predicting power failures with reactive point processes
TRANSCRIPT
Predicting Power Failures with Reactive Point Processes
Seyda ErtekinMIT Sloan School of Management
Massachusetts Institute of TechnologyCambridge, MA 02139, USA
Cynthia RudinMIT Sloan School of Management
Massachusetts Institute of TechnologyCambridge, MA 02139, USA
Tyler H. McCormickDepartment of Statistics
University of WashingtonSeattle, WA 98195, USA
IntroductionWe present a new statistical model for predicting discreteevents continuously in time, called Reactive Point Processes(RPP’s). RPP’s are a natural fit for many domains wheretime-series data are available, and their development wasmotivated by the important problem of predicting seriousevents (fires, explosions, power failures) in the undergroundelectrical grid of New York City (NYC). RPP’s capture sev-eral important properties of this domain:
• There is an instantaneous rise in vulnerability to futureserious events immediately following an occurrence of apast serious event, and this rise in vulnerability graduallyfades back to the baseline level. This is a type of self-exciting property.
• There is an instantaneous decrease in vulnerability due toan inspection or repair. The effect of this inspection fadesgradually over time. This is a self-regulating property.
• The cumulative effect of events or inspections can satu-rate, ensuring that vulnerability levels never stray too farbeyond their baseline level. This captures diminishing re-turns of many events or inspections in a row.
RPP’s are related to Hawkes processes (Hawkes 1971a;1971b), and more specifically, self-exciting point processes(SEPP). Demonstrated to be highly effective for earthquakemodeling (Ogata 1988; 1998), SEPP have been used in avast array of domains, including criminology (Mohler et al.2011), neuroscience (Krumin, Reutsky, and Shoham 2010)and social networks (Crane and Sornette 2008). A commontheme in these applications is that the likelihood of an eventat a certain time is probabilistically dependent on earlierevents, even after controlling for covariates. RPP’s extendthese models to a much broader array of settings where ex-ternal events can cause vulnerability levels to drop or wherediminishing returns can occur.
We apply RPP’s to the critical problem of predictingpower failures in New York City using data from a large-scale ongoing collaboration with Con Edison, New YorkCity’s power utility company (Rudin et al. 2010; 2012).Manholes are access points to NYC’s underground electri-cal grid, and we have features of each manhole that can be
Copyright c© 2013, Association for the Advancement of ArtificialIntelligence (www.aaai.org). All rights reserved.
useful in predicting vulnerability to serious events such asmanhole fires and explosions, which can cause power fail-ures. The RPP model takes into account past events that arecorrelated with increased future vulnerability, past inspec-tions that reduce future risk, and also the manhole’s charac-teristics (e.g., number and age of cables entering the man-hole). The dataset is comprised of processed trouble tickets,inspection reports, and cable information for all manholes inManhattan (total 213,504 records for 53,525 manholes). Wehave data from the year 1995 up to and including 2010.
ModelOur approach models events as being generated from a non-homogenous Poisson process for entity (manhole) j with in-tensity λj(t) where
λj(t) = λ0
[1+g1
( ∑∀te<t
g2(t− te)
)
− g3
( ∑∀ti<t
g4(t− ti)
)+ c11[N(τ)≥1]
]where te are event times and ti are inspection times. λ0 de-notes the baseline vulnerability. The vulnerability level per-manently goes up by c1 if the number of events on the in-terval [0, τ ], N(τ), is more than zero. The constant c1 canbe fitted from the observed data. Functions g2 and g4 are theself-excitation and self-regulation functions, which have ini-tially large amplitudes and fade over time. Note that SEPP’shave only g2, and not the other functions. Functions g1 andg3 are the saturation functions, which start out as the iden-tity function and then flatten farther from the origin. If thetotal sum of the excitation terms is large, g1 will prevent thevulnerability level from increasing too much. Similarly, g3
controls the total possible amount of self-regulation.For the electrical grid data, we used a family of functions
of the form
g1(x) = a11 ×(1− log(1 + e−a11x)
log(2)
), g2(x) =
1
1 + eβx
g3(x) = −a31 ×(1− log(1 + e−a32x)
log(2)
), g4(x) =
−1
1 + e−γx,
where a11, a12, a31, a32, β, and γ are parameters that can beeither constant across all manholes, or can be adapted based
0 200 400 600−2
0
2
4
6
8
Number of Days
g2 values from case controlfitted g2
(a) Fitted g2
0.52 1.57 2.61 3.66 4.700
5
10
15
Bin Ranges
g1 values from case controlfitted g1
(b) Fitted g1
−100
10 −20−10010
−10
0
10
20
Total Num SetsMain PH
Ag
e
(c) KL Divergence
−20−10
010
−1001020
−10
0
10
Age
Main PH Total Num. Sets
(d) Log-Likelihood
Figure 1: Results for the Manhattan Dataset. Figures 1(a) and 1(b) show the fitted functions for the Manhattan dataset fromCase Control. Figures 1(c) and 1(d) show the KL Divergence from ABC and the Log-Likelihood, respectively.
