Stochastic Models

Part of the Green Energy and Technology book series (GREEN)


This chapter proposes the formulation of stochastic models (logistic regression analysis and Markov chains model), which allow analysing the claddings’ service life based on probabilistic distribution functions. These models allow assessing: i) the probability of each façade with a given degradation condition according to its age, its characteristics, and the environmental exposure condition; ii) the period of time with maximum probability of transition from a degradation condition to the next one (more severe); iii) the probability of each case study reaching the end of its service life over a given period of time.


Service Life Maximum Probability Markov Chain Model Akaike Information Criterion Degradation Condition 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. Agresti A (1986) Applying R2 type measures to ordered categorical data. Technometrics 28(2):133–138Google Scholar
  2. Agresti A (2002) Categorical data analysis, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  3. Aikivuori AM (1999) Critical loss of performance—what fails before durability. In: 8th international conference on durability of buildings materials and components, DBMC, Vancouver, Canada, pp 1369–1376Google Scholar
  4. Akaike HA (1979) Bayesian extension of the minimum AIC procedure of autoregressive model fitting. Biometrika 66(2):237–242MathSciNetCrossRefzbMATHGoogle Scholar
  5. Akaike HA (1981) Likelihood of a model and information criteria. J Econometrics 16(1):3–14MathSciNetCrossRefzbMATHGoogle Scholar
  6. Akaike HA (1984) New look at the statistical model identification. IEEE Trans Autom Control 19(6):716–723MathSciNetCrossRefzbMATHGoogle Scholar
  7. Ash A, Shwartz M (1999) R2: a useful measure of model performance when predicting a dichotomous outcome. Stat Med 18(4):375–384CrossRefGoogle Scholar
  8. Augenbroe GLM, Park C-S (2002) Towards a maintenance performance toolkit for GSA. Interim Report submitted to GSA, Georgia Institute of Technology, Atlanta, USAGoogle Scholar
  9. Baik HS, Jeong HS, Abraham DM (2006) Estimating transition probabilities in Markov chain-based deterioration models for management of wastewater systems. J Water Resour Plan Manage 132(1):15–24CrossRefGoogle Scholar
  10. Balaras A, Droutsa K, Dascalaki E, Kontoyiannidis S (2005) Deterioration of European apartment buildings. Energy Build 37(5):515–527CrossRefGoogle Scholar
  11. Baum LE, Petrie T (1966) Statistical inference for probabilistic functions of finite state Markov chains. Ann Math Stat 37(6):1554–1563MathSciNetCrossRefzbMATHGoogle Scholar
  12. Bayaga A (2010) Multinomial logistic regression: usage and application in risk analysis. J Appl Quant Methods 5(2):288–297Google Scholar
  13. Bladty M, Sorensen M (2009) Efficient estimation of transition rates between credit ratings from observations at discrete time points. Quant Finance 9(2):147–160MathSciNetCrossRefGoogle Scholar
  14. Bocchini P, Saydam D, Frangopol DM (2013) Efficient, accurate, and simple Markov chain model for the life-cycle analysis of bridge groups. Struct Saf 40:51–64CrossRefGoogle Scholar
  15. Bogdanoff JL (1978) A new cumulative damage model- Part 1. J Appl Mech 45(2):246–250CrossRefGoogle Scholar
  16. Brandt E, Rasmussen M (2002) Assessment of building conditions. Energy Build 34(2):121–125CrossRefGoogle Scholar
  17. Cargal JM (2013) Discrete mathematics for neophytes: number theory, probability, algorithms, and other stuff. Accessed in 31 Jan 2013. Available online in
  18. Carrington PJ, Scott J, Wasserman S (eds) (2005) Models and methods in social network analysis, 1st edn. Cambridge University Press, New YorkGoogle Scholar
  19. Chew M (2005) Defect analysis in wet areas of buildings. Constr Build Mater 19(3):165–173CrossRefGoogle Scholar
  20. Choi EC (1999) Wind driven rain on building facades and the driving rain index. Wind Eng Ind Aeronaut 79(1–2):105–122CrossRefGoogle Scholar
  21. Choi M, Lee G (2010) Decision tree for selecting retaining wall systems based on logistic regression analysis. Autom Construct 19(7):917–928MathSciNetCrossRefGoogle Scholar
  22. Cox DR, Miller HD (1965) The theory of stochastic processes, 1st edn. Chapman and Hall, LondonzbMATHGoogle Scholar
