A Bayesian network-based data analytical approach to predict velocity distribution in small streams

Water resources and the factors that have an impact on them should be considered for water resource management and conservation. Therefore, active monitoring of flow properties is required for ecological balance and socio-economic reasons. Velocity profile, discharge, energy and momentum coefficients, and shear stress are the main characteristics of the most important flow properties that need to be identified carefully. Particularly, determination of the velocity profile is essential to estimate the other above-mentioned properties in a river cross section. Some difficulties arise during the collection of point velocities in river cross sections since the roughness characteristics and water depth tend to change with time from one section to another along the flow direction. The procedure of velocity measurements also requires time, money, and effort (Ardiclioglu et al. 2012).

Bayesian networks (BNs) are one of the core components of modeling systems frequently used in ecological systems (Watson et al. 2004; Ames et al. 2005; Borsuk et al. 2006; Pollino et al. 2007; Hamilton et al. 2015; Lucena-Moya et al. 2015). These studies in the literature report that Bayesian belief networks (BBNs) provide the incorporation and representation of uncertain information by identifying causal dependencies between model inputs in terms of probabilistic relationships (Borsuk et al. 2004; Hamilton et al. 2015).

Specifically related with the applications of BN in water management related areas (i.e., river management, lake management, etc.), Adriaenssens et al. (2004) performed a study to assess the success of prediction of macroinvertebrate taxa in rivers. Correctly classified instances and Cohen's Kappa (K) were employed to assess the predictive capacity of the classification models. Their results showed that careful feature selection as well as sensitivity analysis (SA) are the main mechanics that can potentially improve models' performances and related outcomes, for practical use in river restoration management. In a similar study, Reckhow (1999) employed BN to predict the surface water quality and assessment. The rationale behind employing BN is their reasonable structure that aims to handle the complex probabilistic behavior of the variables in the ecosystem.

However, the existing machine learning algorithms have not taken into account the relations among the predictor variables, which might be crucially important in understanding the mechanics of the influential factors of velocity distributions. Furthermore, the existing researches have not considered the probability of accuracy for each prediction. In other words, they have not paid enough attention to the risk of not being able to classify the point velocities correctly, although their major concern was the correct classification.

In the proposed study, a tree-augmented network (TAN)-based BBN model has been utilized to not only predict the velocity distributions but also uncover the hidden relations among the attributes as well as model the probability of prediction risk. In this sense, the aim of developing the velocity predictor is to visualize the conditional interdependency among the explanatory variables and to derive case-specific risk measures that are associated with the predictions. Therefore, in this study, a novel approach, which has never been employed in the associated field, has been utilized to reach the aforementioned goals. To the best of our knowledge, the targeted issues have not been questioned and handled in the existing literature.

In the proposed methodology, a probabilistic acyclic graph-based data analytic approach is proposed to predict the velocity distributions in small streams. The procedure starts with a data collection and preparation phase, and the details of this phase are discussed in the subsequent paragraphs. The second step employs a five-fold cross validation approach to prepare five training samples and five mutually exclusive test samples to provide robustness and to prevent the potential bias that can be caused by data splitting issues. In the third step, tree-augmented BN is employed (1) to predict the velocity distribution and (2) to identify the conditional relations between the independent predictors. The fourth step involves a SA, to observe the relative contribution of each variable to the prediction. In the fifth and final step, the prediction results that were obtained through applying BN are compared with the other data mining models that were widely used in similar research problems.

The advantage of the TAN model utilized in the proposed study over the other machine learning models is that it provides a case-specific probability of outcome and the detailed understanding of conditional relations among the independent variables. These two outcomes are two novel contributions of the proposed methodology to the existing literature, as discussed in the ‘Introduction’. These main steps of the proposed study are discussed in detail in the subsequent sections.

