Application of probabilistic neural network to design breakwater armor blocks

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Ocean Engineering 35 (2008) 294–300 www.elsevier.com/locate/oceaneng

Application of probabilistic neural network to design breakwater armor blocks Dookie Kima, Dong Hyawn Kimb,, Seongkyu Changa a

b

Department of Civil and Environmental Engineering, Kunsan National University, Kunsan, Jeonbuk, Republic of Korea Department of Ocean System Engineering, Kunsan National University, Miryong, Kunsan 573 701, Jeonbuk, Republic of Korea Received 3 April 2007; accepted 17 November 2007 Available online 22 November 2007

Abstract This study presents a probabilistic neural network (PNN) technique for predicting the stability number of armor blocks of breakwaters. The PNN is prepared using the experimental data of Van der Meer. The predicted stability numbers of the PNN are compared with those of previous studies, i.e. by an empirical formula and a previous neural network model. The agreement index between the measured and predicted stability numbers by PNN are better than those by the previous studies. The PNN offers a way to interpret the network’s structure in the form of a probability density function and it is easy to implement. Therefore, it can be an effective tool for designers of rubble mound breakwaters. r 2007 Elsevier Ltd. All rights reserved. Keywords: Breakwater; Armor block; Stability number; Probabilistic neural network; Probability density function

1. Introduction Armor units are designed for defending breakwaters from repeated wave loads. Because armor units are decided by the stability numbers, these numbers are very important to design rubble mound breakwaters. Hudson (1958) proposed the formula for estimating the required weight of an individual armor unit in the cover layer of rubble structures. Van der Meer (1988) suggested two stability formulae for breakwater design and most of the Van der Meer tests were performed for deep-water condition. Recently, Mase et al. (1995) examined the applicability of a neural network to analyze stability of rubble-mound breakwaters and compared between predicted stability numbers by neural network and measured ones of Van der Meer (1988) and Smith et al. (1992). Unlike Van der Meer (1988), Van Gent et al. (2003) proposed the formulae for shallow water conditions through physical model tests. Mertens (2007) performed a comparison and analysis of datasets about Van der Meer (1988) and Van Gent et al. Corresponding author. Tel.: +82 63 469 1862; fax: +82 63 463 9493.

E-mail address: [email protected] (D.H. Kim). 0029-8018/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2007.11.003

(2003). Kim and Park (2005) presented several network models to predict the stability number of armor blocks of breakwaters. The same training dataset is used for the neural networks but the structures of neural network and the number of neurons at input and hidden layer differ from those of Mase’s neural networks. They also extended the study of the stability numbers to the reliability analysis. Even if the neural network technique shows better performance than the empirical model-based approach in breakwater design, it has the disadvantage of needing more efforts to determine the architecture of network and more computational time in training the network. In this study, the probabilistic neural network (PNN) is employed to complement the weak point of the neural network. The PNN has been widely used for pattern recognition problems, e.g. in civil/geotechnical engineering fields (Goh, 2002; Aoki et al., 2002; Sinha and Pandey, 2002; Kim et al., 2005). Training and test patterns for the PNN are prepared using the datasets from the experimental data of Van der Meer (1988). To verify the prediction capability of the PNN, the predicted results by PNN are compared with those of an empirical formula and a previous neural network model (Kim and Park, 2005).

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The results of the PNN prove its better performance over the previous studies. 2. Probabilistic neural network The objective of this paper is to apply the PNN technique for predicting stability numbers of breakwater. The overview of the PNN is briefly described as follows. The PNN, a Bayes–Parzen classifier, was first introduced by Specht (1990). The PNN has gained interest because it is easy to implement and it offers a way to interpret the network’s structure in the form of a probabilistic density function (PDF) as follows:   m 1 1X ðX  XAi ÞT ðX  XAi Þ f A ðXÞ ¼ exp  , (1) 2s2 ð2pÞp=2 sp m i¼1 where X is the test vector to be classified, fA(X) the value of the PDF of category A at point X, m the number of training vectors in category A, p the dimensionality of the training vectors, XAi the ith training vector for category A, s the smoothing parameter and T is the transpose matrix. The PNN consists of four layers: input, pattern, summation and output layers (Fig. 1). The input layer passes the inputs to the next layer, in which, five design parameters including the notional permeability of the breakwater (P), the damage level (Sd), the surf similarity parameter (xm), the dimensionless water depth (h/Hs) and the spectral shape (SS) are used as the input neurons for the PNN to predict the stability numbers. The inputs are fully connected with the neurons of the pattern layer. The input neurons are processed by neurons of the pattern layer through an activation function. The exponential function is generally used as the PDF instead of Eq. (1) as

Fig. 2.

