Application of Gamma Process for Deterioration Prediction of Buildings from Discrete Condition Data

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1 Applcaton of Gamma Process for Deteroraton Predcton of Buldngs from Dscrete Condton Data Ruwn Edrsnghe, Sujeeva Setunge and Guomn Zhang RMIT Unversty, Melbourne, Australa ABSTRACT Deteroraton predcton of cvl nfrastructure from dscrete condton data s a challenge faced by many asset managers developng effectve mantenance and renewal strateges. Due to hgh varablty of data, often, determnstc methods are not readly applcable. Some of the relablty based methods adopted are tme dependent relablty analyss, such as Markov chan and more recently gamma process. Whlst such models have been developed for assets wth smaller number components, such as brdges and storm water ppes, for complex systems such as buldngs relablty based methods are less common. The second largest class of nfrastructure assets the local governments own n Australa s the communty buldngs. As the majorty of exstng communty buldngs are maturng, the local governments seek more relable asset management strateges. Condton based forecastng s a major component of such asset management approaches. Ths paper presents development of a relablty based methodology for deteroraton predcton of communty buldngs. Gamma process s consdered to be an approprate approach for predctng buldng element deteroraton due to the temporal varablty of degradaton. The Gamma deteroraton process presented n ths paper s a stochastc process wth ndependent non-negatve ncrements havng gamma dstrbuton wth dentcal scale parameter. Buldng nspecton data from one of the local governments n Vctora are used n the model. Further, analyss of the data and model results are dscussed. Key words: Deteroraton predcton, Gamma process, buldng management. 1. Introducton Infrastructure assets belongng to Vctoran Local Governments (LGs) represent a vast nvestment bult up over many generatons, and are valued at approxmately 23.3 bllon dollars [1]. Out of these, communty buldngs are the second largest class of assets. Buldng management s challengng due to the hgh level of 13

2 complexty of the buldngs unlke the other asset classes such as roads. Above and beyond, the number of agng communty buldng nfrastructure the LGs own s growng exponentally. Consequently, the muncpaltes are n need of relable buldng management strateges. A research project funded by the Australan Research Councl (ARC) enttled A relablty based approach for sustanable management of communty buldngs has sx Vctoran muncpaltes as partners. One of the requrements of the partner councls s to develop a sophstcated and more relable deteroraton predcton mechansm n contrast to the default lnear deteroraton curves they currently use for buldng assets management. Further, currently ths default deteroraton curve s not analyzed aganst each buldng element or the whole buldng rather s used as a snapshot for the whole set of buldngs the councl owns. Moreover, the current decson makng strateges for buldngs most of these councls practce are reactve or on an as-needed bass rather than a proactve strategy wth the understandng of the future needs. Hence, a more relable deteroraton predcton method for ndvdual buldng elements s an ndustry need and a gap to be flled. Deteroraton predcton s a sgnfcant stage n whole of lfe buldng management process [2]. In bref, the whole buldng management process s dvded n to sx stages as Infrastructure System and Elements Herarchy, Condton Ratng Method, Data Collecton Method, Deteroraton Predcton Method, Cost Forecastng and Decson Makng. The STAGE 4 of buldng management process s Deteroraton Predcton Method and t focuses on the deteroraton predcton of the ndvdual elements of the buldng and/or the buldng as a whole nfrastructure system based on the condton data from STAGE 3. The focus of ths paper s to derve and analyze a relablty based deteroraton predcton model based on the buldng condton nspecton data from one partner councl of the research project. The rest of the paper s organzed as follows. Secton 2 presents the lterature revew of the deteroraton predcton mechansms used for varous nfrastructure asset management approaches. Secton 3 dscusses the gamma process model whch s a stochastc deteroraton predcton method that consders the temporal varablty nature of the deteroraton rate of nfrastructure. Secton 4 presents the analyss of the gamma process deteroraton predcton model based on the buldng nspecton data collected by the Vctoran councl. The data preparaton, model parameter 14

