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Modeling and Uncertainty Analysis of Groundwater Level Using Six Evolutionary Optimization Algorithms Hybridized with ANFIS, SVM, and ANN

Akram Seifi, Mohammad Ehteram, Vijay P. Singh, Amir Mosavi

arXiv:2006.16848v1eess.SPcs.LGcs.NEstat.ML

TL;DR

Accurate groundwater-level prediction is important for water management, but prior hybrid AI studies leave uncertainty evaluation and spatial variation modeling insufficiently explored. This study combines six optimization algorithms with ANN, ANFIS, and SVM models; ANFIS-GOA performs best in the reported prediction and uncertainty comparisons.

  • Problem

    Accurate GWL prediction is important for groundwater management, while the authors identify limited study of optimization-based hybrids for prediction, uncertainty evaluation, and spatial variation.

    Source: §1 · pp. 2–3

  • Method

    The study hybridizes ANN, ANFIS, and SVM with six optimization algorithms, applies PCA to lagged inputs, and uses Monte Carlo simulations to quantify prediction uncertainty.

    Source: §2.8 · pp. 13–14, §3.5 · p. 31, §4 · p. 39

  • Results

    ANFIS-GOA outperformed other models in testing at piezometer 6 (RMSE: 1.21, MAE: 0.878, NSE: 0.93, PBIAS: 0.15), and hybrid models generally performed better than standalone models.

    Source: §3.3 · pp. 19–20, §4 · p. 39

  • Takeaways & Limitations

    ANFIS-GOA had the strongest reported uncertainty verification, with p = 0.94 and d = 0.16, and its confidence intervals were smaller than those of other models.

    Source: §3.5 · p. 31

Abstract

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In the present study, six meta-heuristic schemes are hybridized with artificial neural network (ANN), adaptive neuro-fuzzy interface system (ANFIS), and support vector machine (SVM), to predict monthly groundwater level (GWL), evaluate uncertainty analysis of predictions and spatial variation analysis. The six schemes, including grasshopper optimization algorithm (GOA), cat swarm optimization (CSO), weed algorithm (WA), genetic algorithm (GA), krill algorithm (KA), and particle swarm optimization (PSO), were used to hybridize for improving the performance of ANN, SVM, and ANFIS models. Groundwater level (GWL) data of Ardebil plain (Iran) for a period of 144 months were selected to evaluate the hybrid models. The pre-processing technique of principal component analysis (PCA) was applied to reduce input combinations from monthly time series up to 12-month prediction intervals. The results showed that the ANFIS-GOA was superior to the other hybrid models for predicting GWL in the first piezometer and third piezometer in the testing stage. The performance of hybrid models with optimization algorithms was far better than that of classical ANN, ANFIS, and SVM models without hybridization. The percent of improvements in the ANFIS-GOA versus standalone ANFIS in piezometer 10 were 14.4%, 3%, 17.8%, and 181% for RMSE, MAE, NSE, and PBIAS in the training stage and 40.7%, 55%, 25%, and 132% in testing stage, respectively. The improvements for piezometer 6 in train step were 15%, 4%, 13%, and 208% and in the test step were 33%, 44.6%, 16.3%, and 173%, respectively, that clearly confirm the superiority of developed hybridization schemes in GWL modeling. Uncertainty analysis showed that ANFIS-GOA and SVM had, respectively, the best and worst performances among other models. In general, GOA enhanced the accuracy of the ANFIS, ANN, and SVM models.

1. Introduction

The introduction motivates groundwater-level prediction for water management and identifies unresolved questions about optimization, uncertainty, and spatial variation in hybrid AI models. It presents GOA-based hybrids and comparison with other optimizers as a response to those gaps.

1. Introduction

  • Accurate and reliable GWL prediction is presented as essential for sustainable groundwater management, while data-intensive mathematical models are limited in practice by data scarcity.The introduction also names exploitation, pollution, and climate factors among pressures or influences on groundwater levels.Source: §1 · p. 2
  • Standard training algorithms for ANN, ANFIS, and SVM can converge slowly or become trapped in local optima, motivating the use of nature-inspired optimization to tune model parameters.The passage identifies weights and biases in ANN and linear and nonlinear parameters in ANFIS and SVM as tuning targets.Source: §1 · p. 2
  • Prior GWL studies report differing model advantages, while reviews of hybrid approaches describe improved accuracy and efficiency and call for uncertainty analysis.Examples include better results from neuro-fuzzy than ANN, and from wavelet-SVM than wavelet-ANN.Source: §1 · pp. 2–3
  • GOA was selected because its multi-solution search and convergence behavior are described as helping it avoid local optima, while its use for GWL modeling and uncertainty analysis had not been studied.The introduction also notes earlier GOA applications to river flow, soil compression, sediment-rating curves, and concrete slump.Source: §1 · p. 3
  • The study targets the limited evaluation of novel optimization algorithms for GWL prediction, uncertainty, and spatial variation, comparing hybrid ANN, ANFIS, and SVM models and using PCA for input selection.The authors describe comparative uncertainty analysis of the hybrid models as a contribution and novelty.Source: §1 · p. 3

2. Materials and Methods

The study uses monthly GWL records from the Ardebil plain and selects time-series inputs up to a 12-month lag with PCA. Its ANFIS model combines fuzzy-rule processing with parameter tuning by optimization algorithms.

