WAVELET-NEURAL NETWORK CONJUNCTION MODEL IN FLOW FORECASTING OF SUBHIMALAYAN RIVER BRAHMAPUTRA by iaemedu

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									 International Journal of Civil Engineering and Technology (IJCIET), ISSN 0976 – 6308
  INTERNATIONAL JOURNAL OF CIVIL ENGINEERING AND
 (Print), ISSN 0976 – 6316(Online) Volume 3, Issue 2, July- December (2012), © IAEME
                                TECHNOLOGY (IJCIET)
ISSN 0976 – 6308 (Print)
ISSN 0976 – 6316(Online)
Volume 3, Issue 2, July- December (2012), pp. 415-425
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  WAVELET-NEURAL NETWORK CONJUNCTION MODEL IN FLOW
   FORECASTING OF SUBHIMALAYAN RIVER BRAHMAPUTRA

                      Khandekar Sachin Dadu1 and Paresh Chandra Deka2
  1
      Research Scholar, 2Associate Professor, Department of Applied Mechanics and Hydraulics,
          National Institute of Technology Karnataka, Surathkal, Mangalore- 575025, India
                     E-mail: paresh_deka@sify.com, khandekarsd@yahoo.com


 ABSTRACT

         In this current study, the Discrete Wavelet transform was hybridized with ANN
 naming Wavelet Neural Network (WLNN) for river flow forecasting at selective stations
 such as Pandu and Pancharatna of international river Brahmaputra within India, upto 4 time
 steps lead time. The main time series of daily, weekly and monthly discharge data were
 decomposed to multiresolution time series using discrete wavelet transformations and were
 used as input of the ANN to forecast the river flow at different multistep lead time. It was
 shown how the proposed model, WLNN, that makes use of multiresolution time series as
 input, allows for more accurate and consistent predictions with respect to classical ANN
 models. The proposed wavelet model (WLNN) results shows that it is better forecasted and
 consistent than single ANN model because of using multiresolution time series data as inputs.

 Keywords: Artificial neural network; Discrete Wavelet transform; Subtropical; Time series
 forecasting; Brahmaputra River; Hybridization

 1. INTRODUCTION

         In the last decade, wavelet transform has become a useful technique for analysing
 variations, periodicities, and trends in time series (Lu 2002, Xingang et.al, 2003; Coulibaly
 and Burn, 2004;Partal and Kucuk,2006).Smith et al(1998) applied discrete wavelet
 transform(DWT) to quantify streamflow variability and suggested that streamflow could be
 effectively classified into distinct hydroclimatic categories using DWT.The dynamical link
 between streamflow and dominant modes of climatic variability in the Northern Hemisphere
 was explored by Coulibaly and Burn(2004) using wavelet analysis.Labat (2005)reviewed the
 most recent wavelet applications in the earth science field and explained new wavelet
 analysis methods in the field of hydrology.

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         Combination of wavelet transformation with neural network models in time series
forecasting has been reported since few years in the different fields such as ocean
engg,Earthquake,hydrology( Cannas et al.,2005; Rao et al.,2009; Shiri and Kisi ,2010;
Nourani et al., 2011 ;Deka and Prahlada,2012; Wang et al., 2011). Wang and Ding (2003)
used Wavelet-ANN combination in hydrology to predict hydrological time series. A hybrid
wavelet predictor-corrector model was developed by Zhou et.al, (2008) for prediction of
monthly discharge time series and showed that the model has higher prediction accuracy than
ARIMA and seasonal ARIMA.All these studies revealed that Wavelet Transform is a
promising tool for precisely locating irregularly distributed multiscale features of climatic
elements in space and times
        Wavelet analysis is multiresolution analysis in time and frequency domain and is the
important derivative of the Fourier transform.The original signal is represented by different
resolution intervals using Discrete Wavelet Transform(DWT).In other words, the complex
discharge time series are decomposed into several simple time series using a
DWT.Thus,some features of the subseries can be seen more clearly than the original signal
series.These decomposed time series may be given as inputs to ANN which can handle non-
linearity efficiently,higher forecasting accuracy may be obtained. Forecasts are more accurate
than that obtained by original signals due to the fact that the features of the subseries are
obvious. This is why the hybridization of wavelet transformation and neural network can
performs better than single ANN model.
        In this work, an attempt has been made to investigate the potential and applicability of
Hybrid Model by combining Wavelet and ANN with objectives to address the above
mentioned scenarios using time series data of different frequencies for multistep lead time
forecasting. It is expected that this approach can improve the low level model accuracies in
long range (>1 time steps) flow forecasting. For this purpose, wavelet neural network
(WLNN) algorithm was introduced and employed to develop a river flow forecasting model
which has an ability to make forecasts up to 4timesteps leadtime using flow time series
observed data. The results of WLNN model were compared with the results obtained from
single ANN model.Also,the proposed WLNN model performance were evaluated to assess
the model efficiency in the higher lead times alongwith different decomposition levels.