follows (Fig. 2):   ðxi  wi;j ÞT ðxi  wi;j Þ exp  , 2s2

(2)

where xi is the ith variable of the test pattern to be classified, wi,j the ith variable of the jth training pattern, s the smoothing parameter and T denotes the transpose of a vector. The denominator of the exponential function denotes the Euclidean distance. The distance indicates the similarities between the test and training patterns. If the distance is close to zero, it means that they are similar. The results of the pattern layer ranges between 0 and 1. The summation layer accumulates the output of the pattern layer. The last layer is an output layer that classifies a test pattern to one of the classes based on the summation layer output. The procedure of the PNN method for calculating stability numbers is described with a simple example in the Appendix A. More details regarding the PNN that is described in this paper can be found in Kim et al. (2005). 3. Prediction models of stability numbers 3.1. Description of dataset

Fig. 1.

The experimental data obtained by Van der Meer (1988) are applied to the PNN. The ranges of the parameters used in Van der Meer (1988) are shown in Table 1. The stability numbers (Ns) are obtained using the original formula of Van der Meer (1988) as follows: 8  0:2 0:18 pSffiffiffiffiffi > d ffi p1ffiffiffiffiffi > xm oxc ; < 6:2P Nw xm (3) Ns ¼  0:2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > Sd P > ffi : 1:0P0:13 pffiffiffiffiffi Xx ; cot ax x m c m N w

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Table 1 Ranges of the parameters used in Van der Meer (1988) Slope angle Relative density No. of waves Surf similarity parameter

cot a ¼ 1.56 D ¼ 12.1 No7000 zm ¼ 0.77

where P is the notional permeability of breakwater, Sd the damage level, Nw the number of attacking waves, xm the surf similarity parameter and cot a is the slope angle. More details regarding the formula can be found in Van der Meer (1988) and Mase et al. (1995). In Van der Meer’s 641 data, there are only two realizations for the number of attacking waves: 1000 and 3000 wave attacks. In general, 1000 attacks are used mainly in actual design for armor blocks of breakwaters. In this paper, therefore, two PNNs are separately constructed for ease of use: PNN I is for 1000 attacks and is composed of 326 experimental datasets, and PNN II is for 3000 attacks and is composed of 315 experimental datasets. 3.2. Neural network model Mase et al. (1995) first examined the applicability of a neural network to analyze model test data of the stability of rubble-mound breakwaters. They added two design parameters (h/Hs; SS) to improve the prediction accuracy of the neural networks (Mase et al., 1995). Kim and Park (2005) employed five neural network models to predict the stability numbers using the experimental data of Van der Meer (1988). Then, they compared the performance of neural networks according to input parameters and results according to the number of neurons in the input and hidden layers. They also extended the study of the stability numbers to the reliability analysis. In this paper, a neural network model that used the same data with Mase’s input parameter is employed to verify the capability of the PNN. The data used to train the neural network are not used in the prediction of stability numbers. Five design parameters including the notional permeability of the P, the Sd, the xm, the h/Hs and the SS are used as the input sets for the neural network model. The number of hidden layer is one, the number of neurons in the hidden layer is four, the number of training epoch is 1  104, the learning rate is 0.5 and the goal of error is 1  103. 3.3. Probabilistic neural network model In order to apply the PNN to the prediction of stability numbers, the rule base should first be composed by using the so-called training patterns. In this study, two PNNs are used, as stated in Section 3.1. PNN I and PNN II are constructed using 207 and 201 training patterns from experimental datasets, respectively, while the other datasets are used as test patterns.

Permeability Armor grading Stability parameter Damage level

P ¼ 0.10.6 Dn85/Dn15 ¼ 2.5 Hs/DDn15 ¼ 2.5 So30

Five design parameters as mentioned in Section 3.2 are used as the input sets for the PNN, and a value of 0.1 for the s variable is used in Eq. (2). To give an equal weighting factor before implementing the data to the network, all input sets are normalized to 0.1–0.9 using the following equation:   Y i;j  minðY j Þ Y i;j ¼ 0:8 þ 0:1, (4) maxðY j Þ  minðY j Þ where Yi,j is the jth variable of the ith input data, min(Yj) is the minimum value of the jth variable and, max(Yj) is the maximum value of the jth variable. In the implementation, the notional permeability of the P is assumed to be 0.1, 0.5 and 0.6 in the cases of impermeability core, permeability core and homogeneity structure, respectively, while the SS used are 1, 2 and 3 in cases of Pierson Moskowitz, narrow and wide spectrum, respectively. The notional permeability of the breakwater and the SSs are hypothetical values to be used in the PNN model. Table 2 shows the samples for constructing PNN using Van der Meer’s (1988) data. Fig. 3 shows the implementation procedure of the PNN for stability number estimation. First, all the input data of training and test patterns are normalized to 0.1–0.9 using Eq. (4) to give an equal weighting factor (Steps 1 and 2). Euclidian distance between training and test patterns is computed by using Eq. (2) (Step 3). Finally, the output (stability number) is obtained through the non-linear operation (Step 4). The process of implementing the PNN method for stability number calculation is described in detail in the Appendix A. 3.4. Results and discussion In this paper, the PNN is compared with the experimental formula and the previous neural network for verification. The stability numbers are only predicted by untrained experimental data (test patterns). The comparison between the measured and the predicted stability numbers by previous neural network is shown in Figs. 4 and 5. Figs. 6 and 7 show the predicted results using the PNN. From the figures, it is difficult to identify which model more effectively estimates the stability numbers. However, PNN appears to have slightly better results than the neural network model. In order to evaluate the results, the capability of the three models to estimate the stability numbers using the agreement index (Ia) and the correlation coefficient (CC) are shown in Table 3. The Ia represents the error or agreement using the ratio between the mean square