2.1. Case Study and Data

  • PCA selects input combinations from GWL time-series lags extending up to 12 months.The case study is the Ardebil plain in Iran, where groundwater serves drinking, agricultural, and industrial uses.Source: §2.1 · p. 4
  • The dataset comprises 140 monthly records from 2000–2012 at each of nine wells, with a random 80%/20% split for training and testing.The measurements were taken once per month; the nine wells support spatiotemporal analysis.Source: §2.1 · p. 4

2.2. ANFIS Model

  • ANFIS uses fuzzy if-then rules to construct a predictive model, with five layers processing inputs through to the final output.The first layer fuzzifies inputs, and the bell-shaped membership function was selected for smoothness and concise notation.Source: §2.2 · pp. 4–5
  • The ANFIS layers multiply rule strengths, normalize them, compute adaptive consequent outputs, and sum incoming signals to produce the model output.The premise and consequent parameters are tuned by optimization algorithms in the hybrid training scheme rather than the classical least-squares and gradient-descent approach.Source: §2.2 · p. 5

2.3. ANN Model

The ANN receives inputs through input, hidden, and output layers, with neurons connected by weights and biases. The study uses meta-heuristic optimization to address the slow convergence of backpropagation training.

2.3. ANN Model

  • The ANN has input, hidden, and output layers, with connection weights passing signals between neurons and the final layer generating the model output.Hidden neurons use weighted sums and activation functions; bias values also affect neuron outputs.Source: §2.3 · p. 5
  • ANN training prepares the network structure and adjusts connection weights to reduce the error between observed and estimated values.The described backpropagation process modifies weights and biases when outputs differ from observations.Source: §2.3 · p. 5
  • Meta-heuristic optimization algorithms are used in the study to address backpropagation's slow convergence rate.The passage refers to the figure as showing ANN structure and hybridization with intelligence algorithms.Source: §2.3 · pp. 5–6

2.4. SVM Model

The SVM maps training inputs into a high-dimensional feature space to generate a regression decision function. The paper describes kernel-based modeling and grid search for selecting SVM parameters.

2.4. SVM Model

  • SVM maps the selected training input vector into a high-dimensional feature space and generates an optimal decision function for regression estimation.The passage identifies ϕ(x) as the nonlinear mapping function, b as bias, and W^T as the weight vector.Source: §2.4 · p. 6
  • The SVM optimization objective combines model complexity, a penalty parameter C, and an ε-error control function.The passage describes D(f) as the generalized optimal function and ‖w‖2 as model complexity.Source: §2.4 · p. 6
  • Grid search with cross-validation selects C and γ by choosing the parameter combination with the lowest cross-validation error; the SVM uses a radial basis kernel.The methodology framework is described as covering hybrid ANN, ANFIS, and SVM optimization models.Source: §2.4 · p. 6, Figure 2 · p. 7

2.5. Optimization Algorithms

The section describes six population-based optimization algorithms through their search behaviors and update rules. Their candidate solutions are iteratively adjusted or selected according to objective or fitness values.

2.5.1. Grasshoppers Optimization Algorithm (GOA)

  • GOA combines grasshopper social interaction, gravity, and wind in position updates, then corrects the update to converge toward a target while shrinking interaction zones.The social-force function distinguishes repulsion at short distances from a comfort zone farther away.Source: §2.5.1 · p. 8

2.5.2. Weed Algorithm (WA)

  • WA initializes weeds randomly, produces fitness-dependent seeds, and limits population growth by removing poorer-fitness weeds during competitive exclusion.The seed distribution's standard deviation changes over iterations.Source: §2.5.2 · pp. 8–9

2.5.3. Cat Swarm Optimization (CSO)

  • CSO alternates seeking, a local search that changes cats' positions while they remain alert, and tracing, which updates their locations through velocity.Seeking uses the seeking memory pool, selected-dimension range, and number of dimensions changed to govern position changes.Source: §2.5.3 · pp. 10–11

