1 Introduction

Lakes typically provide water for various domestic, industrial and agricultural applications (Vuglinskiy 2009). Forecasting the water-level in lakes is necessary for water resources planning and management, lake navigation, management of tidal irrigation and agricultural drainage canals, etc. The water-level in lakes is a complex phenomenon, which is mainly controlled by the natural water exchange between the lake and its watershed, and thus the level reflects the hydrological changes in the watershed (Karimi et al. 2012; Altunkaynak 2007). For many practical applications, it is necessary to have a model to predict the water-level fluctuations based on previously recorded levels (Karimi et al. 2012). Numerous studies have been performed to predict water level fluctuations in lakes using neural networks, neuro-fuzzy, and genetic programming models (Kisi et al. 2015; Buyukyildiz et al. 2014; Karimi et al. 2012, 2013; Kisi et al. 2012; Shiri et al. 2011; Sulaiman et al. 2011).

Using the measured data from the Urmia lake in northwest of Iran, we use the Extreme Learning Machine (ELM) approach to predict its daily water-level for different prediction intervals.

Soft computing approaches have been used in many disciplines. Neural network (NN), as a well-known soft computing approach, has been employed in many water resources engineering problems. NN is able to solve complex nonlinear problems which may be difficultly solved by classic parametric approaches. There are many algorithms for training NNs, e.g. back propagation (BP), support vector machine (SVM), and hidden Markov model (HMM). One of the shortcomings of NN is its relatively large learning time. Huang et al. (2004) introduced an algorithm (which is known as Extreme Learning Machine (ELM)) for training the single layer feed forward NN. This algorithm is able to solve problems caused by ANNs’ gradient descent based algorithms e.g. back propagation. ELM significantly reduces the training time of NN and generates accurate results (Huang et al. 2003). A number of studies have successfully employed the ELM algorithm in different problems (e.g.Yu et al. 2014; Wang and Han 2014; Ghouti et al. 2013; Sajjadi et al. 2016; Nian et al. 2014; Shamshirband et al. 2015).

In general, ELM is a robust algorithm with faster learning speed and a better performance compared to traditional algorithms such as back-propagation (BP), where it tries to get the smallest training error and norm of weights. To the our knowledge, there is not any published paper that applies the ELM for lake level prediction.

In this study, the ELM algorithm is used to predict the daily water-level in the Urmia Lake. Results indicate that the proposed algorithm can adequately predict the water level in the Urmia Lake for different prediction intervals. ELM results were also compared with those of the genetic programing (GP) and ANN-BP techniques.

This paper is organized as follows: Section 2 explains the Urmia Lake and the collected data. Section 3 presents the description of ELM algorithm and GP and ANN used as benchmark methods. The obtained results are presented in Section 4. Finally, the conclusions are given in Section 5.

2 The Urmia Lake

The Urmia Lake is the second largest saline lake in the world. Daily water-level data in the Urmia Lake in northwest of Iran were used in this study. Table 1 shows the statistical metrics of daily water-level (WL) in the Urmia lake. It can be seen from Table 1 that the train and test data present a more skewed distribution than the validation data.

Table 1 Statistical metrics of the utilized data during the study period
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3 Soft Computing Algorithms

3.1 Extreme Learning Machine

(ELM was introduced by Huang et al. (2004) for the single layer feed-forward neural networks (SLFN) (Annema et al. 1994; Huang et al. 2006). This algorithm chooses the input weights of SLFN randomly, but determines the output weights analytically. The ELM algorithm is most robust with faster learning speed. ELM does not involve too much human intervention, because it determines all the network parameters analytically, and can be run much faster than the conventional algorithms.

3.1.1 Single Hidden Layer Feed-Forward Neural Network

The SLFN function with L hidden nodes might be presented as a network incorporating both additive and RBF hidden nodes in an unified way given (Huang et al. 2006; Liang et al. 2006):

$$ {f}_L(x)={\displaystyle \sum_{i=1}^L}{\beta}_iG\left({a}_i,{b}_i,x\right),\kern3em x\in {R}^n,\kern3em {a}_i\in {R}^n $$
(1)

where a i and b i denote the learning parameters of hidden nodes, and β i stands for the weight connecting the ith hidden node to the output node. The output of the ith hidden node with respect to the input x may be shown by G(a i , b i , x). The additive hidden node with the activation function g(x) : R → R (e.g., sigmoid and threshold), G(a i , b i , x) is (Huang et al. 2006):

$$ G\left({a}_i,{b}_i,x\right)=g\left({a}_i.x+{b}_i\right),\kern4em {b}_i\in R $$
(2)

where x is the inner product of vector a i and x in R n. G(a i , b i , x) can be determined for RBF hidden node with activation function g(x) : R → R (e.g., Gaussian), G(a i , b i , x) as (Huang et al. 2006):

$$ G\left({a}_i,{b}_i,x\right)=g\left({b}_i\left\Vert x-{a}_i\right\Vert \right),\kern4em {b}_i\in {R}^{+} $$
(3)

a i and b i represent the center and impact factor of ith RBF node. The set of all positive real values is indicated by R +. Further details about RBF might be found in e.g. Huang et al. (2006).

