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International Journal of Computer Science and Information Security. IJCSIS invites authors to submit their original and unpublished work that communicates current research on information assurance and security regarding both the theoretical and methodological aspects, as well as various applications in solving real world information security problems. . Frequency of Publication: MONTHLY ISSN: 1947-5500 [Copyright � 2011, IJCSIS, USA & UK]
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(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 12, 2011
An Adaptive Neuro-Fuzzy Inference System
based on
Vorticity and Divergence for Rainfall forecasting
Kavita Pabreja
Research Scholar, Birla Institute of Technology and Science, Pilani, Rajasthan, India
Assistant Professor, Maharaja Surajmal Institute (an affiliate of GGSIP University), New Delhi, India
kavita_pabreja@rediffmail.com
Abstract— A new rainfall forecasting model based on Adaptive provide information and warning of extreme weather events
Neuro-Fuzzy Inference System is proposed in this paper. A for minimizing losses both to human and property. Such data
neuro-fuzzy model inherits the interpretability of fuzzy models consists of a sequence of global snapshots of the Earth,
and learning capability of neural networks in a single system. It typically available at various spatial and temporal intervals
has got wide acceptance for modelling many real world problems including atmospheric parameters over land and ocean (such
because it provides a systematic and directed approach for model
building and gives the best possible design parameters in
as temperature, pressure, wind speed, wind direction, sea
minimum time. The datasets used in this paper for the training of surface temperature, etc.). The NWP models do not produce
Adaptive Neuro-Fuzzy Inference System (ANFIS) are the forecast of rainfall directly. Forecast of weather elements like
European Center for Medium-range Weather Forecasting rain/snow, sky conditions etc. at a place are derived through
(ECMWF) model output products and the gridded rainfall statistical relation popularly known as Model Output Statistics
datasets, provided by Indian Meteorological Department (IMD). (MOS) proposed by National Weather Service [1]. General
To determine the characteristics of ANFIS that best suited the experience is that MOS products show improved skills over
target rainfall forecasting system, several ANFIS models were the raw model output. Basis of MOS is statistical relationship
trained, tested and compared. Different training and checking which requires long term consistent series of NWP products.
data, type and number of membership functions and techniques
Since NWP models get upgraded regularly[2], the series does
to generate the initial Fuzzy Inference Systems were analyzed.
Comparisons of the different models were performed and the not remain consistent.
results showed that the model generated by grid partitioning In view of above limitation of MOS, it has been proposed
using gbellmf membership functions provided the smallest errors to explore other Intelligent techniques like ANFIS so as to
for rainfall forecasting. forecast rainfall based on NWP model output products. In
past, Artificial Neural Networks (ANN) has been applied [3]
to predict the average rainfall over India during summer-
Keywords- NWP model forecast, ECMWF model, rainfall, monsoon i.e. the months of June, July, and August, by
vorticity, divergence, ANFIS
exploring the rainfall data corresponding to the summer
monsoon months of years 1871-1999. It has been found that
I. INTRODUCTION the prediction error in case of ANN is 10.2% whereas the
Weather is not just an environmental issue; it is a major prediction error in the case of persistence forecast is 18.3%.
economic factor. Economic value of weather for Agriculture, A neural network, using input from the Eta Model and
Fishery, Energy, Transportation, Aviation and health area is upper air soundings, has been developed [4] for the probability
immeasurable. With its huge and growing population and low- of precipitation (PoP) and quantitative precipitation forecast
lying coastline and an economy that is closely tied to its (QPF) for the Dallas–Fort Worth, Texas, area. Forecasts from
natural resource base, India is considerably sensitive to two years were verified against a network of 36 rain gauges.
weather and climate. One failure of monsoon can totally upset The resulting forecasts were remarkably sharp, with over 70%
the economic performance of our country. But timely of the PoP forecasts being less than 5% or greater than 95%.
forecasting can help to considerably minimize the adverse Of the 436 days with forecasts of less than 5% PoP, no rain
effect. occurred on 435 days. Of the 111 days with forecasts of
Analysis and forecast of weather data created through greater than 95% PoP, rain always occurred. The application
Numerical Weather Prediction (NWP) models offers an of ANFIS for forecasting of meteorological parameters is very
unprecedented opportunity for predicting weather events, rare and particularly rainfall forecasting has not been
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considered in the studies and hence has been taken up in this between 0 and 1. This process is known as fuzzification and
paper to look for even better accuracy of forecast. takes place in layer 2, the fuzzification layer. Each node in this
layer is adaptive.
