In this study, the adaptive network-based fuzzy inference system (ANFIS) and artificial neural network (ANN) were employed to estimate the wind- and wave-induced coastal current velocities. The collected data at the Joeutsu-Ogata coast of the Japan Sea were used to develop the models. In the models, significant wave height, wave period, wind direction, water depth, incident wave angle, and wind speed were considered as the input variables; and longshore and cross-shore current velocities as the output variables. The comparison of the models showed that the ANN model outperforms the ANFIS model. In addition, evaluation of the models versus the multiple linear regression and multiple nonlinear regression with power functions models indicated their acceptable accuracy. A sensitivity test proved the stronger effects of wind speed and wind direction on longshore current velocities. In addition, this test showed great effects of significant wave height on cross-shore currents' velocities. It was concluded that the angle of incident wave, water depth, and significant wave period had weaker influences on the velocity of coastal currents.

## NOMENCLATURE

*A*_{i},*B*_{i},*C*_{i},*D*_{i},*E*_{i}fuzzy sets

standard deviations of the Gaussian membership functions

mean values of the Gaussian membership functions

*E*mean square error

significant wave height

*h*_{d}water depth

number of training data

number of testing data

*o*_{i},*p*_{i},*q*_{i},*r*_{i},*s*_{i},*t*_{i}the linear consequent parameters

firing strength of rule

*i*degree of membership

observed value

*P*estimated value

*V*_{longshore}velocity of longshore current

*V*_{cross−shore}velocity of cross-shore current

nonlinear antecedent parameter

learning rate

number of training epochs

*P*_{o}potential value

significant wave period

*W*wind speed

wind direction

incident wave angle

distance between the candidate clusters

radii of clusters

quash factor

acceptance ratio

rejection ratio

value of data point

normalized inputs

original inputs

minimum value of input data

maximum value of input data

## INTRODUCTION

Estimation of wave- and wind-induced coastal current velocities is one of the most important issues in the design of coastal and offshore structures. Coastal currents are usually divided into longshore and cross-shore currents. Field measurements by Yamashita *et al.* (1998) for the Joeutsu-Ogata coast of the Japan Sea showed that the longshore currents are mostly produced by winds. They also showed that the cross-shore currents, i.e., reverse seaward-flowing rip currents, are mainly generated waves. Longshore currents are considerably dominant in the extended area of coasts, especially in offshore zones. By contrast, cross-shore currents are significantly generated by breaking waves in nearshore zones (Yasuda *et al.* 1996; Yamashita *et al.* 1998). In other words, strong offshore-going currents are emerging with high waves inside surf zones. Outside surf zones, the generation of coastal currents is a function of wind conditions (Kato & Yamashita 2003).

Thus far, numerous empirical formulas and numerical models have been presented to estimate the coastal current velocities. Empirical formulas are usually based on wave characteristics, water depth, and incident wave angle whereas wind characteristics such as wind speed and wind direction are neglected in these formulas (Horikawa 1978). In spite of their accuracy, numerical models are not economical for the basic design stage. On the one hand, the execution time significantly increases considering the interaction between wind and wave. On the other hand, the models need many input data such as friction coefficient, high-resolution bathymetry data of study areas, etc. (Kato & Yamashita 2000, 2003).

Data-driven approaches such as the adaptive network-based fuzzy inference system (ANFIS) and the artificial neural network (ANN) can be used to deal with the drawbacks of empirical formulas and numerical models. The behavior of a complex phenomenon can readily be investigated by these approaches. In these models, a black box containing some reasoning relations finds interrelationships between the inputs and output variables representing the physics of the phenomenon. To utilize them properly, they should be trained by a series of training and validation data sets. As well, their efficiency is evaluated by testing data set not used in the training process.

ANN-based models have been used to predict many complex nonlinear systems in coastal engineering fields. For instance, Ruchi *et al.* (2005) employed an ANN model to estimate the significant wave height at coastal areas from deep water wave heights. The model involved a common feedforward network trained by the backpropagation algorithms (FFBP). The obtained results showed a higher accuracy of the FFBP network than the RBF and ANFIS models.

Fuzzy inference system (FIS)-based models have been widely used in water engineering for the following: modeling of rainfall–runoff (Sen & Altunkaynak 2004), capturing scour uncertainty around bridge piers (Johnson & Ayyub 1996), predicting scour depth at abutments of armored beds (Muzzammil 2010; Muzzammil & Alam 2011), optimizing water allocation system (Kindler 1992), controlling reservoir operation (Shrestha *et al.* 1996), modeling of water seepage in an unsaturated zone (Bardossy & Disse 1993), analyzing regional drought (Pongracz *et al.* 1999), modeling of time series (Altunkaynak *et al.* 2004a, 2004b), modeling of equilibrium scour at the downstream of a vertical gate or around pipelines (Uyumaz *et al.* 2006; Zanganeh *et al.* 2011), estimating pile group scour (Bateni & Jeng 2007, Bateni *et al.* 2007), predicting stream flow (Ozger 2009), finding scour location at the downstream of a spillway (Azmathullah *et al.* 2009), and estimating critical velocity for slurry transport in pipelines (Azamathulla & Ahmad 2013). Most of the studies indicate the FIS's superiority to regression approaches.

