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Multiscale Change Point Segmentation · Csorgö/Horvath‘97, Chen/Gupta’00,...
Transcript of Multiscale Change Point Segmentation · Csorgö/Horvath‘97, Chen/Gupta’00,...
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 1
Multiscale Change Point Segmentation
Axel Munk 1,2
1 Faculty for Mathematics and Computer Science, Georg-August University Göttingen, Germany
2 Max-Planck-Institute for Biophysical Chemistry, Göttingen, Germany
www.stochastik.math.uni-goettingen.de/munk
Klaus Frick, Buchs NTB, CH
Thomas Hotz, TU Illmenau
Hannes Sieling, Blue Yonder Big Data Analytics, Hamburg
Housen Li, MPI bpc
Florian Pein, UGöttingen
Merle Behr, UGöttingen
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 2
Applications: quality control, electrical engineering, signal processing, quantum optics, financial econometrics, genetics, …
I. The Regressogram/Change Point-Problem
Regressogram (John W. Tukey‘61)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 3
[
nm]
I. The Regressogram/Change Point-Problem
Regressogram (John W. Tukey‘61)
time[s]
[
pA]
(A) 0.3ms of current recordings of a phospholipid bilayer containing recombined protein Tim23 excited at 160mV, sampled at 50kHz (15K data), Meinecke lab Med.Dep. Göttingen
Membrane biophysics:
Channel characteristics multiscale issue
(B) Time trace of one coordinate of an atom of MD T4 lysozyme, de Groot lab, MPIbpc
time[ns]
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 4
Statistical Methods: Wavelet based (multiscale): Antoniadis/Gijbels‘02 (JNPS), Donoho/Johnstone’04 (Biometrika), Fryzlewicz/Nason/von Sachs‘04,07,08, Kolazyk/Nowak‘05 (Biometrika), Killick et al‘13 (EJS),... Kernel based: Müller’92 (AoS), … , Arlot et al.’12 (arXiv),… Aggregation: Rigollet/Tsybakov’12 (Stat. Science) Bayesian approaches: Yao‘84 (AoS), Chib‘98(JoE), Barry/Hartigan‘93 (JASA), Green‘95 (Biom.), Ghoasl et al.‘’99 (AISM), Fearnhead‘06 (SC), Luong et al’12 (arXiv), Du/Kou’15 (JASA), … (Penalized) maximum likelihood: Hinkley’70 (Biometrika), Braun/Braun/Müller’00 (Biometrika), Au, Yao‘89 (Sankhya), Birge/Massart’01 (JEMS), Zhang/Siegmund’07 (Biometrics), …, Boysen et al.’09 (AoS), Harachoui/Levy-Leduc‘10 (JASA), …. Time series: Bai/Perron‘98 (Econometrika), Yao‘93 (Biometrika), Lavielle/Moulines’00 (JTSA), Huskova/Antoch‘03 (TMMP), Mercurio/Spokoiny‘04 (AoS), Preuß et al.‘14 (JASA), … HMM/State space: Fearnhead/Clifford’03 (JRSS-B), Fuh‘04 (AoS), Cappe et al.‘05, … Distributional changes, Online prediction, sequentially, optimal stopping, … Monographs: Ibragimov/Kashminskii‘81, Baseville/Nikivorov‘93, Carlstein et al.’94, Csorgö/Horvath‘97, Chen/Gupta’00, Korostelev/Korosteleva‘11, …
I. The Regressogram/Change Point-Problem
Regressogram (John W. Tukey‘61)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 5
„segmentation of time series“: ~36millions hits Statistically vs Computational efficiency fast (local) search/segmentation methods:
I. The Regressogram/Change Point-Problem
Regressogram (John W. Tukey‘61)
CBS, Ohlsen et al.’04, (Biostatistics), Venkatraman/Ohlsen‘07 (Bioinformatics) , PELT, Killick et al. 12/14 (JASA, JSS)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 6
I. The Regressogram/Change Point-Problem
Hybrid approach: Statistical Multiscale Change Point Estimation (SMUCE): make inference on number of segments We aim for statements like: „With 90% prob. the number of selected segments is correct“ multiscale estimation/detection simultaneous (honest/uniform) confidence statements on jump locations/size/signal computationally fast
Regressogram (John W. Tukey‘61)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 7
I. A Gentle Introduction to
Statistical Multiscale Changepoint Estimation
(SMUCE)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 8
Example: Gaussian white noise, variance = 1
Candidate function (MLE, #jumps=4)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 9
Marginal residuals fit well…
Although global residuals look normal, local residual patterns indicate that the candidate is not a reasonable solution
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 10
Small scale scanning scale size =10
local t-test: H: residual signal = 0 on scale 10
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 11
Small scale scanning scale size =10
Violators: local t-test rejections residual signal on scale 10 not zero
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 12
