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Scour is a natural phenomenon that is created by the rivers streams or the flood which brings about transferring or eroding of bed materials. To have accurate and safe erosion control structures design, maximum scour depth in downstream of the structures gain
s
specific significance. In the current study, M5 model tree as remedy data mining approaches is suggested to estimate the scour depth around the abutments. To do this, Kayaturk laboratory data (2005), with different hydraulic conditions, are used. Then, the results of M5 model were also compared with genetic programming (GP) and pervious empirical results to investigate the applicability, ability, and accuracy of these procedures.
To examine the accuracy of the results yielded from the M5 and GP procedures, two performance indicators (determination coefficient (R2) and root mean square error (RMSE)) were used. The comparison test of results clearly shows that the implementation of M5 technique sound
s
satisfactory regarding the performance indicators (R^{2}
=
0.944 and RMSE
= 0.126) with less deviation from the numerical values. In addition, M5 tree model, by presenting relationships based on liner regression, has good capability to estimate the depth of scour abutment for engineers in practical terms.

Scour is the result of water erosion that causes digging and transferring bed materials and rivers banks. Bridge collapse, due to the total scour in foundation (including abutment and pier), makes the significance of the study about scour prediction and different countermeasures against it completely clear. According to Melville [

Due to lack of enough information in this issue, studies related to scour pattern began from early 1980 (e.g., at Auckland University). Richardson and Richardson [

The exact estimation of scour depth by the help of laboratory studies is a difficult, costly and time-consuming task. Hence, by developing computer software and using in hydraulic research, the estimation of scour depth has been carried out applying these methods by researchers.

Genetic Programming is one of these techniques that are being used in water engineering in recent decades. Azamathulla et al. [

Salmasi et al. [

Recently a new method called M5 decision tree model is presented for solving various problems and predicting output parameters. This model is used to solve engineering problems such as rainfall-runoff modeling [

The purpose of this study is to estimate time development of scour around abutments by the use of M5 and GP liner regression technique. The comparison of the obtained results with the empirical results shows the high capability of this software to estimate time development of scour depth.

The following idea is used by this machine-learning technique: the parameter space is split into areas (subspaces) and in each of them a linear regression model is built. As a matter of fact, the resulting model would be regarded as a modular model, or a committee machine, in which the linear models being specialized on the particular subsets of the input space.

The algorithm called the M5 algorithm is utilized for the sake of inducing a model tree [

A divide-and-conquer method constructs Tree-based models. The set K is either related to a leaf, or some tests are chosen to split K into subsets corresponding to the test outcomes and a similar procedure is applied recursively to the subsets. The splitting criterion used for M5 model tree algorithm depends on treating the standard deviation of the class values that reach a node as a measure of the error at that node, and calculating the expected reduction in this error as a result of testing each attribute at that node. To compute the standard deviation reduction (SDR), the help of this formula seems necessary:

where K indicates a set of examples that reaches the node; K shows the subset of examples that have the ith outcome of the potential set; and sd stands for the standard deviation.

After examining all potential splits, M5 selects the item that enhances the expected error reduction fully. Splitting in M5 ceases when the class values of all the instances that reach a node differ just marginally, or only a few instances are left. The relentless division often creates over-elaborate structures that must be pruned back, namely by substituting a subtree with a leaf. Eventually, a smoothing process is carried out with the aim of compensating for the sharp discontinuities that will unavoidably take place between adjacent linear models at the leaves of the pruned tree, especially for some models constructed from a smaller number of training examples. In this process, the adjacent linear equations are updated in such a way that the projected outputs for the neighboring input vectors corresponding to the different equations are becoming close in terms of value. For more details of this process, Quinlan [

Genetic programming, a branch of the genetic algorithm, is a method for acquiring the most “fit” computer programs by taking the full advantage of artificial evolution [

The great merit of GP for the modeling process lies in its ability to produce models that construct an understandable structure, i.e., a formula or equation. Accordingly, for “data rich, theory poor” instances, GP benefits may outweigh other techniques inasmuch as GP can self-modify, via the genetic loop, a population of function trees so as to ultimately produce an “optimal” and physically interpretable model [

The following expression can analyze the fitness of GP algorithm:

where X_{j} = value returned by a chromosome for the fitness case j and Y_{j} = expected value for the fitness case j. This configuration has been tested for the proposed GP model and has been found adequate [

Kayatork [

where, L_{a} = abutment length, B_{a} = abutment width, U = mean approach flow velocity, y = flow depth, S = slope of the channel, g = gravitational acceleration, ρ = density of the fluid, ρ_{s} = density of the sediment, µ = dynamic viscosity of fluid, d_{50} = median particle grain size, σ_{g} = geometric standard deviation of sediment size distribution, t = time variation of scour when it starts, B = width of channel (

In this study, channel bed sloop, channel width, sediment particle size, flow depth, and consequently Froude number assumed constant. Finally, the above dimensional analysis is summarized as follows:

The first dimensionless relation shows the geometry of the model and represents the fraction of height to weight, and the second dimensionless number is time dimensionless parameter.

SummaryIn this study statistical parameters of correlation coefficient (R^{2}) and root mean square error (RMSE) were used in order to compare the results of two regression methods. The lesser amount of RMSE (0.01) and the larger amount of correlation coefficient (0.0961) introduced GP model better in predicting scour time development in abutments (

One major advantage of M5 model tree is the availability of three simple linear relations (Equations (9), (10) and (11)) which can be easily used to predict the scour around the abutments (

Modeling Approach | RMSE (m) | Correlation Coefficient |
---|---|---|

M5 Model Tree | 0.126 | 0.944 |

Genetic programming | 0.1 | 0.961 |

The temporal variation of local scour depth can be defined as a function of its independent parameters when dimensionless abutment height (L_{a}/B_{a}) is less than 1.12 by the following expressions:

On the other hand, if dimensionless abutment height is greater than 1.12, temporal variation of local scour depth is broken into two parts (Equations (7) and (8))

If time dimensionless parameter

And for

The scour development process aroundthe abutment for different heights was shows in

Non-liner regression relation is represented by Genetic Programming model as follows:

In contrast with the M5 model results, Genetic Programming model trained by dimensionless data was complicated, thus the regression tree has adaptability and capability to predict the scour depth around the abutments.

The potential of M5 model tree in predicting the temporal local scour depth around the abutments was investigated in this paper by using Kayaturk laboratory data [

major conclusion from this study is that M5 model tree works equally well to the Genetic Programming model and provides improved results in comparison to all three empirical relations used in this study. Furthermore, M5 decision tree model, besides simple calculation and equations, has good capability in estimating the depth of time scour in abutment.

Biabani, R., Meftah Halaghi, M. and Ghorbani, Kh. (2016) M5 Model Tree to Predict Temporal Evolution of Clear-Water Abutment Scour. Open Journal of Geology, 6, 1045-1054. http://dx.doi.org/10.4236/ojg.2016.69078