II. ADAPTIVE NEURO-FUZZY INFERENCE SYSTEM Once the locations of the inputs in the fuzzy spaces are
ANFIS is a hybrid of two intelligent systems: Artificial identified, then the product of the degrees to which the inputs
Neural Networks (ANNs) and Fuzzy Inference Systems satisfy the membership functions is found. This product is
(FISs). ANNs map an input space to an output space through a called the firing strength of a rule and is represented by layer
collection of layered processing elements called neurons that
are interconnected in parallel by synaptic junctions. ANNs are
developed by continuously passing real world system data
from its input to output layer. For each pass of data, signals
propagate from the input to output layer to produce an output
which is compared to the desired output. The difference
between these values is then used to adjust the synaptic
connections so that the ANN can mimic the system the data
represents. This procedure gives ANNs the capability of
looking for patterns in the information presented to it, thus
providing it with the advantage of learning about systems.
FISs are based on fuzzy logic (a continuous range of truth
values from 0 to 1), IF-THEN fuzzy rules and fuzzy reasoning
(which can be likened to human reasoning through linguistic
variables such as small, medium, large). These features of FIS
allow it to make inferences using the rules and known facts to 3, the rule layer where each node in this layer is fixed. Each
derive reasonable decisions [5]. Thus the combination of fuzzy space is governed by an ANFIS rule where the
ANNs and FISs to form ANFIS, integrates the benefits of the antecedent of the rule defines a fuzzy space in the input space
individual intelligent systems to form a superior technique that [5]. For ANFIS, there are Mn fuzzy rules where M is the
can optimally model the dynamics of difficult systems. number of membership functions per input and n is the
An example of ANFIS has been explained which is of a 6 number of inputs.
layer feedforward neural network and of the Sugeno FIS type. In layer 4, the normalization layer, the ratio of each rule’s
To understand the structure and operation of ANFIS in firing strength is calculated with respect to the sum of the
forecasting, a 2 input - 1 output ANFIS model is presented and firing strengths of all the rules. Each node in this layer is fixed.
its structure and operation is related to a generalized model. In layer 5, the defuzzification layer, the output of each
Fig. 1 shows the ANFIS structure and Equations 1 to 4 are the node is the weighted consequent value. Layer 6 is the
rules for this model where the IF part of the rule is referred to summation layer and its output which is the sum of all the
as the antecedent and the THEN part is the consequent. outputs of the layer 5 which gives the overall output for the
respective inputs within the fuzzy space. Before the ANFIS
Rule 1: If x is A1 and y is B1, then f1 = p1x + q1y + r1 (1) system can be used for prediction, the parameters of the rules
Rule 2: If x is A2 and y is B2, then f2 = p2x + q2y + r2 (2) are determined by first generating an initial FIS where random
Rule 3: If x is A3 and y is B3, then f3 = p3x + q3y + r3 (3) values are assigned to the parameters and then applying an
Rule 4: If x is A4 and y is B4, then f4 = p4x + q4y + r4 (4) optimization scheme to determine the best values of the
parameters that would provide rules that would idealistically
In general, an n-input, 1-output ANFIS model is an n + 1 model the target system. After training, the rules remain so
dimensional input-output space. Therefore, a 2 inputs-1 output that when new input data is presented to the model, the rules
ANFIS model is a 3-dimensional input-output space. In order provide a corresponding reasonable output.
for ANFIS to be used to model a system, data that is The optimization technique is a learning algorithm which
representative of the target system must be presented to uses data (training data) from the target system to generate
ANFIS. The entry of raw data or crisp inputs from the target signals that propagate backwards and forwards and update the
system into ANFIS corresponds to layer 1 – the input layer in parameters by a process known as training. The learning
Fig. 1. algorithm proposed for ANFIS is a hybrid learning algorithm
Since the Neural Network classifies data and looks for that minimizes the error between the ANFIS model and the
patterns within it, then when the input data is in the 3- real system [5]. ANFIS employs the least squares estimate and
dimensional space, it is classified into groups called fuzzy the gradient descent method in the hybrid learning algorithm.