In coastal engineering, Kazeminezhad *et al.* (2005) employed an ANFIS model to predict wave parameters in the fetch-limited condition. Their results showed superiority of the ANFIS to the so-called Coastal Engineering Manual method at Lake Ontario. Ozger & Sen (2007) applied fuzzy logic to find relationships among the wind speed and previous and current wave characteristics in the Pacific Ocean. Zanganeh *et al.* (2009) developed a genetic algorithm–adaptive network-based fuzzy inference system model (GA-ANFIS) to predict wave parameters at Lake Michigan for the duration-limited condition. Later, Mahjoobi *et al.* (2008) employed the ANN and ANFIS models for wave hindcasting at Lake Ontario. Bakhtyar *et al.* (2008a, 2008b) applied the ANFIS for prediction of wave run-up and longshore sediment transport in swash zones. Recently, Shiri *et al.* (2011) used the ANFIS model to predict the sea level fluctuations at Hillarys Boat Harbour in Perth, Western Australia.

Despite the apparent effects of winds and waves on coastal currents velocities, few studies have been conducted on this issue so far. The aim of the present study is to apply the ANFIS and ANN models to estimate the wind- and wave-induced current velocities. Moreover, the effects of numerous wind and wave variables on the generation of coastal currents are investigated. These models are evaluated using the field observation data of the Joeutsu-Ogata coast of the Japan Sea which is a recognized place for the interaction between wind and wave. Finally, the accuracy of the models is compared with other data-driven approaches, such as the multiple linear regression (MLR) and multiple nonlinear regression with power function (MNLRP) models.

The present paper is set out in seven main sections. Following this section, is a section outlining the study area and its hydrodynamic characteristics. The ANFIS, ANN, MLR and MNLRP models are introduced next. Discussion about the prerequisites to develop the data-driven models follows, and then a detailed discussion on the developed models. This is followed by a section evaluating the developed data-driven models to estimate the current velocities, then finally, concluding remarks are presented.

## BACKGROUND OF THE CASE STUDY

### The study area

^{2}is located between Naoetsu harbors and fishery dock of Kakizaki. Field studies to measure coastal currents, waves and winds were conducted by the Research Centre for Disaster Environment, Disaster Prevention Research Institute at Kyoto University. Figure 1 schematically shows the basic plan of the field study along with recording data stations during 1998–1999. It is clear from the figure that field measurements have been conducted at 13 stations. Stations No. 2, 4, 5, 6, 7, 8, 12, and 13 are placed at the nearshore region whereas stations No. 1, 3, 9, 10, and 11 are located at the offshore zone. At each station about 1,000 data points were measured.

The instruments used in the field study include: (1) high frequency acoustic Doppler current profiler (ADCP, 1,200 Hz) and electro magnetic current meters installed at the sea bottom to measure the current profiles; (2) Wave Hunter was also installed at the same place to measure the incident wave properties; and (3) three-component ultrasonic anemometer installed at the top as TOP to measure local wind characteristics.

### Hydrodynamic characteristics of the study area

*et al.*(1998), the hydrodynamic pattern of the study area during a winter monsoon contains a wide range of high-speed winds (exceeding 10 m/s) and high waves. These circumstances lead to strong coastal currents in the longshore and cross-shore directions of the coast. Figure 2 illustrates the characteristics of observed coastal currents, waves and local winds field during the winter storm from January to the end of February. As shown in the figure during the storm events (circled periods), strong coastal currents are produced by the stormy winds and high wind-driven waves. Inside the nearshore region (5–8 m depth), strong offshore-going currents have a high correlation with wave conditions. However, longshore coastal currents are mostly affected by stormy winds outside the nearshore region (15–20 m depth).

Tidal coastal currents also occur at the Joeutsu-Ogata coast. These currents take place in conjunction with the rise and fall of the tide. The tidal current velocities are generally modest and thought to be less important than those caused by winds and waves in this area. Figure 2 shows the wave and current characteristics observed at the coast during a winter monsoon (Kato & Yamashita 2003). The sketched circles in the figure indicate the period of time when winds and waves are dominant. As seen, the resultant coastal currents are mostly imposed by storms, and other events, such as tides and general circulation, are negligible.

