Final Defense Unified Prediction and Diagnosis in Engineering Systems by means of Distributed Belief Networks Robert H. Dodier Joint Center for Energy Management University of Colorado at Boulder 1
Problem Domain and Proposed Solution Buildings contain lots of equipment, and there are a lots of buildings in the world. Equipment can break down and not be noticed for a while, or simply degrade and never be noticed. Buildings could run more efficiently (cheaper, less resources used) and comfortably, if we could reliably, automatically query the status of the building. I propose that we use belief networks to automate prediction and diagnosis over functionally and geographically distributed systems. 2
Problems not solved by existing methods Existing prediction methods: linear regression, neural networks, locally-weighted regression, first principles, ... These prediction methods don’t tell what to do about missing observations and model uncertainty. Existing diagnostic methods: “statistical inference,” neural networks, rule-based expert systems, fuzzy logic, ... Formal problems: easy to invent situations in which fuzzy logic and certainty-factors logic give incorrect results — dependencies are handled incorrectly. Can’t bet on a degree of membership, a certainty factor, or a confidence level. 3
Unsolved problems, cont’d Need methods that don’t have formal deficiencies. Need methods for combining empirical information with expert knowledge, and prior information with new observations. Need methods for organizing complex structures. 4
* Probability as an extension of logic Classical logic works great with propositions that we consider entirely true or false. We want something similar for “iffy” propositions. Assuming that a reasoning system (i) represents uncertainty by numbers, (ii) is consistent with common sense and ordinary logic, and (iii) is internally consistent, we get a few equations to solve: F ( x, F ( y, z )) = F ( F ( x, y ) , z ) , S ( S ( x )) = x 5
* Probability as extended logic, cont’d The first equation yields the product rule p ( A, B | C ) = p ( A | B, C ) p ( B | C ), and the second yields the negation rule p ( ¬ A | B ) = 1 − p ( A | B ). Since every logical proposition can be built up using conjunction and negation, we’re done. This analysis (originated by R.T. Cox, 1946) shows that it’s meaningful to talk about the probability of any logical proposition. Probability is not limited to repetitive events. 6
Interesting operations on probabilistic models Prediction: Compute p (effects | causes). E.g., What is energy use of an air handling unit when a heat exchanger is fouled? Diagnosis: Compute p (causes | effects). E.g., What’s the quantity of refrigerant within a chiller? Value of information: VOI is an index of the utility of measuring some presently-unknown variable. Which variable shall we measure next? Explanation: Find most likely values of hidden variables, or maybe identify the most influential variables. Explanation is not yet implemented in riso . 7
Belief network = relations graph + conditional probabilities A large probabilistic model can be built up by considering only the conditional distribution of each variable given some others; then the joint distribution is just the product of all the conditionals. It’s convenient to display the model as a directed graph. A graph is easy to think about, and independence can be verified using only the graph and ignoring the numbers. 8
Belief network = graph + probabilities, cont’d p ( A, B, C, D, E ) = p ( E | D ) p ( D | B, C ) p ( C | A ) p ( B | A ) p ( A ) A B C D E 9
Belief networks as building system models Represent heterogeneous information. In engineering problems, variables may be discrete or continuous; distributions may be well-known or special-purpose; relations based on empirical data or expert knowledge. B.n.’s can handle all these kinds of information. Not emphasized before due to limitations of existing b.n. inference algorithms. “Blueprint” of probabilistic relations. The b.n. is separate from the code used to transfer data and compute inferences. The b.n. expresses what’s known about the building — no need to browse the code, read the manual, or phone the last guy who worked on the program. 10
Belief networks as building system models, cont’d Hierarchical organization. • Natural to model functional hierarchies. E.g., Equipment models are parts of a building, buildings are parts of a campus. Use distributed belief networks for distributed systems! • Represent two or more alternative organizations in one belief network — e.g., grouping status variables by building and by function. 11
* Belief networks as building system models, cont’d Temporal dependence. It’s easy to form a complex belief network by linking together instantaneous models. E.g., hidden Markov model. However, the extra dependencies may make computations intractable. Both empirical and first-principles relations. The conditional distribution of a variable can be either empirical or derived from first-principles, or both. “Either-or” is easy; “both” needs some work. E.g., diagnostic models. Empirical data is typically available for normal operation, but failure models must be constructed mostly from prior info. 12
* Details: π - and λ - messages Laws of probability require that we distinguish upstream and downstream evidence. In a simply-connected b.n. on a directed graph (called a “Bayesian network”) the posterior of X is the product of the predictive distribution π X and the likelihood function λ X . The predictive distribution summarizes the π -messages coming down from parents. No evidence ⇒ π X is prior. The likelihood function summarizes the λ -messages coming up from children. No evidence ⇒ λ X is non-informative. 13
* π - and λ - messages at a typical node π U 1 ,X π U 2 ,X U 2 U 1 p X | U 1 ,U 2 X λ Y 2 ,X λ Y 1 ,X Y 2 Y 1 � � π X ( X ) = p ( X | U 1 , U 2 ) π U 1 ,X ( U 1 ) π U 2 ,X ( U 2 ) dU 1 dU 2 λ X = λ Y 1 ,X λ Y 2 ,X p X | e = π X λ X 14
Details: Handling computational problems in riso riso computes an exact result if an exact result is known for a given combination of π - and λ -messages and the node’s conditional distribution. AFAIK this is new — existing schemes are entirely exact or entirely approximate. Otherwise, riso attempts to compute an approximation (a monotone spline or mixture of Gaussians). riso postpones the approximation until it is needed — distributions are represented in the b.n. description in their original forms; keep the b.n. description close to the engineering description. General cases are handled by numerical integrations. 15
“Distributed” meets “belief network” Locating and connecting b.n.’s on different hosts. Each host runs some code that knows how to find a b.n. description and load the software required to run the b.n.; if necessary, the code can be loaded across the Internet. AFAIK riso is the first cross-host d.b.n. system. Communicating π - and λ -messages between b.n.’s. The messages are probability distributions and likelihood functions. A block of data containing the necessary parameters is constructed, sent across the Internet, then reconstituted into a variable in a program. Coping with communication failures. A host might crash or the process running a b.n. might be killed. If a child is lost, the child is removed. If a parent is lost, the prior for that parent is substituted for any π -message. 16
* Publishing information as distributed belief networks B.n. approach lends itself well to time-honored software development policies: break up your problem and get the pieces to talk to each other. Probability shows that messages should be distributions; mechanism could be function calls, or could be Internet data packets — difference is a trivial detail. Prevent wheel-reinvention by making your b.n. publicly available — e.g., weather service; building or equipment database. Connect many b.n.’s together to obtain summary information — e.g., monitoring service offered by consulting firm. 17
* Uncertainty in a 1st-principles model There exists an enormous body of information about buildings in the form of DOE-2, TRNSYS, etc. first-principles models; b.n.’s should make use of this. Probabilistic operations can extend first-principles models: • Propagating uncertainty from model parameters • Taking weather variability into account • Concise representation of distributions over results Relatively easy to show how uncertainties are propagated through commonly-occurring modeling equations (e.g., response factors). 18
Application I: Selecting electricity rates Energy component is easy: mostly need to estimate average electricity use. Demand is a little harder: need maximum electricity use. Use first-principles building model — envelope loads modeled with transfer function (discrete convolution). Transfer fcn. for mean is always stable; variance blows up for massive walls/roofs. Compute expected cost using schedules and distributions over total energy use and maximum energy use. Expected cost is interesting if distributions cross breakpoints in schedules. 19
Recommend
More recommend
