS
method by using the SPI outputs for 1, 3, 6, 9 and
12 months.
2. Then Gamma distribution is fitted to the data set and
thus the observed precipitation probabilities are
defined. Gamma distribution is the best fitting distri-
bution to the climatologic time series. Gamma
distribution is defined by either the frequency distri-
bution or the probability density function
g x 1baC a x
a1ex=b for x [ 0: 2
a([0) is the shape parameter; b([0) is the scale parameter;x([0) is the precipitation amount, and C (a) is the Gammafunction. In the calculation of a and b, maximumprobability solutions are used. According to this:
a 14A
1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 4A3
r
!
3
b xa
4
A ln x P
ln x n
5
3. These probability definitions obtained from the present
data may later be used to determine the cumulative
probability of a value observed at any month. In this
situation, the cumulative probability distribution
function is defined as follows:
G x Z
x
0
g x dx 1baC a
Z
x
0
xa1ex=bdx 6
4. Gamma function is undefined for x = 0 and
precipitation distribution can have zero values. When
this is the case, the cumulative probability distribution
is defined as follows:
H x q 1 q G x 7
In the equation above, n is the number of precipitation
observations, q represents the probability for zero value. If
m is used for denoting the zero values in a precipitation
series then the following definition can be made: q = m/n.
Table 1 Classification according to the SPI values
SPI Drought category
2[ Extremely wet1.991.5 Very wet
1.491.0 Moderately wet
0.99(-0.99) Near normal
(1.0)(-1.49) Moderately dry
(1.5)(-1.99) Severely dry
2\ Extremely dry
Stoch Environ Res Risk Assess (2009) 23:11431154 1145
123
5. The cumulative probability value H(x) is converted to
Z variable with a standard normal random value
denoting the SPI value having zero mean value and
variance equal to 1. H(x) is the value of SPI.
Normalization of SPI values enables the consideration
of the variations of precipitation series of that station
by both time and place (McKee et al. 1993; Guttmann
1998).
3 Adaptive Neuro Fuzzy Inference System (ANFIS)
The FL approach proposed by Zadeh (1965) is based on the
linguistic uncertainly expression rather than numerical
uncertainty. FL approach has become popular and has
been successfully used in various engineering problems
(Mahabir et al. 2000; Liong et al. 2000; Nayak et al. 2005;
Sen 2001). Fuzzy inference system (FIS) is a rule based
system consisting of three conceptual components. These
are: (1) a rule-base, containing fuzzy if-then rules, (2) a
data-base, defining the Membership Function (MF) and (3)
an inference system, combining the fuzzy rules and pro-
duces the system results (Firat and Gungor 2007, 2008; Sen
2001). The main problem with fuzzy logic is that there is
no systematic procedure to define the membership function
parameters and to design of fuzzy rules. In recent years,
ANFIS method, which is integration of ANN and FL
methods, has the potential to capture the benefits of both
these methods in a single framework. ANFIS eliminates the
basic problem in fuzzy system design (defining the mem-
bership function parameters and design of fuzzy ifthen
rules) by effectively using the learning capability of ANN
for automatic fuzzy rule generation and parameter opti-
mization. There are two types of FISs, Sugeno-Takagi FIS
and Mamdani FIS, in literature. In this study, Sugeno-
Takagi FIS is used for drought forecasting. The most
important difference between these systems is definition of
the consequent parameter. The consequence parameter in
Sugeno FIS is either a linear equation, called first-order
Sugeno FIS, or constant coefficient, zero-order Sugeno FIS
(Jang et al. 1997). It is assumed that the FIS includes two
inputs, SPI(t - 1) and P(t - 1) and one output, SPI(t). The
membership functions and the structure of are shown in
Fig. 1. For the first-order Sugeno-Takagi FIS, typical two
rules can be expressed as:
Rule 1: IF SPIt 1 is A1 and Pt 1 is B1THEN f1 p1 SPIt 1 q1 Pt 1 r1