## STRUCTURE OF EMPLOYED MODELS

### ANFIS structure

FIS simulates an ill-defined event generating some linguistic fuzzy IF-THEN rules (Jang 1993). This is the main advantage of the FIS compared to classical learning systems, e.g., ANN models. Several types of FISs proposed in the literature are different in the defuzzification of fuzzy IF-THEN rules consequent part. In this paper, the FIS model introduced by Takagi & Sugeno (1989) (TS) is used to estimate coastal current velocities. In this kind of FIS, there is no systematic way to tune fuzzy IF-THEN rule parameters including the antecedent and consequent parameters. An efficient way to achieve this purpose is to employ an ANN model and then the combined model is termed as an ANFIS model. Consequently, the ANFIS is functionally a TS FIS whose parameters are tuned by a training algorithm.

*mean square error*(

*MSE*) as follows:where is the antecedent parameter, is the learning rate, and

*E*is the

*MSE*defined as follows:where is the th network output at a given output node, is the th target output, is the number of training data. The learning rate during the process is updated as follows:where is the number of training epochs.

#### Subtractive clustering method

Referring to experts is one of the most common methods to extract fuzzy IF-THEN rules in the ANFIS model. This may not be applicable when phenomena have not been experienced yet. A typical solution is benefiting from the clusters obtained by the clustering techniques, such as subtractive clustering method (Chiu 1994).

*D*(

*i*

*=*1,2,….

*D*) dimensions are calculated as follows:where is the potential value of the

*k*th data point, is the clustering radius of the

*i*th dimension of a data point, is the value of the

*i*th dimension of the

*k*th data point, is the value of the

*i*th dimension of the

*j*th data point, and

*K*is the number of data points.

*i*th dimension of the first cluster.

New cluster centers for each step are chosen on the basis of the following two criteria:

A data point with a relative potential value greater than the acceptance threshold () () is directly accepted as a cluster center.

*n*clusters, the Gaussian membership function (represented by the mean and standard deviation) is considered. Then, the

*i*th dimension of the

*m*th (

*m*

*=*

*1, …, n*) cluster center is chosen as the mean value of the

*m*th membership function of the

*i*th dimension. The deviation parameters () are estimated as follows:where is the radius associated with the

*i*th dimension of data points, and is the

*i*th dimension of data points.

It should be noted that the fuzzy IF-THEN rules in FIS and ANFIS models are extracted in order to have their lowest similarities. The number of rules and linguistic variables for each input variable is equal to the number of clusters. To meet minimum similarities in construction of fuzzy IF-THEN rules, only linguistic variables at the same levels are chosen (MATLAB GENFIS 2 command). For example, ‘*A*_{1}’ as the first linguistic variable of input variable *A* makes a rule with the first linguistic variable (‘*B _{1}*’) of input variable

*B*.

### ANN model

*o*, is weight vector, is the input vector for neuron

*i*( = , …, ), is the bias for neuron

*o*, and

*f*is the network transfer function. In this study, the tangent sigmoid is selected as the network transfer function to scale input and output variables between 0 and 1. This function is expressed as follows (Haykin 2009):

### MLR, MNLRP models

In order to verify ANFIS and ANN models, these models are compared with the MLR and MNLRP approaches. The following subsections outline these two competent models.

#### MLR model

#### MNLRP model

## MODEL DEVELOPMENT PREREQUISITES

### Selection of input variables

Based on the hydrodynamic characteristics of the study area and to evaluate the effects of different wind climate on coastal currents, the wind climate is differentiated: the stormy condition is one with wind speeds greater than 10 m/s; otherwise it is the windy condition. It should be noted that the condition in which there is no division between the wind climates is termed ‘general condition’.

Appropriate selection of input variables is an important task in developing any data-driven model. According to the literature, the input variables involved in coastal current velocities' estimation are listed as below (Horikawa 1978):

*V*):(2) for the velocity of cross-shore direction (

_{longshore}*V*)where and are significant wave height and significant wave period, respectively; is water depth,

_{cross−shore}*W*is wind speed, is wind direction with the north, and is the incident wave front angle with the Ogata coastline.

Application of any data-driven approach to predict an event is related to its data sets. In this paper, 9,040 data points collected at the Ogata coast were chosen to identify the relationship between different input variables and the longshore and cross-shore current velocities. Of them, 5,000 data points were chosen randomly as the training data, 700 data points were used as the validation data points, and the remaining 3,340 data points were used as the testing data at the general condition.

Out of 9,040 data points, 2,610 data points were related to the stormy condition while, the remaining 6,430 data points were for the windy condition. 1,500 out of 2,610 data points for the stormy condition were selected randomly as the training data, 200 data points were chosen as the validation data, and the 910 remaining data points were considered as the testing data. At the windy condition, 3,500 out of 6,430 data points were selected randomly as the training data and 500 data points were chosen as the validation data. The 2,430 remaining data points were selected as the testing data set. Table 1 outlines statistical characteristics of the data sets used to develop the estimator models. In this table, the maximum, minimum, average, and range of the training, validation, and testing data are reported for each input variable.

. | . | . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

Training data (numbers = 5,000) | ||||||||

Avg. | 7.112 | 1.53 | 12.4 | 10.80 | −0.321 | 0.191 | 0.011 | 0.0207 |

Min. | 0.40 | 0.17 | 4.98 | 0.24 | −1.55 | 0 | −0.58 | −0.142 |

Max. | 11. 2 | 6.06 | 30.3 | 17.2 | 1.56 | 1.46 | 0.609 | 0.797 |

Range | 10.8 | 5.89 | 25.32 | 16.96 | 3.11 | 1.46 | 1.189 | 0.939 |

Validation data (numbers = 700) | ||||||||

Avg. | 6.88 | 1.832 | 7.92 | 6.73 | 0.220 | 0.262 | 0.0421 | 0.0732 |

Min. | 3.8 | 0.27 | 6.20 | 0.285 | −1.497 | 0.0 | −0.288 | −0.079 |

Max. | 10.5 | 4.18 | 10.00 | 16.15 | 1.560 | 1.239 | 0.460 | 0.532 |

Range | 6.70 | 3.91 | 3.80 | 15.87 | 3.061 | 1.239 | 0.748 | 0.611 |

Testing data (numbers = 3,340) | ||||||||

Avg. | 7.16 | 1.721 | 8.57 | 8.431 | −0.375 | 0.272 | 0.0922 | 0.0841 |

Min. | 2.90 | 0.18 | 4.98 | 0.24 | −1.560 | 0.00 | −0.35 | −0.142 |

Max. | 11.1 | 6.06 | 15.80 | 16.23 | 1.559 | 1.01 | 0.6 | 0.797 |

Range | 8.20 | 5.88 | 10.82 | 15.99 | 3.119 | 1.01 | 0.95 | 0.939 |

. | . | . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|---|

Training data (numbers = 5,000) | ||||||||

Avg. | 7.112 | 1.53 | 12.4 | 10.80 | −0.321 | 0.191 | 0.011 | 0.0207 |

Min. | 0.40 | 0.17 | 4.98 | 0.24 | −1.55 | 0 | −0.58 | −0.142 |

Max. | 11. 2 | 6.06 | 30.3 | 17.2 | 1.56 | 1.46 | 0.609 | 0.797 |

Range | 10.8 | 5.89 | 25.32 | 16.96 | 3.11 | 1.46 | 1.189 | 0.939 |

Validation data (numbers = 700) | ||||||||

Avg. | 6.88 | 1.832 | 7.92 | 6.73 | 0.220 | 0.262 | 0.0421 | 0.0732 |

Min. | 3.8 | 0.27 | 6.20 | 0.285 | −1.497 | 0.0 | −0.288 | −0.079 |

Max. | 10.5 | 4.18 | 10.00 | 16.15 | 1.560 | 1.239 | 0.460 | 0.532 |

Range | 6.70 | 3.91 | 3.80 | 15.87 | 3.061 | 1.239 | 0.748 | 0.611 |

Testing data (numbers = 3,340) | ||||||||

Avg. | 7.16 | 1.721 | 8.57 | 8.431 | −0.375 | 0.272 | 0.0922 | 0.0841 |

Min. | 2.90 | 0.18 | 4.98 | 0.24 | −1.560 | 0.00 | −0.35 | −0.142 |

Max. | 11.1 | 6.06 | 15.80 | 16.23 | 1.559 | 1.01 | 0.6 | 0.797 |

Range | 8.20 | 5.88 | 10.82 | 15.99 | 3.119 | 1.01 | 0.95 | 0.939 |

In addition to the input variable effects on coastal currents' velocities, another feature in the selection of the input variables is their independency. This issue was investigated here by using a correlation matrix (see Table 2). As shown in the table, correlations among the input variables are low enough to consider them as independent input variables.

. | . | . | . | . | . | . |
---|---|---|---|---|---|---|

1 | ||||||

0.481 | 1 | |||||

0.0204 | 0.065 | 1 | ||||

0.0908 | 0.1707 | 0.0007 | 1 | |||

0.033 | 0.111 | 0.0029 | 0.008 | 1 | ||

0.1197 | 0.1453 | 0.0038 | 0.0043 | 0.1080 | 1 |

. | . | . | . | . | . | . |
---|---|---|---|---|---|---|

1 | ||||||

0.481 | 1 | |||||

0.0204 | 0.065 | 1 | ||||

0.0908 | 0.1707 | 0.0007 | 1 | |||

0.033 | 0.111 | 0.0029 | 0.008 | 1 | ||

0.1197 | 0.1453 | 0.0038 | 0.0043 | 0.1080 | 1 |

### Criteria for evaluation of the models

*bias*,

*root mean square error*(

*RMSE*), and correlation coefficient (

*R*) are used to evaluate the performance of estimator models. The

*bias*evaluates whether a model overestimates or underestimates a desired variable by the following equation:where is the th observed value, is the th estimated value, and

*N*is the number of testing data points.