3 estmaton, probablty dstrbuton and deteroraton predcton are dscussed. Fnally, Secton 5 concludes the paper. 2. Related Work Varous nfrastructure deteroraton predcton models have been dscussed n the lterature. The exstng deteroraton models for nfrastructure assets can be broadly classfed nto three categores as determnstc models, statstcal models and Artfcal Intellgence (AI) based models. Determnstc deteroraton predcton models can be of two types: lnear or nonlnear. Determnstc models are often used for phenomenon where relatonshps between components are certan. Tme lnear and power law models are appled for water mans [3] and pavements [4]. Exponental determnstc models are appled for ppes [5-7]. The determnstc models, f can be ftted accurately can be analyzed n mathematcal formulae and the relatonshp between nput and output parameters s straght forward. However, ths approach s too smplstc to reflect the probablstc nature of the deteroraton partcularly wth the data wth random nose. Among the AI based technques, Case-Based Reasonng (CBR), fuzzy set theory and Neural Networks (NNs) are used for modelng the deteroraton of nfrastructure facltes. CBR s used for sewer [8]. Fuzzy set theory s used n bured ppes [9]. NNs are used to model deteroraton of varous nfrastructure assets ncludng sewers [10, 11 ], brdges [12], ol and gas ppelnes [13], concrete structures [14] and storm water ppes [15]. Some of the AI based models are of the data-drven type n whch model structure s determned by data,.e. no assumptons are made regardng the model structure [15]. Even though, AI based approaches are nsenstve to nosy data and can automatcally detect non-lnear underlyng models, they have a demand for large quanttes of data and are less lkely to have an underlyng model that descrbes the process hence, leadng to a danger of over fttng [15]. Statstcal modelng s based on statstcal theory for modelng phenomenon where random nose n components exsts and uncertanty s n-bult. Lkelhood that condton of an nfrastructure faclty wll change from one state to another s probablstc n nature because the nfrastructure deteroraton cannot be predcted wth certanty due to unobserved explanatory varables, the presence of measurement errors and the nherent stochastcty of the deteroraton process [16]. Hence, t s more approprate to base the deteroraton modelng of engneerng structures and nfrastructure n terms of a tme-dependent stochastc process. 15

4 In these tme-dependent models the co-effcents such as average rate of deteroraton per unt tme are random varables [17]. However, these randomvarable models do not take the temporal varablty assocated wth the deteroraton rate n to account. Hence, the need for Markov process based models for deteroraton modelng has been noted n the lterature [18]. Dscrete tme Markov process and contnuous tme Markov process are two varants of Markov process used n statstcal deteroraton predcton modelng. Markov chan s a dscrete tme Markov process and s used to predct deteroraton n brdges [19], sewers [20] and storm water ppes [15, 21] n the lterature. Markov model based decson makng process together wth the buldng weghtng s suggested [22]. Compound posson process and gamma process are varants of contnuous tme Markov process that can be used n deteroraton modelng. Accordng to Sngpurwalla and Wlson, the man dfference between these two s that compound posson process has a fnte number of jumps n fnte tme ntervals, whereas gamma process has an nfnte number of jumps n fnte tme ntervals [23]. The compound posson processes s sutable for modelng usage such as damage due to sporadc shocks [17]. The gamma process n contrast, s more sutable for modelng gradual damage accumulatng over tme n a sequence of ncrements by contnuous use. As deteroraton s generally uncertan and non-decreasng, t can best be regarded as a gamma process [24] whch gves a proper model for random deteroraton wth tme. Gamma process s used to model the uncertanty n the tme to falure (lfetme) and/or the rate of deteroraton. Gamma processes are ftted to data on creep of concrete [25], fatgue crack growth [26], thnnng due to corroson [27] and corroded steel gates [28]. Further, gamma process has been used n brdge deteroraton modelng [29, 30]. Even though Markov chan has been wdely used to model the deteroraton of varous nfrastructure assets, the process has been crtczed and the restrctve statonary assumptons about the tme dependent deteroraton rate n Markov chan are beng argued n the lterature [16, 31]. There are reasons for these arguments. Frstly, n Markov chan the deteroraton s assumed to be a sngle step functon. That s the elements do not transt more than one condton state wth n a gven tme step perod. Secondly, the transton probabltes n Markov chan are not tme 16