2. WAVELET THEORY

        A Wavelet transformation is a signal processing tool like Fourier transformation with
the ability of analysing both stationary as well as non stationary data, and to produce both
time and frequency information with a higher(more than one) resolution, which is not
available from the traditional transformation.
Wavelet means small wave,whereas by contrast,sinus and cosinus are big waves(Percival and
Walden,2000).A function Ψ(.) is defined as wavelet if fulfill

                                         (1),                                  (2)
Commonly, wavelet is functions that have characteristic in equation(1),which is if it is
integrated on (-∞,∞) the result is zero, and the integration of the quadrate of function Ψ (.)
equals to 1 as written in equation (2).There are two functions in wavelet transform,i.e.scale
function(father wavelet) and mother wavelet. These two functions give a function family that
can be used for reconstructing a signal. Some wavelet families are Haar wavelet, which is the

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oldest and simplest wavelet, besides that there are Meyer wavelet, Daubechies wavelet,
Mexican Hat wavelet, Coiflet wavelet, and Last Asymmetric(Daubechies,1992). Ψa,b(t) can
be obtained by translating and expanding Ψ (t):

       Ψ a,b(t)=                       ,    a ϵ R, b ϵ R, a ≠ 0                      (3)

Where a,b(t) is successive wavelet; a is scale or frequency factor, b is time factor; R is the
domain of real number. The successive wavelet transform of finite energy signal or time
series f(t) ϵ L2(R)is defined as

       Wf(a,b)=                               dt                                     (4)

Where (t) is the complex conjugate function of (t); Wf(a,b) is the wavelet coefficient under
different resolution levels(scale) and different time.
In general, the time series are discrete, so the discrete form of eq.4 can be written as:


       Wf(a,b) =                                                                     (5)


    Where N is the number of discrete time step.∆t is the sample time interval.(Zhou et
al,2008)
Here, Wf(a,b) is the output of the time series f(t) or f(k ∆t) through the unit impulse response
filter.The characteristics of the original time series in time(b) and frequency domain (a) at the
same time can be reflected through this output.Frequency resolution of wavelet transform is
low for small value of ‘a’ with high time resolution.Also,for large ‘a’ value,it is just opposite.
There are many discrete wavelet transform algorithms, but Mallat algorithm (1989) has been
adopted due to simplicity and most efficient case for practical purposes for decomposition
and reconstruction of time series.
.It can be expressed as follows:
       cj+1 = Qcj for j=0,1,2……J and dj+1 = Gcj for j =0,1,2…..,J                    (6)
where Q and G are low pass filter and high pass filter respectively.If co represents the original
time series X,then X can be decomposed to d1,d2,….dj and cj through equation(6) where J is
the scale number.cj and dj are the approximated signal and the detail signal of original time
series under the resolution level 2-j,respectively.In flow time series,cj represents the
deterministic components like tendency,period and approximate period,etc.dj represents the
stochastic components and the noise.The basic idea of multiscale decomposition is trend
influences Low frequency(L) components that tend to be deterministic.Whereas,High
frequency(H) component is still stochastic.
The time series is decomposed into one comprising its trend(approximation) and one
comprising the high frequencies and the fast events(details)(Kisi,2009).The decomposition
signals can be reconstructed as follows:( Zhou et al,2008)
       Cj = (Q*)j cj and Dj = (Q*)j-1 G*.dj for j = 0,2,…..J                         (7)