Large scale scanning scale size =100
local t-test on scale 100
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 13
Violators: local t-test rejections residual signal on scale 100 not zero
Large scale scanning scale size =100
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 14
Statistical Multiscale Testing Scan residuals at all scales (= interval lengths) simultaneously (= multiplicity issue)
Sca
le le
ngth
= w
indo
w s
ize
Candidate function (MLE, #jumps = 4)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 15
Sca
le le
ngth
= w
indo
w s
ize
Candidate function (MLE, #jumps = 5)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 16
Sca
le le
ngth
= w
indo
w s
ize
Candidate function (MLE, #jumps = 6)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 17
Sca
le le
ngth
= w
indo
w s
ize
Candidate function (MLE, #jumps = 7)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 18
Statistical Multiscale Scanning selects final candidate which does not violate
any local t-test on each scale
Sca
le le
ngth
= w
indo
w s
ize
Candidate function (MLE, #jumps = 8)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 19
Statistical Multiscale Scanning selects final candidate which does not violate
any local t-test on each scale
Sca
le le
ngth
= w
indo
w s
ize
Candidate function (MLE, #jumps = 8)
Issues: Multiplicity? (Finite sample) error control? How to calibrate scales? How to get candidate functions?
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 20
II. SMUCE: Statistical MUltiscale Change Point Estimator
Statistical multiscale shape constraint for model selection/detection: Testing and confidence set
Estimation: Modify LSE according to SMSC constraint
Combine two different routes
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 21
II. Multiscale Testing in
Change Point Regression
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 22
Some Terminology
Regressogram (John W. Tukey‘61)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 23
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 24
FWER control
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 25
Dümbgen, Spokoiny‘01 (AoS), Dümbgen, Walther‘09 (AoS), Schmidt-Hieber et al.‘13 (AoS)
Based on Dümbgen/Spokoiny‘01 (AoS) FWER control
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 26
black: uncalibrated red: scale calibrated
Excursion: Scale Calibration
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 27
III. SMUCE
Statistical Multiscale Change Point Estimator
Shape Constraint Estimation and Confidence Sets
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 28
Related estimators: Boysen et al.‘09 (AoS), Davies et al.‘12 (CSDA)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 29
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 30
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 31
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 32
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 33
model selection error
Controlling the model selection error
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 34
Controlling the model selection error
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 35
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 36
….. n= 50 --- n= 500 n= 5000
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 37
overestimation error
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 38
IV. SMUCE in Action
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 39
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 40
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 41
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 42
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 43
Can be used to
obtain uniform/honest confidence sets obtain uniform convergence for jump locations (not shown) determine q (later)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 44
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 45 q
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 46
Example: A novel acylated gramicidin A derivative (Diederichsen lab)
Time trace (grey) of conductance for the acylated gA derivative, Vm = 50 mV, 21s, SMUCE (blue solid line).
Zooming in: 52-56s SMUCE (blue solid line).
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 47
(B) Histogram of raw data. (C) Histogram for SMUCE with state boundaries (brown, dashed vertical lines).
Time trace (grey) of conductance for the acylated gA derivative, Vm = 50 mV, 21s, SMUCE (blue solid line).