_{test}*N*is the number of testing data points.

_{test}A criterion like the correlation coefficient is not valuable unless it is properly interpreted. As a rule of thumb, correlation coefficients less than 0.35 are generally considered as weak correlation. Also, correlation coefficients between 0.36 and 0.67 show modest or moderate correlation; whereas the *R* values between 0.68 and 0.9 are high correlation. If the correlation coefficient reaches up to 0.9 or more, that means a very high correlation (Weber & Lamb 1970; Kuma 1984). However, the higher values of correlation coefficient do not merely guarantee the performance of the estimator models.

## MODELS' DEVELOPMENT

In this section, the ANFIS, ANN, MLR, and MNLRP models are developed to estimate coastal current velocities. For our experiments, we used the chosen training, validation, and testing data sets in the section ‘Selection of input variables’. As mentioned before, these three subsets have been selected randomly to have models with acceptable generalization capability.

### Development of ANFIS models

In this sub-section, ANFIS models were developed to estimate the coastal current velocities. In order to develop the models, fuzzy IF-THEN rules are needed. The following expressions outline a sample of Sugeno-type fuzzy IF-THEN rules used to estimate the velocity of longshore currents at the general condition.

*& is B*

_{1}*& is C*

_{1}

_{1}*W*is D

*& is E*

_{1}*& is F*

_{1}*THENRule 2: IF is A*

_{1}*& is B*

_{2}*& is C*

_{2}

_{2}*W*is D

*& is E*

_{2}*& is F*

_{2}*THENRule i: IF is A*

_{2}*& is B*

_{i}*& is C*

_{i}

_{i}*W*is D

*& is E*

_{i}*& is F*

_{i}*THENwhere A*

_{i}*, B*

_{i}*, C*

_{i}*, D*

_{i}*, E*

_{i}*, and F*

_{i}*are fuzzy sets related to significant wave period, significant wave height, water depth, wind speed, wind direction, and incident wave angle, respectively. , , , , , and are consequent parameters of the fuzzy rules.*

_{i}*RMSE*s in the figures ensure a fair selection of input variables. Figures 6 and 7 also show initial and improved membership functions by the ANFIS model for each input variable. As shown in the figures, all membership functions have changed during the process. This confirms the effectiveness of every selected variable on the phenomenon. However, a sensitivity analysis can clarify this effectiveness quantitatively.

The *RMSE* associated with the validation and training data are reported in Tables 3–5 for all conditions. Clustering parameters and epochs in which validation and training errors are minimized simultaneously are also reported in the tables. As is apparent from the tables, all ANFIS models compared to initial FIS models performed well enough. For instance, the *RMSE* obtained by the FIS model for estimating the velocity of longshore currents at the general condition is 0.0932 while it is equal to 0.0895 for the ANFIS model. This shows the efficiency of the training process to tune fuzzy antecedent and consequent parameters. In this model, the appropriate number of fuzzy IF-THEN rules is 4 in accordance with the following clustering parameters:

