Determination of Compressive Strength of Concrete by Statistical Learning Algorithms

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1 Artcle Determnaton of Compressve Strength of Concrete by Statstcal Learnng Algorthms Pjush Samu Centre for Dsaster Mtgaton and Management, VI Unversty, Vellore, Inda E-mal: Abstract. hs artcle adopts three statstcal learnng algorthms: support vector machne (SVM), lease square support vector machne (LSSVM), and relevance vector machne (RVM), for predctng compressve strength (f c ) of concrete. Fly ash replacement rato (FA), slca fume replacement rato (SF), total cementtous materal (CM), fne aggregate (ssa), coarse aggregate (ca), water content (W), hgh rate water reducng agent (HRWRA), and age of samples (AS) are used as nput parameters of SVM, LSSVM and RVM. he output of SVM, LSSVM and RVM s f c. hs artcle gves equatons for predcton of f c of concrete. A comparatve study has been carred out between the developed SVM, LSSVM, RVM and Artfcal eural etwork (A). hs artcle shows that the developed SVM, LSSVM and RVM models are practcal tools for the predcton of f c of concrete. Keywords: Support vector machne, least square support vector machne, relevance vector machne, compressve strength, concrete. EGIEERIG JOURAL Volume 17 Issue 1 Receved 18 July 01 Accepted 16 September 01 Publshed 1 January 013 Onlne at

2 1. Introducton In the last years, a number of effcent statstcal learnng algorthms, e.g. support vector machne (SVM) [1, ], least square support vector machne (LSSVM) [3], and relevance vector machne (RVM) [4] have been proposed. Successful applcatons of statstcal learnng algorthms have been reported for varous felds [5-7]. hs artcle adopts SVM, LSSVM and RVM for determnaton compressve strength (f c ) of concrete. SVM was developed by Vapnk and hs coworkers n 1995, and t s based on the structural rsk mnmzaton (SRM) prncple. LSSVM s proposed by takng wth equalty nstead of nequalty constrants to obtan a lnear set of equatons nstead of a quadratc programmng (QP) problem n the dual space [3, 8]. RVM s a sparse method for tranng generalzed lnear models [4]. It can be seen as probablstc verson of SVM. hs study uses the database collected by Pala et al. [9]. able 1 shows the dataset. he database contans nformaton about fly ash replacement rato (FA), slca fume replacement rato (SF), total cementtous materal (CM), fne aggregate (ssa), coarse aggregate (ca), water content (W), hgh rate water reducng agent (HRWRA), age of samples (AS) and f c. A comparatve study has been carred out between the developed SVM, LSSVM and RVM models. he developed SVM, LSSVM and RVM provde equatons for the predcton of f c. able 1. Dataset used n ths study. FA (%) SF (%) CM (kg/m 3 ) ssa (kg/m 3 ) ca (kg/m 3 ) W (lt/m 3 ) HRWRA (lt/m 3 ) Age (days) f c (MPa) EGIEERIG JOURAL Volume 17 Issue 1, ISS (

3 EGIEERIG JOURAL Volume 17 Issue 1, ISS ( 113

4 Detals of SVM SVM uses the followng expresson for the predcton of output varable (y): y w. where x expresses the hgh-dmensonal feature space whch s nonlnearly mapped from the nput space x, b s bas and w s weght. hs artcle adopts FA, SF, CM, ssa, ca, W, HRWRA, and AS as nput varables. he output of SVM s f c. hus x b x FA, SF, CM, ssa, ca, HRWRA, AS (1) 114 EGIEERIG JOURAL Volume 17 Issue 1, ISS (

