# GMDH algorithms applied to turbidity forecasting

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## Abstract

By applying the group method of data handling algorithm to self-organization networks, we design a turbidity prediction model based on simple input/output observations of daily hydrological data (rainfall, discharge, and turbidity). The data are from a field test site at the Chiahsien Weir and its upper stream in Taiwan, and were recorded from May 2000 to December 2008. The model has a regressive mode that can assess the estimated error, i.e., whether a threshold has been exceeded, and can be adjusted by updating the field input data. Consequently, the model can achieve accurate estimations over long-term periods. Test results demonstrate that the 2006 turbidity prediction model was selected as the best predictive model (RMSE = 5.787 and CC = 0.975) because of its ability to predict turbidity within the acceptable error range and 90 % required confidence interval (50NTU). 70(3,1,1) is the optimum modeling data length and variable combinations.

## Keywords

GMDH Turbidity forecast Nanhua Reservoir Chiahsien Weir Over-basin diversion## Introduction

Water consumption in Taiwan has increased significantly in recent years. The Water Resources Agency and the Taiwan Water Corporation have raised certain issues regarding the quantity and quality of water. According to statistical data, the island’s average annual rainfall is approximately 2515 mm. Despite its abundance, rainfall is unevenly distributed in terms of both time and space. Because of the island’s steep natural terrain, short river flows, and geological weaknesses, the majority of rainwater flows out to sea before it can be harnessed for public use. Thus, reservoirs are an essential means of realizing effective water usage. From the viewpoint of water resource management, both the availability and quality of water are a concern.

One of the water resources of the Nanhua Reservoir is the discharge of the Cishan River, which is diverted through a tunnel from the Chiahsien Weir. The majority of water diversion occurs during the annual wet period, from June to October, which is also the typhoon season. Because of the adverse effect of soil degradation in the upstream catchment area, heavy rainstorms rapidly and significantly increase the Cishan River discharge; they also increase the sand content and turbidity. If this flow is allowed to persist and enter the Nanhua Reservoir, the level of reservoir sediment will undoubtedly increase, potentially shortening the lifespan of the reservoir and creating problems for the operation of the Nanhua water-treatment plant.

This study examines the relevant hydrological data variables that influence water turbidity in the Chiahsien Weir. A unique group method of data handling (GMDH) multilayer algorithm is used to deduce the relationship between groups of input variables and output functions. The result is combined into a suitable set of higher-order nonlinear equations that engender a simple turbidity-forecasting model. This enables the prediction of water turbidity, and provides pertinent reference turbidity information for the Chiahsien Weir water diversion operation.

## Methodology

The GMDH algorithm introduced by Ivakhnenko (1968) is a heuristic self-organization process that establishes an input–output relationship within a complex system. It utilizes a multilayered conceptual structure, similar to a feed-forward multilayer neural network. Ikeda et al. (1976) added a recursive procedure to the GMDH algorithm to utilize updated observation data and to modify parameters within the nodes of each layer, enabling time-variable modeling. They subsequently applied the enhanced model to the prediction of daily river flows. Tamura and Kondo (1980) utilized the prediction of sum-of-squares or Akaikes’s information criterion as parameter selection indicators. Because the algorithm can easily generate high-level nonlinear terms, this nonlinear dynamic system can be well defined; however, its practicality would be seriously reduced. In response, Yoshimura et al. (1982) improved the model with a stepwise regressive procedure, returning the complex final system to a low-level nonlinear system, thereby increasing its applicability.

The GMDH algorithm enables the automatic selection of input variables during model construction, as well as a hierarchical polynomial regression of necessary complexity (Farlow 1984). Specific functional dependence between the input and output variables is unnecessary, as the dependence has been incorporated into the modeling structure. The GMDH algorithm has been applied in various fields, e.g., weather modeling, pattern recognition, physiological experiments, cybernetics, medical science, education, ecology, safety science, economics, and hydraulic field engineering systems (Lebow et al. 1984; Ivakhnenko et al. 1994; Kondo et al. 1999; Chang and Hwang 1999; Sarycheva 2003; Pavel and Miroslav 2003; Hwang et al. 2009; Tsai et al. 2009; Najafzadeh et al. 2013, 2014, 2015; Najafzadeh 2015). Nevertheless, few studies have explored turbidity modeling.

