Prediction of Local Scour around Bridge Piers Using Hierarchical Clustering and Adaptive Genetic Programming

Figures & data

Table 1. Field scour data sources in literature that are used in this study

Field Data	Live-bed	Clear-water
Bata and Todorovic (1960)	2	–
Benedict and Caldwell (2006)	4	–
Benedict and Caldwell (2009)	16	–
Boehmler and Olimpio (2000)	–	9
Breusers, Nicollet, and Shen (1977)	1	–
Butch (1991)	16	35
Chang (1980)	6	–
C. Watt, (USGeologicalSurvey 2001 (in 2014 press))	7	8
Davoren (1985)	10	–
Gao, Posada, and Nordin (1993)	249	237
Hayes (1996)	37	80
Hodgkins and Lombard (2002)	1	11
Holnbeck (2011)	1	8
Hopkins, Vance, and Kasraie (1980)	1	–
NBSD (USGeologicalSurvey 2001)	143	98
Muller, Miller, and Wilson (1994)	17	–
Muller and Wagner (2005)	140	96
Neil (1965)	1	–
Shen (1975)	–	3
Southard (1992)	3	–
Williamson (1993)	2	6
Wilson (1995)	36	–
Zhuravlyov (1978)	82	1
Total	775	592

Table 2. Experiment scour data sources in literature that are used in this study

Experiment Data	Live-bed	Clear-water
Aksoy et al. (2017)	–	16
Chabert and Engeldinger (1956)	12	81
Chee (1982)	36	1
Chiew (1984)	92	9
Coleman (unpublished)	3	3
Dey, Bose, and Sastry (1995)	–	18
Dey and Raikar (2005)	–	16
Ettema (1980)	1	96
Ettema et al. (2006)	–	6
Ettema (1976)	–	19
Graf (1995)	–	3
Jain and Fischer (1979)	30	4
Jones (unpublished)	1	16
Lan¸ca et al. (2013)	–	38
Melville (1997)	–	17
Melville and Chiew (1999)	–	27
Mignosa (1980)	–	13
Oliveto and Hager (2002)	15	73
Pandey et al. (2018)	–	4
Pandey et al. (2019)	–	85
Shen, Schneider, and Karaki (1969)	21	2
Max and Miller (2006)	20	4
Max, Odeh, and Glasser (2004)	2	12
Yanmaz and Altinbilek (1991)	–	33
Total	233	596

Figure 1. Local scour around bridge pier.

Figure 2. An overview of the method, with each major step is enclosed in dotted rectangles. “Data Preparation” step is applied to all of the categories in the previous step, “Clustering” is applied to every set in the previous step, and “Genetic Programming” is applied to every cluster in the previous step.

Table 3. Number of observations in each data set using 5-tuple and 6-tuple parameters

	Field		Laboratory
	5-tuple	6-tuple	5-tuple	6-tuple
Clear Water	592	399	596	580
Live-bed	775	456	233	232

Table 4. The resulting clusters for eight data sets. F represents field data, L represents laboratory data, C represents clear-water, and B represents live-bed scour type

Data Set	C. No	C. Size		Data Set	C. No	C. Size		Data Set	C. No	C. Size
FC5	1	76		FB5	1	46		LC5	1	46
	2	38			2	49			2	127
	3	37			3	89			3	84
	4	66			4	46		LC6	1	72
	5	115			5	57			2	44
	6	47			6	38			3	287
FC6	1	35			7	104		LB5	1	69
	2	62			8	74			2	64
	3	56		FB6	1	52			3	100
	4	87			2	168		LB6	1	53
					3	37			2	89

Table 5. Our GP uses the following primitives to form its expression tree. We have modified some of the mathematical functions to control the output

Primitives for GP	Description
$F_r$, $R_e$, $\frac{y}{b}$, $I,$ $\frac{b}{D_{50}}$, ${\sigma }_g$	5-tuple or 6-tuple values acquired from the data sets
Random constants	Float values in the range ${\sigma }_g$
$+,-,\times$	Elementary arithmetical functions
$/$	Division returns 1 if the divisor is 0.
$\sqrt{\left|n\right|}$	Uses absolute value
$min\left(a,b\right)$	Returns the minimum of two values
$tanh$	Returns the hyperbolic tangent
${\left|a\right|}^{\left|b\right|}$	Uses absolute values. Returns 1 if results is greater than 1.