Rule 2: IF SPIt 1 is A2 and Pt 1 is B2THEN f2 p2 SPIt 1 q2 Pt 1 r21Input notes (Layer 1) Each node in this layer generates
membership grades of the crisp inputs and each nodes
output O1i is calculated by:
O1i lAi SPIt 1 for i 1; 2;O1i lBi2Pt 1 for i 3; 4
8
where, SPI(t - 2) is the SPI value at time (t - 2) to the
node i, P(t - 1) is the actual precipitation at time (t - 1),
A1 B1 B1Membership Degree
B2
P (t-1) Rule Base and Inference System
Membership Degree
A2
SPI (t-2)
1111 )1(*)1(* rtPqtSPIpf ++=
2222 )1(*)1(* rtPqtSPIpf ++=
P (t-1)
SPI (t-2)
)1(),1((1 tPtSPIw
N
N
)1(),1((2 tPtSPIw
11 fw
22 fw
Layer 1 Layer 2 Layer 3 Layer 4 Layer 5
21
11 ww
ww +=
21
22 ww
ww
+=
1A
2A
1B
2B
)1(),1((1 tPtSPIf
)1(),1((1 tPtSPIf
)1(),1((2 tPtSPIf
Fig. 1 The scheme of AdaptiveNeuro-Fuzzy Inference System
1146 Stoch Environ Res Risk Assess (2009) 23:11431154
123
SPI(t) is the SPI value at time (t) to the node i, Ai and Bi are
the linguistic labels, pi, qi and ri are the consequence
parameters, lAi and lBi are the MFs for Ai and Bi linguisticlabels, respectively and in this study, the Gauss MF is used,
as
O1i lAix eSPIt1c2
2r2 : 9
Rule nodes (Layer 2) The outputs of this layer, called
firing strengths O2i , are the products of the corresponding
degrees obtained from layer 1, named as w as follows:
O2i wi lAi SPIt 1lBi Pt 1; i 1; 2 10
Average nodes (Layer 3) Main target is to compute the
ratio of firing strength of each ith rule to the sum firing
strength of all rules. The firing strength in this layer is
normalized as
O3i wi wiP
i wii 1; 2 11
Consequent nodes (Layer 4) The contribution of ith rule
towards the total output or the model output and/or the
function defined is calculated by Eq. (12)
O4i wifi wipi SPIt 1 qiPt 1 ri i 1; 212
Output nodes (Layer 5) This layer is called as the output
nodes in which the single node computes the overall output
by summing all incoming signals
Q5i f SPIt 1;Pt 1 X
i
wi fi wif1 wif2
P
i wifiP
i wi14
where wi is the ith node output from the previous layer as
demonstrated in the third layer. ANFIS applies the hybrid-
learning algorithm, which consists of the combination of
the gradient descent and the least-squares methods to
determine the input and output model parameters. The task
of the learning algorithm is to tune all the antecedent and
consequence parameters to make the ANFIS response
match the training data. The gradient descent method is
used to assign the nonlinear antecedent parameters and the
least-squares method is employed to identify the linear
consequent parameters. All these parameters are updated
using this hybrid learning algorithm until acceptable error
is reached. The details and mathematical background of
these algorithms can be found in Jang et al. (1997) and in
Nayak et al. (2004).
4 Study area and data
The temperature difference between summer and winter is
high, the precipitation generally occurs in spring and winter
and dry periods dominate summers. This climate is experi-
enced in Central, East, Southeast Anatolia and Trakya
region. Climate of Central Anatolia has the following
properties: The weather in the summer is a little hot and
winters are cold. The severity of cold weather increases
towards the east parts of Central Anatolia. Natural flora
consists of steppes in the lower regions and dry forests in the
higher regions because of summer droughts. Mean temper-
ature of January, the coldest month, is 0.7C and it is 22C inJuly, the hottest month. Annual mean temperature is 10.8C.Mean annual precipitation is 413.8 mm and most of the
precipitation occurs in winter and spring seasons. The per-
cent of summer rains among the annual total is 14.7%. The
annual mean proportional moisture in the region is 63.7%.
Observed monthly rainfall data records from ten meteoro-
logical stations (Aksaray, Ankara, Cankr, Eskisehir,Karaman, Kayseri, Konya, Krsehir, Nevsehir and Yozgat)located in Central Anatolia, Turkey, have been selected for
this study. The length of available records at these stations is
between 1964 and 2006. The SPI for this study have been
calculated on the basis of these rainfall data.