### GMDH algorithm

The GMDH algorithm is a kind of feed-forward network, normally classified as a special type of neutral network. The model’s underlying concept resembles animal evolution or plant breeding, as it adheres to the principle of natural selection. The multilayer criteria preserve superior networks for successive generations, eventually yielding an optimal network. This network (equation) more closely describes the physical phenomena that the model is intended to simulate. The self-organization algorithm can be classified as GMDH, SGMDH (stepwise regressive GMDH), and recursive/sequential GMDH. These model types are described below.

*t*) is the output variable,

*X*(

*x*

_{1},

*x*

_{2},

*…*,

*x*

_{ m }) is the vector of input variables, and

*A*(

*a*

_{1},

*a*

_{2},

*…*,

*a*

_{ m }) gives the vector coefficients or weights.

The GMDH model based on heuristic self-organization was developed to overcome the complexity of large-dimensional problems. It first pairs variables that might affect the system, and sets a default threshold to eliminate variables that cannot achieve a certain level of performance. This procedure describes a self-organization algorithm; it is a fundamental concept of derivative hierarchical multilevel models. The GMDH was built according to the following steps:

#### Step 1: Divide the original data into training and test sets

The original data are separated into training and test sets. The training data are used to estimate certain characteristics of the nonlinear system, and the test data are then applied to determine the complete set of characteristics.

#### Step 2: Generate combinations of input variables in each layer

*r*input variables are generated for each layer. The number of combinations is given by:

*m*is the number of input variables and

*r*is usually set to two (Ivakhnenko 1971).

#### Step 3: Optimization principle for elements in each layer

*r*

_{ i }is the RMS,

*t*= 1, 2,…

*n*,

*n*represents the length of the measurement data,

*y*(

*t*) is the measured value at moment

*t*; and

*Z*

_{ i }

^{ k }(

*t*) is the output value of element

*i*in layer

*k*.

#### Step 4: Stopping rule for multilayer structure generation

By comparing the index value of the current (competent) layer with that of the next layer to be generated, further layers are prevented from being developed if the index value does not improve or falls below a certain objective default value; otherwise, Steps 2 and 3 are repeated until the value matches the limited condition set above.

After the above steps have been completed, all competent elements in each layer are recombined as an optimum high-level nonlinear equation. This is utilized as the final model for turbidity forecasting.

### Stepwise regressive GMDH algorithm

The process of the stepwise regressive GMDH algorithm is very similar to that of the original GMDH algorithm. The key difference is that the least-squares method is replaced by a stepwise regressive procedure in Step 2. This procedure evaluates the optimum forward state, and determines whether it is more accurate than the next variable to be introduced. If so, it is incorporated into the model; otherwise, it is deleted to ensure the most precise simplified system equation. The assessment method employs the *F*-test for statistical analysis.

### Recursive/sequential GMDH algorithm

*n*-set of data, and the parameters (\(\theta\)) of the newly composed equations of each layer are forecast as \(Y_{n} = X_{n} \times \theta\). When the

*n*+ 1 data point is added, the system parameter \(\theta\) can be updated to \(\theta^{*}\) according to:

## Establishment and assessment of the turbidity forecast model

### Establishment of a turbidity-forecasting model

- 1.
Obtain turbidity-related historical data, such as turbidity, rainfall, and discharge, at specific stations.

- 2.Select the input variables.
- (1)
Assume the output variable is

*Y*, which represents the forecast turbidity. - (2)
Assume the input variables are

*X*_{1},*X*_{2},*X*_{3},*…*,*X*_{ m }, which represent turbidity, rainfall, discharge, and so on. - (3)
Establish a nonlinear equation

*Y*=*f*(*X*_{1},*X*_{2},*…*,*X*_{ m }).

- (1)
- 3.
Determine the optimum number of modeling data and variable combinations to establish a forecast model through trial-and-error.