spaces. To do this, the crisp inputs are compared with Once input-output data is presented to ANFIS, in one epoch
membership functions in the antecedent of the rules of ANFIS, the data is propagated forwards from one layer to the next
to determine the degree to which the inputs, in this case, X1 until the fourth layer, and the least squares estimate is
and X2 belong to fuzzy sets Ai and Bi respectively. The degree employed to update the linear or consequent parameters. An
to which the inputs lie within the fuzzy space is given a value error is calculated and this is propagated backwards and the
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gradient descent is used to update the non-linear or premise
parameters [5].
III. DATASETS USED
Two different datasets provided by IMD have been used for
the study. First one is the forecasts by NWP model (ECMWF
model) and the other is the observed values of rainfall
datasets. These dataset files and pre-processing applied on
them are explained in section a and b respectively.
(a) ECMWF T-799 model forecasts and its pre-processing
The datasets produced as forecast by ECMWF model are
in GRIB format which is a mathematically concise data format
commonly used in meteorology to store historical and forecast
weather data. It is standardized by the World Meteorological
Organization's Commission for Basic Systems. The forecast
datasets of T-799 model includes values for 87variables
(including all atmospheric pressure levels), for latitude -10° to
50° and longitude 50° to 110° at a grid spacing of 0.25°,
making it equal to 241X241 grid points i.e. forecast of 87
variables at 58081 grid points. Finally it becomes a huge
datasets of 50,53,047 (approx. 5million) values for just one
forecast of a particular time.
The GRIB files have been converted to (.csv) format by
using National Digital Forecast Database - NDFD GRIB2
decoder program of NOAA downloaded from Internet. The
model does not provide vorticity and divergence directly,
which are important determinant of rainfall, so this has been
derived by using vertical (v) and horizontal (u) component of
wind as forecasted by model, using the formulas given below:-
Divergence formula: ∂u ∂v
∂x ∂y Figure 2. Data pre-processing of forecast by ECMWF model
Vorticity formula: ∂v ∂u
∂x ∂y
where v denotes meridian wind flow
u denotes zonal wind flow
x denotes longitude
y denotes latitude
These steps of data pre-processing have been shown in
Fig. 2. For the purpose of this study, we have calculated
vorticity and divergence at atmospheric pressure level of
850hPa, on 0000GMT 29 July 09 with initial conditions of
0000GMT 28 July 09 for the model, as input parameters for
training of ANFIS. A small sample of this datasets has been
shown in table I.
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TABLE I – FORECAST OF VORTICITY AND DIVERGENCE MADE TABLE II - RAINFALL FOR YEAR 2009
ON 0000GMT 28JULY 09 VALID FOR 0000GMT 29JULY09
Latitude Longitude
Latitude Longitude Vorticity Divergence (°N) (°E) Rainfall in mm
(°N) (°E) (X10-5 per sec ) (X10-5 per sec )
24.5 94.5 14.1
24.5 94.5 2 -6 28 94.5 22.7
28 94.5 2 -14 24 88.5 6.1
24 88.5 4 -4 26.5 89 56.6
26.5 89 4 -4 30 78.5 16.4
30 78.5 6 -8 26.5 90 8.5
26.5 90 6 -18 21.5 84 1.2
21.5 84 8 -6 26 92.5 5.4
26 92.5 8 -14 27.5 94 12.8
27.5 94 8 -8 27.5 81.5 24.1
27.5 81.5 10 -4 25 85.5 18.4
25 85.5 10 -10 22.5 86 10.4
22.5 86 10 -10 28.5 80.5 40.7
28.5 80.5 14 -8 27.5 84 27
27.5 84 16 -10 26.5 86 1.2
26.5 86 16 -10 29 79.5 2.5
29 79.5 20 -14 23 92.5 42
(SOURCE: AS A RESULT OF PRE-PROCESSING RF2009.GRD PROVIDED BY IMD)
23 92.5 20 -10
(b) Rainfall datasets and its pre-processing Finally the two different datasets of model forecast and
A high resolution (0.5° × 0.5°) daily rainfall (in mm) rainfall datasets location-wise have been merged, as shown in
dataset for mesoscale meteorological studies over the Indian table IV using the Rainfall category as explained in table III,
region has been provided by IMD and described by [6]. The so that they can be presented to ANFIS model for training and
dataset is in .grd format, a control file describing the structure obtaining rules that correlate the vorticity and divergence as
of .grd file provided by IMD. antecedents with rainfall category as consequent.