2.5.4. Particle Swarm Optimization (PSO)

  • PSO randomly initializes particles that search adjacent solutions by updating position and velocity, with personal and neighbor-derived global best positions in the velocity rule.The passage identifies easy implementation and computational efficiency as PSO advantages.Source: §2.5.4 · p. 12

2.5.5. Genetic Algorithm (GA)

  • GA selects fitter chromosomes, exchanges genes through crossover, and mutates genes to generate candidate solutions, repeating until convergence criteria are met.The stated drawback is that GA requires a high number of iterations.Source: §2.5.5 · p. 12

2.5.6. Krill Herd Algorithm (KHA)

  • KHA updates krill locations using three behaviors: motion induced by other krill, foraging motion, and physical diffusion.The foraging terms include food attraction and the effect of each krill's best fitness.Source: §2.5.6 · p. 12

2.6. Principal Component Analysis (PCA)

PCA transforms observations into linearly uncorrelated principal components to reduce the number of inputs while retaining most of the original data's variance.

2.6. Principal Component Analysis (PCA)

  • PCA applies an orthogonal transformation to produce uncorrelated components and reduce inputs while retaining most of the original data's variance.The passage describes components using eigenvectors and inputs, with the variance-covariance matrix and eigenvalues in the associated equation.Source: §2.6 · p. 13

2.7. Taguchi Model

The Taguchi method is used to choose random-parameter settings for optimization algorithms by comparing parameter combinations using their signal-to-noise ratios.

2.7. Taguchi Model

  • The method treats random optimization parameters as influential on model outputs and tests parameter levels to determine settings.The passage distinguishes controllable parameters from uncontrollable noise parameters.Source: §2.7 · p. 13
  • Parameter combinations with higher signal-to-noise ratios are regarded as the best combinations.The stated fitness function is obtained using the Taguchi model.Source: §2.7 · p. 13
  • For a four-parameter, three-level PSO example, the population size, acceleration coefficient, and inertia coefficient are tested across levels.The passage describes testing acceleration and inertia coefficients at levels 1, 2, 3, and 4.Source: §2.7 · p. 13

2.8. Hybrid ANN, ANFIS, and SVM Models with Optimization Algorithms

The paper uses optimization algorithms to tune ANN, ANFIS, and SVM parameters for groundwater-level forecasting. Each model's optimization process defines its decision variables and an error-based objective.

2.8. Hybrid ANN, ANFIS, and SVM Models with Optimization Algorithms

  • For hybrid ANN models, optimization agents represent network weights and biases, and their locations are updated to decrease prediction error iteratively.The initial agent positions supply the initial weights and biases used to train the models.Source: §2.8 · p. 13
  • For hybrid ANFIS models, optimization algorithms tune premise and consequent parameters to minimize RMSE after the ANFIS model is trained.The procedure initializes agents and adjusts their locations while minimizing the objective function.Source: §2.8 · p. 13
  • For hybrid SVM models, agents optimize the C and kernel-function parameters using RMSE as the objective to minimize model error.The passage identifies these parameters as significantly affecting SVM accuracy.Source: §2.8 · p. 14
  • The described workflow uses PCA to select input combinations, then uses hybrid and standalone models with optimized parameters to forecast monthly groundwater levels.The passage states that optimization algorithms minimize an error function to select model-parameter values.Source: §2.8 · p. 14

2.9. Uncertainty Analysis of Soft Computing Models

The framework treats input data and model structure as sources of uncertainty and evaluates input-data uncertainty using an error model. For each soft-computing model, the error model’s mean and variance are estimated with dynamically dimensioned search.

2.9. Uncertainty Analysis of Soft Computing Models

  • Input data and model structure are identified as sources of uncertainty in the framework.The framework is described as evaluating both sources.Source: §2.9 · p. 14
  • Combined Bayesian uncertainty is used to estimate the contribution of input data uncertainty, which is represented by an input error model.The error model is used to account for input-data uncertainty.Source: §2.9 · p. 14
  • For each soft-computing model, the input error model’s mean and variance are added to the system and estimated using dynamically dimensioned search.The error model describes adjusted groundwater level relative to observed groundwater level with a normal distribution.Source: §2.9 · p. 14

Mode Structure Uncertainty

Bayesian model averaging (BMA) is used to represent model uncertainty by combining predictions from selected models. The section also describes estimating model weights and uncertainty indices, alongside performance measures for comparing predictions.

