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1 Da, Wujao and Lu, Bn and Meng, Xaoln and Huang, D. () Spato-temporal modellng of dam deformaton usng ndependent component analyss. Survey Revew, 6 (9). pp. 7-. ISSN 7-76 Access from the Unversty of Nottngham repostory: %Deformaton%Usng%ICA-.pdf Copyrght and reuse: The Nottngham eprnts servce makes ths work by researchers of the Unversty of Nottngham avalable open access under the followng condtons. Ths artcle s made avalable under the Unversty of Nottngham End User lcence and may be reused accordng to the condtons of the lcence. For more detals see: A note on versons: The verson presented here may dffer from the publshed verson or from the verson of record. If you wsh to cte ths tem you are advsed to consult the publsher s verson. Please see the repostory url above for detals on accessng the publshed verson and note that access may requre a subscrpton. For more nformaton, please contact eprnts@nottngham.ac.uk

2 SPATIO-TEMPORAL MODELLING DAM DEFORMATION USING INDEPENDENT COMPONENT ANALYSIS Wujao Da, Bn Lu, Xaoln Meng, Dawe Huang ) Dept. of Surveyng Engneerng & Geo-nformatcs, Central South Unversty, Changsha, Chna ) Nottngham Geospatal Insttute, the Unversty of Nottngham, UK wjda@csu.edu.cn( * correspondng author) ABSTRACT Modellng dam deformaton based on the montorng data plays an mportant role n the assessment of a dam s safety. Tradtonal dam deformaton modellng methods generally utlze sngle montorng ponts. It means t s necessary to model for each montorng pont and the spatal correlaton between ponts wll not be consdered usng tradtonal modellng methods. Spato-temporal modellng methods provde a way to model the dam deformaton wth only one functonal expresson and analyze the stablty of dam n ts entrety. Independent Component Analyss (ICA) s a statstcal method of Blnd Source Separaton (BSS) and can separate orgnal sgnals from mxed observables. In ths paper, ICA s ntroduced as a spato-temporal modellng method for dam deformaton. In ths method, the deformaton data seres of all ponts were processed usng ICA as nput sgnals, and a few output ndependent sgnals are used to model. The real data experment wth dsplacement measurements by wre algnment of Wuqangx Dam was conducted and the results show that the output ndependent sgnals are correlated wth physcal responses of causatve factors such as temperature and water level respectvely. Ths dscovery s benefcal n analyzng the dam deformaton. In addton, ICA s also an effectve dmenson-reduced method for spato-temporal modellng n dam deformaton analyss applcatons. Keywords: Dam Deformaton Analyss; Independent Component Analyss; Spato-temporal modelng INTRODUCTION Analyss of montorng data plays an mportant role n the assessment of a dam s safety (Ardto et al, 8; Mata, ; Szostak-Chrzanowsk et al, ; X et al, ). Tradtonal dam deformaton modellng methods, ncludng statstcal analyss and structural dentfcaton, are mostly for sngle montorng pont,.e., one pont, one model (Yu et al, ). It needs to model for each montorng pont and the spatal correlaton between ponts wll not be consdered. But actually, as a whole deformaton body, the dsplacements of each montorng pont are closely lnked. Furthermore, wth the development of modern deformaton montorng technologes, the deformaton data becomes enormous and complex, whle ncludng more useful nformaton. So, new and more effectve analyss tools are now n actve demand for dam deformaton montorng.