The rainfall datasets under study are for year 2009. The
TABLE III - CATEGORY AND CODE FOR RAINFALL
data is for the geographical region from longitude 66.5 ºE to CORRESPONDING TO RAINFALL (IN MM)
100.5 ºE and latitude 6.5 ºN to 38.5 ºN for each day of the
year. There are 4485 grid points readings every day and Rainfall value (in mm) Category Code
rainfall record for 122 days (June to September) per year are 1-15 very low 1
selected for analysis i.e 5,47,170 records out of a total of
15.1 – 40 low 2
16,37,025 records for one year of rainfall. Steps followed for
pre-processing of the .grd so that an intelligent system can be 40.1-75 good 3
applied, are mentioned below: 75.5 - more heavy 4
1. The .grd file has been converted to .dat file using a
FORTRAN programme. This dataset is very huge in
size.
2. The .txt files have been exported to Excel worksheet
and then to Access database. The data looks like as if
a rectangular grid is filled with values of rainfall in
mm.
3. Using a Visual Basic program to organize data in
tabular format, as shown in table II.
4. Finally exporting the dataset into .xls format for
analysis, by Matlab.
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avoid curse of dimensionality problem [7]. Therefore we have
TABLE IV - FORECASTED VALUE OF VORTICITY AND
opted for Grid partitioning as we have just 2 input parameters
DIVERGENCE BY ECMWF MODEL AND OBSERVED VALUE OF
RAINFALL CATEGORY viz. vorticity and divergence.
Once the grid partitioning technique is applied at the
beginning of training, a uniformly partitioned grid which is
Vorticity Divergence
(X10-5 per second ) (X10-5 per second ) Rainfall code defined by membership functions (MFs) with a random set of
parameters is taken as the initial state of ANFIS. During
training, this grid evolves as the parameters in the MFs
2 -6 1
change. With the grid partitioning technique, the number of
4 -10 1 MFs in the premise part of the rules must be determined.
6 -18 1 Negnevitsky et al. [8] stated that a larger number of MFs
better represents a complex system and therefore should
8 -14 1
produce better results. However, a large number of inputs or
18 -10 1 MFs in the premise part of the rules can produce a large
20 -14 1 number of fuzzy rules which can cause the learning
complexity of ANFIS to suffer an exponential explosion,
20 -4 1
called the curse of dimensionality which can adversely affect
2 -18 2 the performance of ANFIS [7, 9, 10].
8 -8 2 We have generated 5 different ANFIS models by grid-
partitioning. The idea was to explore the ANFIS generation
10 -10 2
first with different shapes of membership functions, keeping
10 -4 2 the dataset for training, checking fixed at original values (i.e.
16 -10 2 no normalization done) and number of membership functions
fixed. Next it was decided to explore the ANFIS generation
18 -8 2 with increase in membership functions. Finally, it was decided
26 -18 2 to normalize datasets [-1 1] for training and checking and
44 -8 2
observing the FIS outputs after training. All these parameters
for different models are explained in table V.
4 -4 3 The number of MFs was increased with one of the ANFIS
14 -8 3 models (number 1 in table V) to get a greater understanding of
the impact on the performance of ANFIS with this change. In
16 -4 3
generating the Rainfall forecasting FIS, by grid partitioning, the
20 -10 3 bell-shaped MF was favored over the other types since it
-8 4 4
offered more parameters which provided a greater number of
degrees of freedom. The generalized bell-shaped MF is
standard for ANFIS because of its smoothness and concise
IV. GENERATION OF ANFIS notation [7, 8, 9]. Other function such as Gaussian was used as
well to evaluate the performance with different types of MFs.
