Structure and m are the numbers of neurons

Structure of neural
networks include 3 separate layers: 1) input layer which is responsible for introducing the data to the model, 2) hidden
layer (s) where the data are processed, and 3) the output layer to produce
results. Each layer comprises one or multiple elements known as a neuron. A schematic view of a neural network is
demonstrated in Fig. 2. A number of
neurons in the input, hidden, and output
layers depend on the problem type and are determined based on the difficulty level of
the problem. In case an insufficient number of neurons is selected, the network
may not demonstrate an appropriate degree of freedom for training purposes. On
the other hand, in case of selecting a large
number of neurons for the hidden layer, the learning process can take a
considerably long time to complete. A number
of neurons in input and output layers is constant and depends on the number of
input and output parameters. Gamma test can be used to determine optimal
parameters for the input layer. Although the number of neurons in the hidden
layers is determined through trial and error (Salehnia et
al., 2013), it is suggested that the number of neurons
in the hidden layer should be within the range n-m, where n and m are the numbers
of neurons in the input and output layers, respectively.

 

Table 1: Basic statistics of the measured water quality
variables in Karun River, Iran

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Kurtosis

Skewness

SDc

Mean

Maxb

Mina

Unit

Variable

Number for Embedding

26.743

4.077

45.76

40.00

344.00

2.00

NTU

Turbidity

1

26.496

4.443

53.68

64.00

420.00

20.00

mg/L

SS

2

0.492

.226

14.50

168.49

206.00

133.00

mg/L

TA

3

8.503

4.024

0.02

0.02

0.12

0.01

mg/L

PO4

4

-1.066

.010

5.70

22.41

35.70

12.00

?C

Temperature

5

3.777

.074

1.65

6.65

13.53

1.92

mg/L

NO3

6

9.539

.398

0.006

0.01

0.05

.010

mg/L

NO2

7

-0.641

-.496

0.14

0.51

0.85

0.28

mg/L

NH4

8

-1.833

-.282

46540.76

61887.83

110000.00

2100.00

U/100mL*

Total coliform bacteria

9

-0.323

2.112

366.88

1711.51

2585.00

856.00

mg/L

TDS

10

-0.294

.351

573.79

2692.04

4040.00

1300.00

S/m?

EC

11

0.280

-1.631

0.22

7.60

8.00

6.90

pH

12

0.625

-.110

2.26

9.24

15.51

2.83

mg/L

SO4

13

3.233

-.240

0.39

3.24

4.34

1.76

mg/L

HCO3

14

-0.616

-.485

4.46

15.47

27.00

7.55

mg/L

Cl

15

6.986

.112

1.63

7.71

15.60

3.37

mg/L

Ca

16

0.000

-.472

1.30

4.64

7.80

1.51

mg/L

Mg

17

-0.585

.431

4.43

15.81

26.48

7.04

mg/L

Na

18

1.804

1.669

101.45022

618.05

905.00

268.50

mg/L

TH

19

0.558

.675

0.96

3.3115

6.22

1.08

mg/L

BOD

-0.314

.078

5.09

15.81

28.40

8.40

mg/L

COD

0.558

-.248

1.22

7.42

10.00

3.80

mg/L

DO

N=74.

a Min: minimum.

b Max: maximum.

c SD: standard deviation.

*=Unit is count per 100 mL

 

Fig. 2: A typical artificial neural network

 

Employing artificial
neural networks (ANNs) is the most common method to solve complex, nonlinear
mathematical problems. Similarly, multilayer perceptron (MLP) is the most
widely used types of neural network in solving such problems. In order to
create an MLP neural network, the appropriate threshold function, weight, and
bias should be determined for each neuron. During training of neural networks,
the weight and bias of each neuron are altered until their favorable values are
obtained. The most important threshold functions used in the development of MLP models include Gaussian,
sigmoid, and tangent sigmoid.

(1)

 

(2)

 

(3)

 

In this study,
parameters of calcium and magnesium were selected as the input, and parameters
of nitrate and nitrite were selected as the output. According to the literature
studies, no randomization was conducted on the data of water quality.
Therefore, in order to predict the water quality of Karun river, the data of
water quality were divided into two categories according to (Basant et al., 2010). These categories included training and validation data, each comprising
50 items (80 percent) and 24 items (20 percent) of the total data. Regarding
the first station, the two categories included 29 and 8 items of the data, and
for the second station, 31 and 8 data were included.