5 Drought forecasting by ANFIS
5.1 Input variables
Different time steps like 3, 6, 12, 24 and 48 months are
determined as (1) for monitoring the variations in the
indexes by considering the effect of precipitation lack on
different water resources. In this study, the values of SPI
and precipitation in the previous months are used for
generating a drought estimation model with ANFIS
f x; y w1f1 w2f2w1 w2
w1 SPIt 1;Pt 1f1SPIt 1;Pt 1 w2SPIt 1;Pt 1f2SPIt 1;Pt 1w1SPIt 1;Pt 1 w2SPIt 1;Pt 1
13
Stoch Environ Res Risk Assess (2009) 23:11431154 1147
123
method. For this, the SPI outputs for 1, 3, 6, 9 and
12 months were considered. In the construction of esti-
mation models, again, different models were generated for
each of the SPI output for 1, 3, 6, 9 and 12 months. The
data sets for all stations were divided into two subsets,
training and testing data set. The training data set includes
data records measured between 1964 and 1986 years. In
order to get more reliable evaluation and comparison,
models are tested by evaluating a data set which was not
used during the training process. Testing data set consists
of data records observed between 1987 and 2006 years.
The statistical parameters, minimum value, maximum
value, mean, standard deviation, variance, skewness coef-
ficient and Kurtosis for training and testing data sets are
calculated and given in Tables 2 and 3 to see a comparison
of the training and testing data sets.
5.2 Model structures
One of the most important steps in developing a satisfac-
tory forecasting model is the selection of the input
variables. Because, these variables determine the structure
of forecasting model and affect the weighted coefficient
and the results of the model. Here, different estimation
models were constructed for each phase. The models for 1,
3, 6, 9 and 12 months were named as SPI-1, SPI-3, SPI-6,
SPI-9 and SPI-12, respectively. Here, SPI-1, SPI-3 and
SPI-6 were considered as the index for short term or sea-
sonal variation, SPI-9 for short term drought and SPI-12
was considered as the drought index for long term. 20
models with different input numbers and structures were
constructed for each phase by using these variables. In this
study, forecasting models based on various combinations
of antecedent values of actual precipitations and SPI values
were constructed (Table 4). In each model every input
variable must be clustered into several class values in layer
1 to build up fuzzy rules. And each fuzzy rule would be
constructed through several parameters of membership
function in layer 2. As the number of parameters increases
with the fuzzy rule increment, the model structure becomes
more complicated. In this study, the subtractive fuzzy
clustering function was used to establish the fuzzy rule
based on the relationship between the inputoutput vari-
ables. In order to determine the nonlinear input and linear
output parameters, the hybrid algorithm was used. The
Table 2 The statistical parameters for training data sets (19641986)
Min. Max. Mean SD Variance Skewness Kurtosis
Aksaray 0.0 110.1 28.96 23.17 536.98 0.775 0.142
Ankara 0.0 121.5 34.86 26.08 680.54 0.799 -0.005
Cankr 0.0 137.7 34.62 26.64 709.96 1.012 0.856
Eskisehir 0.0 128.2 34.07 25.92 671.92 0.954 0.949
Karaman 0.0 144.1 29.15 26.88 722.91 1.182 1.469
Kayseri 0.0 133.2 30.50 23.70 562.01 0.927 0.869
Konya 0.0 112.2 28.74 23.47 551.11 0.876 0.433
Krsehir 0.0 145.8 31.84 26.49 702.12 0.820 0.430
Nevsehir 0.0 116.7 34.29 26.56 705.89 0.668 -0.107
Yozgat 0.0 192.3 48.79 38.68 1496.22 0.876 0.475
Table 3 The statistical parameters for testing data sets (19872006)
Min. Max. Mean SD Variance Skewness Kurtosis
Aksaray 0.0 101.3 28.47 23.21 539.14 0.813 0.100
Ankara 0.0 122.4 32.33 25.92 672.28 0.953 0.630
Cankr 0.0 149.8 32.76 27.46 754.28 1.381 2.175
Eskisehir 0.0 129.7 29.16 22.36 500.28 1.329 2.823
Karaman 0.0 121.8 26.27 22.60 510.87 0.956 0.802
Kayseri 0.0 164.7 33.62 27.56 760.07 1.078 1.663
Konya 0.0 124.0 24.94 23.87 570.18 1.540 2.909
Krsehir 0.0 121.0 31.39 25.42 646.24 0.916 0.764
Nevsehir 0.0 148.8 33.91 28.14 792.11 1.053 1.318
Yozgat 0.0 172.0 50.01 37.61 1414.92 0.819 0.245