For the consequent part of the rules the MFs responsible for
ANFIS in this study was trained and simulated using defuzzification were the Sugeno type of first order. The output
Matlab 7.0 (matrix laboratory) designed and developed by MF is chosen to be linear for the rainfall forecasting models
Math Works Inc. The fuzzy inference commonly used in since, the higher the order of output MFs, the greater is the
ANFIS is first order Sugeno fuzzy model because of its likelihood of ANFIS fitting the target system [11].
simplicity, high interpretability, and computational efficiency,
built- in optimal and adaptive techniques. A typical TABLE V - DIFFERENT ANFIS MODELS USED IN THE STUDY
architecture of an ANFIS has already been shown in Fig. 1. ANFIS Type of membership function Number of Number of
Among many FIS models, the Sugeno fuzzy model is the most
Model for input parameters membership membership
widely applied one for its high interpretability and
computational efficiency, and built-in optimal and adaptive functions for functions for
techniques. Vorticity Divergence
Generation of ANFIS involves selecting a structure for I gbellmf (original data) 5 3
the ANFIS model by determining the number of membership
functions per input, type/shape of the membership functions II gaussmf (original data) 5 3
for the premise part of the rule and the output membership III gbellmf (original data) 7 5
functions for the consequent part of the rule. MATLAB 7.0
offers two methods for generating the initial FIS: Grid IV gaussmf (normalized data) 5 3
Partitioning and Subtractive Clustering. Subtractive V gbellmf (normalized data) 5 3
partitioning is used if number of inputs is more than 6 so as to
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V TRAINING THE ANFIS
Training involves the selection of optimization
technique, error tolerance and the number of epochs. The
ANFIS toolbox provided two optimization methods: hybrid
and backpropagation. To develop the ANFIS rainfall
forecasting models, the hybrid technique was used since it is
more popularly used with ANFIS than the backpropagation
because it is a combination of least-squares and back-
propagation gradient descent method [5,8]. In addition, it is
regarded as the faster of the two techniques [5]. We have
trained five models with 80% of the datasets for training and
Figure 4 target output (in red) and ANFIS predicted output (in blue) for the
20% as checking data. The number of epochs has been 2input (5 bell MF for vorticity input, 3 bell MF for divergence input) , 1
changed from 50 to 100 depending on the shape of training output ANFIS model
and checking curves. In accordance to the approach provided
by J.S.R. Jang [5], different models were created by changing
2. The checking error curves for the model with 5 gaussian
some part of its structure or parameters, and each was
MFs for vorticity input and 3 gaussian MFs for divergence
compared to the previous models created to determine if the
input, remains almost constant from the first epoch till last
changed characteristic provided better results. If the model
epoch, as shown in Fig. 5.
produced better results, then these characteristics were kept
and if not, the model was retrained with one of the
characteristics of its structure changed. After which, one
feature of the chosen model: type of input data, size of training
or checking data, type of membership functions or the number
of membership functions per input was changed one at a time.
The chosen structures were trained with datasets mentioned in
section 3, once trained they were evaluated using the
performance metrics: RMSE.
VI RESULTS
The findings from these five models trained using the
grid partitioning technique provided following important Figure 5 Training and checking error curves for the 2input (5 gaussian MF for
results:- vorticity input, 3 gaussian MF for divergence input) , 1 output ANFIS model
1. The checking error curves for the model with 5 bell-shaped
MFs for vorticity input and 3 bell-shaped MFs for divergence This model when compared for the actual checking data verses
input, decreases from the first epoch. This was trained for output generated by ANFIS model demonstrated very poor
50epochs which resulted in almost same value for training and results as shown in Fig. 6.
testing error, as shown in Fig. 3.
Figure 6 target output (in red) and ANFIS predicted output (in blue) for the
2input (5 gaussian MF for vorticity input, 3 gaussian MF for divergence input)
Figure 3 Training and checking error curves for the 2input (5 bell MF for , 1 output ANFIS model
vorticity input, 3bell MF for divergence input) , 1 output ANFIS model
3.The testing/checking error curves for the model with 7 bell-
This model when compared for the actual checking data verses shaped MFs for vorticity input and 5 bell-shaped MFs for
output generated by ANFIS model demonstrated good results divergence input, decreases from the first epoch. This was
as shown in Fig. 4. trained for 50epochs after which model started overfitting, as
shown in Fig. 7.