3 Two man methods, statstcal analyss and structural dentfcaton, are usually used n the area of dam deformaton montorng. From the result of comparson between statstcal analyss and structural dentfcaton (De Sorts and Paolan, 7), the statstcal model has the advantages of smplcty for functonal expresson, fast executon and sutablty to any knd of correlaton between the governng and dependent parameters. But the statstcal parameters do not have any physcal meanng, whch s not conducve to nterpretng the dam deformaton. The method of blnd source separaton (BSS) was used to separate contrbutons of external loads to the dsplacements from the deformaton data of several ponts on the dam (Popescu, ). Popescu s work showed that the all ponts, one model (.e. spato-temporal model) wth physcal meanng parameters s possble for dam statstcal deformaton modellng. Independent component analyss (ICA) s a method of blnd source separaton proposed n 99s, whch transforms the observed mxed sgnals nto a seres of sgnals whose components are mutually ndependent n statstcal sense. Snce ndependent component ndcates some physcal meanng n some case, ICA can be taken as a data mnng tool. In ths paper, we appled ICA to extract the ndependent dsplacement components from the montorng data of ponts measured by wre algnment on Wuqangx Dam and analyzed the correlaton between the ndependent dsplacement components and causatve factors such as temperature and water level. Then, a spato-temporal dsplacement model of Wuqangx Dam was establshed usng the extracted ndependent components and the correspondng spatal response values to the montorng ponts. In ths paper, the fundamental theory of ICA and the FastICA algorthm are ntroduced n Secton. The deformaton montorng data and the ndependent dsplacement components of Wuqangx Dam are analyzed n Secton. The steps of spato-temporal modellng usng ICA are descrbed n detal n Secton. The spato-temporal dsplacement model of Wuqangx dam deformaton s establshed and the result analyss s descrbed n Secton. Fnally, the conclusons are presented n Secton 6. Basc model of ICA INDEPENDENT COMPONENT ANALYSIS (ICA) ICA s a useful method for blnd source separaton. Its fundamental prncple can be llustrated usng Fgure. Suppose that there are M observatons X, [ (t),, T X X, from N ndependent components S ( t ),,,, N X (t) (t)] M have:, we X (t) AS(t) ; M N () Fg.. The fundamental prncples of ICA

4 Wthout any other pror nformaton about matrx A or source sgnals, ICA ams to obtan a separatng matrx W to separate the orgnal sgnals S (t) n Eq. based on some optmzaton crtera and learnng methods. Generally, the process of calculatng W can be dvded nto two steps: ) Whten the observed sgnals X (t) by a whtenng matrx B, to let T Z BX and E( ZZ ) I ( I s a unt matrx). ) Calculate the rotaton matrx by the specfc ndependence optmze rule, to let Y(t ) UZ, where Y (t) s the best approxmaton vector of (t) FastICA Algorthms S. ICA algorthms can be dvded nto two man categores, and both of them are based on the non-gaussanty and ndependence of the source sgnals. The FastICA s a fast optmzaton teratve algorthm wth a good stablty (Hyvärnen, 999; Hyvärnen and Oja, ). It s based on the negentropy whch s a common quanttatve measure of the non-gaussanty of a random varable. The stronger the non-gaussanty of a random varable s, the greater the negentropy wll be. The detaled steps are as follows:. Centralze and whten the observed data.. Choose an ntal weght vector of unt norm (random) w. T T. Update w through w( k ) E[ xg( w ( k) x)] E[ g'( w ( k) x)] w.. Normalzate w by w( k ) w ( k ). w( k ). Go back to step () f not converged.. INDEPENDENT DISPLACEMENT COMPONENTS OF WUQIANGXI DAM Wuqangx Dam and Its Montorng Data The Wuqangx Dam, bult n 99, s located n the man stream of Yuanshu Rver n Hunan provnce, Chna. The rver s about 7km gong through the cty of Yuanlng. The dam s equpped wth the automated montorng system of wre algnment, nverted plumb, hydrostatc levelng, seepage montorng, uplft pressure montorng, water level measurng, and so on.