This study focuses on relatively small and shallow streams, where the discharges, Q, vary between 0.29 and 3.93 m³/s. The maximum water depths, H_max, change between 0.26 and 0.86 m in measured cross sections and the water surface width, T, values vary between 2.3 and 9.0 m. The value of the mean velocity, U_m, cross-section area, A, aspect ratio, T/R, and slope of the water surface, S_ws, in the measured cross section are summarized in Table 1. Froude, Fr, and Reynolds, Re, numbers which are presented in Table 1 demonstrate that all the flow measurements were performed under subcritical and turbulent flow conditions. All maximum and minimum values of these mentioned properties are in bold font in Table 1.

An acoustic Doppler velocimeter (ADV), which can record instantaneous three-dimensional (x, y, and z) flow velocities at a single point with a relatively high frequency, is used to measure point velocities during flow measurements. These flow measurements are performed by measuring the velocity of particles in a remote sampling volume based upon the Doppler shift effect (Voulgaris & Trowbridge 1998; Mclelland & Nicholas 2000). Based on the water surface width, measured cross sections were split into n number of slices, which are given in Table 1, column 12. The point velocities were determined in the vertical direction starting about 4 cm from the streambed for each vertical. These transactions were repeated every 2 cm from this point to free water surface for every vertical slice. The ADV was kept in a fixed position while field studies were carried out.

As was mentioned in earlier sections, the variable of interest in the proposed method is point velocities. The existing studies in the literature have forecast point velocity by handling it as a continuous variable (Genc & Dag 2016). BN models can be applied as a prediction method only if the outcome is a discrete variable. Therefore, in the proposed method, point velocity values are converted into discrete classes. ‘Discretization’ is the term that is used for the process of converting a continuous variable into a limited number of classes. Discrete values, as opposed to continuous one, are preferred by many researchers in prediction problems because of various advantages such as the following:

• Discrete values are easy to understand, use, and explain (Liu et al. 2002).

• Many algorithms spend less time on learning discrete values rather than continuous ones (Dougherty et al. 1995).

• The data set, itself, can be simplified and the data size can be reduced by conversion from continuous to discrete values.

Also, depending on the problem to be solved, the number of classes can be another important issue that needs to be determined. There are cases where the binary classification does not provide so much information (as in our case), since the outcome should be predicted as close as possible to its actual value. Therefore, in our study, the outcome variable is converted into ten classes using the following breakpoints (Table 2).

By applying the above-mentioned procedure, the problem has now become a multinomial classification problem, where the outcome variable has ten possible ordinal values, ranging from 1 to 10. Therefore, the learning algorithm will predict which class the point velocity belongs to by taking the eight predictor variables' values into account.

The feasible alternatives to the ‘five-fold cross validation’ could have been obtained by changing the number of folds. There is a trade-off between the number of samples (number of cases) and the number of folds employed. For example, if ‘three-fold cross validation’ had been employed in such a study, the learning algorithms would have learned the existing classes in a much deeper manner since each class would be represented with more learning samples. However, the risk of having bias among the folds would be higher when compared to the ‘five-fold cross validation’ case. Similarly, if ‘ten-fold cross validation’ had been employed in such a study, the learning algorithms would have learned the existing classes in a more superficial manner since each class would be represented with less learning samples. However, the risk of having bias among the folds would be less when compared to the ‘five-fold cross validation’ case. Therefore, based on our past experience (depending on the sample size and number of classes used in the study) and the domain expert knowledge in the related field, it has been decided that employing ‘five-fold cross validation’ would have been the most reasonable option in such a case.

A BBN is a directed acyclic graph that consists of interconnected variables, which present the probabilistic relations/dependencies among the predictor variables as well as the outcome variable. The nodes in the network represent the variables and the arcs represent the causal relations between these variables (Pearl 1985). It is a widely used data mining technique to model the complex relations among the attributes as well as to help reasoning under uncertainty (Anderson 1986). BBNs have been extensively used in prediction problems such as medicine, finance, civil engineering, etc. (Hearty et al. 2009; Landuyt et al. 2013; Sanford & Moosa 2013; Dag et al. 2015).