Mode Structure Uncertainty

  • BMA estimates model uncertainty using posterior model probabilities and averages predictions from the best models with model weights.Each model contributes a point prediction plus noise, weighted across models.Source: Mode Structure Uncertainty · p. 14
  • The log-likelihood function is used to estimate model weights and standard deviations, with MCMC simulations used to compute that function.The weights and standard deviations are needed for accurate application of BMA.Source: Mode Structure Uncertainty · p. 14
  • The workflow selects models, assigns prior probabilities, defines an input error model, and obtains posterior distributions for input-error models and model parameters.It then provides predetermined GWL samples for each model and estimates model variances and weights.Source: Mode Structure Uncertainty · pp. 14–15
  • The uncertainty-index definitions identify the number of observations, prediction bounds, standard deviation, coverage probability, and average interval width.The text specifies that p is bracketed by 95% of predicted uncertainties and defines d as the distance between bounds.Source: Mode Structure Uncertainty · p. 15

2.10. Statistical Indices for Evaluation of Different Models

  • RMSE and MAE are best at zero, NSE at one, and PBIAS at zero for evaluating agreement between observed and predicted values.The section lists RMSE and NSE and defines the observed and predicted values used in the evaluation indices.Source: §2.10 · p. 15

3. Results and Discussion

Across the evaluated piezometers, ANFIS-GOA was the strongest reported model, and hybrid models generally outperformed standalone ANN, ANFIS, and SVM. The section also reports PCA-based input selection, uncertainty comparisons, and spatial variation results.

3.1. Inputs Selection by PCA

  • The first four principal components explained 91% of variance, and loadings reduced the monthly GWL inputs from 12 lags to five, H(t-1) through H(t-5).Those four components were selected for the hybrid models because the first five lags had higher factor loadings.Source: §3.1 · p. 16, Table 1 · p. 16

3.2. Selection of Random Parameters by the Taguchi Model

  • Taguchi S/N analysis selected optimizer parameter levels; GOA's selected population size was 300, with l = 1.5 and f = 0.5.The highest S/N ratio determined the best level for each parameter.Source: §3.2 · p. 16, Table 2 · p. 16

3.3. Results of Hybrid ANN, ANFIS, and SVM Models

  • Across piezometers 6, 9, and 10, ANFIS-GOA led reported comparisons, and hybridized models generally outperformed standalone models.For piezometers 6 and 9, the text explicitly identifies ANFIS-GOA as best; for piezometer 10 it reports better RMSE, MAE, and PBIAS than other models.Source: §3.3 · pp. 18–20, Table 3 · p. 19, Table 4 · p. 19

3.4. Analysis of Scatterplots of Soft Computing Models

  • For piezometers 9 and 10, testing R2 for ANFIS-GOA was 0.94, and the plotted predictions were reported as closer to observations than competing models.The reported testing R2 sequence for piezometer 9 also places ANFIS-GOA ahead of the other listed ANFIS variants.Source: §3.4 · p. 21, Figure 7 · p. 30

3.5. Uncertainty Analysis of Soft Computing Models

  • ANFIS-GOA had the strongest reported uncertainty performance, with p = 0.94 and d = 0.16, and its prediction bounds were narrower than other models'.The study describes 95% uncertainty bounds and reports that standalone models had higher d and lower p than hybrid models.Source: §3.5 · p. 31, Table 6 · p. 32, Figure 8 · p. 35

3.6. Spatiotemporal Variation of GWL

  • Across the heterogeneous Ardebil aquifer, ANFIS-GOA had lower reported RMSE than ANN-GOA and SVM-GOA in the spatial-temporal comparison.The reported RMSE ranges were 1.2–2.2 m for ANFIS-GOA and 1.7–2.7 m for ANN-GOA and SVM-GOA.Source: §3.6 · p. 36, Figure 10 · p. 37, Figure 9 · p. 37

4. Conclusions

The conclusion reports ANFIS-GOA as the strongest model overall, including the best testing performance at piezometer 6. It also reports accurate spatial maps across the heterogeneous Ardebil aquifer.

4. Conclusions

  • At piezometer 6 in testing, ANFIS-GOA achieved RMSE 1.21, MAE 0.878, NSE 0.93, and PBIAS 0.15, outperforming the other models.The conclusion also reports ANFIS-GOA as the most dominant model overall.Source: §4 · p. 39
  • The study combined ANFIS, ANN, and SVM with six optimization algorithms, using PCA to reduce time-series inputs for three Ardebil plain piezometers.The input time series extended to 12-month lags.Source: §4 · p. 39
  • In uncertainty analysis, ANFIS-GOA outperformed other models, while ANFIS-GA was the weakest ANFIS variant, with p = 0.87 and d = 0.21.Source: §4 · p. 39
  • Spatial-variation maps showed high ANFIS-GOA accuracy for the heterogeneous Ardebil aquifer; future work could evaluate the models under climate-change conditions.The authors propose simulating future temperature and rainfall as model inputs.Source: §4 · p. 39
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