The ANN modeling framework has been used increasingly in various aspects
of science and engineering because of its ability to model both linear
and nonlinear systems without the need to make any assumptions as are
implicit in most traditional statistical approaches.
It has been shown by Hornik et al. (1989) that
an ANN with sufficient complexity is capable of approximating any smooth function
to any desired degree of accuracy.
Some of the hydrologic problems ANN has been used for include rainfall-runoff
modelling (Hsu et al., 1995; Smith
and Eli, 1995; Achela et al., 1998), scheduling
of hydroelectric power systems (Saad and Bigras, 1996)
and river flow prediction (Karunanithi et al., 1994;
Zhu and Fujita, 1994). The ANN methodology has been applied
also to forecast rainfall; for example, French et al.
(1992) used synthetically generated rainfall storms to calibrate an ANN
model and then generated plausible rainfall sequences that could occur over
a catchments using a physically based rainfall to validate the ANN. The main
advantage of ANN is in cases where intrinsic nonlinearities in the dynamics
prevent the development of exactly solvable models. Note that, in meteorology,
all of these criteria are present in the sense that the dynamics are inherently
nonlinear and prediction is one of the main goals. ANN has been used successfully
to model rainfall in meteorological applications (Navonne and Ceccatto, 1994).
Xiao and Chandrasekar (1997) have developed a three-layer
perceptron ANN based algorithm for rainfall estimation from reflectivity of
radar observations. The objective of this investigation is forecasting precipitation
with artificial neural networks.
The artificial neural networks approach: An ANN is composed of
an input layer of neurons, one or more hidden layers and an output layer.
Each layer comprises multiple units connected completely with the next
layer (Fig. 1), with an independent weight attached
to each connection. The number of nodes in the output layer is equal to
the number of values to be predicted.
|| Network model for precipitation forecasting
Back propagation algorithm: One of extensively used ANN models for system
modeling application is the multilayer perceptron model based on the BP (Erb,
1993). Multilayer perceptrons (MLPs) are layered feed forward networks typically
trained with static back propagation. These networks have found their way into
countless applications requiring static pattern classification. Their main advantage
is that they are easy to use and that they can approximate any input/output
map. The key disadvantages are that they train slowly and require lots of training
data (typically three times more training samples than network weights). In
the BP algorithm learning based, the initial weights of connections are randomly
chosen. Suppose we have N learning examples with each example having n inputs
and l outputs, the input vector can be described as Xj = (X1j,
, Xnj) and the output vector as Aj = (B1j
, Blj), 1 ≤ j ≤ N. The learning process takes place
using the following two steps.
Forward propagation: The input vector Xj is fed into the input
layer and an output vector Oj = (O1j,
is generated on the basis of the current weights W = (W1l,...,
Wnl). The value of Bj is compared with the actual
output Aj and the output differences are summed to generate an error function
E defined as:
Error back propagation: In this step, the error from Eq.
1 is back propagated by performing weights update using gradient descent
where, 0 < η < 1 is a parameter controlling the convergence rate
of the algorithm. The process of forward propagation and error back propagation
continues until E converges to a predefined small value. ΔWnl is
accumulated and updated after a complete run of all the N examples for our research.
The basic BP algorithm described above is often very slow to converge in real
practice. In order to overcome the obstacles of these problems, BP algorithm
has adopted many improvement techniques. Initializing the BP weights with GA
contains many merits hence attracts the scholars and researchers interesting
(Sexton et al., 1998). And its modern form is
derived mainly from Hollands study in the 1960s.
|| Flowchart of micro algorithm genetic
As has been shown in Fig. 2, GA is capable of solving wide range of complex optimization
problems only using three simple genetic operations (selection, crossover and
mutation) on coded solutions (strings, chromosomes) for the parameter set, not
the parameters themselves in an iterative fashion. GA considers several points
in the search space simultaneously, which reduces the chance of convergence
to a local optimum. At first a random population is generated. This population
feeds the population memory, which is divided in two parts. The non-replaceable
portion of the population memory will never change during the entire run and
is meant to provide the required diversity for the algorithm. The initial population
of the micro-GA at the beginning of each of its cycles is taken (with a certain
probability) from both portions of the population memory as to allow a greater
diversity. During each cycle, the micro-GA undergoes conventional genetic operation:
tournament selection, two-point crossover, uniform mutation and elitism (only
one nondominated vector is arbitrarily selected at each generation and copied
intact to the following one). After the micro-GA finishes one cycle (i.e., when
nominal convergence is achieved), two nondominated will chose vectors from the
final population (the first and last) and compare them with the contents of
the external memory (this memory is initially empty). If either of them (or
both) remains as nondominated after comparing against the vectors in this external
memory, then they are included there. All the dominated vectors are eliminated
from the external memory. These two vectors are also compared against two elements
from the replaceable portion of the population memory (this is done with a loop,
so that each vector is only compared against a single position of the population
memory). If either of these vectors dominates to its match in the population
memory, then it replaces it. The idea is that, over time, the replaceable part
of the population memory will tend to have more nondominated vectors. Some of
them will be used in the initial population of the micro-GA to start new evolutionary
cycles (Carlos et al., 2001).