Table 4 The structures of forecasting models
Model Input structure Output
M1 SPI(t - 1) SPI(t)
M2 SPI(t - 1), SPI(t - 2) SPI(t)
M3 SPI(t - 1), SPI(t - 2), SPI(t - 3) SPI(t)
M4 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4) SPI(t)
M5 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4),SPI(t - 5)
SPI(t)
M6 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4),SPI(t - 5), SPI(t - 6)
SPI(t)
M7 R(t - 1) SPI(t)
M8 R(t - 1), R(t - 2) SPI(t)
M9 R(t - 1), R(t - 2), R(t - 3) SPI(t)
M10 R(t - 1), R(t - 2), R(t - 3), R(t - 4) SPI(t)
M11 R(t - 1), R(t - 2), R(t - 3), R(t - 4), R(t - 5) SPI(t)
M12 R(t - 1), R(t - 2), R(t - 3), R(t - 4), R(t - 5),R(t - 6)
SPI(t)
M13 SPI(t - 1) R(t - 1) SPI(t)
M14 SPI(t - 1), SPI(t - 2) R(t - 1) SPI(t)
M15 SPI(t - 1), SPI(t - 2) R(t - 1), R(t - 2) SPI(t)
M16 SPI(t - 1), SPI(t - 2), SPI(t - 3) R(t - 1) SPI(t)
M17 SPI(t - 1), SPI(t - 2), SPI(t - 3) R(t - 1),R(t - 2)
SPI(t)
M18 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4)R(t - 1)
SPI(t)
M19 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4)R(t - 1), R(t - 2)
SPI(t)
M20 SPI(t - 1), SPI(t - 2), SPI(t - 3), SPI(t - 4),SPI(t - 5) R(t - 1)
SPI(t)
1148 Stoch Environ Res Risk Assess (2009) 23:11431154
123
learning procedure and the construction of the rules were
provided by this algorithm. The performance of ANFIS
models for training and testing data sets were evaluated
according to statistical criteria such as, Correlation Coef-
ficient (CORR), Efficiency (E), and Root Mean Square
Error (RMSE). The CORR is a commonly used statistic
and provides information on the strength of linear rela-
tionship between the observed and the computed values.
The E is one of the widely employed statistics to evaluate
model performance. The values of CORR and E close to
1.0 indicate good model performance. The RMSE statistic
indicates a models ability to predict a value away from the
mean.
As it is impossible to show the model results for each
phase having 20 models because of space restrictions, only
the results for SPI-6 at Ankara station (Ankara is the capital
of Turkey and it is one of the cities where water shortage
and drought is severely experienced) are presented. The
testing performances of ANFIS models for SPI-6 are given
in Fig. 2.
When the results of the ANFIS models are compared, it
is seen that the performances of models composed of
precipitation values belonging to the previous time step are
lower than the performances of the other models. The
results of models in which SPI is used, show that the
performances of the models at all stations are close to
each other and that the model defined as M5 has a better
performance than the others. A general decrease in per-
formance was observed in all models when the values at
(t - 6) time step were used. When the results of the models
consisting of only the precipitation values are evaluated, it
is seen that M11 is the model with the best performance for
all stations. It was also observed that the model (M7)
composed of precipitation values at the (t - 1) time step
has the lowest performance. The figure shows that the
model (M12) generated by using the values at (t - 6) time
step generally have lower performances. On the other side,
the investigation of the results given in the graphs for SPI-6
show that the models generated with the previous values of
SPI and precipitation data have a better performance. By
using the precipitation and SPI variables together for all
stations, an improvement has been achieved in the model
performances. According to the criteria, the model defined
as M20 ANFIS for Aksaray, Ankara, Karaman, Kayseri,
Krsehir, Konya and Yozgat stations, had the best resultsover the other models. On the other hand, while the best
results are obtained from the M5 model for Eskisehir and
Cankr stations, it was determined that M14 ANFIS modelhad the best performance for Nevsehir station. As a result,
the performances of the best fit ANFIS models for SPI-6 at
all stations (after the analysis of all stations, only perfor-
mances of the models giving the most suitable results are
presented.) are shown in Table 5.