5 Fg.. Pcture of Wuqangx Dam Two tenson wre algnments are manly used to montor the horzontal dsplacements of the Wuqangx Dam. The dsplacement data of dfferent montorng ponts n the second tenson wre algnment was selected for the spato-temperal modellng experment, n whch ponts are used to model and another pont s used to check the accuracy of the model. The measurements of water level (the dfference between the water level of upstream and downstream) and ar temperature are also collected for the modellng experment. All the data are measured daly. The dsplacement data seres of the ponts are shown n Fg. and the data seres of causatve factors, ncludng ar temperatures and water level, are shown n Fg.. ex-(mm) ex-(mm) ex-8(mm) ex-(mm) ex-(mm) ex-(mm) ex-9(mm) ex-(mm) ex-(mm) ex-7(mm) ex-(mm) Fg.. Dsplacement data seres of the montorng ponts 6 Temperature( ) Water level(m) Fg.. Data seres of water level and ar temperature Common Dsplacement Components Separaton Before processng the dsplacement data usng ICA, all the data seres of ponts have been centralzed by subtractng the mean values whch are taken as the constant dsplacements of each pont. And then the FastICA algorthm was appled to extract dsplacement components from the centralzed dsplacement montorng data. Three ndependent components (ICs), ncludng almost 99.9% nformaton of the observed data, have been determned and are shown n Fg..

6 IC(mm) IC(mm) IC(mm) Fg.. Independent components extracted from the data of dam dsplacements To probe the relatonshp between ICs and causatve factors, comparsons are made between some ICs and ar temperature and water level as shown n Fg. 6. All the data X X have been standardzed wth mean and varance by Z ( X s the mean S value and S s the varance) and adjusted n the same sgn n order to make clear comparsons. It can be noted that, the common components of each pont extracted by ICA have strong correlaton wth the ar temperature and water level. The data seres of IC has the smlar varaton wth the data seres of ar temperature, and a lag effect exsts at the same tme, whch s consstent wth the effect of ar temperature to the dam deformaton (He, ). The data seres of IC has a smlar varaton wth the data seres of water level, whch means IC represents the common water level dsplacement response of each pont. IC has no obvous features and t has a lttle spatal response to each pont of the dam. We guess t may be due to the other unknown external loads or some mnor combned effects of water level and ar temperature on the dam deformaton. From the above results and analyss, t can be concluded that ICA can extract the ndependent dsplacement components whch can be correlated wth the causatve factors respectvely wthout a pror knowledge. IC and Temperature - (a) temperature IC IC and Water Level - (b) w ater level IC Fg. 6. Comparson between ICs and envronmental factorsspato-temporal Model based on ICA As we can see from the concluson n the thrd secton, ICA can effectvely extract the common dsplacement components of the all ponts caused by ar temperature and water level. It means that ICA can provde a method to nvestgate relatonshp between

7 dsplacements over an entre structure (.e. spato-temporal model) and to descrbe ts global behavor wth only a few ndependent components. Furthermore, each ndependent component s related to only one causatve factor. When extractng the ndependent dsplacement components usng ICA, we can also get the spatal response values of ICs for each pont to the dam dsplacements from the mxng matrx. The spatal response values of ICs to each pont are shown n Fg.7, from whch t seems that the response values may be related to the structure of the dam. spatal response of ICs Fg. 7. IC IC IC spatal poston of each pont (m) Spatal response values of ICs to each pont From the dsplacement measurements of each pont, we can see that the dsplacement responses to the external loads are dfferent. From the physcal vew, t s due to dfferent structure features and external loads n the dfferent postons. However, as an entre structure, there wll be an entre dsplacement response to external loads. Ths entre dsplacement can be measured by all montorng ponts although t hdes n the dsplacement data of the ponts. The three ndependent dsplacement components extracted from data of the ponts usng ICA can be nterpreted as the entre dsplacement responses to hydrostatc load, thermal effect and tme effect or other unknown external loads. The spatal response values of ICs reflect the dfferent dsplacement responses to external loads n dfferent postons. From the fundamental prncple of ICA, the dsplacement of a pont s the entre dsplacement response multplyng the correspondng spatal response value. It means that the spato-temporal modellng procedures can be dvded to spatal modellng wth spatal response values and temporal modellng wth dsplacement ICs respectvely. The steps of spato-temporal modellng dam deformaton based on ICA are shown as follows:. Extract the ndependent components (ICs) from the observed montorng data X usng FastICA algorthms and the ICs and the mxng matrx A can be obtaned. Then X A IC s, s,,.. Model each ndependent component wth sutable methods (dam statstcal modellng such as HHT and HTS or geometrcal modellng such as curve fttng).. Get the spatal response values of ICs to each pont from the mxng matrx A, and model the spatal response values usng space fttng methods. In ths paper, snce