An optimization problem arises to find the best tree where the objectives are to maximize the logarithmic likelihood of ⁠, and to build a maximum likelihood tree to find a maximal weighted spanning tree in a graph (Chow & Liu 1968). The TAN construction algorithm can be presented below as shown in Drakos et al. (2007).

• I. Compute conditional mutual information function for each (i, j) pairs:

  

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The information amount that provides about when the class variable is known is provided with this function:

• II. Construct a complete undirected graph and use the conditional mutual information function to annotate the weight of an edge that connects to ⁠.

• III. Build a maximum weighted spanning tree.

• IV. Convert the undirected graph to a directed one by choosing a root variable and setting the direction of all edges to be outward from it.

• V. Construct a TAN model by adding a vertex labeled by C and adding an arc from C to each ⁠.

Variable selection is a critical issue that needs to be handled carefully in employing data mining models for prediction problems. In building a BN, the number of the variables plays a crucial role in that BBN uses acyclic graph to visualize the relations among these attributes. Therefore, in the cases where the data set has a crowded set of potential predictors, the variables that should be incorporated in the model should be decided carefully to simplify the visual network. This selection can be made either by the domain expert or the use of variable selection algorithms. In our case, the eight predictor variables that need to be collected are carefully selected by the field experts who have many years of experience in the associated field. After having decided on the variables that need to be incorporated in the model, the network can be learnt and built without any explicit knowledge of domain experts by using a comprehensive data set (as in this current case). Therefore, researchers have paid a considerable amount of interest to learning from the data itself (Lucas et al. 2004). Our objective in the proposed study is to learn a TAN network, where predictor variables (⁠⁠) (n = 8) can be defined as a set of attributes/predictors. These predictors were obtained through a data collection procedure that was discussed in detail in the section ‘Data sources and study area’.

In binary classification problems, there are several performance criteria that can be measured to evaluate the performance of the models employed. These metrics are sensitivity, specificity, area under the receiver operating characteristic curve, F measure, G measure, etc., while for predicting the continuous target, R and R² measures are commonly employed for comparison. As for the multinomial classification problems, where the outcome has more than two classes to be predicted, there is no such standard evaluation criterion.

In our proposed study, accuracy rate (AR) was used as the prediction performance of the BBN model. It represents the ratio of total correct classifications to total number of samples in a model. Therefore, the bigger AR the better classification performance the model has. However, the performance of the model should be compared to the expected percentage of correct classification if the prediction was made in a random manner. For our case, if the assignment was made randomly per each case, the expected correct classification rate would be 10%, since the outcome has ten different classes. Therefore, one should judge the prediction performance with respect to the number of classes that the outcome has.

Based on the discussion in the ‘Methodology’ section, a BBN (TAN) model has been employed to predict the value of the point velocities on a multi-nominal scale. The results that were obtained through using BBN have also been compared with some of the popular prediction techniques that have been widely used in the literature. The remainder of this section is organized as follows. In the following section ‘BN results’, the multi-nominal classification results are presented by using a ten-by-ten matrix, since the dependent variable has ten categories. In this subsection, the network structure that visualizes the interactions among the independent factors is presented in addition to SA results. Finally, in the section ‘Comparison to other methods’, the results that were obtained through using other prediction models were compared with BN results.

The notations used in this study can be presented in the following forms. y/T is the ratio of lateral distance from channel wall (y) to the water surface width (T). z/H indicates the ratio of vertical distance from channel bed (z) to the water depth (H). T/H shows the ratio of water surface width (T) to the water depth (H). T/R is the aspect ratio, the ratio of surface width (T) to hydraulic radius (R). z/T denotes the ratio of vertical distance from the channel bed (z) to the water surface width (T). z/y represents the ratio of vertical distance from the channel bed (z) to the distance from channel wall (y). S_ws is the water surface slope. U_sh is the water surface velocity. u shows the measured point flow velocity (target variable).