  23. Cox DR, Snell EJ (1989) The analysis of binary data, 2nd edn. Chapman and Hall, LondonzbMATHGoogle Scholar
  24. Cox DR, Wermuch N (1992) A comment on the coefficient of determination for binary responses. Am Stat 46(1):1–4Google Scholar
  25. Das PC (1998) New developments in bridge management methodology. Struct Eng Int 8(4):299–302Google Scholar
  26. Dekker R, Nicolai RP, Kallenberg LCM (2008) Maintenance and Markov decision models. Encyclopedia of statistics in quality and reliability. Wiley, LondonGoogle Scholar
  27. Ellingwood BR (2005) Risk-informed condition assessment of civil infrastructure: state of practice and research issues. Struct Infrastruct Eng Maintenance Manage Life-Cycle Des Perform 1(1):7–18CrossRefGoogle Scholar
  28. Frangopol DM, Neves LC (2004) Probabilistic maintenance and optimization strategies for deteriorating civil infrastructures. In: Topping BHV, Mota Soares CA (eds) Progress in computational structures technology. Saxe-Coburg Publishers, Stirling, pp 353–377Google Scholar
  29. Frangopol DM, Kallen M-J, Noortwijk JMV (2004) Probabilistic models for life-cycle performance of deteriorating structures: review and future directions. Prog Struct Mat Eng 6(4):197–212CrossRefGoogle Scholar
  30. Freitas VP, Sousa M, Abrantes V (1999) Survey of the durability of facades of 4000 dwellings in northern Portugal. In: 8th DBMC, international conference on the durability of building materials and components; Ottawa, Canada, pp 1040–1050Google Scholar
  31. Gao S, Shen J (2007) Asymptotic properties of a double penalized maximum likelihood estimator in logistic regression. Stat Probab Lett 77(9):925–930MathSciNetCrossRefzbMATHGoogle Scholar
  32. Garavaglia E, Gianni A, Molina C (2004) Reliability of porous materials: two stochastic approaches. J Mater Civ Eng 16(5):419–426CrossRefGoogle Scholar
  33. Gaspar P (2009) Service life of constructions: development of a method to estimate the durability of construction elements. Application to renderings of current buildings (in Portuguese). Doctor thesis in sciences of engineering, Instituto Superior Técnico, University of Lisbon, PortugalGoogle Scholar
  34. Gu MG, Kong FH (1998) A stochastic approximation algorithm with Markov chain Monte Carlo method for incomplete data estimation problems. Nat Acad Sci USA 95(13):7270–7274MathSciNetCrossRefzbMATHGoogle Scholar
  35. Hair JF, Black WC, Babin B, Anderson RE, Tatham RL (2007) Multivariate data analysis, 6th edn. Prentice-Hall Publishers, Englewood CliffsGoogle Scholar
  36. Handel RV (2008) Hidden Markov models. Lecture Notes, Princeton University, New Jersey, USAGoogle Scholar
  37. Henry NW (1971) The retention model: a Markov chain with variable transition probabilities. J Am Stat Assoc 66(334):264–267CrossRefGoogle Scholar
  38. Higham D, Higham N (2005) MATLAB guide, 2nd edn. Society for Industrial Mathematics, PhiladelphiaCrossRefzbMATHGoogle Scholar
  39. Horn R, Johnson C (1985) Matrix analysis. Cambridge University Press, CambridgeGoogle Scholar
  40. Hosmer DW, Lemeshow S (1980) Goodness of fit tests for the multiple logistic regression model. Commun Stat Part A Theory Methods 9(10):1043–1069CrossRefzbMATHGoogle Scholar
  41. Hosmer DW, Lemeshow S (2000) Applied logistic regression, 2nd edn. Wiley, New YorkCrossRefzbMATHGoogle Scholar
  42. Hosmer DW, Hjort NL (2002) Goodness-of-fit processes for logistic regression: simulation results. Stat Med 21(18):2723–2738CrossRefGoogle Scholar
  43. Hossain KMA, Lachemi M, Şahmaran M (2009) Performance of cementitious building renders incorporating natural and industrial pozzolans under aggressive airborne marine salts. Cement Concr Compos 31(6):358–368CrossRefGoogle Scholar
  44. Howell DC (2002) Statistical methods for psychology, 5th edn. Wadsworth Publishing, Pacific GroveGoogle Scholar
  45. Hu B, Shao J, Palta M (2006) Pseudo-R2 in logistic regression model. Statistica Sinica 16(3):847–860MathSciNetzbMATHGoogle Scholar
  46. Johnson JT (1987) Continuous-time, constant causative Markov chains. Stoch Process Appl 26:161–171MathSciNetCrossRefzbMATHGoogle Scholar
  47. Kalbfleisch JD, Lawless JF (1985) The analysis of panel data under a Markov assumption. J Am Stat Assoc 80(392):863–871MathSciNetCrossRefzbMATHGoogle Scholar