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Table 2 The samples of training patterns for construction of PNN Output (class) (Ns)

Input (normalized data) Nw

P

Sd

xm

h/Hs

SS

PNN I 1000 1000 1000 1000 1000 1000 1000 1000 1000

0.1(0.1) 0.5(0.74) 0.6(0.9) 0.1(0.10) 0.1(0.10) 0.1(0.10) 0.5(0.74) 0.1(0.1) 0.6(0.9)

1.21(0.13) 0.68(0.11) 1.13(0.12) 13.77(0.48) 4.55(0.22) 6.71(0.28) 10.54(0.39) 11.84(0.42) 10.64(0.39)

2.78(0.34) 4.95(0.60) 2.58(0.32) 3.65(0.45) 2.19(0.28) 1.57(0.20) 2.99(0.37) 1.44(0.19) 2.02(0.26)

10.87(0.57) 10.26(0.54) 7.93 (0.43) 7.58 (0.41) 7.14 (0.39) 5.75 (0.32) 4.91 (0.28) 4.80 (0.27) 4.25(0.24)

3(0.9) 1(0.1) 1(0.1) 3(0.9) 1(0.1) 2(0.5) 1(0.1) 2(0.5) 1(0.1)

1.25(16) 1.41(28) 1.74(62) 1.80(68) 1.96(84) 2.37(128) 2.81(161) 2.84(163) 3.24(188)

PNN II 3000 3000 3000 3000 3000 3000 3000 3000

0.1(0.10) 0.1(0.10) 0.1(0.10) 0.1(0.74) 0.5(0.74) 0.1(0.10) 0.6(0.90) 0.5(0.74)

0.65(0.10) 1.87(0.12) 7.01(0.21) 18.12(0.10) 7.29(0.22) 17.00(0.39) 18.76(0.42) 16.96(0.39)

5.37(0.68) 3.71(0.48) 3.28(0.42) 2.16(0.63) 3.71(0.48) 1.44(0.20) 2.02(0.27) 2.88(0.37)

17.24(0.90) 13.58(0.72) 10.09(0.54) 6.55(0.55) 5.11(0.29) 4.80(0.27) 4.25(0.24) 3.65(0.21)

1(0.1) 3(0.9) 2(0.5) 3(0.1) 1(0.1) 2(0.5) 1(0.1) 1(0.1)

0.79(1) 1.00(4) 1.35(24) 2.08(92) 2.13(97) 2.84(158) 3.29(179) 3.97(193)

5 4.5

Predicted Ns (neural network)

4 3.5 3 2.5 2 1.5 1 0.5 Fig. 3.

0 0

error (MSE) and the potential error (PE) (Eq. (5)). The PNN used in this study shows the lowest error among the three models. It means that the PNN is very effective in predicting the stability numbers. PE margins are determined by the training patterns. If abundant and dense training patterns are used, the error will be decreased. The Ia and the CC are calculated as follows (Willmott, 1981):  yi Þ 2 MSE , ¼1 I a ¼ 1  Pn 2 PE i¼1 ðjxi  my j þ jyi  my jÞ

1

1.5

2 2.5 3 Measured Ns

3.5

4

4.5

5

Fig. 4.

CC ¼

ð1=nÞ½ðxi  mx ÞT ðyi  my Þ sxy ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . sx  sy ð1=nÞðxi  mx Þ2  ð1=nÞðyi  my Þ2 (6)

Pn

i¼1 ðxi

0.5

(5)

In Eq. (5), x and y denote the estimated and the measured stability numbers, respectively, my is the average

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5

5

4.5

4.5

4

4

3.5

3.5 Predicted Ns (PNN)

Predicted Ns (neural network)

298

3 2.5 2

3 2.5 2

1.5

1.5

1

1

0.5

0.5

0

0 0

0.5

1

1.5

2 2.5 3 Measured Ns

3.5

4

4.5

0

5

0.5

1

2

2.5

3

3.5

4

4.5

5

Measured Ns

Fig. 5.