As has been discussed in previous sections, our BBN model aims to predict the point velocities in one of the ten multinominal categories. The performance of the model is measured through using two accuracy metrics. The first one is a conservative metric that represents the exact and correct classified cases. For our situation, where the dependent variables are categorized into ten classes, the field engineers would be satisfied if they could predict within one on either side. Therefore, the 1-away correct classification rates are also reported as the second measure, which is not as conservative as the first one. Our aggregated five-fold cross validation BBN results in a confusion matrix which is presented in Table 3. A confusion matrix is widely used in presenting the classification results. Actual and predicted classes are represented in columns and rows, respectively. The samples that are exact-correctly classified are represented in the diagonal cells from the upper left corner to lower right, as they are highlighted in Table 3. For example, the number of cases that are classified as class-1 while in fact they belong to class-1 are denoted in the upper left corner. Therefore, the numbers in diagonal cells represent the correct classification while others represent the misclassifications. In Table 3, the cells that are immediate neighbors to the diagonal cells represent the misclassified samples by only a single class, which were used in calculating 1-away classification accuracies. In Table 3, the summary statistics on the accuracy measurements (both for exact and 1-away) of both for each individual category and the overall prediction accuracy are presented separately.

Table 3

Confusion matrix for the aggregated five-fold BBN classification results

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Since the dependent variable has been categorized into ten classes, it is expected that a random guess (prediction) about a sample has a 10% chance to correctly classify that sample. In other words, if a modeler randomly predicts the outcomes without using any mathematical model, the expected average for exact-correctly classified cases is 0.10, while in our model the average is 0.38, as can be seen from Table 3. These results indicate that the performance that the BBN model achieved has improved about 300% over a random guess. Not only the exact-correctly classified cases but also 1-away prediction results have shown a great performance in classifying the point velocities into ten multinomial categories by an average of 0.79. This basically means, almost eight out of ten predictions are either exactly-correct or only one class away from the target class.

As is emphasized in various sections of the proposed study, the main goal of this study is not only to predict the velocity distributions but also to investigate: (1) which of these predictors have conditional relations with each other, and if we can visualize these relations and (2) if we can derive a score for each predicted sample that gives the probability of accurate prediction?

To achieve the aforementioned goals, TAN has been employed since its structure perfectly fits the projected goals. The only way to apply such method is that the target outcome should be a discrete outcome, which was not the case in our original data set. Therefore, there were two options from which we could choose: (1) converting the continuous outcome to a binary variable; however, for those cases that need to be predicted as closely as possible, this option was not reasonable; and (2) converting the continuous outcome to a number of intervals such that these intervals should be satisfying the field engineers. Therefore, dividing the target outcome into ten intervals has created ten classes. Thus, two performance metric criteria that can handle such a situation have been employed accordingly, and are explained in great detail in ‘Performance metrics’.

Having said that, to our knowledge, there is not any single study that has applied TAN to the related field. The studies in the existing literature (Adriaenssens et al. 2004; Ames et al. 2005; Kisi et al. 2012; Azamathulla et al. 2013; Genc et al. 2014) have rather handled the related problems by considering the target as a continuous outcome. Therefore, the performance criteria that were employed by the existing literature are measures such as R, MAE (mean absolute error), R², etc., which are widely used for continuous predictions. Therefore, the performance criteria that have been employed in the proposed method are not comparable with the existing studies on velocity predictions in streams.

SA is a method to observe the cause and effect relationship between the predictors and the outcome that have been widely applied in machine learning models. It is used to identify the relative contribution of each variable to the prediction performance of the model. The basic idea in performing SA is to compare the model's classification performance when it includes a specific variable with the case when it does not include that specific variable. Therefore, the corresponding change in the output provides the amount of change in the model performance (Principe et al. 2000). This process is repeated for each and every input variable. A more detailed explanation about SA can be found in Dag et al. (2015).