5 varant and cannot capture the temporal varablty assocated wth the evoluton of the deteroraton. Thrdly, the unt of tme or the nspecton perod whch s the tme between two consecutve nspectons s same n the Markov chan method. 3. Gamma Process Deteroraton Model As dscussed above, the gamma process s the most approprate mechansm to model the monotonc and gradual deteroraton occurrng n buldng components. The gamma process s a stochastc process wth ndependent, non-negatve ncrements havng a gamma dstrbuton wth an dentcal scale parameter. 3.1 Non statonary Gamma Process In the gamma process deteroraton model, the cumulatve deteroraton X s a random quantty. At tme t t has a gamma dstrbuton wth the shape functon a ( t) > 0 and the scale parameter b > 0 whch s a constant. The Probablty Densty Functon (PDF) of X s gven by a( t) b a( t) 1 bx Ga( x a( t), b) = f X ( t) ( x) = x e I (0, ) ( x) (1) Γ( a( t)) Where, I A ( x) = 1for x A and I A ( x) = 0for x A and Γ (u) s the gamma functon for u > 0 whch s gven by u 1 u Γ( u) = v= 0 v e du (2) The functon a(t) must be an ncreasng, rght-contnuous real-valued functon of tmet, wth a( 0) 0 to facltate the monotonc nature of deteroraton over tme. { X ( t), t > 0}, the contnuous-tme stochastc gamma process has ndependent ncrements..e. Let the cumulatve deteroraton at tme t 1 and t2 are X ( t 1 ) and X ( t 2 ), respectvely. The deteroraton ncrement from tme t 1 to t 2 be ndependent of the cumulatve deteroraton at tme t 1 and t s a non-negatve quantty..e. X ( t2 ) X ( t1) s ndependent of X ( t 1 ). Further, X ( t 2 ) X ( t1) ~ Ga( a( t 2 ) a( t1), b) (3) The gamma Cumulatve Dstrbuton Functon (CDF) of the damage s denoted as y GA( y a( t), b) = Ga( x a( t), b) dx (4) 0 17

6 The mean, varance and coeffcent of varaton of the cumulatve deteroraton at tme t are gven as a( t) a( t) µ X ( t ) =, σ X ( t) =, b b and v X ( t ) = 1 a( t) (6) Where coeffcent of varaton s a tme-dependent functon and t s nversely proportonal to the tme. 4. Data Analyss 4.1 Condton Data The councl buldng condton data of nspectons conducted n years 2007 and 2009 are used n our model. The element herarchy used composed of two levels. The frst level buldng elements are Essental Servces, Fnshes, Fttngs, Rsks, Servces, Substructure and Superstructure. The condton ratng used n buldng nspecton s 1-10 ratng where 1 corresponds to near new asset and 10 corresponds to an asset that has faled and s no longer servceable. The condton audts were conducted n 2007 on 139 buldngs and n 2009 on 209 buldngs. Out of whch there are 77 buldngs of whch nspecton data are common to both years and be used n the model. The elements are categorsed n to three groups accordng to the nspecton data. Let the element condton n most recent nspecton be C c (current condton) and the condton n prevous nspecton be C p (prevous condton). Then an element can be n one of the followng three states S 1, S2 or S 3 whch are mutually exclusve and exhaustve states defned based on two consecutve element condtons. The states are S 1, where Cc C p ; S 2, where C c < C p and S 3, where C c = C p. S2 represents a state where the element has been renewed or refurbshed as the current condton s mproved compared to the prevous condton. S 3 represents a state where the element has not been deterorated as the condton remans the same. S 1 represents a state where the element has been actually deterorated as the current condton s poor compared to the prevous condton. The gamma process s used to model the ncremental deteroraton over tme. Hence, the non-ncremental datasets 18

7 corresponds to states S 2 and S 3 are omtted n ths study. The dataset corresponds to the state S 1 that exhbts the postve ncrements are used n our gamma process model. There are 339 buldng element data ponts common to two years, 2007 and 2009 that corresponds to the crtera of condton state S 1. The buldng element Superstructure s used n the gamma process model that has 151 common data ponts Deteroraton Data As the gamma process model s used for the damage or the deteroraton, the nspecton data are to be adjusted to derve the deteroraton of the buldng element under consderaton. Hence, the condton data are adjusted so that, Accumulate ddeteroraton = condton 1 (7) For example, a brand new element whch s n condton 1 has the accumulated deteroraton 0 and a faled element whch s n condton 10 has the accumulated deteroraton 9. Hence, from ths pont onwards the paper refers to the deteroraton rather than the condton of the element Determnstc Approach In 2002 a fnancal valuaton and buldng audt was carred out on the buldngs and the estmated lfe and remanng lfe of the buldngs were recorded. The component age s not an nput parameter n ths study for the focus of gamma process. However, to hghlght the motvaton of ths study, these fnancal estmates are used to derve the age of the superstructure components. The scatter of deteroraton vs. age s shown n Fgure 01. It s noted that the determnstc approach ether lnear or non-lnear cannot be used to predct the deteroraton of the superstructure wth a suffcent accuracy. In addton, there are assumptons about the age of the elements that affect the accuracy of the predcton. 19