- 4.
Establish an input–output relationship with both the GMDH and SGMDH algorithms; derive the model layer-by-layer until optimality is achieved, and then return, layer-by-layer, to the inertial input layer to establish a GMDH or SGMDH forecast equation.

- 5.
Input the variables and begin model forecasting.

- 6.
Output the forecast results.

- 7.
Consider whether there is a temporal impact. If so, a recursive/sequential structure is necessary.

- 8.
Generate a final optimum turbidity-forecasting model.

*t*) at time

*t*, where

*t*represents the time period. The input variables are the daily turbidity

*T*(

*t*−1) ~

*T*(

*t*−

*m*) for the period 1 ~

*m*, daily rainfall

*R*(

*t*−

*1*) ~

*R*(

*t*−

*n*) for the period 1 ~

*n*, and daily discharge of the Cishan River

*Q*(

*t*−

*1*) ~

*Q*(

*t*−

*k*) for the period 1 ~

*k*. The forecast relation is presented below:

### Model efficiency evaluation

*N*represents the total number of observations in the data set. RMSE values approaching 0 and CC values approaching 1 signify better forecast performance.

## Case studies

In this section, we compare the results given by our forecast model with real-world data. We first describe the study area and the data set used for comparison; then, we present the forecast results and evaluate the model’s performance.

### Study area description

^{2}. Figure 2 illustrates the reservoir location.

### Selection of research data

This paper explores turbidity changes in the Nanhua Reservoir prior to over-basin diversion (i.e., turbidity changes at the diversion tunnel entrance of the Chiahsien Weir). Numerous variables, such as storms, human activities, and complex natural processes, affect turbidity. These influencing factors closely match the nonlinear structural model of the GMDH algorithm.

Those factors that have the greatest impact on turbidity were utilized as input parameters. Thus, turbidity, rainfall, and discharge were chosen as the domain input parameters. The turbidity at the entrance to the diversion tunnel of the Chiahsien Weir was selected as the main parameter. Rainfall data from the Jiashian rainfall station (the only rainfall station upstream of the diversion channel) and Cishan River discharge data were used as secondary parameters. Using the aforementioned nonlinear system, a predictive turbidity model was built, calibrated, and verified.

### GMDH and SGMDH calibrated result comparison

#### Selection of best algorithm

Comparison of evaluation indicators of GMDH and SGMDH forecast efficiency

Modeling event | GMDH forecast result | SGMDH forecast result | ||||
---|---|---|---|---|---|---|

Modeling data length (variable combination) | RMSE (NTU) | CC | Modeling data length (variable combination) | RMSE (NTU) | CC | |

2000 | 70 (3, 0, 1) | 57.920 | 0.211 | 70 (4, 3, 1) | 52.905 | 0.126 |

2001 | 70 (2, 1, 0) | 37.574 | 0.619 | 70 (4, 1, 2) | 33.149 | 0.609 |

2002 | 70 (3, 1, 1)** | 24.598 | 0.929 | 70 (3, 1, 1) | 32.806 | 0.891 |

2003 | 60 (4, 1, 1) | 22.750 | 0.939 | 70 (2, 1, 1) | 39.962 | 0.518 |

2004* | 70 (3, 0, 1) | 36.395 | 0.615 | 70 (3, 1, 1) | 67.892 | 0.107 |

2005 | 40 (5, 0, 0) | 15.053 | 0.952 | 50 (5, 1, 1) | 19.142 | 0.949 |

2006 | 70 (3, 1, 1) | 5.787 | 0.975 | 70 (4, 1, 1) | 13.477 | 0.956 |

2007 | 70 (3, 0, 0) | 11.026 | 0.962 | 70 (3, 0, 0) | 7.770 | 0.965 |

2008 | 60 (4, 1, 1) | 60.892 | 0.209 | 70 (4, 1, 1) | 54.773 | 0.121 |

Average | – | 29.450 | 0.724 | – | 31.748 | 0.642 |

Regression parameters of all segments by GMDH method

Export module | a0 | a1 | a2 | a3 | a4 | a5 | |
---|---|---|---|---|---|---|---|

First layer | \(Z_{9}^{1} [Q(t - 1),R(t - 1)]\) | 16.62866 | 0.53860 | −0.00147 | 0.31608 | −0.00088 | 0.00200 |