Some drawbacks might
be observed in the performance of the neural network due to the difference
between the maximum and minimum ranges
for each parameter as well as the different type of each variable. Therefore,
it seems necessary to convert the parameters into a dimensionless interval so
as to standardize them. The general formula for standardization within the
interval (a, b) is as follows:

(4)

 

where xs and xo are the original and normalized observational parameters,
respectively. a and b represent the upper and lower limits of standardization. xmin and xmax
indicate the maximum and minimum values of parameter x, respectively. Since a
and b are considered zero and one in the present study, respectively, the
formula is further simplified as:

(5)

 

Moreover, Marquardt algorithm was used to
train the neural network, since according to literature studies, this method is
more powerful and faster than the other existing methods. An optimal number of hidden layers was obtained
through trial and error and based on the proposed domain by
Ehteshami (2014) for Karun river.

2.3. Gamma test

As is
described in the previous section, in order
to determine the optimum neuron of the input layer, it is helpful to use gamma test
(GT). This method is one of the most important procedures to select a useful predictor from a database. Since GT has
been used in many studies in the field of ANN (Tian et al., 2016), it was then used in the presented study.

A formal proof of GT
was extended by Chang et al. (2010). By supposing a set of data observation in the following form:

(6)

Where,

 are input vectors
confined to some closed bounded set

and,

 are corresponding
outputs. The system of GT can be expressed in
the following form:

(7)

Where f is a smooth function and r is a random variable representing
noise. In general, the mean of the distribution of r is assumed as 0 and the
variance of the noise (Kim and Kim, 2008) is bounded. The gamma statistic

 is the main parameter,
which can estimate the model’s output variance.

For each vector xi,

the

 are the kth

 nearest neighbors

 

(8)

Where,

 denotes Euclidean
distance, and the corresponding Gamma function of the output values:

(9)

Where y is the corresponding y-value for the kth nearest
neighbor of xi in Equation (8). In order to compute

 the

 points

 are calculated by
univariate linear regression equation with least-squares:

(10)

The value of

is the intercept of the Equation (9). A is a gradient of a line
that describes the complexity of the
model. The high value of A show
more complexity and low one indicate less complexity. Another term that can
describe invariant noise called Vratio:

              

(11)

Where,

is the variance of output y. According to the
definition of Vrario,
the value of Vrario
close to 0 indicate a high degree of predictability of given output y.
In addition, the estimation of noise variance on the given output can be more
credible if the standard error (SE) is close to 0.

GT estimate the mean square error (MSE) of noise variance
which cannot be modeled by the smoothest
possible model (Goyal et al., 2013).

2.4. Performance evaluation of models

In order to assess the generated neural networks, four metrics, namely
RMSE, MAE, and R2 were employed. RMSE metric represents the error of
the model and is defined according to Relation 6. MAE metric determines over-
and underestimation. The coefficient of
determination (R2) represents the percentage of the variables which
can be estimated by the model, and is calculated as follows:

(6)

(7)

(8)

 

In the above equation,

,

 and n are respectively the
representatives of  predicted values,
observed values and the number of data.

3. Results
and discussion

In this study, gamma test was used to omit less-effective
parameters. However, this procedure also reduced the number of input parameters
of the neural network. Table 2 shows the results of gamma test for BOD
simulation, using Scenario 1, Scenario 2 and Scenario 3. In row “embedding” in
the table, different types of input parameters for each station are determined.
Here, 0 is assigned to the parameter which is
not considered as an input for ANN model and 1 is assigned for the considered parameter as an ANN model input.  The ordering
of 0, 1 given in Table 2-4 (1st row) is the same as the ordering of
parameters given in Table 1.

