Development of Adaptive Neuro Fuzzy Inference System for Estimation of Evapotranspiration

IJSRD - International Journal for Scientific Research & Development| Vol. 1, Issue 4, 2013 | ISSN (online): 2321-0613
Abstract— The accuracy of an adaptive neurofuzzy
computing technique in estimation of reference
evapotranspiration (ETo) is investigated in this paper. The
model is based on Adaptive Neurofuzzy Inference System
(ANFIS) and uses commonly available weather information
such as the daily climatic data, Maximum and Minimum Air
Temperature, Relative Humidity, Wind Speed and Sunshine
hours from station, Karjan (Latitude - 22°03′10.95″N,
Longitude - 73°07′24.65″E), in Vadodara (Gujarat), are used
as inputs to the neurofuzzy model to estimate ETo obtained
using the FAO-56 Penman–Monteith equation. The daily
meteorological data of two years from 2009 and 2010 at
Karjan Takuka, Vadodara, are used to train the model, and
the data in 2011 is used to predict the ETo in that year and to
validate the model. The ETo in training period (Train- ETo)
and the predicted results (Test-ETo) are compared with the
ETo computed by Penman-Monteith method (PM-ETo)
using “DailyET” Software. The results indicate that the PM-
ETo values are closely and linearly correlated with Train-
ETo and Test- ETo with Root Mean Squared Error (RMSE)
and showed the higher significances of the Train- ETo and
Test- ETo. The results indict the feasibility of using the
convenient model to resolve the problems of agriculture
irrigation with intelligent algorithm, and more accurate
weather forecast, appropriate membership function and
suitable fuzzy rules.
Keywords: ANFIS model, Fuzzy Inference, Neural Network,
Penman-Monteith, Predicting, Reference evapotranspiration
(ETo), Weather Forecast.
I. INTRODUCTION
Evaporation is the process whereby liquid water is
converted to water vapor and removed from the evaporating
surface. Transpiration consists of the vaporization of liquid
water contained in plant tissues and the vapor removal to the
atmosphere, the combination of the two separate processes
whereby water is lost on the one hand from the soil surface
by evaporation and on the other hand from the crop by
transpiration is referred to as evapotranspiration (Allen et al.
1998). Knowledge of crop evapotranspiration (ET) is very
important, because it allows optimization of the irrigation
water use in arid and semiarid regions where water shortage
is a problem. The estimation of evapotranspiration is of
great importance for agricultural, hydrological, and climatic
studies, as it constitutes a major part of the hydrological
cycle (Sobrino et al. 2005). Numerous methods have been
proposed for modeling evapotranspiration as described by
Brutsaert (1982) and Jensen et al. (1990), In general,
combination of energy balance/ aerodynamic equations
“provides the most accurate results as a result of their
foundation in physics and basis on rational relationships”
(Jensen et al. 1990). The Food and Agricultural
Organization of the United Nations (FAO) assumed the ET
definition from Smith et al. (1997) and adopted the FAO
Penman–Monteith as the standard equation for estimation of
ET (Allen et al. 1998; Naoum and Tsanis 2003).
Neural networks approaches have been
successfully applied in a number of diverse fields, including
water resources. In the hydrological context, recent
experiments have reported that artificial neural networks
(ANN) may offer a promising alternative for modeling
hydrological variables (e.g., rainfall–runoff, streamflow,
suspended sediment) (Minnes and Hall 1996; Jain et al.
1999). However, the application of ANN to
evapotranspiration modeling is limited in the literature.
Kumar et al. (2002) used a multilayer perceptron (MLP)
with backpropagation training algorithm for estimation of
ETo. They used various ANN architectures and found that
the ANN gave accurate ETo estimates. Sudheer et al.
(2003b) used radial basis ANN in modeling ETo using
limited climatic data. The objective of this study is to use
the free general weather forecasting messages to determine
daily ETo value by the analytical method (AM) and adaptive
neural network and fuzzy inference system (ANFIS), and
made comparison between (1) main variables with values
estimated using AM and PM method, (2) Penman -
Monteith ETo values calculated from complete daily weather
records (PM- ETo) and values calculated by AM (AM- ETo)
and fuzzy method (Train- ETo and Test- ETo) using daily
general weather forecasting messages.
The objectives of this study are to, (1) Made
comparison between Penman-Monteith ETo values
calculated from “DailyET” software and neuro-fuzzy
method (Train- ETo and Test- ETo) using daily general
metrological data and (2) Make fuzzy and neuro fuzzy
model and their results will be evaluated. The fuzzy rule-
based approach is applied for the construction of fuzzy
models. To remove the weaknesses of fuzzy models that are
not trained during the modeling, adaptive neuro-fuzzy
inference system (ANFIS) with given input/output data sets
will be use for neuro-fuzzy model. To do this, Fuzzy models
was studied with input parameters such as daily maximum
and minimum temperature, relative humidity percent,
sunshine hours, wind speed. As well as output of this fuzzy
system such as Evapotranspiration will studied in this
research. After determining effective parameters in
modeling, data sets will be divided into two parts: the first
Development of Adaptive Neuro Fuzzy Inference System for Estimation of
Evapotranspiration
Dhruv Patel1
Dr. T. M. V. Suryanarayana2
1
Student M. E. (Civil) 2
Assistant Professor
1,2
Faculty of Technology and Engineering
1, 2
Water Resources Engineering and Management Institute, Samiala
1,2
The Maharaja Sayajirao University of Baroda, Vadodara.