3.1.2 Principle of ELM

  1. Theorem 1:

    (Liang et al. 2006) Let an SLFN with L additive or RBF hidden nodes and an activation function g(x) which is infinitely differentiable in any interval of R be given. Then for arbitrary L distinct input vectors {x i |x i  ∈ R n, i = 1, …, L} and {(a i , b i )} L i = 1 randomly produced by any continuous probability distribution, respectively, the hidden layer output matrix is invertible with probability one, the hidden layer output matrix H of the SLFN is invertible and ‖Hβ − T‖ = 0.

  2. Theorem 2:

    (Liang et al. 2006) Given any small positive value ε > 0 and activation function g(x) : R → R which is infinitely differentiable in any interval, there exists L ≤ N such that for N arbitrary distinct input vectors {x i |x i  ∈ R n, i = 1, …, L} for any {(a i , b i )} L i = 1 randomly produced based upon any continuous probability distribution ‖H N × L β L × m  − T N × m ‖ < ε with probability one.

Since the hidden node parameters of ELM should not be tuned throughout training and because they are easily assigned with random values, Eq. (5) becomes a linear system and the output weights can be estimated as (Huang et al. 2006):

$$ \beta ={H}^{+}T $$
(4)

where H + is the Moore-Penrose generalized inverse (Singh and Balasundaram 2007) of the hidden layer output matrix H which can be computed via several approaches consisting orthogonal projection, orthogonalization, iterative, singular value decomposition (SVD), etc. (Singh and Balasundaram 2007). The orthogonal projection method can be utilized only when H T T is non-singular and H + = (H T T)− 1 H T. Owing to the use of searching and iterations, orthogonalization method and iterative method have limitations. Implementations of ELM uses SVD to compute the Moore-Penrose generalized inverse of H, because it can be utilized in all situations. ELM is thus a batch learning method.

3.2 Artificial Neural Networks

The multilayer feed forward network with a backpropagation learning algorithm is one of the most popular neural network architectures; it has been deeply studied and widely used in many fields (Yu et al. 2014). One of the most commonly used functions is the sigmoid function, which is monotonic increasing and ranges from zero to one. Details on ANNs can be found in e.g. Karunanithi et al. (1994). The parameters used for ANN are determined iteratively (see Table 2).

Table 2 User-defined parameters for ANN
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3.3 Genetic Programming

GP is an evolutionary algorithm based on Darwinian theories of natural selection and survival to approximate the equation, in symbolic form, that best describes how the output can relate to the input variables. The algorithm considers an initial population of randomly generated programs (equations), derived from the random combination of input variables, and random numbers and functions. The population of potential solutions is then subjected to an evolutionary process, then the ‘fitness’ (a measure of to which degree they can solve the problem) of the evolved programs is assessed. Then, the individual programs which produce the most accurate fit of data are selected from the initial population. The programs which gives the best fit are then selected to exchange part of the information between them to produce better programs through ‘crossover’ and ‘mutation’, which mimics the natural world’s reproduction process. More details on GP can be found in e.g. Koza (1992). Table 3 summarizes the parameters used per run of GP.

Table 3 Parameters employed in GP modelling
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4 Results and Discussion

Predictive models of water levels were established for 1- as well as 7-days ahead time intervals. The utilized input parameters consist of previous water levels from 1 to 6 days in the past. When developing multi-step ahead prediction models, two approaches are possible:

  • Iterated prediction approach,

  • Direct prediction approach.

In the iterated prediction scheme, one step ahead prediction is used for building the subsequent predictions (i.e. for p steps ahead), while in direct prediction approach, separate (direct) prediction models are developed for each prediction horizon.

The advantages/shortcomings of each approach have been extensively discussed in literature (e.g. Martin 1989; Farmer et al. 1987; Marcellino et al. 2006). In the present paper, direct approach is employed. The main advantage of this approach is its intuitive manner (Chevillon and Hendry 2005), where there is no accumulation of forecasting error, in contrary to the iterated scheme. Nevertheless, the superiority of direct approach has been confirmed in electricity load forecasting (McSharry et al. 2005; Ramanathan et al. 1997; Fan and Hyndman 2012), where the domain is quite similar to one discussed here.