It can be stated that the model performances of ANFIS
models for all stations are at an acceptable level for SPI-6.
Figure 3 shows the performances of ANFIS models at
Ankara station for the data from 1 month to 12 months
(SPI-1 to SPI-12). In this figure, the variations of CORR, E
and RMSE criteria for SPI-1 to SPI-12 at Ankara station
during the testing period are demonstrated.
The results of other stations that are not presented here
due to space restrictions indicate that the ANFIS models
for SPI-12 have shown the best performance at all sta-
tions. It is seen that the performances of ANFIS model for
SPI-12 at Ankara station is better than those of other
models. The values of CORR and E of ANFIS models for
SPI-1 are lower than those of other models. The reason
for the ANFIS models developed by using SPI outputs of
12 months to show a better performance is that the SPI
Ankara Station
0.21 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1716 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1716 18 19 20
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Model
CO
RR
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
E
CORRE
Ankara Station
0.5
0.6
0.7
0.8
0.9
1.0
1.1
Model
RM
SE
RMSE
Fig. 2 Comparison ofperformances of ANFIS Models
for SPI-6 at Ankara station
Table 5 The performances of the best fit models for SPI-6 at allstations
Station Testing set Training set
CORR E RMSE CORR E RMSE
Aksaray (M20) 0.837 0.686 0.628 0.810 0.656 0.514
Ankara (M20) 0.824 0.685 0.549 0.876 0.754 0.526
Cankr (M5) 0.773 0.599 0.644 0.870 0.767 0.508
Eskisehir(M5) 0.825 0.710 0.547 0.879 0.781 0.471
Karaman (M20) 0.826 0.714 0.578 0.860 0.741 0.521
Kayseri (M20) 0.846 0.712 0.601 0.855 0.731 0.462
Krsehir (M20) 0.804 0.642 0.615 0.828 0.694 0.541
Konya (M20) 0.815 0.68 0.603 0.872 0.761 0.443
Nevsehir (M14) 0.841 0.701 0.608 0.810 0.710 0.470
Yozgat (M20) 0.818 0.667 0.578 0.821 0.704 0.549
Stoch Environ Res Risk Assess (2009) 23:11431154 1149
123
values calculated for a long term include dry and wet
periods for longer duration. Short term periods like 1 or
3 months may include a wet or a dry period for a short
time. For example, in 3 months period, drought occurs
more frequently and for a shorter time and when the
period increases the duration of drought increases but its
frequency decreases. This means that for shorter periods
the SPI values may contain 1 month dry and 1 month wet
period and this causes instability. Passages between
positive and negative values occur more frequently and
this also results with instability. For this reason, the
ANFIS estimation models constructed with the SPI values
calculated for shorter periods, cannot catch dry and wet
periods and give unsuccessful results. Besides, the SPI
outputs for 12 months have a more stable run. Thus, the
ANFIS models developed by using SPI outputs for
12 months can catch dry and wet periods and give better
results. Figure 4 shows the results of ANFIS models for
Ankara station from SPI-1 to SPI-12.
In order to evaluate the results of ANFIS models, the
best fit models for Ankara station (SPI-1 to SPI-12) have
also been tested by Feed Forward Neural Networks
(FFNN) and Multiple Linear Regression (MLR). The
FFNN models have been trained and tested using the same
data sets. The error back propagation algorithm and tangent
activation function is used for training/testing of the FFNN
models. The number of hidden layers and the hidden
neurons in this layer, the learning rate, the coefficient of
momentum and epochs were selected by trial and error
method during the training. The results of FFNN and MLR
models for SPI-12 at Ankara station are shown in Table 6
and Fig. 5.
Comparing performances of ANFIS models for Ankara
station, it is seen that the performance of the ANFIS model
for SPI-12 are better than other ANFIS models for SPI-1 to
SPI-9. As a result, it is said that ANFIS can be successfully
applied and provide high accuracy and reliability for
drought forecasting. On the other hand, comparing the
results of ANFIS and FFNN forecasting models for Ankara
station (SPI-1 to SPI-12), it can be seen that the RMSE
values of the ANFIS models are lower than that of FFNN
model. In addition, the values of E and CORR of the
ANFIS model are also higher than those of FFNN models.