Development of Adaptive Neuro Fuzzy Inference System for Estimation of Evapotranspiration
(IJSRD/Vol. 1/Issue 4/2013/0031)
All rights reserved by www.ijsrd.com 940
part included 70% of the data used for training the model.
The second part consisted of 30% of data used for testing of
the model. The result concluded from ANFIS model and the
result concluded from Fuzzy Logic & ANN will be
compared.
Fig. 1: Two input first-order Sugeno fuzzy model with two
rules
II. ADAPTIVE NEUROFUZZY INFERENCE SYSTEM
The adaptive neuro fuzzy inference system (ANFIS), first
introduced by Jang (1993), is a universal approximator and
as such is capable of approximating any real continuous
function on a compact set to any degree of accuracy (Jang et
al. 1997). ANFIS is functionally equivalent to fuzzy
inference systems (Jang et al. 1997). Specifically the ANFIS
system of interest here is functionally equivalent to the
Sugeno first-order fuzzy model (Jang et al. 1997; Drake
2000). Below, the hybrid learning algorithm, which
combines gradient descent and the least-squares method, is
introduced. As a simple example we assume a fuzzy
inference system with two inputs x and y and one output z.
The first-order Sugeno fuzzy model, a typical rule set with
two fuzzy If–Then rules can be expressed as
Rule 1: If x is A1 and y is B1, then f1 = p1x + q1y + r1. . . . (1)
Rule 2: If x is A2 and y is B2, then f2 = p2x + q2y + r2. . . .(2)
Here, the output z weighted average of the individual rule
outputs and is itself a crisp value. The corresponding
equivalent ANFIS architecture is shown in Fig. 2. Nodes at
the same layer have similar functions. The node function is
described next. The output of the ith
node in layer l is
denoted as Ol,i.
Layer 1
Every node i in this layer is an adaptive node with node
function
( )
Or
( )
Where x (or y) input to the ith node; and Ai (or Bi-2) is a
linguistic label (such as “low” or “high”) associated with
this node. In words, Ol,i is the membership grade of a fuzzy
set A (=A1, A2, B1, or B2) and it specifies the degree to
which the given input x (or y) satisfies the quantifier A. The
membership functions for A and B are generally described
by generalized bell functions, e.g.
( )
[( )] ]
Where {ai , bi , ci } is the parameter set. As the values of
these parameters change, the bell-shaped function varies
accordingly, thus exhibiting various forms of membership
functions on linguistic label Ai. In fact, any continuous and
piecewise differentiable functions, such as commonly used
triangular-shaped membership functions, are also qualified
candidates for node functions in this layer. Parameters in
this layer are referred to as premise parameters. The outputs
of this layer are the membership values of the premise part.
Layer 2
Fig. 2: Equivalent ANFIS structure
This layer consists of the nodes labeled П, which multiply
incoming signals and sending the product out. For instance
( ) ( )
Each node output represents the firing strength of a rule.
Layer 3
In this layer, the nodes labeled N calculate the ratio of the ith
rule’s firing strength to the sum of all rules’ firing strengths
̅̅̅
The outputs of this layer are called normalized firing
strengths.
Layer 4
This layer’s nodes are adaptive with node functions
̅̅̅ ̅̅̅( )
Where ̅̅̅ = output of Layer 3; and {pi, qi , ri} parameter set.
Parameters of this layer are referred to as consequent
parameters.