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Figure 7 Training and checking error curves for the 2input (7 bell MF for Figure 10 target output (in red) and ANFIS predicted output (in blue) for the
vorticity input, 5 bell MF for divergence input) , 1 output ANFIS model 2input (5 gaussian MF for vorticity input, 3 gaussian MF for divergence
input), 1 output ANFIS model with normalized data
This model when compared for the actual checking data verses
output generated by ANFIS model demonstrated results as 5. With the normalized datasets, again a new ANFIS model
shown in Fig. 8. was generated with 5 bell-shaped MFs for vorticity and 3 bell-
shaped MFs for divergence in order to get better results for
checking error. This model was trained for 100epochs after
which model became stable, as shown in Fig. 11.
Figure 8 target output (in red) and ANFIS predicted output (in blue) for the
2input (7 bell MF for vorticity input, 5 bell MF for divergence input) , 1
output ANFIS model
4. It was experimented to train the model with Gaussian Figure 11 Training and checking error curves for the 2input (5 bell MF for
membership functions for representation of the inputs but the vorticity input, 3 bell MF for divergence input) , 1 output ANFIS model with
response of the model was very poor so the datasets for input – normalized data
output were normalized so that they fall in the range [-1 1].
With these datasets, it was observed that the testing/checking
This model when compared for the actual checking data verses
error curves with 5 gaussian MFs for vorticity input and 3
output generated by ANFIS model did not produce better
gaussian MFs for divergence input, trained on 80% of rainfall results than when trained with actual original datasets, as
data produced good results. This was trained for 50epochs shown in Fig. 12.
after which model started overfitting, as shown in Fig. 9.
Figure 12 target output (in red) and ANFIS predicted output (in blue) for the
Figure 9 Training and checking error curves for the 2input (5 gaussian MF for 2input (5 bell MF for vorticity input, 3 bell MF for divergence input) , 1
vorticity input, 3 gaussian MF for divergence input) , 1 output ANFIS model output ANFIS model with normalized data
with normalized data The values of root mean square errors for training and
This model when compared for the actual checking data verses checking datasets for all these five ANFIS models are
output generated by ANFIS model demonstrated the results as tabulated in table VI.
shown in Fig. 10.
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TABLE VI - ROOT MEAN SQUARE ERROR FOR THE ANFIS MODELS [4] Hall T., Brooks H.E., Doswell C.A. Precipitation Forecasting
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if we increase number of membership functions or normalize
the antecedents and consequent data variables, it does not AUTHORS PROFILE
cause any improvement in the RMSE and hence predicting the Ms. Kavita holds more than 17 years of experience with
Educational institution and Industry. She is currently Assistant Professor -
value of rainfall. The rules diagram of the best ANFIS model Computer Society, Maharaja Surajmal Institute, an affiliate of GGS
has been shown in Fig. 13. Indraprastha University. She has teaching experience of over a decade and she
has worked for more than 5 years with Indian as well as USA MNC. These
ACKNOWLEDGEMENT companies include Rockwell International Overseas Corp., Parekh
Microelectronics (I) Ltd., HCL Hewlett Packard Ltd. and Shyam Telecom
This study is based on the datasets made available by courtesy Ltd.
of Indian Meteorological Department, India. The author She is M.S.(Software Systems) from BITS, Pilani; AMIETE (eq.
would also like to deeply acknowledge the support and B.E. (Electronics and Telecommunication Engg.)) from IETE. She holds
guidance of Dr. Rattan K. Datta, Former Advisor – Deptt. of membership of many professional bodies viz. Senior Member of Computer
Society of India, Member of Institute of Electronics and Telecommunication
Science & Technology, Former President - Indian Engineers, Member of Indian Meteorological Society and Member of IACSIT,
Meteorological Society and Computer Society of India. Singapore.
She has designed and developed Workbooks and textbooks for the
REFERENCES ICT Project, Punjab undertaken by Educational Consultants India Ltd. She
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in rainfall forecasting, Proceedings of the International conference
on IT, HIT, March 2007, 523-526
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ISSN 1947-5500
(IJCSIS) International Journal of Computer Science and Information Security,
Vol. 9, No. 12, 2011
Figure 13 Graphical illustration of a set of rules and their contribution to the final results in case of Model I
53 http://sites.google.com/site/ijcsis/
ISSN 1947-5500
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