5 and f c y. he value of w and b have been estmated by mnmzng the regularzed rsk functon, where L y, f L 1 Mnmze: w C L y f 1 1 x 0 y f x y, f x () y f x others x s ε-nsenstve loss functon and ε s error nsenstve zone. o mnmze the effect of nose data, postve slack varables ( and ) have added n Eq. (). By ntroducng kernel functon followng way [1]: 1 1 Mnmze: w C y w. x 1 Subjected to: wx b b. y 0 0 (3) K x, x j, the above optmzaton problem (3) can be wrtten n the 1 Maxmze:. Kx, x y Subject to: j j j 1 1 j (4), 0 C, where, are Lagrange multplers. he fnal equaton of SVM takes the followng form: y K x, x 1 b (5) o develop the SVM, the data have been dvded nto the followng groups: ranng dataset: hs s requred to construct the SVM model. hs artcle uses the same tranng dataset as used by Pala et al. [9]. estng Dataset: hs s requred to verfy the developed SMV. hs artcle uses the same testng dataset as used by Pala et al. [9]. hs study adopts the radal bass functon: x xx x k k Kx, xk exp, k, l = 1,, EGIEERIG JOURAL Volume 17 Issue 1, ISS ( 115

6 where σ s the wdth of radal bass functon and s transpose) as kernel functon. he data have been normalzed between 0 and 1. he program of SVM has been constructed by usng MALAB. 3. Detals of RVM RVM uses the followng equaton for the predcton of output (y). n x a Kx, x y a a (6) where x s nput, K(x, x ) s kernel functon, n s number of data and a s weght. In ths study, and 1 x FA, SF, CM, ssa, ca, HRWRA, AS y he lkelhood of the complete data set can be wrtten as f c 0 p y a,σ πσ n 1 exp y aφ σ (7) o prevent overfttng, automatc relevance detecton (ARD) pror s set over the weghts. p n j exp (8) 0 0 n 1 a 0, where α s a hyperparameter vector that controls how far from zero each weght s allowed to devate [10]. he posteror dstrbuton over the weghts s thus gven by: p p y a, pa a y, py, n , exp a a (9) where the posteror covarance and mean are respectvely: A1 (10) y For unform hyperprors over α and σ, one needs only maxmze the term y α,σ p : p y, py a, pa n I A exp y I A y 1 (11) Maxmzaton of ths quantty s known as the type II maxmum lkelhood method [11, 1] or the evdence for hyper parameter [13]. Hyper parameter estmaton s carred out n teratve formulae, e.g., gradent descent on the objectve functon [14]. he outcome of ths optmzaton s that many elements of 116 EGIEERIG JOURAL Volume 17 Issue 1, ISS (

7 ths go to nfnty such that w wll have only a few nonzero weghts that wll be consdered as relevant vectors. hs study adopts the same tranng dataset, testng dataset, kernel functon and normalzaton technque for the RVM as used by the SVM. MALAB has been used to develop RVM. 4. Detals of LSSVM LSSVM adopts the followng equaton for predcton of output (y) x y w b (1) where w s weght, b s bas, x s nput varable and φ(x) s non-lnear mappng functon. In ths study, and x FA, SF, CM, ssa, ca, HRWRA, AS LSSVM uses the followng optmzaton problem determnaton of w and b: Mnmze: y 1 f c 1 w w Subject to: k k. ek k 1 y x w x b e, k = 1,,. (13) where e k s the random errors and γ s a regularzaton parameter n determnng the trade-off between mnmzng the tranng errors and mnmzng the model complexty. he followng equaton has been obtaned by solvng the above optmzaton problem and t has been used for predcton of f c [15, 16]: where x f c y k 1 K x, x b (14) K, x k s kernel functon and α k s lagrange multplers. hs study adopts the same tranng dataset, testng dataset, kernel functon and normalzaton technque for the LSSVM as used by the SVM and RVM. he program of LSSVM has been constructed by MALAB. 5. Results and Dscusson For SVM, the desgn value of C, ε and σ have been determned by a tral and error approach. he desgn values of C, ε and σ are 100, 0.01 and respectvely. he best SVM produces 115 support vectors. he performance of tranng and testng dataset has been determned by usng the desgn values of C, ε and σ. Fgure 1 shows the performance of tranng and testng for the SVM. hs artcle employs coeffcent of correlaton (R) to assess the performance of SVM. For a good model, the value of R should be close to one. It s observed from Fg. 1 that the value of R s close to one for tranng as well as testng dataset. So, the developed SVM predcts f c farly well. he developed SVM presents the followng equaton (by substtutng x k x xk x Kx, xk exp, σ =, b = 0 and = 130 n Eq. (5) for the predcton of f c ): k k EGIEERIG JOURAL Volume 17 Issue 1, ISS ( 117