  48. Kallen MJ (2009) A comparison of statistical models for visual inspection data. Safety, reliability and risk structures, infrastructures and engineering systems. In: Tenth international conference on structural safety and reliability (ICOSSAR), Osaka, Japan, pp 3235–3242Google Scholar
  49. Kobayashi K, Do M, Han D (2010) Estimation of Markovian transition probabilities for pavement deterioration forecasting. KSCE J Civil Eng 14(3):343–351CrossRefGoogle Scholar
  50. Koch A, Siegesmund S (2004) The combined effect of moisture and temperature on the anomalous expansion behaviour of marble. Environ Geol 46(3–4):350–363Google Scholar
  51. Kus H, Nygren K (2000) Long-term exposure of rendered autoclaved aerated concrete: measuring and testing programme. In: RILEM/CIB/ISO international symposium—integrated life-cycle design of materials and structures, Helsinki, Finland, pp 415–420Google Scholar
  52. Kus H, Nygren K (2002) Microenvironmental characterization of rendered autoclaved aerated concrete. Build Res Inf 30(1):25–34CrossRefGoogle Scholar
  53. Kus H, Nygren K, Norberg P (2004) In-use performance assessment of rendered autoclaved concrete walls by long-term moisture monitoring. Build Environ 39(6):677–687CrossRefGoogle Scholar
  54. Kuss O (2002) Global goodness-of-fit tests in logistic regression with sparse data. Stat Med 21(24):3789–3801CrossRefGoogle Scholar
  55. Lacasse MA, Kyle BR, Talon A, Boissier D, Hilly T, Abdulghani K (2008) Optimization of the building maintenance management process using a markovian model. In: 11th international conference on the durability of building materials and components, Istanbul, Turkey, pp 1–9 (T72)Google Scholar
  56. Lee S, Hershberger S (1990) A simple rule for generating equivalent models in covariance structure modeling. Multivar Behav Res 25(3):313–334CrossRefGoogle Scholar
  57. Lethanh N, Adey B (2012) A hidden Markov model for modeling pavement deterioration under incomplete monitoring data. In: International conference on mathematical, computational and statistical sciences, and engineering 2012, Stockholm, Sweden, pp 722–729Google Scholar
  58. Lewis KN, Heckman BD, Himawan L (2011) Multinomial logistic regression analysis for differentiating 3 treatment outcome trajectory groups for headache-associated disability. Pain 152(8):1718–1726CrossRefGoogle Scholar
  59. Li P (2007) Hypothesis testing in finite mixture models. Doctoral thesis in Statistics, University of Waterloo, Ontario, CanadaGoogle Scholar
  60. Li Z, Wang W (2006) Computer aided solving the high-order transition probability matrix of the finite Markov chain. Appl Math Comput 172(1):267–285MathSciNetzbMATHGoogle Scholar
  61. Liu T (2010) Application of Markov chains to analyze and predict the time series. Mod Appl Sci 4(5):162–166CrossRefzbMATHGoogle Scholar
  62. Ljung L (1987) System identification—theory for the user. Prentice-Hall, Englewood CliffsGoogle Scholar
  63. Lounis Z, Vanier DJ (2000) A multi-objective and stochastic system for building maintenance management. Comput Aided Civil Infrastruct Eng 15(5):320–329CrossRefGoogle Scholar
  64. Lounis Z, Lacasse MA, Siemes AJM, Moser K (1998) Further steps towards a quantitative approach to durability design. Materials and technologies for sustainable construction, construction & environment. CIB World Building Congress, Gävle, Sweden, pp 315–324Google Scholar
  65. Madanat S, Mishalani R, Ibrahim WHW (1995) Estimation of infrastructure transition probabilities from condition rating data. J Infrastruct Syst 1(2):120–125CrossRefGoogle Scholar
  66. Maroco J (2007) Statistical analysis using SPSS. (in Portuguese), 3rd edn. Sílabo Editions, LisbonGoogle Scholar
  67. Marteinsson B, Jónsson B (1999) Overall survey of buildings—performance and maintenance. In: 8th DBMC, international conference on the durability of building materials and components, Ottawa, Canada, pp 1634–1654Google Scholar
  68. Mc Duling JJ (2006) Towards the development of transition probability matrices in the Markovian model for the predicted service life of buildings. PhD thesis in civil engineering, Faculty of Engineering, Built Environment and Information Technology, University of Pretoria, PretoriaGoogle Scholar
  69. McFadden D (1973) Conditional logit analysis of qualitative choice behavior. In: Zarembka P (ed) Frontiers of econometrics. Academic Press, New York, pp 105–142Google Scholar