8 the ponts are on one lne of wre algnment, the spatal response models of the ICs are curve functons R s (x), where s,, and x s the postons of the ponts.. Space ft the constant dsplacements usng a surface functon (n two dmenson case) or a curvlnear functon (n one dmenson case) D cons(x).. Multply the temporal models of ICs and the spatal response functons R s (x) and add the spatal constant dsplacement functon Dcons(x) to get the spatal-temporal dsplacement model of the dam D(x) IC s R (x) D (x), where s,, and x s const s the postons of the ponts. As ndcate above n step ), the three dsplacement component need to be modeled usng statstcal modellng or geometrcal modellng methods. Accordng to the analyss before, IC s related to ar temperature and IC s related to water level. So we establsh the models of IC and IC usng the temperature and water level components n the dam HHT model respectvely. The functon model of IC s Equ.. where IC a a T () T means the average temperature of -, -7, 8- and -6 days before because of the lag effect between the temperature of dam and the envronment. The functon model of IC s Equ.. b IC b H () where H denote the dfference of water level between upstream and downstream. Snce the physcal meanng of IC s not clear, we a curve fttng method wth equaton () to model IC. IC a sn( bt c ) a a sn( b t c ) sn( b t c ) a sn( b t c ) SPATIO-TEMPORAL MODEL OF WUQIANGXI DAM AND ITS STATISTICAL ANALYSIS Three common dsplacement components have been extracted from eleven montorng ponts n a tenson wre algnment of Wuqangx Dam n the thrd secton. Based on the modellng method n secton, the three ICs models are establshed. Fg. 8 compares extracted and computed the three dsplacements components. The results ndcate that the common dsplacement component from ICA can be modeled usng Equ. (), () and () very accurately and also confrm that ICA can separate the dsplacement components caused by dfferent external loads. ()

9 Dsplacement(mm) - IC IC (a) fttng value (b) fttng value (c) 7 9 Dsplacement(mm) Dsplacement(mm) - IC fttng value Fg. 8. The fttng results of external load of ar temperature (a), water level (b) and other factors(c) after modellng the ICs. Spatal response values of each dsplacement component are obtaned from the mxng matrx, wth whch the three spatal response functon models are establshed usng curvlnear fttng method. Equ. (), (6) and (7) are the spatal response functons of IC, IC and IC respectvely. The fttng results are shown n Fg. 9. R (x) p p x () x b ( ) c e x b ( ) c x b ( ) c R (x) a a e a e (6) R (x) p p x (7) The constant dsplacements n the ponts are ftted usng a curvlnear functon as Equ. (8) and the fttng results are shown n Fg.. D (x) a (a cos ( x w) b sn ( x w)) (8) const spatal response functon of ICs IC IC IC spatal poston (m) Fg. 9. The spatal response functon of ICs

10 constant dsplacements(mm) - obseved values fttng values spatal postons (m) Fg.. The fttng results of the constant dsplacements At last, the dam dsplacement spato-temporal model s establshed as Equ. (9). D(x) IC IC R (x) IC R (x) D const R (x) where x s the poston n the tenson wre algnment lne. As we known, one of the man purposes of modellng the dam dsplacement s to predct the dsplacement of dam. In order to verfy the effectveness of spato-temporal model shown as Equ. (9), the predcted dsplacements of days for all ponts usng spato-temporal model and tradtonal sngle pont models are compared. The results shown n table ndcate that both models can predct dsplacement wth a hgh accuracy, but the predcton accuracy of sngle pont model s hgher than the one of spato-temporal model. However, from the predcted dsplacement of pont ex-6 whose data hasn t been used to establsh the model, spato-temporal model stll can predct the dsplacement wth a hgh accuracy. Obvously, compared to the sngle pont model the advantage of the spato-temporal model can predct the dsplacement of any poston of the dam no matter where there s a montorng pont. The results of fttng and predcton of the spato-temporal model are shown n Fg. and Fg.. Table. The RMS values of predcted dsplacement error and modellng error of each pont usng dfferent models (x) (9) Sngle Pont Model Spto-temporal Model Modellng Predctve Modellng Predctve Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex Ex