8 (-) Predcted ormalzed f c ranng Dataset(R=0.975) estng Dataset(R=0.97) Acutal=Predcted Actual ormalzed f c Fg. 1. Performance of the SVM. f c xkxxk x exp he value of s shown n Fg (15) Fg.. Values of for the SVM ranng Dataset In RVM, the tral and error approach has been adopted for determnng the desgn value of σ. he developed RVM gves best performance at σ = 1. herefore, the desgn value of σ s 1. Fgure 3 shows performance of RVM model. It s observed from Fg. 3 that the value of R s close to one for tranng as well as testng dataset. So, the developed RVM has the capablty for predctng f c. he developed RVM gves the followng equaton for predcton of f c : 118 EGIEERIG JOURAL Volume 17 Issue 1, ISS (

9 x xx x 130 k k f c a exp (16) 1 6. Concluson hs study successfully appled SVM, RVM and LSSVM for the predcton of f c of concrete. 130 datasets have been utlzed to develop the models. User can use the developed equatons for practcal purposes. he developed RVM, SVM and LSSVM gve almost the same performance. he obtaned varance from the RVM can be used to determne uncertanty. SVM and RVM produce sparse solutons. In summary, t can be concluded that SVM, RVM and LSSVM can be examned for solvng dfferent problems n concrete. References [1] V.. Vapnk, he ature of Statstcal Learnng heory, ew York: Sprnger-Verlag, [] B. Schölkopf, Support Vector Learnng, R. Oldenbourg Verlag,M unchen. Doktorarbet, U Berln. Onlne at: [3] J. A. K. Suykens, and J. Vandewalle, Least squares support vector machne classfers, eural Process. Lett., vol. 9, pp , [4] M. E. ppng, he relevance vector machne, Advances n eural Informaton Processng Systems, vol. 1, pp , 000. [5] D. Zhao, W. Zou, and G. Sun, A fast mage classfcaton algorthm usng support vector machne, Proceedng nd Internatonal Conference on Computer echnology and Development, pp , 010. [6] H. Suetan, A. M. Ideta, and J. Mormoto, onlnear structure of escape-tmes to falls for a passve dynamc walker on an rregular slope: Anomaly detecton usng mult-class support vector machne and latent state extracton by canoncal correlaton analyss, Proceedng IEEE Int. Conference on Intellgent Robots and Sys., pp , 011. [7] D. Matć, F. Kulć, M. Pneda-Sánchez, and I. Kamenko, Support vector machne classfer for dagnoss n electrcal machnes: Applcaton to broken bar, Expert Syst. Appl., vol. 39, no. 10, pp , 01. [8] J. A. K. Suykens,. Van Gestel, J. De Brabanter, B. De Moor, and J. Vandewalle, Least Squares Support Vector Machnes, Sngapore: World Scentfc, 00. [9] M. Pala, E. Özbay, A. Öztaş, and M. I. Yuce, Apprasal of long-term effects of fly ash and slca fume on compressve strength of concrete by neural networks, Constr. Buld. Mater., vol. 1, no., pp , 007. [10] B. Schölkopf, and A. J. Smola, Learnng wth Kernels: Support Vector Machnes, Regularzaton, Optmzaton, and Beyond, Cambrdge: MI Press, 00. [11] J. O. Berger, Statstcal Decson heory and Bayesan Analyss, nd ed., ew York: Sprnger, [1] G. A. Wahba, Comparson of GCV and GML for choosng the smoothng parameters n the generalzed splne-smoothng problem, Ann. Statst., vol. 13, no. 4, pp , [13] D. J. MacKay, Bayesan methods for adaptve models, Ph.D. hess, Calforna Insttute of echnology, 199. Onlne at: [14] M. E. ppng, Sparse Bayesan learnng and the relevance vector machne, J. of Machne Learnng Research, vol. 1, pp , 001. [15] V.. Vapnk, Statstcal Learnng heory, ew York: Wley, [16] A. J. Smola, and B. Schölkopf, On a kernel based method for pattern recognton, regresson, approxmaton and operator nverson, Algorthmca, vol., pp EGIEERIG JOURAL Volume 17 Issue 1, ISS ( 119

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