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# Changepoint Detection with Bayesian Inference

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An overview of the application of Bayesian Inference in the detection of changepoints in noisy time series data, applied to three different and diverse domains.

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### Changepoint Detection with Bayesian Inference

1. 1. Change  Point  Detec.on   with  Bayesian  Inference   By  Frank  Kelly   Py  data   6th  January  2015
2. 2. Overview   •  Nigeria,  oil  wells  &  drilling   •  Noisy  data   •  Some  maths   •  Python  implementaDon   •  Examples  in  diﬀerent  domains
3. 3. FPSO  (oil  plaIorm  picture)
4. 4. 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.
5. 5. The  problem   •  Poor  bit  rate  and   resoluDon   •  Time  consuming   analysis
6. 6. 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”
7. 7. FrequenDst  vs.  Bayesian
8. 8. 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
9. 9. Modelling  the  problem   µ2 1µ m N
10. 10. 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
11. 11. 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
12. 12. The  maths
13. 13. More  maths   •  Integrate  w.r.t.  (and  thereby  remove)   nuisance  parameters