4.1 Input Variables for Model Building

The whole data set, comprising the observations for the period between 1972 and 2003 was divided into three blocks, including train set (50 % of observations), test set (25 % of observations), and validation test (25 % of observations). Figure 1 illustrates the daily water levels for the study period. A linear trend line was also fitted to each data set as depicted in the figure.

Fig. 1
figure 1

Time series of the observed water level for a train set, b test set and c validation set

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4.2 Evaluating Accuracy of Proposed Models

The performances of the proposed models were assessed using root means square error (RMSE), coefficient of determination (R2) and Pearson correlation coefficient (r), defined as:

  1. 1)

    Root-mean-square error (RMSE)

$$ RMSE=\sqrt{\frac{{\displaystyle \sum_{i=1}^n{\left({P}_i-{O}_i\right)}^2}}{n}}, $$
(5)
  1. 2)

    Pearson correlation coefficient (r)

$$ r=\frac{n\left({\displaystyle \sum_{i=1}^n{O}_i\cdot {P}_i}\right)-\left({\displaystyle \sum_{i=1}^n{O}_i}\right)\cdot \left({\displaystyle \sum_{i=1}^n{P}_i}\right)}{\sqrt{\left(n{\displaystyle \sum_{i=1}^n{O}_i^2}-{\left({\displaystyle \sum_{i=1}^n{O}_i}\right)}^2\right)\cdot \left(n{\displaystyle \sum_{i=1}^n{P_i}^2}-{\left({\displaystyle \sum_{i=1}^n{P}_i}\right)}^2\right)}} $$
(6)
  1. 3)

    Coefficient of determination (R2)

$$ {R}^2=\frac{{\left[{\displaystyle \sum_{i=1}^n\left({O}_i-\overline{O_i}\right)\cdot \left({P}_i-\overline{P_i}\right)}\right]}^2}{{\displaystyle \sum_{i=1}^n\left({O}_i-\overline{O_i}\right)\cdot {\displaystyle \sum_{i=1}^n\left({P}_i-\overline{P_i}\right)}}} $$
(7)

where P i and O i are the observed and forecasted values of water level, respectively, and n is the total number of data patterns.

4.3 Performance Evaluation of the Proposed ELM Algorithms

Figure 2 presents the scatterplots of the observed vs. predicted water levels by ELM during the train, test and validation periods for the 1-day ahead prediction interval. From the scatters, it is seen that the most of the points are located along the diagonal line, so, it follows that the prediction results of ELM are in very good agreement with the corresponding measured water levels. The corresponded R2 values (in the scatters) also confirm this statement.

Fig. 2
figure 2

Scatter plots of the observed vs. predicted water levels using ELM method for 1 day ahead: a train set, b test set, and c validation set

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The scatter plots of the observed and 7-days ahead predicted water levels have been presented in Fig. 3. Analyzing the scatterplots it is clear that the ELM algorithm give promising results in predicting 7-days ahead water levels, but, expectedly, the prediction accuracy has been reduced (to a less extent) compared to 1-day ahead predictions.

Fig. 3
figure 3

Scatter plots of actual and predicted lake level values using ELM method for 7-days ahead for a training set, b testing set and c validation set

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Figure 4 shows the time series variations of the water level predictions (using ELM) and corresponding observed values for the validation period. Although clear differences can be seen between the results of two prediction intervals (where the outcomes of 1-day ahead prediction is much better than those of the 7-days ahead prediction), ELM is able to predict the water levels in both the prediction intervals rigorously.

Fig. 4
figure 4

Predicted water levels (suing ELM) against corresponding measured data for a 1-day ahead for b 7-days ahead

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5 Performance Comparison of ELM, ANN and GP

In order to assess the proposed ELM approach with respect to the GP and conventional ANN methods, a comparison was also made among the results of these models. Table 4 summarizes the prediction accuracy of each model for train, test and validation periods.

Table 4 Statistical parameters of the ELM, ANN and GP models
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For training data set, all three methods provide similar results. In case of 1 day ahead water level forecasting, ELM performs better than the ANN and GP models in validation period. In case of 7 days ahead prediction, however, the ELM shows better performance than GP. The ANN seems to be slightly better than the ELM model in this case.

6 Conclusion

The ability of ELM model was investigated in prediction of daily water levels in Urmia Lake. Observations collected in Urmia Lake (Northwestern Iran) during from 1972 to 2003 were utilized for model building and evaluation. Two predictive models, for different prediction horizons (1 day and 7 days ahead), were created using the ELM method. A comparison of ELM method with GP and ANN was performed in order to assess the prediction accuracy. Comparison of the results in terms of RMSE, r and R2, indicated that the ELM model is superior to the GP and ANN approaches in 1 day ahead water level prediction. The results suggested that the proposed ELM model can be successfully used in water level forecasting in lakes.