MATERIALS AND METHODS
Study area: The city of Tehran, the capital city of Iran, is located
in Tehran Province. This city having more than 7 million people is the
most populated city in the country. The city is surrounded by Alborz Mountains
in the Northeast. Figure 3 shows the geographic location
of study area.
Data used: In this study, monthly precipitation data obtained
from Tehran Meteorology Center for Tehran station from 1950 to 2003.
Method: The samples were divided into two sets. The first set
is the learning set for the ANN training, while the other set represents
the holdout set for pollution prediction to verify the efficiency and
correctness of the cutting force model. With these collected data, the
ANN system is ready to launch its training scheme. TanhAxon function that
is a most important function in B.P method is used as irritates function.
Neural solutions software is used for training artificial network. To
reduce forecasting error, train and error on the network parameters carried
out and at the end parameters arranged as has been shown in Table
||Network parameters are trained after train and error
|| The geographic location of study area that highlighted
with black color
Momentum is one of the main parameters in training that show the fraction
of before weights effect to new weights. Neuron is smallest processing
unit of hidden layer and learning coefficient control the speed of collation
the weights among neurons.
RESULTS AND DISCUSSION
The objective of this investigation was forecasting precipitation with
artificial neural networks. To achieve this objective two models in accordance
with the methods of determining the connection weights of BP and hybrid
of GA-BP are compared with the real experimental results. The results
of trained network without GA showed in Fig. 4.
There is a good coordination between real data (precipitation) with ANN
forecast. There is a nonlinear correlation between monthly precipitation
that is one of the reasons of increase artificial networks accuracy rather
than classic models such as regression models that are linear.
Figure 5 shows minimum training error of network without
combined with GA. Minimum training error of network after training equal
Correlation coefficient presents relation rate between network outputs
with real data (precipitation).
||Comparing amount of real precipitation data with ANN
||Minimum training error of network with ANN
Correlation coefficient equal to 1, shows fulfill coordination between
real data (precipitation) with ANN forecast and -1 show no coordination
between them. Figure 6 shows correlation and adjust
coefficient between real data (precipitation) with ANN forecast that equal
As described above the basic BP algorithm is often very slow to converge
in real practice. In order to overcome the obstacles of these problems,
the BP weights with GA was used. The result of this combination showed
in Fig. 7. Figure 8 shows the minimum
training error of network. Minimum training error of network rate after
training approximately equal to 0/03.
||Correlation between real precipitation data with ANN
||Comparing amount of real precipitation data with ANN-GA
||Minimum training error of network in combination with
||Correlation between real precipitation data in combination
with ANN-GA forecast
Figure 9 showed Correlation between real data (precipitation)
with combination ANN-GA forecast. The adjust coefficient equal to 0/83.
Classic or liner models are used for trends that with increase a parameter
another parameter increases or decreases. Thus using of these models for nonlinear
trends increase mistake rate. The neural networks can be a useful tool to model
the relation between variables because we can consider the ANNs as being very
general forms of non-linear regression models. In fact, they can model time
series very effectively (Chakraborty et al., 1992)
and are employed when the classic methods do not work. There is a nonlinear
trend for precipitation. For forecasting this nonlinear trend must use nonlinear
models. One of these nonlinear models is artificial neural networks forecasting
model. The results after testing network with hidden layers and different learning
coefficients showed that using artificial neural networks with one perceptron
with 1 hidden layers, 0/1 learning coefficient and 0/7 momentum result in a
relative better model and after retraining and testing network with different
hidden layers and learning coefficients in combination with Genetic algorithm
cause decrease error and increase the speed of calculations and present a better
||Artificial networks forecast the nonlinear trend of
||The combinations of algorithm genetic with artificial network increase
the speed of analyzing and processing accuracy. In other hand decrease
The authors appreciate Professor Seyyed Abolfazl Massodian (Department
of Climatology, Isfahan University, Iran) for reviewing the manuscript
and for his useful comments.