\(Z_{10}^{1} [T(t - 4),T(t - 3)]\) | 3.60373 | 0.92624 | −0.00036 | 0.50314 | 0.00275 | −0.00903 | |

\(Z_{5}^{1} [Q(t - 1),T(t - 4)]\) | 0.00000 | 0.29692 | 0.00000 | 1.04566 | 0.00148 | 0.01170 | |

\(Z_{8}^{1} [Q(t - 1),T(t - 1)]\) | 24.90101 | 0.08174 | −0.00022 | 0.42458 | −0.00260 | 0.00479 | |

Second layer | \(Z_{2}^{2} \left( {Z_{9}^{1} ,Z_{10}^{1} } \right)\) | −4.54640 | 1.64904 | −0.00720 | −0.86180 | 0.00757 | 0.00280 |

\(Z_{3}^{2} \left( {Z_{9}^{1} ,Z_{5}^{1} } \right)\) | 6.83660 | 1.20849 | 0.00150 | −0.78898 | 0.01028 | −0.00896 | |

\(Z_{4}^{2} \left( {Z_{8}^{1} ,Z_{10}^{1} } \right)\) | 20.20312 | 0.22685 | 0.00340 | −0.22952 | 0.00681 | −0.00292 | |

Third layer | \(Z_{4}^{3} \left( {Z_{4}^{2} ,Z_{3}^{2} } \right)\) | −4.75467 | 2.95573 | −0.01610 | −1.83673 | 0.00561 | 0.01000 |

\(Z_{2}^{3} \left( {Z_{2}^{2} ,Z_{3}^{2} } \right)\) | 20.80679 | 1.70620 | −0.00304 | −1.54077 | 0.01046 | −0.00323 | |

Fourth layer | \(Z_{1}^{4} \left( {Z_{2}^{3} ,Z_{4}^{3} } \right)\) | 16.08485 | −1.98537 | 0.03366 | 2.35709 | 0.00260 | −0.03360 |

#### Choices of modeling data length and variable combinations

Calibration RMSE results (NTU) of the best annual forecast model given by trial-and-error

Algorithm | Modeling data length | 2000 | 2001 | 2002 | 2003 | 2005 | 2006 | 2007 | 2008 |
---|---|---|---|---|---|---|---|---|---|

GMDH | 20 | – | – | – | 112.172 | 241.589 | – | 63.565 | – |

30 | – | 112.723 | 127.826 | – | 639.366 | 158.908 | 31.363 | – | |

40 | 68.251 | 65.930 | 56.117 | 52.822 | 15.053 | 9.308 | 29.677 | 210.145 | |

50 | – | 182.696 | 83.799 | – | 17.230 | 7.715 | 37.756 | 330.65 | |

60 | 62.713 | 147.584 | 49.732 | 22.750 | 21.551 | 6.310 | 39.408 | 60.892 | |

70 | 57.920 | 37.574 | 24.598 | 36.395 | 24.455 | 5.787 | 11.026 | 91.777 | |

SGMDH | 20 | 60.927 | 498.712 | 90.844 | 251.413 | 42.926 | 111.513 | 69.136 | 239.910 |

30 | – | 207.838 | 248.777 | 173.750 | 92.572 | 47.650 | 27.351 | 222.800 | |

40 | 64.072 | 98.749 | 105.783 | 72.218 | 20.264 | 130.401 | 26.117 | 108.854 | |

50 | 64.087 | 761.013 | 122.730 | 84.200 | 19.142 | 18.948 | 29.234 | 77.314 | |

60 | 53.575 | 100.265 | 73.950 | 43.982 | 24.871 | 13.538 | 33.044 | 62.374 | |

70 | 52.905 | 33.149 | 32.806 | 39.962 | 19.738 | 13.477 | 7.770 | 54.773 |

The best annual input variables

Modeling event | Input variables | Modeling data length | RMSE (NTU) |
---|---|---|---|