As shown in Table 2, the best inputs to develop
ANN model to estimate BOD for station 1 are
Turbidity, SS, TA, Temperature, NO2, Total coliform bacteria, TDS, EC, pH, SO4, HCO3,
Cl. In this station, the number of neural network inputs decreased from
19 to 13 parameters. In station 2, the number of input parameters to achieve the best
gamma was reduced to 11. In this station, Turbidity, TA, PO4, NO3, NO2, NH4, TDS, EC, pH, SO4, and HCO3 were determined as the most
optimal input parameters for ANN model to estimate BOD. Furthermore, if data in
both stations 1 and 2 are simultaneously used in gamma test, the number of
input parameters will reduce to 10. Applying gamma test for the data collected
from station 1, station 2 showed that TA, NO2, EC, pH, SO4, and HCO3 must be selected as input parameters in all three scenarios.
Magnesium, sodium,
and pH were not thus selected as inputs, at all.

.

 

Table 2: The best selective masks and their performance
criteria for BOD

Both of station

Station 2

Station 1

Parameters

1011001110111100000

1011011101111100000

1110101011111111000

Embedding

0.0550

0.0001

0.0242

Gamma statistic

0.0963

0.0843

0.0700

Gradient

0.0359

0.0239

0.0692

Standard error

0.2200

0.0005

0.0970

V ratio

 

 

Table 3: The best selective masks and their
performance criteria for COD

Both of station

Station 2

Station 1

Parameters

0011110010000101000

1111110100011010000

0111010010100000100

Embedding

0.0379

0.0001

0.0001

Gamma statistic

0.1394

0.0902

0.1377

Gradient

0.0350

0.0349

0.0264

Standard error

0.1518

0.0001

0.0001

V ratio

 

 

Table 4: The best selective masks and their
performance criteria for DO

Both of station

Station 2

Station 1

Parameters

1101000111101000000

1001000111101010000

1111100000001001000

Embedding

0.0440

0.0145

0.0001

Gamma statistic

0.1782

0.1985

0.1642

Gradient

0.0302

0.0609

0.0516

Standard error

0.1761

0.0582

0.0001

V ratio

 

The Results of gamma test
for COD estimation are shown in Table 3. Applying
gamma test for data from station 1, showed that the best parameters for COD
simulation include SS, TA, PO4,
Temperature, NH4, TDS and Mg. While, gamma test results implied that Turbidity,
SS, TA, PO4, Temperature, NO3, NH4, pH, SO4,
and Cl are the best input parameters for COD simulation considering data in
station 2. Furthermore, Aggregation of data in station 1 and 2 for gamma test
demonstrated that TA, PO4, Temperature, NO3, Total coliform bacteria, HCO3 and
Ca should be selected as input for COD
simulation (Table 3). Moreover, Table 4 shows different inputs for Do simulation
due to applying gamma test under three scenarios.

 

 

Table 5: Results of ANN to predict BOD, COD and
DO

Correlation coefficient

MAE (mg/L)

RMSE (mg/L)

Stage

Station

Variable

0.89

0.0411

0.0090

Training

St. 1a

BOD

0.91

0.0395

0.0112

Testing

0.88

0.0452

0.0084

Training

St. 2b

0.84

0.0421

0.0142

Testing

0.81

0.0357

0.0093

Training

Both station

0.80

0.0574

0.0132

Testing

0.89

0.0573

0.0112

Training

St. 1

 
COD
 

0.85

0.0596

0.0089

Testing

0.89

0.0695

0.0183

Training

St. 2

0.87

0.0609

0.0297

Testing

0.81

0.0984

0.0401

Training

Both station

0.79

0.0594

0.0114

Testing

0.82

0.0325

0.0085

Training

St. 1

DO

0.81

0.0338

0.0086

Testing

0.84

0.0500

0.0184

Training

St. 2

0.86

0.1606

0.0528

Testing

0.82

0.0626

0.0086

Training

Both station

0.84

0.0770

0.0125

Testing

a St. 1: station 1.

b St. 2: station 2.

 

Based on gamma test analysis, it can be
inferred that SS, TA, and temperature are
the common parameters for COD simulation for all three scenarios. While phosphate was also the only common parameter for DO simulation under three
scenarios. In a study on the Karun River, Emamgholizadeh et al. (2014) investigated the sensitivity of MLP model to
input parameters using omitting them one by one. Although they used fewer
parameters; however, they reported similar results regarding the little impact of parameters such as Ca and Mg
in predicting BOD, COD and DO. Phosphate and turbidity were used in most
scenarios to predict BOD, COD and DO and their effectiveness in determining these
parameters can be expressed. These results correspond with findings of Emamgholizadeh et al. (2014) and Singh et al.
(2009). Phosphate plays an important role in
oxidation as well as the energy-release process and its increment, increases
the number of microorganisms (Singh et al.,
2009). Turbidity is an important parameter in
determining the self-purification and the amount of dissolved oxygen in the
river (Talib and Amat,
2012). Therefore, it plays an important role in the simulation of the quality of the Karun River
water.