Layer 5
This layer’s single fixed node labeled ∑ computes the final
output as the summation of all incoming signal
∑ ̅̅̅
∑
∑
Thus, an adaptive network is functionally equivalent to a
Sugeno first-order fuzzy inference system. The learning rule
specifies how the premise parameters (see Layer 1) and
consequent parameters (see Layer 4) should be updated to
minimize a prescribed error measure E. The error measure is
a mathematical expression that measures the difference
between the networks actual output and the desired output,
such as the squared error. The steepest descent method is
used as the basic learning rule of the adaptive network. In
this method the gradient is derived by repeated application
of the chain rule. Calculation of the gradient in a network
structure requires use of the ordered derivative denoted as
as opposed to the ordinary partial derivative ∂. This
technique is called the back propagation rule. The core of
this learning rule involves how to recursively obtain a
gradient vector in which each element is defined as the
derivative of an error measure with respect to a parameter.
The update formula for simple steepest descent for the
generic parameter is α is
Where = learning rate.
While the back propagation learning rule can be used to
identify the parameters in an adaptive network, this method
is slow to converge. The hybrid learning algorithm, which
combines back propagation and the least-squares method
can be used to rapidly train and adapt the equivalent fuzzy
inference system. It can be seen from the Fig. 2 that if the
premise parameters are fixed, the overall output can be
given as a linear combination of the consequent parameters.
The output f can be written as
. . . . . . . (9)
This is linear in the consequent parameters p1, q1, r1, p2,
q2, and r2. Then we have S = set of total parameters; S1 =
set of premise (nonlinear) parameters; and S2 = set of
consequent (linear) parameters. Given values of S1, we can
plug P training data into the Eq. (9) and obtain the matrix
equation,
= y
Where = unknown vector whose elements are parameters
in S2, the set of consequent (linear) parameters. Then the set
S2 of consequent parameters can be identified with the
standard least-squares estimator (LSE)
( )
Where AT
= transpose of A; and ( ) =
pseudo inverse of A if AT
A is nonsingular. The recursive
least-square estimator (RLS) can also be used to calculate
(Jang 1993). Backpropagation and LSE can now be
combined to update the parameters of the adaptive network.
In the forward pass of the hybrid learning algorithm, node
outputs go forward until the final layer (Layer 4 in Fig. 2)
and the consequent parameters are identified by the least-
squares method. In the backward pass, the error signals
propagate backward and the premise parameters are updated
by gradient descent.
III. STUDY AREA & DATA USED
For this study, daily climatic data are required. The daily
climatic data of weather stations Karjan which is situated at
Vadodara District in Gujarat state of India (Latitude -
22°03′10.95″N, Longitude - 73°07′24.65″E) were utilized in
the calculation. The elevation is 29 m. from the mean sea
level. The location is shown in Fig 3.
It is about 1095 daily metrological data such as
Maximum and Minimum Air Temperature, Relative
Humidity, Wind Speed and Sunshine hours which are
extracted from three years data (January 1, 2009 to
December 31, 2011)
To calculate the ETo by Penman-Monteith method,
“DailyET” software was used for the same data.
Fig. 3: Location of study area (Karjan, Vadodara).
IV. THEORY AND METHODOLOGY
Adaptive Network-Based-Fuzzy Inferences System
(ANFIS) approach was employed in this study. The ANFIS
KARJAN

architecture consists of fuzzification layer, inferences
process, defuzzification layer, and summation as final output
layer. Typical architecture of ANFIS is shown by Figure 2.
The process flows from layer 1 to layer 5. It is started by
giving a number of sets of crisp values as input to be
fuzzyfied in layer 1, passing through inference process in
layer 2 and 3 where rules applied, calculating output for
each corresponding rules in layer 4 and then in layer 5 all
outputs from layer 4 are summed up to get one final output.
The main objective of the ANFIS is to determine the
optimum values of the equivalent fuzzy inference system
parameters by applying a learning algorithm using input-
output data sets. The parameter optimization is done in such
a way during training session that the error between the
target and the actual output is minimized. Parameters are
optimized by Backpropagation. The parameters to be
optimized in ANFIS are the premise parameters which
describe the shape of the membership functions, and the
consequent parameters which describe the overall output of
the system. The optimum parameters obtained are then used
in testing session to calculate the prediction. A number of
730 data were utilized during training session and 365 data
were used during testing session.
In this study, basic model is constructed by 5 inputs
and 1 output. The inputs are Maximum and Minimum Air
Temperature, Relative Humidity, Wind Speed and Sunshine
hours while for the output is Evapotranspiration (ETo). The
basic model then varied in 3 different number of
membership functions.
The membership function and fuzzy rules of the
inference system always determine the ANFIS- ETo in good
performance or not. In training period, some different
membership functions (trimf, gbellmf, gaussmf, guass2mf
and smf) in Fuzzy Logic Toolbox of MATLAB were tried to
find which would be the most suitable one. The Gauss
distribution function (gaussmf) would be best one for the
presented ANFIS- ETo, with speedy running and relative
small error in 100 training epochs. The fuzzy rules were
assessed based on the knowledge of the regression
relationships between each input and PM- ETo and the
relative expert knowledge.
The Architecture of the above described ANFIS
Model is shown in figure 4 which uses Backpropagation and
Grid Partition.
V. STATISTICAL INDICES USED TO EVALUATE
PERFORMANCE
The following statistical indices were used to evaluate the
agreement of main actual meteorological parameters and
PM- ETo with estimated values.
A. Root Mean Square Error (RMSE)
It yields the residual error in terms of the mean square error
expressed as
n
XX
RMSE
n
i idelmoiobs 