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Where Q* and G* are dual operators of Q and G respectively.The reconstruction of
decomposition signals will increase the number of signals.The wavelet transform series
{D1,D2,…DJ,CJ},obtained by reconstructing d1,d2,…dj and cj,has the same length with the
original time series X and X = D1+D2+…DJ+CJ.

3. STUDY AREA

        The study area is located in the international river Brahmaputra main stream within
India. The two consecutive discharge gauging sites namely Pandu(u/s) and Pancharatna(d/s)
are selected for the study as these two stations recorded heavy discharge flows in the main-
stream of the Brahmaputra river causing frequent floods to the downstream area.
        The Brahmaputra originates in Tibet region in China is the fourth largest river in the
world in terms of average discharge at mouth, with a flow of 19,830 cumecs
(Goswami,1985). The hydrologic regime of the river responds to the seasonal rhythm of the
monsoons and to the freeze-thaw cycle of the Himalayan snow. The discharge is highly
fluctuating in nature. Discharge per unit drainage area in the Brahmaputra Basin River is
among the highest of major rivers of the world. The basin lies between latitudes 24013´ and
31030´9 North and longitudes 820 and 96049´ East. The catchment area upto pandu
station(u/s) is 500,000 sqkm and upto Pancharatna(d/s) is 532,000 sqkm. The location of the
two discharge gauging stations namely Pandu and Pancharatna are shown in the figure 1
below.




                                   Figure.1: Location of the study area


4. METHODOLOGY

       Here, considering the dominance of persistence in the flow time series, future river
flow to be forecasted from the past/previous flow data. River flow upto previous four time
steps were taken as predictor variables.The input scenarios formed by various predictor
configurations are;

[ 1] Q(t) [2] Q(t),Q(t-1) [3] Q(t),Q(t-1),Q(t-2) [4] Q(t),Q(t-1),Q(t-2),Q(t-3).

Where Q(t)—current discharge .Also Q(t-1),Q(t-2),Q(t-3) are onetime step,two time step and
three time step past discharge respectively. The predictand is Q (t+n) where n is the lead time.
These input and output scenarios are same for both ANN and WLNN model.Here, wavelet

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and ANN techniques are used together as a combined method.While wavelet transform is employed to
decompose discharge time series into their spectral bands, ANN is used as a predictive tool that
relates predictand (output) and predictors (inputs).
          The original non-stationary time series were decomposed into a certain number of stationary
time series through discrete wavelet transform such as, periodic properties,non-linearity and
dependence relationship of the original time series were separated.Hence, each wavelet transform
series has obvious regularities. Then the ANN model was used to simulate the wavelet transform
series in the form of approximations and details coefficients. Therefore, the prediction accuracy was
expected to improve.

4.1. Artificial neural network
         ANN is a flexible mathematical structure having an interconnected assembly of simple
processing elements or nodes, which emulates the function of neurons in the human brain. It possesses
the capability of representing the arbitrary complex non-linear relationship between the input and
output of any system. Mathematically, an ANN can be treated as universal approximators having an
ability to learn from examples without the need of explicit physics.
         In this study, A single layer feed forward network with a back propagation learning algorithm
has been selected for the ANN model .Here, TRAIN LM (Levenberg-Marquardt) learning function,
Tangent Sigmoid as transfer function has been chosen and the analysis was carried out for different
input scenarios of previous time steps discharge data. The optimal structure of the ANN is selected
based on mean square error during training. The ANN model implementation was carried out using
MATLAB routines.

