c r o penalties using the r package - advance · r package version 2.0, 2015 (to appear very soon)....
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Statistics and Operational ResearchDoctoral Training CentreLancaster University
Changepoints for a Range Of PenaltieS using thechangepoint R package
Kaylea HaynesSupervised by: Idris Eckley, Paul Fearnhead and Ralph ManssonuseR! 2015 Aalborg
July 2015
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Introduction CROPS Example
Overview
I changepoint: An R package for changepoint analysisI Version: 2.0I Author: Rebecca Killick [aut, cre], Kaylea Haynes [aut], Idris Eckley
[aut] and Paul Fearnhead [ctb]
I Multiple changepoint search methodsI Binary SegmentationI Pruned Exact Linear Time (PELT)I Changepoints for a Range of Penalties (CROPS)
(Killick et al. 2014, Killick and Eckley 2014)
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Introduction CROPS Example
Changepoint Detection
For data y1, ..., yn, a changepoint is a location τ where the statisticalproperties of y1, ..., yτ differ from those of yτ+1, ..., yn.
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Introduction CROPS Example
Segment Costs
Common costs
m+1∑i=1
[Cy(τi−1+1):τi ] + βm
I cpt.mean(data,...)
I cpt.var(data,...)
I cpt.meanvar(data,...)
test.stat: "Normal", "Gamma", "Exponential" and
"Poisson".
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Introduction CROPS Example
Optimal Partitioning
Recursion
F (t) = minτ∈τt
{m+1∑i=1
[C(y(τi−1+1):τi ) + β]
}= min
s∈{0,...,t−1}{F (s) + C(y(s+1):n) + β}
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Introduction CROPS Example
Optimal Partitioning
ttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss0
5
10
15
0 100 200 300
Index
Dat
a
tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttt0
50
100
150
200
0 100 200 300
Index
Last
Cha
nge
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Introduction CROPS Example
Optimal Partitioning
4
8
12
16
0 100 200 300
Index
Dat
a
0
50
100
150
200
0 100 200 300
Index
Last
Cha
nge
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Introduction CROPS Example
PELT
tttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttttsssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssssss0
5
10
15
0 100 200 300
Index
Dat
a
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Introduction CROPS Example
PELT
Pruning
If there exists a constant K such that for all s < t < T
C(y(s+1):t) + C(y(t+1):T ) + K ≤ C(y(s+1):T ),
and for t > s, if
F (s) + C(y(s+1):t) + K ≥ F (t),
then at a future time T > t, s can never be the optimal last changepointprior to T .
cpt.var(data, penalty, pen.value, test.stat, method =
"PELT", minseglen)
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Introduction CROPS Example
Penalty
cpt.var(data, penalty, pen.value, test.stat, method,
minseglen)
I “None”
I “SIC” or “BIC”
I “MBIC”
I “AIC”
I “Hannan-Quinn”
I “Asymptotic”
I “Manual”: pen.value = ?
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Introduction CROPS Example
Penalty
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16
0 100 200 300
Index
Dat
a
Penalty = 2
4
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0 100 200 300
Index
Dat
a
Penalty = 100
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Introduction CROPS Example
Changepoints for a Range of PenaltieS(CROPS).
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Introduction CROPS Example
Step 1: Run algorithm* for βmin and βmax
If both the segmentations have the same number of changepoints:STOP.The optimal number of changepoints decreases as β increases. So allvalues of β ∈ [βmin, βmax ] will have the same optimal segmentation.
* Optimal Partitioning (Jackson et al 2005) or PELT (Killick et al(2012)).
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Introduction CROPS Example
Step 2: Compare the number of changepoints
If the number of changepoints differs by 1, m(βmin) = m(βmax) + 1:STOP.
5.0 5.5 6.0 6.5 7.0 7.5 8.0
510
1520
25
beta
(Pen
alis
ed)
log−
likel
ihoo
d
X
X
3
2
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Introduction CROPS Example
Step 2: Compare the number of changepoints
Otherwise calculate the point of intercept β∗, and run CPD algorithmwith this value.
5 6 7 8 9 10
510
1520
25
beta
(Pen
alis
ed)
log−
likel
ihoo
d
X
X
3
1
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Introduction CROPS Example
Step 3: Check if new Optimal Segmentation
If the optimal segmentation for β∗ is the same as for βmin or βmax :STOP.
5 6 7 8 9 10
510
1520
25
beta
(Pen
alis
ed)
log−
likel
ihoo
d
X
X
3
1
X3
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Introduction CROPS Example
Step 3: Check if new Optimal Segmentation
Otherwise, treat the intervals [βmin, β∗] and [β∗, βmax ] separately and
process using Step 2.
5 6 7 8 9 10
510
1520
25
beta
(Pen
alis
ed)
log−
likel
ihoo
d
X
X
3
1
X2
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Introduction CROPS Example
Final Solution
10
11
12
13
14
15
16
789
775
800
825
850
6 7 8 9 10 11Beta
Co
st
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Introduction CROPS Example
CROPS
cpt.mean(data, penalty, pen.value = c(0,10), test.stat,
method = "PELT", minseglen)
if (length(pen.value == 2){return(CROPS(data, pen.value, test.stat, minseglen, func=”mean”))}
(Haynes et al. (2014))
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Introduction CROPS Example
CROPS - Example output
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Introduction CROPS Example
CROPS - Example output
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Introduction CROPS Example
Example
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Introduction CROPS Example
Example
I Changes in nuclear magnetic response as a probe is lowered down abore-hole into the Earth’s surface.
I Change in mean with σ̂2 = median absolute distance
100000
110000
120000
130000
140000
0 1000 2000 3000 4000index
Well−log data
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Introduction CROPS Example
Example
100000
110000
120000
130000
140000
0 1000 2000 3000 4000
index
PELT with SIC (16.564)
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Introduction CROPS Example
Example
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●
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●
●
●
●
●
2e+10
4e+10
6e+10
8e+10
5 10 15 20 25
Number of Changepoints
Cos
t
100000
110000
120000
130000
140000
0 1000 2000 3000 4000
index
611.5720 − 1055.205
100000
110000
120000
130000
140000
0 1000 2000 3000 4000
index
1055.205 − 1116.312
100000
110000
120000
130000
140000
0 1000 2000 3000 4000
index
1116.312 − 1545.416
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Introduction CROPS Example
Summary
I changepoint package has been updated to version 2.0.
I Penalty choiceI Default changed to MBICI Inclusion of CROPS - more appropriate to use a range of
segmenations rather than just using a single choiceI Popular choices such as the SIC lack robustness in the presence of
model mis-specification
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Introduction CROPS Example
Thank you for listening.
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Introduction CROPS Example
References
K. Haynes, I.A. Eckley and P. Fearnhead.Efficient penalty search for multiple changepoint problems.In Submission, arXiv:1412.3617, (2014)
B. Jackson and J.D. Sargle.An Algorithm for optimal partitioning of data on an interval.IEEE, Signal Processing Letters, 12(2):105–108, 2005.
R. Killick, P. Fearnhead and I.A. Eckley.Optimal detection of changepoints with a linear computational cost.JASA, 107, 1590-1598, 2012.
R. Killick, and I.A. Eckley.changepoint: An R package for changepoint analysis.Journal of Statistical Software, 58(3):1–19, 2014.
R. Killick, I.A. Eckley and K. Haynes.changepoint: An R package for changepoint analysis.R package version 2.0, 2015 (To appear very soon).
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