 

 
 
 

 
 
 

 
 

 
 
 
 

 
 
 

 
 
 
 
 

 

Fig. 3: Scatter plots of observed and predicted BOD using
the data of station 1 (top panel), station 2 parameters (middle panel) and both
station (bottom panel): a training and b testing.

 

 

ANNs results for simulation of BOD, COD and DO
are shown in Table 5. Statistics of RMSE, MAE and coefficient correlation were
used to compare simulation results with observed data.
RMSE and MAE values show that the neural
network could well predict these parameters. These results correspond to those
of Emamgholizadeh et al.’s study (2014). However, the presented study has improved RMSE
and MAE. In the present study, two phases of “training” and “testing” were
employed. Comparing corresponding results of each phase shows that the
current networks has sufficient accuracy
for simulation of desired parameters. The Values of RMSE
and MAE gave in Table 5 indicate the ANNs
for simulation BOD have more appropriate performance than the other ANNs under
three scenarios.

However, All the ANNs have the acceptable performance to simulate BOD, COD and
DO in training and testing phases. Figures 3 to 5 show that how well the
predicted values of BOD, COD and DO match measured values of those parameters.
As shown in these figures all developed ANN models present acceptable
simulations.  The

, RMSE and
MAE for BOD simulation models indicate that ANNs
models developed using stations 1 and 2 data will lead to better results
compared to the models developed using aggregated data of two stations. The
values of V-ratio and gamma justify these results. Due to the long distances
between stations 1 and 2 and the entry of various kinds of pollutants to the
river between the two sampling stations, significant changes affect water
quality parameters. This results in significant increases or decreases in a different parameter in comparison with station
1.Therefore aggregation of both stations data reduces the coefficient of
determination and the accuracy of MLP model in the prediction process. For COD
simulation,

 values were 89%, 89% and 81% in training phase
under three scenarios respectively. In addition, , RMSE values for all the scenario
were 0.0112, 0.0183 and 0.0401 mg / L respectively and MAE values were 0.0573,
0.0695 and 0.0984 mg / L in the training phase. According to the results, in
the training” phase, the use of data from station 1 (scenario1) relatively well
estimate COD relating to use of station 2 data or aggregated data. The same
results can be inferred for the “testing phase”. Results of DO simulation
also showed that separate use of stations 1 and 2 data will lead to more
accurate results.

 

 

 
 
 
 

 
 
 

 
 

 
 

 

 

Fig. 4: Scatter plots of observed and predicted COD using
the data of station 1 (top panel), station 2 parameters (middle panel) and both
station (bottom panel): a training and b testing.

 

 
 
 

 
 

 
 
 
 
 

 
 

 
 

 

 

Fig. 5: Scatter
plots of observed and predicted DO use
the data of station 1 (top panel), station 2 parameters (middle panel) and both
station (bottom panel): a training and b testing.

4. Conclusion

In this study, artificial neural network and sensitivity analysis -using
gamma test- were used to estimate the quality of the Karun River. In this
regard, BOD, COD and DO were selected as
significant parameters to be estimated using 19 predictor parameters. In doing so,
recent data obtained from the two water
quality stations located before and after the city of Ahvaz, by Khuzestan Water
and Electricity Authority were used. Results
showed that in BOD simulation, TA, NO2, EC, pH, SO4,
and HCO3 were the most significant parameters; however,
the parameters of magnesium, sodium, and
pH have no significant effect on ANN model.
In Addition, phosphate has the most effect on DO Simulation. While phosphate and temperature were a more sensitive parameter in COD simulation
compared to other parameters. Low RMSE and MAE values and high correlation
coefficient values showed that neural network can well simulate parameters of
BOD, COD and DO in Karun River due to the