11 ex-(mm) ex-(mm) ex-8(mm) ex-(mm) ex-(mm) ex-(mm) ex-9(mm) ex-(mm) ex-(mm) ex-7(mm) ex-(mm) ex-6(mm) observed values fttng values 9-9 Fg.. Fttng results of the ponts and an checkng pont (ex-6) usng the spato-temporal model ex-(mm) ex-8(mm) ex-(mm) ex-(mm) - // /6/ // /6/ // /6/ // /6/ Date ex-(mm) ex-(mm) ex-9(mm) ex-(mm) - // /6/ // /6/ // /6/ // /6/ Date ex-(mm) ex-7(mm) ex-(mm) ex-6(mm) // /6/ observed values // /6/ // /6/ predcted values // /6/ Date Fg.. Predcted results of the ponts and an checkng pont (ex-6) usng the spato-temporal model CONCLUSION. ICA can effectvely extract the common dsplacement components caused by dfferent external loads such as water level and temperature. Ths s benefcal to the physcal nterpretaton of dam deformaton.. Spatal correlaton between the ponts can be reflected by the spatal response values of ICs.. Spato-temporal modellng procedures can be dvded to spatal modellng wth spatal response values and temporal modellng wth dsplacement ICs

12 respectvely. So, ICA can be used as an effectve spato-temporal modellng tool.. The spato-temporal model usng ICA provdes a way to model the dam deformaton wth only one functonal expresson and analyze the stablty of dam n ts entrety.. Spato-temporal model can predct the dsplacement of any poston of the dam no matter where there s a montorng pont. ACKNOWLEDGMENT Ths work was supported by the Natonal Natural Scence Foundaton of Chna (Grant No. 7) and the State Key Development Program of Basc Research of Chna (Grant No. CB7). The authors would lke to thank Dr. Oluropo Ogundpe for polshng the wrtng. References. Ardto, R., Maer, G., and Massalongo, G., 8. Dagnostc analyss of concrete dams based on seasonal hydrostatc loadng. Engneerng Structures, (), pp He J. P.,. Dam Safety Montorng Theory and Its Applcaton, Chna Water Power Press. 6 pages. (In Chnese). Hyvärnen, A., 999. Fast and robust fxed-pont algorthms for ndependent component analyss. IEEE Trans on Neural Networks, (), pp Hyvärnen, A., and Oja, E.,. Independent component analyss: algorthms and applcatons. Neural Networks, (-), pp. -.. Mata, J.,. Interpretaton of concrete dam behavour wth artfcal neural network and multple lnear regresson models. Engneerng Structures, (), pp Popescu, T. D.,. A new approach for dam montorng and survellance usng blnd source separaton. Internatonal Journal of Innovatve Computng, Informaton and Control, 7 (6), pp Sorts, A. De., and Paolan, P., 7. Statstcal analyss and structural dentfcaton n concrete dam montorng. Engneerng Structures, 9 (), pp Szostak-Chrzanowsk, A., Chrzanowsk, A., and Masséra, M.,. Use of deformaton montorng results n solvng geomechancal problems case studes. Engneerng geology, 79 (-), pp X, G. Y., Yue, J. P., Zhou, B. X., and Tang, P.,. Applcaton of an artfcal mmune algorthm on a statstcal model of dam dsplacement. Computers and Mathematcs wth Applcatons, 6 (), pp Yu, H., Wu, Z. R, Bao, T. F., and Zhang, L.,. Multvarate analyss n dam montorng data wth PCA. Scence Chna Technologcal Scences, (), pp