Model type . | FIS . | ANFIS . |
---|---|---|

Longshore direction | ||

Training error (m/s) | 0.0932 | 0.0895 |

Validation error (m/s) | 0.0951 | 0.0893 |

Number of rules | 4 | 4 |

Desirable epoch | 125 | |

Clustering parameters | = [0.56, 0.6, 0.3, 0.6, 0.6, 0.6, 0.6, 2] | |

Cross-shore direction | ||

Training error (m/s) | 0.0676 | 0.0575 |

Validation error (m/s) | 0.0563 | 0.0545 |

Number of rules | 4 | 4 |

Desirable epoch | 104 | |

Clustering parameters | = [0.56, 0.56, 0.3, 0.6, 0.6, 0.6, 0. 6, 2] |

Model type . | FIS . | ANFIS . |
---|---|---|

Longshore direction | ||

Training error (m/s) | 0.0932 | 0.0895 |

Validation error (m/s) | 0.0951 | 0.0893 |

Number of rules | 4 | 4 |

Desirable epoch | 125 | |

Clustering parameters | = [0.56, 0.6, 0.3, 0.6, 0.6, 0.6, 0.6, 2] | |

Cross-shore direction | ||

Training error (m/s) | 0.0676 | 0.0575 |

Validation error (m/s) | 0.0563 | 0.0545 |

Number of rules | 4 | 4 |

Desirable epoch | 104 | |

Clustering parameters | = [0.56, 0.56, 0.3, 0.6, 0.6, 0.6, 0. 6, 2] |

Model type . | FIS . | ANFIS . |
---|---|---|

Longshore direction | ||

Training error (m/s) | 0.1034 | 0.0990 |

Validation error (m/s) | 0.0970 | 0.0935 |

Number of rules | 5 | 5 |

Desirable epoch | 19 | |

Clustering parameters | = [0.4, 0.6, 0.5, 0.5, 0.5, 0.6, 0.6, 2] | |

Cross-shore direction | ||

Training error (m/s) | 0.1015 | 0.0763 |

Validation error (m/s) | 0.0769 | 0.0653 |

Number of rules | 3 | 3 |

Desirable epoch | 32 | |

Clustering parameters | = [0.56, 0.56, 0.56, 0.6, 0.6, 0.6, 0. 6, 2] |

Model type . | FIS . | ANFIS . |
---|---|---|

Longshore direction | ||

Training error (m/s) | 0.1034 | 0.0990 |

Validation error (m/s) | 0.0970 | 0.0935 |

Number of rules | 5 | 5 |

Desirable epoch | 19 | |

Clustering parameters | = [0.4, 0.6, 0.5, 0.5, 0.5, 0.6, 0.6, 2] | |

Cross-shore direction | ||

Training error (m/s) | 0.1015 | 0.0763 |

Validation error (m/s) | 0.0769 | 0.0653 |

Number of rules | 3 | 3 |

Desirable epoch | 32 | |

Clustering parameters | = [0.56, 0.56, 0.56, 0.6, 0.6, 0.6, 0. 6, 2] |

Model type . | FIS . | ANFIS . |
---|---|---|

Longshore direction | ||

Training error (m/s) | 0.1071 | 0.1012 |

Validation error (m/s) | 0.1028 | 0.0989 |

Number of rules | 4 | 4 |

Desirable epoch | 19 | |

Clustering parameters | = [0.3, 0.5, 0.36, 0.56, 0.5, 0.5, 0.56, 2] | |

Cross-shore direction | ||

Training error (m/s) | 0.0588 | 0.0497 |

Validation error (m/s) | 0.0534 | 0.0500 |

Number of rules | 4 | 4 |

Desirable epoch | 29 | |

Clustering parameters | = [0.3, 0.3, 0.5, 0.56, 0.36, 0.56, 0.56, 2] |

Model type . | FIS . | ANFIS . |
---|---|---|

Longshore direction | ||

Training error (m/s) | 0.1071 | 0.1012 |

Validation error (m/s) | 0.1028 | 0.0989 |

Number of rules | 4 | 4 |

Desirable epoch | 19 | |

Clustering parameters | = [0.3, 0.5, 0.36, 0.56, 0.5, 0.5, 0.56, 2] | |

Cross-shore direction | ||

Training error (m/s) | 0.0588 | 0.0497 |

Validation error (m/s) | 0.0534 | 0.0500 |

Number of rules | 4 | 4 |

Desirable epoch | 29 | |

Clustering parameters | = [0.3, 0.3, 0.5, 0.56, 0.36, 0.56, 0.56, 2] |

*=* [0.56, 0.6, 0.3, 0.6, 0.6, 0.6, 0.6, 2]

According to the radii and quash factor, the ANFIS model has simultaneous minimum of the training and validation errors at epoch 125.

As reported in Table 3, the error of developed ANFIS models to estimate the velocity of cross-shore currents is lower than that of the models developed for the velocity of longshore currents. The desirable epoch number in which the training and validation errors are simultaneously minimized is 104.

At the stormy and windy conditions, the same situations were experienced. The error of ANFIS models to estimate the velocity of cross-shore currents is lower than that of the models developed for the velocity of longshore currents. As reported in Table 4, at the stormy condition the desired epoch number is 19 whereas the number for cross-shore currents estimator model is 32. In the windy condition, as seen from Table 5, the desired epoch number for the longshore estimator model is 19 versus 29 for the cross-shore currents estimator model.

*RMSE*is investigated by eliminating the variable from the selected input variables. These results are shown in Figure 8 for the general condition. It can be concluded from Figure 8(a) that the wind direction () and wind speed (

*W*) had stronger effects on the velocity of longshore currents. Furthermore, incident wave angle (), waves characteristics (,), and water depth () have lower influences on the velocity of longshore coastal currents. In addition, as shown in Figure 8(b), for the velocity of cross-shore currents, significant wave height () had stronger effects on the event whereas the significant wave period (), wind speed (

*W*), water depth (), wind direction (), and incident wave angle () had lower effects. These results indicate that although coastal currents in both longshore and cross-shore directions were affected by waves, the effect of local wind speed might not be ignored. A hint to prove the ability of the developed ANFIS models for capturing the complexity of the phenomenon is their higher sensitivity to the wind direction than to the wave. This finding is in accordance with Yamashita

*et al.*(1998).