  70. Menard S (2000) Coefficients of determination for multiple logistic regression analysis. Am Stat 54(1):17–24Google Scholar
  71. Mitchell DS (2007) The use of lime & cement in traditional buildings. Technical Conservation, Research and Education Group, Historic Scotland, Longmore House, Salisbury Place, Edinburgh, Scotland, 8pGoogle Scholar
  72. Mittlbock M, Schemper M (1996) Explained variation for logistic regression. Stat Med 15(19):1987–1997CrossRefGoogle Scholar
  73. Monika SO (2005) Statistical inference and hypothesis testing for Markov chains with interval censoring—application to bridge condition data in the Netherlands. Master thesis in civil engineering, Delft University of Technology, DelftGoogle Scholar
  74. Morcous G (2006) Performance prediction of bridge deck systems using Markov chains. J Perform Construct Facil 20(2):146–155CrossRefGoogle Scholar
  75. Morcous G, Lounis Z, Mirza MS (2003) Identification of environmental categories for Markovian deterioration models of bridge decks. J Bridge Eng 8(6):353–361CrossRefGoogle Scholar
  76. Moser K (2003) Engineering design methods for service life planning—state of the art. In: WMDBP 2003, international workshop on management of durability in the building process, Politecnico di Milano, Milan, Italy, paper 40Google Scholar
  77. Myung IJ (2003) Tutorial on maximum likelihood estimation. J Math Psychol 47(1):90–100MathSciNetCrossRefzbMATHGoogle Scholar
  78. Nagelkerke NJD (1991) A note on a general definition of the coefficient of determination. Biometrika 78(3):691–693MathSciNetCrossRefzbMATHGoogle Scholar
  79. Neves LC, Frangopol DM, Cruz PJS (2006) Lifetime multi-objective optimization of maintenance of existing steel structures. In: 6th international symposium steel bridges, European Convention for Construction Steelwork, Prague, Czech Republic, pp 206–215Google Scholar
  80. Newsom JT (2011) Logistic regression. Course of Data analysis II, Institute on Aging, School of Community Health, Portland State UniversityGoogle Scholar
  81. Ng S, Moses F (1996) Prediction of bridge service life using time-dependent reliability analysis. In: 3rd International conference on bridge management, University of Surrey, Guildford, UK, pp 26–33Google Scholar
  82. Norris JR (1997) Markov chains, 1st edn. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  83. Parnham P (1997) Prevention of premature staining of new buildings. E & FN Spon, LondonCrossRefGoogle Scholar
  84. Parzen E (1962) Stochastic processes. Holden Day, San FranciscozbMATHGoogle Scholar
  85. Plackett RL (1983) Karl Pearson and the Chi-squared test. Int Stat Rev 51(1):59–72MathSciNetCrossRefzbMATHGoogle Scholar
  86. Robelin C-A (2006) Facility-level and system-level stochastic optimization of bridge maintenance policies for Markovian management systems. PhD thesis in civil and environmental engineering, University of California, California, USAGoogle Scholar
  87. Robelin CA, Madanat S (2007) History-dependent bridge deck maintenance and replacement optimization with Markov decision process. J Infrastruct Syst 13(3):195–201CrossRefGoogle Scholar
  88. Roh Y-S, Xi Y (2000) A general A general formulation for transition probabilities of Markov model and the application to fracture of composite materials. Probab Eng Mech 15(3):241–250CrossRefGoogle Scholar
  89. Schouenborg B, Grelk B, Malaga K (2007) Testing and assessment of marble and limestone (TEAM) - Important results from a large European research project on cladding panels. ASTM Int 4(5):10–22Google Scholar
  90. Schwarz GE (1978) Estimating the dimension of a model. Ann Stat 6(2):461–464MathSciNetCrossRefzbMATHGoogle Scholar
  91. Shohet I, Rosenfeld Y, Puterman M, Gilboa E (1999) Deterioration patterns for maintenance management—a methodological approach. In: 8th international conference on durability of building materials and components, Vancouver, Canada, pp 1666–1678Google Scholar
  92. Shtatland ES, Kleinman K, Cain EM (2002) One more time about R2 measures of fit in logistic regression. In: 15th annual Nesug conference; statistics, data analysis and econometrics, Issue M, pp 1–6Google Scholar
  93. Silva A, de Brito J, Gaspar PL (2012) Application of the factor method to maintenance decision support for stone cladding. Autom Constr 22(3):165–174CrossRefGoogle Scholar