4.2 Wavelet Neural Network (WLNN)
            In the proposed (WLNN) model, the Discrete Wavelet Transformation discretizes the
input data (Q) in to number of sub signals in the form of approximations and details and henceforth,
these sub signals has been used as input to ANN. The schematic diagram of proposed WLNN model
is shown in figure 2. The proposed Hybrid model which uses multiscale signals as input data may
present more probable forecasting rather than a single pattern input.
        The objectives of WLNN model is to forecast multitime steps ahead discharge from previous
time steps discharge.Here,future discharge are taken as predictand and past discharges as
predictor.After decomposing the time series into several resolution levels,each level subseries
predictand data is estimated from its corresponding separated predictor level. The proposed WLNN
model focused on improving the precision and prolonging the forecasting time period.
        Here, ANN part was constructed with appropriate sub-series belongs to different scales as
generated by DWT.These new series consists of details and approximations were used as input to
ANN .In the proposed WLNN model,only input signals were decomposed into wavelet coefficients so
that ANN was exposed to large number of weights attached with higher input nodes during training
.Hence,the higher adaptability can be achieved for input- output mapping.




                Figure.2: Schematic diagram of the proposed WLNN model


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4.3 Performance Indices
       The conventional performance evaluation such as correlation coefficient is seems to
be unsuitable for model evaluation (Legates and McCabe, 1999). However, Mean Absolute
Percentage Error and Mean Square Error can be used for better evaluation of model
performance. In this study, following performance indices based on goodness of fit are used.
                               2

          MSE =
                  ∑ (X − Y )             MAPE =
                                                  1   N

                                                      ∑
                                                             X −Y
                                                                    × 100
                      N              ,            N   i =1    X

Where, X=observed values, Y=predicted values, N = total number of values,

4.4 Data division
          Daily, weekly average and monthly average discharge data for 20 years has been
collected for both the gauging stations of Pandu and Pancharatna are divided in training and
testing sets. Initial 70% of time series flowdata were used for training and remaining 30%
time series data were used for testing. The analysis is carried out adopting Artificial Neural
Network using varying input scenarios. Later, hybrid model combination of Wavelet-ANN
was proposed to minimise the errors obtained in multistep lead time forecasting. The software
used for analysis is MATLAB (2009) using ANN and Wavelet Toolboxes.

5. RESULTS AND DISCUSSSION

        In this study, a number of ANN models has been developed and the best model
(optimised structure) out of various input combinations were selected.The best ANN model
testing results obtained for input three (3rd scenarios) with seven (7) neurons in the hidden
layer based on various performances indices were presented in Table 1 and Table 2 for Pandu
station and Pancharatna station respectively.It can be seen from the Table 1 and Table 2 that
MSE and MAPE values for ANN model are more than WLNN model using daily flow
data.Although the predictive performance of ANN model within acceptable accuracy such as
MAPE is 4.34% for Pandu and 3.81% for Pancharatna but well below than the WLNN
performance.The mean squared error (MSE) also followed the similar trend with high error
for ANN model.
                     Table 1 Testing results at station Pandu(1day ahead)

Decom                                     WLNN                                      ANN
positio            DB-4                COIFLET-2                 SYMHLET-4
n Level       MSE      MAPE           MSE    MAPE                MSE   MAPE      MSE      MAPE
                 6                       6                          6
             (x10 )     (%)          (x10 )   (%)               (x10 )  (%)     (x106)     (%)
           (cumec)2                (cumec)2                   (cumec)2         (cumec)2
   1          0.96      1.22          8.11    4.01              15.19   4.05     23.40    4.34
   2          7.20      3.96          7.29    3.97              13.64   4.72
   3          2.77      1.38          7.42    3.92              13.72   4.74
   4          3.03      1.95          8.21    4.03               8.73   2.97
   5          2.10      1.52          7.59    4.06              13.63   4.73

This may be due to significant fluctuations of the data around mean value such as skewness
and standard deviation are high,where short term regression between data is minimised.

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.From the Table 3, it is clear that the ANN model doesnot perform well in longterm
streamflow forecasting.The results of weekly forecast are relatively weaker than those of
daily forecasts and the monthly forecast are quite weak with higher MAPE and MSE.