2000 | | 70 | 57.920 |

2001 | | 70 | 37.574 |

2002 | | 70 | 24.598 |

2003 | | 60 | 22.750 |

2005 | | 40 | 15.053 |

2006 | | 70 | 5.787 |

2007 | | 70 | 11.026 |

2008 | | 60 | 60.892 |

### Turbidity forecasting

#### Permissible errors

We adopted the safety concepts applied in general engineering construction projects, allowing a maximum error range of only 10 %. The Taiwan Water Corporation is able to treat water with a turbidity of up to 500 NTU. As such, 50 NTU (10 % of 500 NTU) was chosen as the index of turbidity prediction accuracy. According to standard normal distribution and confidence interval calculations, the results for each year were between 51–66 NTU. An error of only 50 NTU is more restrictive, and was, thus, used as the study threshold.

#### Verification and analysis of forecast results

#### Best forecast model

Comparison of RMSE (NTU) value of annual forecast

Year | 2000 | 2001 | 2002 | 2003 | 2005 | 2006 | 2007 | 2008 |
---|---|---|---|---|---|---|---|---|

2000 | 57.920 | 49.897 | 62.349 | 47.164 | 58.767 | 43.168 | 32.435 | 54.553 |

2001 | 32.161 | 37.574 | 20.959 | 22.114 | 10.090 | 2.365 | 19.161 | 77.564 |

2002 | 44.535 | 38.891 | 24.598 | 60.393 | 28.335 | 31.021 | 30.288 | 31.996 |

2003 | 39.588 | 14.628 | 27.213 | 22.749 | 13.014 | 14.163 | 24.257 | 20.253 |

2005 | 71.141 | 120.520 | 76.395 | 116.597 | 15.053 | 44.511 | 61.079 | 1064.024 |

2006 | 41.340 | 90.357 | 54.057 | 95.049 | 56.392 | 5.787 | 60.938 | 45.879 |

2007 | 565.116 | 114.259 | 84.570 | 180.731 | 42.723 | 93.137 | 11.026 | 72.055 |

2008 | 25.018 | 23.243 | 20.767 | 71.414 | 43.168 | 31.513 | 32.836 | 60.892 |

Average | 109.602 | 61.171 | 46.364 | 77.026 | 33.442 | 33.208 | 34.003 | 178.402 |

### Recursive/sequential turbidity forecast model

The recursive/sequential GMDH algorithm incorporates temporal variability once the variance between the predicted and newly observed turbidity exceeds an acceptable range at a certain time. This newly observed value is then added to the model, with previous data being deleted to maintain the same data length. The updated forecasting model then retains its accuracy for later turbidity forecasts. If the updated forecast model does not produce valid output, the steps for adding newly observed values are repeated to enable the system to auto-adjust. Using these procedures, the actual turbidity trend can be observed over any given time period.

## Conclusion

Turbidity is the most important index for public water supply. High turbidity inflow causes harassment on treatment of public water supply, even bringing the need to cut off the water supply. To avoid high turbidity water inflow, it is important to strengthen the catchment’s conservation, protect the water resources territory, and predict the inflow turbidity concentration before the treatment operation.

A local historical turbidity, rainfall, and discharge database was constructed to develop a turbidity prediction model based on the GMDH algorithm. The results from a cross-validation revealed that GMDH was more appropriate than SGMDH for this case study. The majority of predictive turbidity values were within a confidence interval of 90 % or approaching 90 %. Using the recursive GMDH algorithm, the model can be modified to generate better predictions and improve forecast accuracy. The test results indicate that this turbidity prediction model is feasible and reliable for turbidity forecasting. Even with complex environmental factors, the model remains applicable.

## Notes

### Acknowledgments

We deeply appreciate the assistance of the Taiwan Water Corporation, which provided us with data for hydrological findings, as well as the generous aid from its engineer Mr. Wang Ying-Ming in finishing our research. In addition, the authors are also indebted to reviewers for their valuable comments and suggestions.

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