The results suggest that the ANFIS method is superior to
the FFNN method in the forecasting of drought. It may be
noted that a trial and error procedure has to be performed
for FFNN models to develop the best network structure
while such a procedure is not required in developing an
ANFIS model. Figure and table indicate that the best result
was obtained from the models developed for SPI-12 as
in the ANFIS method. Comparing the performances of
ANFIS and MLR models, it can be seen that the values of
E and CORR of the ANFIS model are also higher than
those of MLR models. The NRMSE values of ANFIS
model are also lower than those of MLR models. The
results suggest that the ANFIS method is also superior to
the MLR method in the drought forecasting. The results
show that ANFIS method can be successfully applied to
establish accurate and reliable drought forecasting models.
6 Conclusions
SPI is one of the most widely used methods related to
drought and SPI should be estimated accurately and reli-
ably. Traditional methods like regression analysis and
autoregressive moving average models are commonly used
in the estimation of hydrological processes.
In this paper, Adaptive Neuro-Fuzzy Inference System
(ANFIS) was proposed as an alternative drought forecast-
ing tool to the traditional methods. The main contribution
of ANFIS method is that it eliminates the basic problems
in fuzzy modeling (defining the membership function
parameters and design of fuzzy ifthen rules) by using
the learning capability of ANN for automatic fuzzy rule
generation and parameter optimization.
To illustrate the applicability of ANFIS method in
drought forecasting, 10 rainfall gauging stations located in
Central Anatolia, Turkey were selected as study area.
Different ANFIS forecasting models for SPI-1, SPI-3,
SP-6, SPI-9 and SPI-12 were trained and tested. When the
results of the ANFIS models are compared, it is seen that
only the performances of models composed of precipita-
tion values belonging to the previous time step are lower
Ankara Station
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 3 6 9 12
Month
CO
RR
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
E
CORRE
Ankara Station
0.2
0.4
0.6
0.8
1
1.2
1 3 6 9 12
Month
RM
SE
RMSE
Fig. 3 The performances ofANFIS models for SPI-1, SPI-3,
SPI-6, SPI-9 and SPI-12 at
Ankara station
1150 Stoch Environ Res Risk Assess (2009) 23:11431154
123
Ankara (SPI-1)
-6.0
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
MonthSP
I
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Ankara (SPI-3)
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
Month
SPI
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Ankara (SPI-6)
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
Month
SPI
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Ankara (SPI-9)
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
Month
SPI
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Ankara (SPI-12)
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
Month
SPI
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Fig. 4 The results of ANFISmodels for for Ankara station
(SPI-1 to SPI-12)
Stoch Environ Res Risk Assess (2009) 23:11431154 1151
123
than the performances of the other models. The results of
models in which only the SPI is used, show that the
model named M5 has a better performance than the other
models. It was also observed that when the SPI value at
(t - 6) time step is used, there is a decrease in perfor-
mance for all stations generally. On the other hand, when
the results of models containing only the precipitation
values were investigated, it was found that M11 has
shown the best performance for all stations. The model
defined as M7 composed of the precipitation value at time
step (t - 1) had the lowest performance. The results
indicate that the models (M12) generated by using the
precipitation values at time step (t - 6) generally have a
lower performance. By using the precipitation and SPI
variables together for all stations, an improvement was
achieved in the model performances. According to the
Table 6 Comparison of performances of ANFIS, FFNN and MLR models for Ankara station
Station Testing set Training set
CORR E RMSE CORR E RMSE
M20 ANFIS (for SPI-1) 0.392 0.371 1.016 0.573 0.502 0.968
M20 ANFIS (for SPI-3) 0.490 0.422 0.707 0.584 0.569 0.654