                    Table 2 Testing results at Pancharatna station (1day ahead)

 Decomp                                 WLNN                                          ANN
 osition          DB-4                COIFLET-2           SYMHLET-4
  Level       MSE    MAPE            MSE    MAPE          MSE   MAPE          MSE       MAPE
             (x106)    (%)          (x106)   (%)         (x106)  (%)         (x106)      (%)
            (cumec)                (cumec)              (cumec)             (cumec)
                2                      2                   2                      2

     1        3.52         1.72     3.94       2.03      4.24       2.34      9.21          3.81
     2        4.42         2.93     4.83       2.97      4.98       3.52
     3        8.91         2.73     9.23       3.12      10.02      3.73
     4       10.81         3.49     11.45      3.56      11.56      3.64
     5        6.30         2.48     7.12       2.75      7.85       2.85

        Also the best input combinations are not same for all prediction intervals.Input
combination 3 is coming best for daily forecast and combination 2 is best for weekly and
monthly forecast.For multistep leadtime forecasting, again optimal combinations were
different which consists of more lagged discharge.
        In the second stage, for hybrid wavelet neural network (WLNN) model, pre-processed
discharge time series data were given to ANN model to improve the model accuracy by
adopting proper selection of wavelet type and decomposition levels. For these objectives,
Discrete Wavelet Transformation (DWT) was used and various type of wavelets such as
Daubechies wavelet order-4 (DB-4), COIFLET-2, SYMHLET-4(Daubechies, 1992; Mallat,
1998) were selected as a mother wavelet considering the shape similarity with time series
signal. The selected mother wavelets are of exact reconstruction possibilities and are
compactly supported and Asymmetric in shape.
        Similar to ANN models, here also a number of WLNN models were developed using
different input combinations (mentioned earlier) with different ANN architecture. The best
results in terms of performance indices were obtained for third input scenarios (three inputs)
for various decomposition levels and results are presented for all three type of mother wavelet
such as DB-4,COIFLET-2 and SYMHLET-4 in Table 1 and Table 2 for one day leadtime for
both the stations.
        In this work, the effects of various decomposition levels on model efficiency have
also investigated to optimize the result. The output result from the discrete wavelet
transformation in the form of ‘approximations’ and ‘details’ sub signals at different levels are
presented in figure 3 for db-4.The mechanism inside the network was somewhat transparent
in WLNN. When coefficients are used as inputs, as the number of input layers increases
accordingly number of weights also increases. The analysis has been done for different
decomposition levels from level 1 to 5 to obtain optimal results. In each case, as the
decomposition level increases, the number of input layers also increases and the network was
trained and tested accordingly.


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         Figure 3: DB-4 Sub signals after data decomposition through DWT at level-5

        The results from the above model (WLNN) for different decomposition levels clearly
revealed the better performance of the proposed model (WLNN) both in low as well as higher
lead time compared to ANN ( Table 3 ), considering various performance indices. The basic
WLNN model of decomposition level 1 (L-1) with DB-4 mother wavelet was performing
better than best ANN model and other type of wavelets considering coefficient of efficiency
and least error criteria. Also, other WLNN improved models based on different
decomposition levels (L-1,L-2,L-3, L-4, and L-5) performed better than ANN model.Also,the
other type of wavelets are performing similar to DB-4 but well above ANN performance.
         For shorter lead times, performances of WLNN models were almost similar to ANN
and observed no significant variations. But in the higher lead time forecast, significant
variations were observed among the performance of WLNN models. For low lead time with
low decomposition levels, the model is performing in a better way than in higher lead times.
        Again from the time series plot in figure 4 for one day leadtime prediction, it was
observed that the ANN and WLNN model results were closely following the observed data

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both in low and high flow.The scatter diagram shows close agreements between WLNN
model results and observed flow as shown in figure 5. As lead time increases, the
performances of ANN decreases drastically but, WLNN performance decreases gradually as
the variation of MAPE for different lead time forecasting was presented for ANN and WLNN
in Table 3. Similarly for the Pancharatna station, WLNN model performance were similar to
Pandu station as the statistical behaviour were different but capturing the hidden knowledge
as presented in Table 1 and Table 2 in flow forecasting.

