Changepoint Detection with Bayesian Inference
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This document discusses using Bayesian inference to detect change points in noisy time series data. It provides an overview of Bayesian statistics and Bayes' theorem. Mud pulse telemetry data from oil drilling is used as an example case study, where change point detection can identify different rock types or when oil is reached. The document outlines modelling the problem statistically and developing a Python implementation of a single change point detector based on calculating the posterior probability of change point locations. Several other potential applications are also mentioned, including financial data and web traffic analysis.
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Changepoint Detection with Bayesian Inference
- 1. Change Point Detec.on with Bayesian Inference By Frank Kelly Py data 6th January 2015
- 2. Overview • Nigeria, oil wells & drilling • Noisy data • Some maths • Python implementaDon • Examples in different domains
- 3. FPSO (oil plaIorm picture)
- 6. Mud pulse telemetry • InformaDon encoded digitally, transmiOed via pressure pulses through mud fluid. • Alert drillers that they have reached oil, detect rock types and general monitoring.
- 7. The problem • Poor bit rate and resoluDon • Time consuming analysis
- 8. Approaches to staDsDcs • FrequenDst – Data gathered is a repeatable random sample. “Frequency” – Underlying parameters are constant – Fisher’s 0.05 • Bayesian – Data are, fixed and observed from the realised sample – Parameters unknown and described probabilisDcally – Introduce “subjecDvity”
- 10. The Theory: Bayesian inference • Methodology of mathemaDcal inference: – Choosing between several possible models – ExtracDng parameters for these models • Bayes’ Theorem: Rev Thomas Bayes 1702 -‐ 1761 p(w | D) = p(D | w)p(w) p(D) Likelihood Prior Probability Posterior Probability Evidence -‐ Remove nuisance parameters by marginalisaDon -‐ InteresDng ones remain
- 11. Modelling the problem µ2 1µ m N
- 12. 0 20 40 60 80 100 120 140 160 180 200 0.5 1 1.5 2 2.5 data = model + noise • a sequence of N samples of data from a piecewise constant source with added Gaussian noise. • Noise independent of mean, idenDcally distributed and S.D. = σ • Heterogenous: divide into two homogenous segments µ2 ⎩ ⎨ ⎧ + + = i i i e e d 2 1 µ µ Nim mi ≤< ≤ 1µ Nm
- 13. Single changepoint detector: How does it work? • SubsDtute likelihood into Bayes’ Law – Simple model-‐ consider Ockham’s Razor • Interested in changepoint locaDon m, integrate w.r.t. the nuisance parameters (µ1, µ2 and σ)…rearrange this… • …get a BIG expression for p({m}|dI), code in Python • On running obtain most likely changepoint locaDon Ockham’s razor: hOp://www.jstor.org/discover/10.2307/29774559?sid=21105568247973&uid=3738032&uid=4&uid=2
- 14. The maths
- 15. More maths • Integrate w.r.t. (and thereby remove) nuisance parameters
- 20. “Google’s algorithm is the “secret sauce recipe” that has enabled it to dominate search.” -‐ FT.com 16th Sept 2014 hOp://www.p.com/cms/s/0/9615661c-‐3ce1-‐11e4-‐9733-‐00144feabdc0.html? siteediDon=uk#axzz3DSwXYAW8 Any business with an online presence today open struggles to accurately evaluate: ● The quality of their website and associated linking pages, as perceived by Google ● The robustness of their website to a sudden change in Google’s search algorithm
- 21. Web traffic 30000 35000 40000 45000 50000 55000 60000 raw daily google search-‐sourced pageviews
- 22. Web traffic (2) 30000 35000 40000 45000 50000 55000 60000 smoothed data using moving average
- 23. Web traffic (3) 30000 35000 40000 45000 50000 55000 60000 smoothed data with cyclicality removed
- 24. Web traffic (4) -‐838 -‐837.5 -‐837 -‐836.5 -‐836 -‐835.5 -‐835 -‐834.5 -‐834 -‐833.5 -‐833 30000 35000 40000 45000 50000 55000 60000 likelihood of change in data plo>ed over .me day removed likelihood CP
- 26. number of tropical storms per year in the North AtlanDc Data obtained from ibtracs database: hOps://www.ncdc.noaa.gov/ibtracs/
- 27. "Amo Dmeseries 1856-‐present" by Rosentod, Marsupilami -‐ hOp://www.cdc.noaa.gov/CorrelaDon/amon.us.long.data. Licensed under Public Domain via Wikimedia Commons -‐ hOp://commons.wikimedia.org/wiki/File:Amo_Dmeseries_1856-‐present.svg#mediaviewer/ File:Amo_Dmeseries_1856-‐present.svg
- 29. Other applicaDons / possibiliDes • Financial markets and poliDcal events • Combine with frequenDst staDcal methods: – Use of GLR in online (moving window) detecDon applicaDon • Your own data/ ideas !
- 30. Thank you • Link to Python code on github: hOps://github.com/swhustla/pydata-‐bayes-‐changepoint – Single changepoint detector (as seen tonight) – Dual changepoint detector – Ramp detector • Further reading: – Numerical Bayesian Methods Applied to Signal Processing (StaDsDcs and CompuDng) by Fitzgerald, O’Ruanaidh, 1996 : hOp://www.amazon.co.uk/Numerical-‐Bayesian-‐Processing-‐ StaDsDcs-‐CompuDng/dp/0387946292 – Bayesian Inference on Change Point Problems (2007) hOp://www.cs.ubc.ca/~murphyk/Students/Xuan_MSc07.pdf TwiOer: @norhustla Email: frank.kelly@cantab.net
- 31. Thank you • AddiDonal links: – Google Algo updates: hOp://moz.com/google-‐algorithm-‐change – Mathsight -‐> insights into algorithm changes hOp://mathsight.org – AtlanDc mulD-‐decadal oscillaDon spaDal paOern: hOp://commons.wikimedia.org/wiki/File:AMO_PaOern.png – NaDonal climaDc data center hOps://www.ncdc.noaa.gov/ibtracs/ – Ockham’s Razor and Bayesian Inference: hOp://www.jstor.org/discover/10.2307/29774559? sid=21105568247973&uid=3738032&uid=4&uid=2 – ConverDng from Matlab to Python: hOp://mathesaurus.sourceforge.net/matlab-‐numpy.html TwiOer: @norhustla Email: frank.kelly@cantab.net