Table 6. The hyperparameters for GP

Population
	Initial Population Size	2000
	Maximum Tree Depth	8 or 16
	Maximum Number of Generations	2000
Selection
	Method	Tournament
	Tournament Size	256
Crossover
	Crossover Probability	0.9
	Terminal Node Probability	0.4
Mutation
	Mutation Probability	0.3

Table 7. Performance output of our method using MSE, RMSE, MAE, MAPE, R² which denotes the score of the training set, $R_v^2$ which denotes the score of the validation set, and $R_t^2$ which denotes the score of the testing set. The clusters denoted by a $\star$ use a maximum tree depth of 16. Values close to 1 are good for $R^2$ and values close to 0 are good for other metrics. Also, $R_v^2>R^2$ and $R_t^2>0.8$ are expected to reduce overfitting

Data Set	C. No	MSE	RMSE	MAE	MAPE	$R_t^2>0.8$	$R_v^2$	$R_v^2$
FC5	1	0.010548	0.102704	0.082777	0.173974	0.800275	0.901659	0.806172
	2	0.006767	0.082261	0.062611	0.284608	0.886301	0.990725	0.944551
	3	0.014459	0.120244	0.087787	0.248155	0.818108	0.957943	0.952453
	4	0.015177	0.123196	0.094176	0.233933	0.800042	0.911575	0.822485
	$5^{\star }$	0.021113	0.145303	0.099401	0.226149	0.804223	0.813475	0.850256
	6	0.004274	0.065378	0.048102	0.096455	0.800589	0.94044	0.952424
FC6	1	0.016168	0.127153	0.104367	0.183802	0.829914	0.839057	0.802505
	2	0.022309	0.149361	0.11143	0.225084	0.800074	0.905491	0.820097
	3	0.020204	0.142142	0.111425	0.476774	0.801273	0.801296	0.898177
	4	0.005677	0.075348	0.058403	0.119834	0.801458	0.962697	0.836493
FB5	1	0.021751	0.147481	0.095046	0.358539	0.805511	0.917735	0.874503
	2	0.006076	0.077947	0.056641	0.146761	0.801219	0.938426	0.822956
	3	0.007157	0.084598	0.050618	0.124501	0.801597	0.815069	0.933436
	4	0.013893	0.117869	0.086634	0.262181	0.840243	0.861084	0.86661
	5	0.033078	0.181872	0.117953	0.201597	0.805651	0.988427	0.997166
	6	0.040691	0.201721	0.162625	0.188316	0.950066	0.988944	0.814211
	7	0.090752	0.30125	0.219491	0.275207	0.803619	0.876843	0.89899
	8	0.069585	0.263789	0.216059	0.484883	0.884063	0.894925	0.978398
FB6	1	0.070473	0.265467	0.184078	0.29092	0.804799	0.805718	0.875073
	2*	0.031252	0.176783	0.129594	0.277199	0.800156	0.902358	0.878799
	3	0.016938	0.130147	0.09608	0.251957	0.806265	0.930083	0.985587
LC5	1	0.046859	0.21647	0.172735	0.117959	0.806122	0.936091	0.808625
	2	0.039697	0.19924	0.127225	0.120902	0.800543	0.89693	0.877788
	3	0.033191	0.182185	0.128039	0.255181	0.802268	0.867877	0.871716
LC6	1	0.018447	0.135819	0.089953	0.110392	0.954166	0.973316	0.94131
	2	0.042362	0.205821	0.140339	0.108058	0.906428	0.938249	0.948942
	3	0.060059	0.245069	0.18656	0.217209	0.800005	0.864921	0.817699
LB5	1	0.025306	0.159079	0.125686	0.110948	0.807244	0.826339	0.95947
	2	0.020862	0.144435	0.094551	0.061326	0.837667	0.842955	0.990569
	3*	0.023271	0.152547	0.107668	0.073573	0.802954	0.884066	0.898402
LB6	1	0.01592	0.126175	0.098589	0.067734	0.8101	0.837749	0.928907
	2	0.017996	0.134151	0.10424	0.068116	0.801582	0.897684	0.94088

Figure 3. These plots of large clusters show how similar the predicted values are to the observed values. The $y=x$ diagonal line shows perfect fit to data. The plots show that laboratory observations are more controlled and almost evenly distributed while the field data accumulates at a range between 0 and 2, while having larger values occasionally.