        Table 3 Testing results at both the stations (weekly and monthly data)
    StationModel       1 week lead        4 week lead       1 week lead     4 week lead
            type      MSE MAPE MSE MAPE MSE MAPE MSE MAPE
                     (x106)    (%)      (x106)     (%)    (x106)    (%)   (x106)    (%)
   PANDU WLNN 5.24             3.66      6.78     4.22     5.76    7.14    7.11    9.27
           ANN        6.76     3.93     11.23     4.97     8.21   14.21 12.34 21.79
  PANCHA WLNN 5.42             3.57      6.87     4.12     5.67    5.45    6.17    8.27
           ANN        7.67     3.89     11.32     5.79     8.12    7.21   12.53 22.97




               Figure.4: Model performance in testing for 1day leadtime at Pandu




       Figure.5: Scatter plot of Observed and WLNN for 1 day leadtime at Pandu
       The main reason for this improvement is that the WLNN model can extract the
behaviour of discharge variation processes through decomposing the nonstationary time
series of daily, weekly and monthly discharge into several stationary time series.These

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stationary time series can exhibit the finer details of the discharge time series and also
reducing the interference between the deterministic components and the stochastic
components. Hence the stability of the data variation has increased which results in improved
prediction accuracy.

5.1 Effects of decomposition level and type of wavelet
        In the WLNN, the results obtained for different lead times has undergone different
decomposition levels starting from 1 to 5 for various type of mother wavelets. In each lead
time analysis, there was an increasing trend in performance error from low decomposition
levels towards higher decomposition levels as presented results shown in Table 1and Table
2.This may be because of higher decomposition levels lead to a large number of parameters
with more complex nonlinear relationships in the ANN . This follows the net errors results
reducing the model performance. In this study,level-1 may be considered as best and
appropriate decomposition level alongwith DB-4 mother wavelet for the given data sets and it
was considered as the best model among the WLNN models.
        Based on the results, it is noticed that the number of decomposition levels has little
impact on the predictive performance of WLNN models. Since the random parts of original
time series were mainly in the first resolution level, the prediction errors were also mainly in
the first resolution level. Thus the errors were not increased proportionately with the
resolution number. Again for higher lead time forecast, higher model efficiency was obtained
at selected decomposition levels. These may be due to the effect of correlation of more
smoothened signals with flattened variability between the inputs and output.

6. CONCLUSIONS

        In this study, a hybrid model of wavelet and ANN (WLNN) has been developed to
forecast discharge for higher lead times such as daily, weekly and monthly at two gauging
stations of India. The accuracy of WLNN model has been investigated for forecasting river
discharge in the present study by adopting various decomposition levels with respect to
different type of mother wavelets. The WLNN models were developed by combining two
techniques such as ANN and DWT.The WLNN model results were also compared with
single ANN model in the study. The WLNN and ANN model performance were tested by
applying to different input scenarios of past discharge data at the two gauging stations of the
river Brahmaputra in Assam within India. The accuracy of WLNN models was found to be
much better than ANN model in modeling for all time steps flow discharge value.The
irregular and asymmetric shaped DB-4 wavelet provides better results than other models for
all the decomposition levels showing superiority at the first level. The proposed hybrid
WLNN model plays an important role in improving the precision and prolonging the
forecasting time period of hydrological time series.The appropriate selection of mother
wavelet and decomposition levels also remains as partially conclusive as further analysis are
required with more lengthy and more stations data.The suggested strategy can be adopted to
other selective hydrological time series of similar statistical behaviour.

ACKNOWLEDGEMENT

       The authors greatly acknowledged the support provided by Water Resources
department, Govt. of Assam, India for providing the necessary data for the analysis.

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