  94. Silva A, de Brito J, Gaspar P (2013) Probabilistic analysis of the degradation evolution of stone wall cladding (directly adhered to the substrate). J Mater Civil Eng 25(2): 227–235Google Scholar
  95. Silva A, Neves LC, Gaspar PL, de Brito J (2015) Probabilistic transition of condition: render facades. Build Res Inf doi: 10.1080/09613218.2015.1023645 Google Scholar
  96. Singer B (1981) Estimation of nonstationary Markov chains from panel data. Sociol Methodol 12:319–337CrossRefGoogle Scholar
  97. Smyth GK (2003) Pearson’s goodness of fit statistic as a score test statistic. In: Goldstein DR (ed) Science and atatistics: a festschrift for terry speed. IMS lecture notes—monograph series, vol. 40. Institute of Mathematical Statistics, Hayward, CaliforniaGoogle Scholar
  98. Soroka I, Carmel D (1987) Durability of external renderings in a marine environment—durability of building materials. Elsevier Science Publishers, Amsterdam, pp 61–72Google Scholar
  99. Stelzl I (1986) Changing causal relationships without changing the fit: some rules for generating equivalent LISREL models. Multivar Behav Res 21(3):309–331CrossRefGoogle Scholar
  100. Straub A (2003) Using a condition-dependent approach to maintenance to control costs and performances. Facil Manage 1(4):380–395CrossRefGoogle Scholar
  101. Suthar V, Tarmizi RA, Midi H, Adam MB (2010) Students’ beliefs on mathematics and achievement of university students: logistics regression analysis. In: International conference on mathematics education research (ICMER 2010). Procedia—Social and Behavioral Sciences, vol 8, pp 525–531Google Scholar
  102. Thomasson F (1982) Les enduits monocouches à base de liants hydrauliques. UNITECNA, France, Paris, 129pGoogle Scholar
  103. Trexler JC, Travis J (1993) Nontraditional regression analysis. Ecology 74(6):1629–1637CrossRefGoogle Scholar
  104. Uchwat C, MacLeod D (2012) Case studies of regression and Markov chain models. In: Conference of the transportation Association of Canada, New Brunswick, Canada, Session: pavement performance case studies (SES), 19pGoogle Scholar
  105. Veiga MR (2000) Methodology to evaluate the cracking susceptibility of mortars. Selection criteria of rendering and repointing mortars for ancient buildings. Seminar “Malte a vista com sabie locali nella conservazione degli edifici storici”, Torino, Italy, Politecnico di Torino, July, pp 1–16Google Scholar
  106. Wald A (1941) Asymptotically most powerful tests of statistical hypotheses. Ann Math Stat 12(1):1–19MathSciNetCrossRefzbMATHGoogle Scholar
  107. Wald A (1943) Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans Am Math Soc 54(3):426–482MathSciNetCrossRefzbMATHGoogle Scholar
  108. Wang Y (2005) A multinomial logistic regression modeling approach for anomaly intrusion detection. Comput Secur 24(8):662–674CrossRefGoogle Scholar
  109. Westergren A, Karlsson S, Andersson P, Ohlsson O, Hallberg IR (2001) Eating difficulties, need for assisted eating, nutritional status and pressure ulcers in patients admitted for stroke rehabilitation. J Clin Nurs 10(2):257–269CrossRefGoogle Scholar
  110. White JL (2013) Logistic regression model effectiveness: proportional chance criteria and proportional reduction in error. J Contemp Res Educ 2(1):4–10Google Scholar
  111. Witten IH, Frank E (2005) Data mining: Practical machine learning tools and techniques, 2nd edn. Morgan Kaufman, BostonzbMATHGoogle Scholar
  112. Wuensch KL (2011) Binary logistic regression. East Carolina University, Department of PsychologyGoogle Scholar
  113. Xie XJ, Pendergast J, Clarke W (2008) Increasing the power: a practical approach to goodness-of-fit test for logistic regression models with continuous predictors. Comput Stat Data Anal 52(5):2703–2713MathSciNetCrossRefzbMATHGoogle Scholar
  114. Yokota H, Furuya K, Hashimoto K, Hanada S (2012) Reliability of deterioration prediction with Markov model for mooring facilities. In: IALCCE—International Association for Life-Cycle Civil Engineering, 3rd international symposium on life-cycle civil engineering, Vienna, Austria, pp 523–530, 3–6 Oct 2012Google Scholar

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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  2. 2.Instituto Superior TécnicoUniversidade de LisboaLisbonPortugal
  3. 3.Faculty of ArchitectureUniversidade de LisboaLisbonPortugal

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