Fig. 7.

5

Table 3 Performance of stability models with only untrained patterns

4.5 4

Ia CC

3.5 Predicted Ns (PNN)

1.5

a

VMa I

VMa II

ANN I

ANN II

PNN I

PNN II

0.926 0.875

0.927 0.877

0.928 0.904

0.930 0.897

0.948 0.905

0.954 0.913

VM is results by experimental formula of Van der Meer (1988).

3 2.5 2 1.5 1 0.5 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Measured Ns Fig. 6.

of the measured stability numbers, and T denotes the transpose of a matrix. If Ia is close to one, the predicted set agrees well with the measured set. In Eq. (6), mx and my are the corresponding mean values of the stability numbers, sxy is the covariance between x and y, and sx and sy are their respective standard deviations.

Figs. 8 and 9 present the trend of estimation capability of the stability number for the PNN according to the number of training patterns. It generates oscillation because the training patterns of the PNN as pattern classifiers are not composed densely. In other words, most of the stability numbers in each condition are only one in Van der Meer’s experimental data. Therefore, the dense training patterns are not composed because the data for training and test are chosen randomly when dividing the data. Insufficient data causes the oscillation as in Figs. 8 and 9. From the results, it can be found that the Ia is generally increased as the training patterns are increased. The PNN generally shows good results for interpolation methods. In other words, the PNN model is applicable for parameters within the range of the training patterns. However, if the test data is not within the same range as the training pattern, the prediction capability is decreased. 4. Conclusions In this study, the PNN model is applied to predict the stability number of breakwaters and compared with the experimental formula and the previous neural network

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0.98 0.96 0.94 Ia

0.92 0.90 0.88 0.86 0.84 80

100

120

140

160

180

200

220

180

200

220

Number of training data Fig. 8.

0.98 0.96 0.94 Ia

0.92 0.90 0.88 0.86 0.84 80

100

120

140

160

Number of training data Fig. 9.

method to verify its effectiveness. The notional permeability of the breakwater (P), the damage level (Sd), the surf similarity parameter (xm), the dimensionless water depth (h/Hs) and the spectral shape (SS) are used as inputs for the PNN. From the results, it is found that the estimation performance of the proposed method is better than those of an empirical model and the previous neural network model. In order to identify the capability trend for estimating the stability number, the PNN is performed according to the number of the training patterns. Although results show oscillations, it is found that the trend of agreement index is also generally increased as the training patterns are increased. This means that PE margins are determined by the training patterns. If abundant and dense training patterns are used, the error will be decreased. In this paper, only two realizations for the number of attacking waves are considered: 1000 and 3000 wave attacks. However, for future studies, the continuous attacking waves need to be considered in the PNN for a better algorithm. Acknowledgements This work is a part of ‘‘Development of Reliability Based Design for Harbor Structures’’ in High-tech Harbor Research Program supported by MOMAF in Korean Government and KIMST. The financial support is gratefully acknowledged.

Appendix A. Procedure for calculating stability numbers using the PNN The step-by-step procedure for the PNN is explained using a simple problem (Table A.1), wherein we use only the damage level (Sd) and the dimensionless water depth (h/Hs) for a simplified process.



Step 1 and Step 2: Normalize training and test patterns Input and output data are normalized to have values between 0.1 and 0.9 using the following equation: 0 B Xi ¼ @



X i  minðX jth i Þ j

maxðX jth i Þ j



1 C

A minðX jth i Þ j

0:8 þ 0:1.

Step 3: Calculate the distance between training and test patterns The denominator in Eq. (2) vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 2  uX

0:86 train test 2 t dist ¼ ¼ . Xi  Xi 0:28 i¼1

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300 Table A.1 Input

Output

Sd

h/Hs

Ns

Training pattern

0.86 4.84

11.92 4.79

1.14 2.27

Test pattern

4.04

6.87

2.00

Table A.2 Normalized input and output data Input

Output

Sd

h/Hs

Ns (class)

Training pattern

0.1 0.9

0.9 0.1

1.14(1) 2.27(2)

Test pattern

0.74

0.33

2.00

The numerator in Eq. (2) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   8:33 1  logð0:5Þ  ¼ ¼ . 0:1 8:33 1 The dot products between the denominator and the  7:16 numerator are n ¼ . 2:33



Step 4: Calculate the exponential equation  output ¼ expðn  nÞ ¼

5:44e  23 0:0044

.

The PNN selects the largest among values obtained by non-linear function. This value 0.0044 becomes the class

of the training pattern. Therefore, the PNN selects a second training pattern (class 2) in Table A.2. In conclusion, the stability number for test pattern is 2.27.

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