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Evaluation of neuro-fuzzy GMDH-based particle swarm optimization to predict longitudinal dispersion coefficient in rivers

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Abstract

In the present research, neuro-fuzzy-based group method of data handling (NF-GMDH) has been applied to evaluate the longitudinal dispersion coefficient in rivers. The NF-GMDH model has been improved through particle swarm optimization algorithms (PSO). Effective parameters on the longitudinal dispersion coefficient including flow depth, channel width, cross-sectional average velocity, and bed shear velocity were selected to characterize a correlation between input and output variables. Field and experimental data sets have been collected from different studies. The efficiency of the proposed NF-GMDH-PSO model for both training and testing stages has been investigated. The performance of the NF-GMDH-PSO model were compared with those obtained from the differential evolutionary (DE), model tree (MT), genetic algorithm (GA), artificial neural network (ANN), and traditional empirical equations. Results analysis showed that among the artificial intelligence approach-based equations, DE and GA methods performed better than the other methodologies. The most accurate empirical equations were also introduced. NF-GMDH-PSO network also predicted the longitudinal dispersion coefficient properly and can be considered as an alternative to the aforementioned successful formulas.

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Abbreviations

A :

Cross-sectional area

\(a_{kj}\) :

Constant value for the corresponding fuzzy rule

\(a_{k,j}^{pm}\) :

Parameter of the kth Gaussian function that is utilized for the jth input variable from the mth model and pth layer

B :

Channel width

BIAS:

Bias of the predicted values with respect to the measured ones

\(b_{kj}\) :

A constant value for the corresponding fuzzy rule

\(b_{k,j}^{pm}\) :

Parameter of the kth Gaussian function that is utilized for the jth input variable from the mth model and pth layer

C :

Cross-sectional average concentration

DR:

Discrepancy ratio

E :

Error parameter of the network

\(F_{kj} (x_{j} )\) :

Gaussian membership function of the kth fuzzy rule in the domain of the jth input value x j

f :

A function

H :

Flow depth

h :

Local flow depth

i :

Counter of input data set

j :

Counter of input value or input value

K :

Number of fuzzy rules or Gaussian functions

k :

Counter of fuzzy rule or Gaussian function

k x :

Dispersion coefficient in longitudinal direction

\(k_{{xi\left( {\text{Actual}} \right)}}\) :

ith measured k x value

\(k_{{xi\left( {\text{Model}} \right)}}\) :

Prediction of the ith measured k x value

\(\overline{{k_{x} }}_{{\left( {\text{Actual}} \right)}}\) :

Mean value of the \(k_{{xi\left( {\text{Actual}} \right)}}\) values

\(\overline{{k_{x} }}_{{\left( {\text{Model}} \right)}}\) :

Mean value of the \(k_{{xi\left( {\text{Model}} \right)}}\) values

k y :

Dispersion coefficient in lateral direction

k z :

Dispersion coefficient in orthogonal direction to the bed

M :

Number of partial descriptions in each layer or total number of the measurements

m :

Counter of model

p :

Counter of layer

R :

Correlation coefficient

RMSE:

Root means square error

U :

Cross-sectional average flow velocity

\(U_{*}\) :

Bed shear velocity

\(u'\) :

Deviation of local mean flow velocity from the cross-sectional mean flow velocity

u k :

Compatibility degree of the premise part of the kth fuzzy rule

w k :

Real value for kth fuzzy rule

\(w_{k}^{pm}\) :

Weight parameter of the kth Gaussian function in mth model and pth layer

x :

Coordinate in streamwise direction

x j :

jth input value

y :

Coordinate in lateral direction or final output parameter

\(y^{ * }\) :

Observed value

\(y^{pm}\) :

Output of the mth model in pth layer

\(\varepsilon_{t}\) :

Transverse mixing coefficient

\(\varepsilon_{t0}\) :

Dimensionless transverse mixing coefficient

\(\mu_{k}^{pm}\) :

kth Gaussian function in the mth model and pth layer

\(\prod {}\) :

Algebraic product operator

\(\varSigma\) :

Summation operator

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Authors and Affiliations

  1. Department of Civil Engineering, Graduate University of Advanced Technology-Kerman, P.O. Box 76315-116, Kerman, Iran

    Mohammad Najafzadeh

  2. Dipartimento di Ingegneria Civile, Università della Calabria, Via P. Bucci, Cubo 42b, 87036, Rende (CS), Italy

    Ali Tafarojnoruz

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Correspondence to Mohammad Najafzadeh.

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Najafzadeh, M., Tafarojnoruz, A. Evaluation of neuro-fuzzy GMDH-based particle swarm optimization to predict longitudinal dispersion coefficient in rivers. Environ Earth Sci 75, 157 (2016). https://doi.org/10.1007/s12665-015-4877-6

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