8 Fgure 1: Superstructure Deteroraton wth age 4.3. Parameter Estmaton of the Gamma Process Modellng the temporal varablty of the deteroraton wth a gamma process, the mean deteroraton at tme t s often proportonal to a power law as shown below. β a( t) αt µ X ( t) = = (8) b b Where µ X ( t ) = α > 0, β > 0. The gamma process s statonary f the mean deteroraton s lnear n tme,.e. when β = 1. The gamma process s non-statonary when β Method of Maxmum Lkelhood We used method of maxmum lkelhood to estmate the parameters α and b of the statonary gamma process (where β = 1). That s the logarthm of the lkelhood functon of the ndependent deteroraton ncrements are maxmzed to obtan α and 20

9 b. The lkelhood functon L( α, b) = f ( δ ) n whch f s the dstrbuton of δ where = x x 1 δ n tme ntervalθ = t t 1. Parameters are obtaned by computng the frst partal dervatves of the log lkelhood functon of the ncrements wth respect to α and b, and solvng the followng equatons: L( α, b) = b α ( θ ) L( α, b) = 0; α. δ α ( θ ) 1. e bδ L( α, b) = 0; (9) b / Γ( αθ ) maxmzng log L, and usng Equaton (9), followng equatons are obtaned [25] for optmal values of lkelhood estmates) for α and b. θ = b δ α and ' logb. θ + θ logδ = θγ ( αθ )/ Γ Where Γ '( z ) = d( Γ( x)) / dx ( αθ ) α and b (maxmum Because the last nspecton contans the most nformaton, the expected deteroraton at tme t can be derved based on the last nspectons as below [17]. ( t ) b n t n (10) E ( X ( t)) = x / ( Analyss of Data and Results α and b are derved usng the parameter estmaton dscussed above. α value s and b s The PDFs at varous ponts n tme t, are derved and shown n Fgure 2. Correspondng CDFs are shown n Fgure 3. Fgure 2: Gamma PDF of deteroraton X(t) 21

10 The Fgure 4 llustrates the predcted deteroraton over tme. Deteroraton at Year 1 corresponds to the mean value of the accumulated deteroraton n Mean accumulated deteroraton predcted for years from 2010 onwards correspond to the values of years from year 2 onwards. Ths predcton can be used for fnancal forecastng [2]. In addton, varous procurement and management decson makng can be done usng ths deteroraton/condton based predcton as a base and ncorporatng other nfluencng factors such as socal, economc, envronmental and functonal whch s out of scope of ths paper. Fgure 3: Gamma CDF of deteroraton X(t) Fgure 4: Predcted deteroraton over tme 22

11 5. Conclusons The paper has demonstrated the dffcultes assocated wth fttng a determnstc deteroraton predcton models to real condton data collected from councl buldngs. Relablty based methods of deteroraton predcton have been revewed and gamma process has been selected as a possble model to be examned. Applcaton of the gamma process for forecastng condton deteroraton of councl buldngs has been demonstrated wth a real set of condton data collected form councl buldngs and usng data from superstructure of buldngs as an example. The ablty of the Gamma process to forecast deteroraton from condton data of hgh scatter has been demonstrated wth the work reported here. Acknowledgement The authors gratefully acknowledge the nsghtful dscussons and gudance provded by Dr. Malhe Abdollahan at school of Math & Geospatal Scences at RMIT Unversty. References: 1. ABS. The Australan Bureau of Statstcs estmates [cted 2010 March ]. 2. Edrsnghe, R., et al. Councl Buldng Management practces, Case studes and Road Ahead. n The 5th World Congress on Engneerng Asset Management (WCEAM) Brsbane, Australa. 3. Klener, Y. and B. Rajan, Comprehensve Revew of Structural Deteroraton of Water Mans: Statstcal Models. Urban Water Journal, : p Lou, Z., et al., Applcaton of Neural Network Model to Forecast Short-Term Pavement Crack Condton: Florda Case Study. Journal of Infrastructure Systems, ASCE, (4): p Wrahadkusumah, R., M.D. Abraham, and T. Iseley, Challengng Issues n Modelng Deteroraton of Combned Sewers. Journal of Infrastructure Systems, ASCE, (2): p Morcous, G., H. Rvard, and A.M. Hanna, Modelng Brdge Deteroraton Usng Case-Based Reasonng. Journal of Infrastructure Systems, ASCE, (3): p Mshalan, R.G. and S.M. Madanat, Computaton of Infrastructure Transton Probabltes Usng Stochastc Duraton Models. Journal of Infrastructure Systems, ASCE, (4): p Hahn, M., R.N. Palmer, and M.S. Merrll. Prortzng Sewer Lne Inspecton wth an Expert System. n the 26th Annual Water Resources Plannng and 23

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