1. Condition State Transitions and Deterioration Models H. Scott Matthews March 10, 2003

2. Announcement  Midterm Grades will be posted today  Based on first 2 homework sets  Mostly As, a few Bs, C  Intended to convince you to keep effort level consistent for rest of semester

3. Recap of Last Lecture  Threats, Vulnerabilities, and Risks  Risk Assessment and Management  Classic and Modern Defense Models  Critical Infrastructure Protection  Focused on physical attacks from terrorists

4. Linear Regression (in 1 slide)  Arguably simplest of statistical models, have various data and want to fit an equation to it  Y (dependent variable)  X: vector of independent variables   : vector of coefficients   : error term  Y =  X +   Use R-squared, related metrics to test model and show how ‘robust’ it is

5. Markov Processes  Markov chain - a stochastic process with what is called the Markov property  Discrete and continuous versions  Discrete: consists of sequence X 1,X 2,X 3,.... of random variables in a "state space", with X n being "the state of the system at time n".  Markov property - conditional distribution of the "future" X n+1, X n+2, X n+3,.... given the "past” (X 1,X 2,X 3,...X n ), depends on the past only through X n.  i.e. ‘no memory’ of how X n reached  Famous example: random walk

6. Markov (cont.)  i.e., knowledge of the most recent past state of the system renders knowledge of less recent history irrelevant.  Markov chain may be identified with its matrix of "transition probabilities", often called simply its transition matrix (T).  Entries in T given by p ij =P(X n+1 = j | X n = i )  p ij : probability that system in state j "tomorrow" given that it is in state i "today".  ij entry in the k th power of the matrix of transition probabilities is the conditional probability that k "days" in the future the system will be in state j, given that it is in state i "today".

7. Markov Applications  Markov chains used to model various processes in queuing theory and statistics, and can also be used as a signal model in entropy coding techniques such as arithmetic coding.  Note Markov created this theory from analyzing patterns in words, syllables, etc.

8. Infrastructure Application  Used to predict/estimate transitions in states, e.g. for bridge conditions  Used by Bridge Management Systems, e.g. PONTIS, to help see ‘portfolio effects’ of assets under control  Helps plan expenditures/effort/etc.  Need empirical studies to derive parameters  Source for next few slides: Chase and Gaspar, Journal of Bridge Engineering, November 2000.

9. Sample Transition Matrix T = [ ]  Thus p ii suggests probability of staying in same state, 1- p ii probability of getting worse  Could ‘simplify’ this type of model by just describing vector P of p ii probabilities (1 - p ii ) values are easily calculated from P  Condition distribution of bridge originally in state i after M transitions is C i T M

10. Superstructure Condition  NBI instructions:  Code 9 = Excellent  Code 0 = Failed/out of service  If we assume no rehab/repair effects, then bridges ‘only get worse over time’  Thus transitions (assuming they are slow) only go from Code i to Code i-1  Need 10x10 matrix T  Just an extension of the 5x5 example above

11. Empirical Results  P = [0.71, 0.95, 0.96, 0.97, 0.97, 0.97, 0.93, 0.86, 1]  Could use this kind of probabilistic model result to estimate actual transitions

12. More Complex Models  What about using more detailed bridge parameters to guess deficiency?  Binary : deficient or not  What kind of random variable is this?  What types of other variables needed?

13. Logistic Models  Want Pr(j occurs) = Pr (Y=j) = F(effects)  Logistic distribution:  Pr (Y=1) = e  X / (1+ e  X )  Where  X is our usual ‘regression’ type model  Example: sewer pipes