The variables and parameters, which are presented in this figure, can be identified as y/T, where y is the lateral distance from channel wall to measured point and T is the water surface width. z/h: z shows the vertical distance from channel bed and h indicates the water depth in measured vertical. T/H represents the ratio of water surface width (T) to the water depth (H). z/T and z/y are the non-dimensional parameters which were derived using observational values. T/R, S_ws, and U_sh are the variables that were presented in Table 1. As was explained in the related section, the presence of an arrow from an independent variable (parent) to another (child) implies that the impact of the child predictor on the point velocity is dependent on the value of the parent. Also, the dependent variable has many children while it has no parents in the network. There are several relationships that can be discovered that intuitively make a great deal of sense. There are several interesting patterns that could be inferred from the figure by closely examining the parent–child relationships in the given figure.

For example, the impact of every predictor on the outcome is somehow affected by either one or more predictor(s). In other words, there is no predictor in the system such that its impact on the outcome is independent. Another expected inference could be that the contribution of predictor T/H in predicting the outcome variable depends on predictor y/T. This means that the interrelation between these two predictors would have an effect on the point velocity. This inference makes absolute sense since the prediction becomes more challenging in the case of wide and shallow streams. Therefore, the relation between T and H is influential on the prediction capability of the model. Also, the number of measured verticals used in splitting the cross sections is in direct proportion to the prediction. The arrow from z/T to z/H implies that the value of z/H has a particular impact on the point velocity. However, this impact is also dependent on the value of z/T. This inference makes absolute sense for the same reason. Similarly, the contribution of predictor U_sh has conditional interdependency with the predictor S_ws. The reason for that is, as the S_ws (slope of free water surface) increases, predicting the point velocity value would be more difficult since high S_ws causes high uncertainty in predicting the outcome.

There are several other inferences that can be made from the given figure, which makes absolute sense to field engineers. Also, we should note that the machine learning algorithms cover the hidden, unexpected, and uncovered patterns, as well. Therefore, there might be some unexpected findings that should be investigated further in a prospective study.

The information that was extracted from the network can be further analyzed in a more detailed manner by a future study. More specifically, the findings can provide field engineers with significant insights on the variables that have impact on point velocity.

As was emphasized in the earlier sections, the main goal of the current study is not only to predict the point velocities in small streams but rather to investigate the relations among the attributes and derive a probabilistic score for each prediction. Having said that, we have compared the BBN model's prediction results with other widely used data mining models that could be utilized for similar problem scenarios. Specifically, we have employed ANN, multinomial logistic regression (MLR) and decision trees due to the high performance that they have shown in preliminary analysis. What follows is a brief description for these methods.

ANNs are based on a computational model that is inspired by the human nervous system. The basic element of an ANN is a neuron. A neural network is a structure of many such neurons connected in a systematic way (Modeler 2012). Neural networks are typically organized in layers. Input variables are loaded into the network via the input layer. ANNs combine input variables in order to process information. An ANN accepts many inputs, learns them, applies a (usually nonlinear) transfer function in hidden layers, and presents the result in the output layer. Each layer is connected to the previous and the following layers. The weights of connections between the neurons are associated with layers, which obtain the strength of influence one neuron has on another (Modeler 2012). In recent studies, many researchers used ANNs in water resources, hydraulics, and hydrology (Yang & Chang 2005; Kisi et al. 2013; Genc et al. 2014). In our ANN model, the multilayer perceptron algorithm has been utilized using one hidden layer due to high accuracy achieved in the preliminary analysis.

MLR is a statistical technique that generalizes logistic regression to multiclass problems based on values of input fields (Greene 1993). Models can be produced by MLR when the target field is a set field with more than two possible values. In logistic regression, each data set is recoded as a group of numeric fields. MLR was utilized to estimate the probabilities of the different possible outputs of a categorically distributed dependent variable, given a data set of independent variables (Modeler 2012). MLR has been commonly employed in hydraulics and water resources in recent years by researchers (Mamitimin et al. 2015; Phillips et al. 2015). A stepwise MLR has been employed in our model via making trial and error based experiments.