on the manhole’s features. For instance, our experimentalanalysis covers both constant β across all manholes as wellas an adaptive β, modeled for manhole k as:
βk = log(
1 + e−∑n
i=1 θifik)
where θi is the coefficient of feature i and fik is the value offeature i for manhole k. For our dataset, the features includethe number of main phase cables, age of the oldest cable,and the total number of cable sets inside the manhole.
Model Fitting and ExperimentsWe will introduce several methods for fitting RPP’s below.To verify that the methods are working properly, we usedsimulated data where ground truth was available (not shownhere) to demonstrate that the underlying function could berecovered.
Case ControlThe Case Control method allows us to nonparametricallytrace out the shapes of g1, g2, g3, and g4, using only rele-vant manholes to estimate specific points of each function.Figures 1(a) and 1(b) show g2 and g1 from Manhattan man-holes, traced out (in blue) using the Case Control method,and parameterized (in red) afterwards. For the g1 curve, therange of observed
∑g2 values in the data was discretized
into equal sized bins, and the likelihood of an event if thesum value falls within the range of a bin was used to fit g1.
Bayesian and Maximum Likelihood ApproachesThe Case Control method is useful for finding parameteriza-tions of the functions, but there are not enough data to permitits use when we would like to customize g1, g2, g3, and g4 toeach individual manhole. Based on insights gleaned from theCase Control method we explore both Bayesian and Maxi-mum Likelihood approaches to model fitting.
For our Bayesian approach, we use the model for the βk’swith covariates described above, where each coefficient hasa Gaussian prior distribution. We fit the model using Ap-proximate Bayesian Computation (ABC). To do this, weevaluated the suitability of simulated data generated usinga given candidate parameter via two statistics, one consist-ing of the number of events and another related to the dis-tribution of time between events. Since RPP’s model associ-ation between event times, the second statistic is important.
We use KL Divergence to measure differences in the dis-tribution of time between events for the observed data andthe distribution of time between events generated using ourcandidate parameter values. In Figure 1(c) the blue bound-ary corresponds to the region that yields the minimum diver-gence, and the optimum parameter values can be derived byfitting a manifold on that region.
We also experimented with model fitting using MaximumLikelihood techniques. Figure 1(d) demonstrates the regionof maximum likelihood (shown in dark red), where the sim-ulated vulnerabilities are in closest proximity of actual vul-nerabilities.
Prediction Performance Evaluation
We assessed the predictive performance of our RPP modelby comparing it against Cox Proportional Hazards Model(Cox 1972), which is a commonly used technique in sur-vival analysis. We trained both an RPP model and a Coxmodel (available in the survival package in R) on the histor-ical event records up to the year 2010 and used the modelsto predict vulnerabilities of each manhole on each day of2010. We then generated a ranked list of manholes for eachday with the most vulnerable manhole ranked at the top. Foreach manhole that had an event during a day, we comparedthe rank of that manhole in the two ranked lists. Aggregatedover days in 2010 where an event was recorded, the RPPmodel ranked the manholes with events significantly higherthan the Cox model in its lists, indicating that the RPP modelmore accurately measures short term vulnerabilities.
Understanding the Impact of Inspection Policies
Beyond identifying the vulnerable manholes to be targetedfor repair and inspection, we used the RPP model to sim-ulate the effect of different broad inspection policies. Oursimulations show that there is a natural inverse relationshipbetween the number of events and the number of inspec-tions: more inspections lead to fewer events. Using our sim-ulations, we can quantify an optimal trade off between theanticipated number of events and the cost of inspections.This can assist Con Edison to make decisions between can-didate inspection policies.
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