M20 ANFIS (for SPI-6) 0.824 0.685 0.549 0.876 0.754 0.526
M20 ANFIS (for SPI-9) 0.851 0.733 0.507 0.920 0.847 0.401
M20 ANFIS (for SPI-12) 0.893 0.808 0.425 0.930 0.865 0.375
M20 FFNN (for SPI-1) 0.314 0.298 1.254 0.487 0.451 1.026
M20 FFNN (for SPI-3) 0.417 0.402 0.916 0.561 0.524 0.845
M20 FFNN (for SPI-6) 0.752 0.625 0.652 0.833 0.694 0.575
M20 FFNN (for SPI-9) 0.813 0.674 0.577 0.858 0.738 0.525
M20 FFNN (for SPI-12) 0.851 0.722 0.512 0.887 0.794 0.453
M20 MLR (for SPI-1) 0.306 0.257 1.291 0.380 0.302 1.354
M20 MLR (for SPI-3) 0.411 0.398 1.096 0.576 0.528 0.885
M20 MLR (for SPI-6) 0.719 0.584 0.669 0.829 0.687 0.582
M20 MLR (for SPI-9) 0.804 0.600 0.621 0.894 0.799 0.462
M20 MLR (for SPI-12) 0.811 0.593 0.619 0.904 0.822 0.435
Ankara FFNN Model (SPI-12)
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
Month
SPI
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Ankara MLR Model (SPI-12)
-4.0
-2.0
0.0
2.0
4.0
1 14 27 40 53 66 79 92 105 118 131 144 157 170 183 196 209 222 235
Month
SPI
ForecastedObserved
-4
-2
0
2
4
-4 -2 0 2 4
Observed
For
ecas
ted
Fig. 5 The results of FFNN andMLR models for SPI-12 at
Ankara station
1152 Stoch Environ Res Risk Assess (2009) 23:11431154
123
criteria given in the figures, the model defined as M20
ANFIS model, which consists of the combination of the
antecedent values of the rainfall and SPI variables, for
Aksaray, Ankara, Karaman, Kayseri, Krsehir, Konya andYozgat stations, had the best results over the other
models. Moreover, while the best results are obtained
from the M5 ANFIS model, which includes the anteced-
ent values of SPI variable, for Eskisehir and Cankrstations, it was determined, that M14 ANFIS model had
the best performance for Nevsehir station. Comparing the
performances of ANFIS models for SPI-1, SPI-3, SP-6,
SPI-9 and SPI-12 at 10 stations during the testing period,
it was seen that the performances of the models for SPI-
12 at all stations are better than those of other models.
The reason for the ANFIS models to show a better per-
formance is that the SPI values calculated for long
periods contain longer periods of dry and wet periods.
This means that for shorter periods the SPI values may
contain 1-month dry and 1-month wet period and this
causes instability. Passages between positive and negative
values occur more frequently and this also results with
instability. For this reason, the ANFIS estimation models
constructed with the SPI values calculated for shorter
periods, cannot catch dry and wet periods and give
unsuccessful results. Besides, the SPI outputs for
12 months have a more stable run. Thus, the ANFIS
models developed by using SPI outputs for 12 months can
catch dry and wet periods and give better results. In order
to evaluate the results of ANFIS models, the best fit
models for Ankara station (SPI-1, SPI-3, SP-6, SPI-9 and
SPI-12) have also been trained and tested by FFNN
method. The FFNN models have been trained and tested
using the same data sets. Comparing the results of ANFIS
and FFNN forecasting models for Ankara station, it can
be seen that the RMSE values of the ANFIS models were
lower than that of FFNN model. In addition, the values of
E and CORR of the ANFIS model were also higher than
those of FFNN models. To get more reliable evaluation of
performance of ANFIS model, the best fit models for
Ankara station were compared to MLR model. It can be
seen that the NRMSE value of ANFIS models were lower
than those of MLR models. The values of E and CORR
of ANFIS models were also higher than those of MLR
models.
The results suggest that the ANFIS method is superior to
the FFNN and MLR methods in the forecasting of drought.
Moreover, the result showed that ANFIS method can be
successfully applied to establish accurate and reliable
drought forecasting models.
Acknowledgments The authors are grateful for editors and anon-ymous reviewers for their helpful and constructive comments on an
earlier draft of this paper.
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Adaptive Neuro-Fuzzy Inference System for drought forecastingAbstractIntroductionStandard Precipitation Index (SPI)Adaptive Neuro Fuzzy Inference System (ANFIS)Study area and dataDrought forecasting by ANFISInput variablesModel structures
ConclusionsAcknowledgmentsReferences
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