Table 8. The centers of the clusters after hierarchical clustering

Data Set	Cluster No	${\sigma }_g$	$F_r$	$R_e$	$y/b$	$u/u_c$	$b/D_{50}$
FC5	1		0.22523	9460.9	0.93748	0.76859	840.37
	2		0.43685	101600	1.3305	0.72798	63.334
	3		0.39784	82812	2.0848	0.85888	155.98
	4		0.26706	50300	3.5868	0.57963	187.46
	5		0.12023	18350	2.8768	0.35992	569.81
	6		0.25666	6881.5	0.86875	0.86132	453.31
FC6	1	2.088	0.086631	5487.4	4.8355	0.76803	1134.4
	2	2.0313	0.1758	33514	4.7772	0.56	599.14
	3	4.495	0.19933	22775	1.1289	0.52884	2172.9
	4	0.66437	0.32738	15149	0.84542	0.76955	361.14
FB5	1		0.18635	699.28	1.001	2.1132	44797
	2		0.23587	2917.1	1.1627	1.2118	936.14
	3		0.45762	41529	0.99893	1.6898	2338.6
	4		0.3796	41685	2.1273	1.196	356.82
	5		0.14767	730.46	5.1409	1.8741	1381.9
	6		0.34404	710.79	1.6204	5.853	14416
	7		0.18932	861.14	5.8938	3.8194	6617.9
	8		0.16381	659.78	2.3986	2.045	9951.1
FB6	1	0	0.22107	1247	1.0164	2.4522	18002
	2	2.6317	0.16538	1203.6	4.2282	1.8365	1807.4
	3	0.20432	0.41924	8624.2	0.48098	1.9522	3285.3
LC5	1		0.14475	205.95	4.1297	0.73695	157.23
	2		0.19582	230.61	2.0522	0.71364	132.87
	3		0.27564	191.53	0.68275	0.78205	269.4
LC6	1	1.2557	521.34	923.42	1.3144	0.6365	61.063
	2	1.2618	298.05	438.91	2.8188	0.56056	40.873
	3	1.3749	106.99	204.69	2.4483	0.72297	202.75
LB5	1		0.54032	1056.3	1.2347	1.3139	312.3
	2		0.81302	717.62	1.7719	3.0048	283.25
	3		0.46511	594.87	4.5693	1.7203	78.192
LB6	1	1.2881	0.51422	310.19	2.679	2.1087	173.97
	2	1.591	0.51357	765.73	3.8135	1.5889	53.494

Table 9. The performance comparison of our method with formulas HEC-18 (Richardson and Davis 2001), Melville (Melville 1997), Jain and Fischer (Jain and Fischer 1979)

Data Set	Cluster No	HEC-18	Melville	Jain and Fischer	This Study
FC5	1	0.331507971	0.277580098	0.014565934	0.800275
	2	0.062945935	0.086026014	0.106400447	0.886301
	3	0.3497482	0.243307859	0.158143031	0.818108
	4	0.082890735	0.00319219	0.029988405	0.800042
	5	0.294976311	0.290508087	0.011480782	0.804223
	6	0.157357114	0.014570134	0.303799706	0.800589
FC6	1	0.191933014	0.271378606	0.100487313	0.829914
	2	0.090716458	0.012302472	0.023148688	0.800074
	3	0.048153021	0.001489986	0.026724057	0.801273
	4	0.011271844	0.048136217	0.000632862	0.801458
FB5	1	0.003662855	0.00000541	0.063332313	0.805511
	2	0.001912554	0.002051145	0.00875351	0.801219
	3	0.019361782	0.104417575	0.0016195	0.801597
	4	0.000212602	0.027951078	0.128290568	0.840243
	5	0.069550409	0.022266545	0.028556313	0.805651
	6	0.145290394	0.004444581	0.136717185	0.950066
	7	0.129009955	0.020443494	0.09620118	0.803619
	8	0.011322782	0.066984863	0.0000001	0.884063
FB6	1	0.498485523	0.287938708	0.26357265	0.804799
	2	0.027278085	0.009814272	0.280456651	0.800156
	3	0.078048793	0.328535146	0.362830688	0.806265
LC5	1	0.027976806	0.022102936	0.015052039	0.806122
	2	0.22501321	0.074365415	0.144046591	0.800543
	3	0.085320422	0.061751552	0.090877131	0.802268
LC6	1	0.468782091	0.702613603	0.79191282	0.954166
	2	0.191146756	0.454842754	0.145084758	0.906428
	3	0.029543504	0.146357597	0.097010229	0.800005
LB5	1	0.14901361	0.254375712	0.546943456	0.807244
	2	0.380280717	0.263218018	0.413500722	0.837667
	3	0.001834544	0.013692274	0.003130038	0.802954
LB6	1	0.614636767	0.398204689	0.640002654	0.8101
	2	0.029094086	0.026969666	0.008632601	0.801582

Table 10. Comparison of $R^2$ with and without ${\sigma }_g$ parameter for clusters with 6-tuples

Data Set	Cluster No	R^2 for 5-tuple	R^2 for 6-tuple
FC6	1	0.8911	0.8413
FC6	2	0.7891	0.8138
FC6	3	0.8711	0.8238
FC6	4	0.7946	0.8596
FB6	1	0.9023	0.8199
FB6	2	0.8019	0.8207
FB6	3	0.9254	0.8971
LC6	1	N/A	0.9551
LC6	2	0.9311	0.9222
LC6	3	0.8503	0.8132
LB6	1	0.6736	0.8370
LB6	2	0.1221	0.8315

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