Classification and regression tree (C&R-T) model is one of the tree-based characterization and estimation methods (Genc et al. 2015). The training set is divided into similar parts in the application of this model. The main aim of utilizing the C&R-T is to have subgroups with similar output variables. Some type of the node impurity measure is employed to measure the similarity. The data are partitioned into two subsets in C&R-T. Each subset should be more homogeneous than in the previous subset. It also allows one to specify the prior probability distribution in a classification problem (Modeler 2012).

In the subsequent section, the results obtained through using MLR, C&RT, and ANNs are presented by incorporating five-fold cross validation into each model by also using the same performance criteria, which were discussed in ‘Performance metrics’. The aggregated five-fold cross validated results obtained are presented in Table 4.

Table 4

Confusion matrixes for data mining models representing the aggregated five-fold cross validated results

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The main goal of the proposed research methodology is to develop a machine learning based mathematical model to observe the conditional relations among explanatory variables as well as to obtain a probabilistic score for prediction accuracy. To achieve these goals, a TAN network based machine learning method was developed by incorporating the five-fold cross validation concept, to classify the point velocities on a multinomial scale. A large data set that was collected between 2005 and 2010 from four different cross sections in Turkey, which includes eight potential predictors, was utilized in our study. By analyzing the data through the methodology used in this study, the following research questions have been addressed:

• (a) What are the important contributors in predicting the point velocities in small streams?

• (b) Which of these predictors have conditional relations with each other, and can we visualize these relations?

• (c) Can we derive a score for each predicted sample that gives the probability of accurate prediction?

Therefore, the proposed method considers a probabilistic graphical network model, which has not been investigated in the relevant literature. This graphical network not only predicts the velocity but also observes the relations among the variables. Although the main goal is not only to predict but rather to observe the relations, TAN network has outperformed widely used popular prediction algorithms except for ANN. The findings which have been obtained through probabilistic graphical approach can provide valuable insights to field engineers for them to utilize in decision-making processes.

The methodology offered by the current research can be applied to any related field. The reason for that is that a very similar procedure can be applied to achieve such similar goals. The sequential steps, from data acquisition to classification of the samples, can be employed for a related problem where the outcome needs to be classified and the conditional relations need to be visualized.

It should be noted that using different training and test samples, the results obtained could have been different. This causes uncertainty and, in turn, becomes a concern that needs to be handled. However employing k-fold cross validation concept would decrease the possible differences between the performance levels. Recall, there are k different results, which have been obtained from k different test sets, which have been averaged while computing the final classification measure. Since, there are five different sets utilized in the proposed study, the potential risk of having different scores has been decreased satisfactorily.

A number of limitations in our data analytical study need to be mentioned. We could have incorporated a ten-fold cross validation concept instead of five-fold, which could increase the robustness and unbiasedness across the models. The main reason for that is each fold should have a sufficient number of classes for each of them to be trained well enough by the prediction algorithms. (Recall that, we have ten classes (from 1 to 10) to classify.) Owing to the limited number of samples collected (2,184 samples), employing five-fold cross validation would provide more samples for each fold, which in turn would prevent the over-fitting issues in the training phases. Second, in the comparison section, we could have presented the results from other machine learning algorithms that we have employed in the preliminary data analysis phase. However, presenting them all would have been unnecessary for readers and would take up a great deal of space. Third, more variables could have been incorporated to improve the model accuracy. However, based on the field engineers' suggestions as well as the relevant literature published, the eight variables that were used in this study have been found to be the most relevant and common ones.

A novel approach to predict the point velocities in small streams on a multinomial scale has been developed. Our approach can be easily extended to other relevant problem scenarios such as forecasting discharge and shear stress in small streams.

Potential research that could be carried out in the near future is that a graphical user interface can be developed that automatically predicts the point velocities by running machine learning based classification models as the background. Another possible improvement that could be made is to develop voting based ensembling techniques, which classify the target outcome based on the votes obtained from various models, that can be employed to classify the point velocities.

The corresponding author would like to thank the Scientific and Technological Research Council of Turkey (TUBITAK), since this paper was written during the period of time in which he was supported by this council to pursue a one-year visiting scholarship program at Auburn University, Auburn, Alabama/USA.