 1
2
,, )(
B. Coefficient of Correlation (r)
The correlation coefficient a concept from statistics is a
measure of how well trends in the predicted values follow
trends in past actual values. It is a measure of how well the
predicted values from a forecast model fit with the real-life
data. It is expressed as







n
i i
n
i i
n
i ii
yyxx
yyxx
r
1
2
1
2
1
)()(
)()(
VI. RESULTS
Figure 5 shows the RMSE when the model is run for 100
epochs, using Grid Partition Algorithm and
Backpropagation which is 0.38001, quite acceptable to
estimate Evapotranspiration. Figure 6 shows the comparison
of training output and predicted output when the model is
plotted against Training data set. Rule viewer and input
output surface view with two different sample inputs [NaN
15.5 3.165 47 NaN] and [NaN 15.5 NaN 47 5.65] are shown
in figure 7, figure 8 and figure 9 respectively, Where NaN is
the value that varies according to the variable which is
plotted as input. Finally, the model is plotted against testing
data set and the graph shown in figure 10 that compares
between Predicted FIS output and Testing Data Set.
As far as the significance of individual
meteorological parameters is concerned, the study revealed
that the highest value of correlation coefficient and least
value of root mean square error were obtained for
evapotranspiration with air temperature, followed by using
sunshine hours and relative humidity (Table 1). While the
lowest correlation coefficient was obtained with wind speed,
which mean wind speed alone does not appear to influence
the evapotranspiration significantly. The effect of air
temperature, wind speed and sunshine hours was found to be
positive; whereas a negative correlation exists between
evapotranspiration and relative humidity (that is
evapotranspiration decreases with increase in relative
humidity). It is a natural fact that the
climatic/meteorological factors in general act in concert.
Therefore, it is pertinent to take into account the combined
influence of all the meteorological parameter on
evapotranspiration. By various trials it was suggested that a
combination of temperature, wind speed, sunshine hour and
humidity provides a maximum value of correlation
coefficient with minimum values of root mean square error
in comparison to other inputs combinations, by ANFIS.
Sr
.No
Data
Maximu
m
Minimu
m
Correlation Coefficient
with evapotranspiration
1
Air temperature
(°C)
45.50 8.50 0.94
2
Relative humidity
(%)
94 0.20 -0.44
3 Wind speed (m/s) 6.40 0.30 0.21
4
Sunshine hours
(hour)
11.30 0 0.36
5
Evapotranspiration
(mm/day)
9.08 1.00 1.00
Table (1): Statistical analysis of the total daily weather data

Fig (4): ANFIS Architecture
Fig 5:Training of ANFIS Backpropagation model with 100
epochs
Fig 6: ANFIS Model on Training Data Set
Fig. 7: Rule Viewer of a sample input of ANFIS
Fig. 8: Surface View with Reference input:
[NaN 15.5 3.165 47 NaN]
Fig. 9: Surface View with Reference input:
[NaN 15.5 NaN 47 5.65]
Fig.10 Comparison of Predicted and observed value
VII. CONCLUSION
The Most important advantage of using ANFIS model to
predict Evapotranspiration is that this model is much
adaptive, easy and effective to every natural behavior of
weather system. Implementation of ANFIS model is less
complicated as compared to other neural network models.
Non-linear behavior of Temperature, Humidity, Wind Speed
and Sunshine Hours can be efficiently handled by ANFIS
model as there is no need to define membership function
manually which may be quite erratic, and time consuming
process. Presently used Model (3-3-3-3-3-1 architecture)
gives very good results when applied on training and testing
data set of Karjan, Vadodara, India. The Root Mean Squared

Error thus obtained is 0.38001 when we compared the
predicted output and compared output. This paper tries to
contribute its part to estimate evapotranspiration so that
decision can be taken in advance in agriculture operation
and planning so as to make the best use of favorable weather
conditions and make adjustment for adverse weather.
The ANFIS technique could be of use in design of
reservoirs and various other hydrological analyses where
other models may be inappropriate. These neuro fuzzy
models can be embedded as a module for estimating ETo
data in hydrological modeling studies. The study used data
from only one station and further studies using more data
from various areas may be required to reinforce the
conclusions drawn from this study.
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