### Development of the ANN models

To develop the FFBP ANN estimator models, first the training and validation data sets used in the ANFIS models were gathered. Then, the validation data associated with the ANN models were selected randomly as the ratios chosen in the ANFIS models. Since the Levenberg–Marquardt training algorithm produces reasonable results for the majority of ANN applications, in this study this training algorithm is used to update weights and bias values.

Following the selection of the ANN models' prerequisites, six ANN models were developed to estimate coastal current velocities for the three conditions of interest. The main factor in the FFBP is the number of hidden neurons (NHN) and is reported in Table 6. To tune this parameter, several numbers of hidden neurons were examined. Note that the *RMSE* of both training and validation data sets are reported in the table along with desired epochs.

. | Condition . | Desired epoch . | NHN . | Validation error (m/s) . | Training error (m/s) . |
---|---|---|---|---|---|

General | 11 | 200 | 0.0534 | 0.0501 | |

General | 23 | 200 | 0.0823 | 0.0797 | |

Stormy | 9 | 200 | 0.0631 | 0.041 | |

Stormy | 12 | 200 | 0.1031 | 0.117 | |

Windy | 7 | 200 | 0.0489 | 0.0434 | |

Windy | 10 | 200 | 0.0545 | 0.0672 |

. | Condition . | Desired epoch . | NHN . | Validation error (m/s) . | Training error (m/s) . |
---|---|---|---|---|---|

General | 11 | 200 | 0.0534 | 0.0501 | |

General | 23 | 200 | 0.0823 | 0.0797 | |

Stormy | 9 | 200 | 0.0631 | 0.041 | |

Stormy | 12 | 200 | 0.1031 | 0.117 | |

Windy | 7 | 200 | 0.0489 | 0.0434 | |

Windy | 10 | 200 | 0.0545 | 0.0672 |

### Development of the multiple regression models

To identify linear relationships between input and output variables for estimating coastal current velocities, the LSE method was used. The obtained linear relationships for the three conditions of interest are reported as follows.

## EVALUATION OF MODELS

*R*

*=*0.866 and

*R*= 0.751 for the cross-shore and longshore velocities, respectively). As well, the estimated velocities by the ANN model are identical to the observed ones (

*R*= 0.962 and

*R*

*=*0.796 for the cross-shore and longshore velocities, respectively). The merits of the developed models can be captured from Figures 9 to 14. As depicted in the figures, the models are able to estimate the velocity of cross-shore currents in both coastward and seaward. This may show that the models can predict the so-called undertow and rip currents. However, assigning of these features in the numerical models is computationally an expensive process.

At the stormy condition, the ANFIS model performs well with a correlation coefficient of 0.933 to estimate the velocity of cross-shore currents. This model estimates the velocity of longshore currents with a correlation coefficient of 0.784. These correlation values for the ANN model are, respectively, equal to 0.9137 and 0.737. The findings prove that both ANFIS and ANN models are able to estimate the coastal current velocities with high accuracy.

At the windy condition, the ANFIS model performs well to estimate the velocity of cross-shore currents with a correlation coefficient of 0.794, whereas this model estimates the velocity of longshore currents with a correlation coefficient of 0.571. These values for the ANN model are 0.817 and 0.5709, respectively.

These results show that the ANFIS and ANN models can accurately estimate coastal current velocities, although the ANFIS and ANN models at the windy condition give lower correlations. This shows that at the windy condition, the coastal currents are perhaps affected by other currents, such as tidal ones. Nevertheless, the obtained correlation coefficients for the stormy condition show that coastal currents are more influenced by storms, and other currents such as tidal ones are insignificant. Performances of the ANFIS and ANN models are compared in Table 7; it can be seen that the data-driven models outperform the MLR and MNLRP models.

. | . | Method . | |||
---|---|---|---|---|---|

Parameter . | Condition . | ANN . | ANFIS . | MNLR . | MLR . |

General | 0.796 | 0.751 | 0.569 | 0.531 | |

General | 0.962 | 0.866 | 0.583 | 0.546 | |

Stormy | 0.737 | 0.784 | 0.512 | 0.503 | |

Stormy | 0.9137 | 0.933 | 0.643 | 0.326 | |

Windy | 0.5709 | 0.571 | 0.455 | 0.343 | |

Windy | 0.817 | 0.794 | 0.578 | 0.541 |

. | . | Method . | |||
---|---|---|---|---|---|

Parameter . | Condition . | ANN . | ANFIS . | MNLR . | MLR . |

General | 0.796 | 0.751 | 0.569 | 0.531 | |

General | 0.962 | 0.866 | 0.583 | 0.546 | |

Stormy | 0.737 | 0.784 | 0.512 | 0.503 | |

Stormy | 0.9137 | 0.933 | 0.643 | 0.326 | |

Windy | 0.5709 | 0.571 | 0.455 | 0.343 | |

Windy | 0.817 | 0.794 | 0.578 | 0.541 |

Since a high correlation coefficient does not necessarily guarantee the efficiency of a model, Table 8 presents statistical indexes of the estimated coastal current velocities by the ANFIS and ANN models (the *bias* and *RMSE* criteria). As seen from the table, the ANFIS estimations were slightly biased. From the *RMSE* calculated by the models it can be concluded that ANN models are more accurate than the ANFIS models. In other words, the ANN models estimate both longshore and cross-shore current velocities with an acceptable accuracy. The error of the ANN models as well as the ANFIS models at the windy condition was higher than that at the stormy condition.

. | ANN . | ANFIS . | ||
---|---|---|---|---|

RMSE(m/s)
. | bias(m/s)
. | RMSE(m/s)
. | bias(m/s)
. | |

General condition | ||||

0.451 | −0.0137 | 0.242 | −0.0124 | |

0.0743 | −0.0123 | 0.0604 | −0.0162 | |

Stormy condition | ||||

0.568 | −0.0192 | 0.434 | −0.0188 | |

0.1196 | −0.0168 | 0.1232 | −0.0212 | |

Windy condition | ||||

0.9508 | −0.0193 | 0.353 | −0.0137 | |

0.1278 | 0.0026 | 0.145 | 0.0043 |

. | ANN . | ANFIS . | ||
---|---|---|---|---|

RMSE(m/s)
. | bias(m/s)
. | RMSE(m/s)
. | bias(m/s)
. | |

General condition | ||||

0.451 | −0.0137 | 0.242 | −0.0124 | |

0.0743 | −0.0123 | 0.0604 | −0.0162 | |

Stormy condition | ||||

0.568 | −0.0192 | 0.434 | −0.0188 | |

0.1196 | −0.0168 | 0.1232 | −0.0212 | |

Windy condition | ||||

0.9508 | −0.0193 | 0.353 | −0.0137 | |

0.1278 | 0.0026 | 0.145 | 0.0043 |

As mentioned before, the most important contribution in estimating coastal currents by the ANFIS and ANN models is their ability to deal with numerous input and output variables. These models are able to learn and build black box reasoning to estimate coastal current velocities, while the physical behavior of the event is not well understood.

To create a sound conclusion, the sensitivity here is repeated on the testing data sets by using the ANN models. To achieve this, the sensitivity of the ANN models' *RMSE* to the inputs was explored by the one-at-a-time elimination of the input variables. The ANN models are used for the new sensitivity analysis due to their higher accuracy than the ANFIS models. As reported in Table 9, in this kind of sensitivity analysis, like the previous one, the wind direction and speed exert more influences on coastal current velocities in the longshore direction. However, this analysis ensures the wave height effectiveness on cross-shore current velocities.

. | No. . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|

0.242 | 0.264 | 0.268 | 0.253 | 0.251 | 0.251 | 0.246 | |

0.0604 | 0.0654 | 0.0579 | 0.0704 | 0.0727 | 0.0691 | 0.0637 |

. | No. . | . | . | . | . | . | . |
---|---|---|---|---|---|---|---|

0.242 | 0.264 | 0.268 | 0.253 | 0.251 | 0.251 | 0.246 | |

0.0604 | 0.0654 | 0.0579 | 0.0704 | 0.0727 | 0.0691 | 0.0637 |

## SUMMARY AND CONCLUSIONS

The ANFIS and ANN models are data-driven techniques allowing a relatively simple process of building regression (numerical prediction) models, whereas employing the conventional (physically based) numerical modeling methods could be quite complicated and time-consuming. In this study, the ANFIS and ANN models were developed to estimate coastal current velocities at the Joeutsu-Ogata coast of the Japan Sea. Final evaluations of the developed models confirm outperformance of the models compared to the MLR and MNLRP models. In addition, it was concluded that the ANN models were more accurate than the ANFIS models. In addition, the sensitivity analysis showed the wind speed and wind direction having stronger effects on coastal current velocities at the longshore direction. However, water depth, wave characteristics, and incident wave angle had relatively lower effects on these currents. At the cross-shore direction, wave height had more influences on the current velocities compared to the wind speed, wind direction, and water depth.

## ACKNOWLEDGEMENTS

This study was partially supported by the Deputy of Research at Golestan University (GU) and the first author sincerely appreciates their continuous support during the study. Also, the authors thank very much Dr Mahmood Hajiani, Dr Hossein Karimian and Dr Mohsen Lashkarbolok for their constructive comments on the manuscript.