Example: A novel acylated gramicidin A derivative (Diederichsen lab)
Hotz et al., 2013, (IEEE Trans.NanoBioscience)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 48
Current method: Semiautomatic
Sabine Bosk and Conrad Weichbrodt (Steinem Lab)
8.000 clicks per hand „ClampFit“
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 49
Comparison: log10(event length)
⇒ manual analysis overestimates event lengths due to many missed events ⇒ automatic analysis suggest 2 dynamics
Magenta: Overlap of manual and SMUCE
average open time manual: 3.48 s SMUCE: 0.71 s
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 50
V. Remarks
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 51
Multiscale detection of vanishing signals I
Ingster‘93, Dümbgen, Donoho/Jin‘04 (AoS), Walther’08 (AoS), Frick et al.‘14 (JRSS-B), Enikeeva et al.‘15 (arXiv)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 52
Multiscale detection of vanishing signals I
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 53
The needle in a haystack
Multiscale detection of vanishing signals I
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 54
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 55
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 56
VI. Extensions
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 57
(Pein et al.‘15)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 58
PoreB channel
SMUCE (const variance)
H-SMUCE
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 59
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 60
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 61
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 62
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 63
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 64
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 65
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 66
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 67
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 68
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 69
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 70
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 71
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 72
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 73
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 74
Summary SMUCE: Multiscale Change Point Estimator in EFs: - -minimisation under multiscale local likelihood constraint - model selection step + constraint estimation for „multiscale regressogram“
mean, normal
binary Poisson
variance, normal
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 75
Summary SMUCE: Multiscale Change Point Estimator in EFs: - -minimisation under multiscale local likelihood constraint - model selection step + constraint estimation for „multiscale regressogram“ Computationally feasible: linear to quadratic time
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 76
Summary SMUCE: Multiscale Change Point Estimator in EFs: - -minimisation under multiscale local likelihood constraint - model selection step + constraint estimation for „multiscale regressogram“ Computationally feasible: linear to quadratic time Bounds for under/overestimation of K - controls model selection error
- guide for thresholding - allows to incorporate prior information - sequentially honest confidence sets
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 77
Summary SMUCE: Multiscale Change Point Estimator in EFs: - -minimisation under multiscale local likelihood constraint - model selection step + constraint estimation for „multiscale regressogram“ Computationally feasible: linear to quadratic time Bounds for under/overestimation of K - controls model selection error
- guide for thresholding - allows to incorporate prior information - sequentially honest confidence sets Obeys good performance confirmed by simulations (not shown) - optimal detection on (essentially) all scales - adapts automatically to sparseness (p=n not p >> n) - (up to log) optimal estimation rates (not shown)
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 78
Summary Extensions to Heterogeneous data (H-SMUCE), Pein et al.‘15 Higher selection power (FDR-based), Li et al.‘14 Inference for TDA: we have some answers for d=1 TDA then relates to mode hunting direct estimation of KS signatures possible confidence statements for KS signatures/persistent barcodes computationally fast Open issues: Much is unexplored: How does ITDA transfer to d>1? Conceptually, computationally?
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 79
R-package StepR www.stochastik.math.uni-goettingen.de/smuce www.stochastik.math.uni-goettingen.de/fdrs
Boysen, L., Kempe, A., Munk, A., Liebscher, V., Wittich, O. 2009. Consistencies and rates of convergence of jump penalized least squares estimators. Ann. Statist. 37, 157- 183.
Frick, K., Munk, A., Sieling, H. 2014. Multiscale change point inference, arXiv:1301.7212v2, Journ. Royal Statist. Soc., Ser. B 76, 495-580. With discussion and rejoinder.
Hotz, T., Schütte, O., Sieling, H., Polupanow, T., Diederichsen, U., Steinem, C., Munk, A. 2013. Idealizing ion channel recordings by jump segmentation and statistical multiresolution analysis, IEEE Trans. NanoBioscience 12, 376-386.
Futschik, A., Hotz, T., Munk, A., Sieling, H. 2014. Multiresolution DNA partitioning: Statistical evidence for segments, Bioinformatics 30, 2255-2262.
Li, H., Munk, A., Sieling, H., 2014. FDR control in multiscale change point segmentation, arXiv:1412.5844.
Pein, F., Munk, A., Sieling, H. 2015. Heterogeneous change point inference, arXiv:1505.04898.
Enikeeva, F., Munk, A., Werner, F. 2015. Bump detection in heterogeneous Gaussian regression. arXiv:1504.07390.
Bauer, U., Munk, A., Sieling, H., Wardetzky, M. 2015. Persistence barcodes versus Kolmogorov signatures: Detecting modes of one-dimensional signals. Found. of Comput. Math., arxiv.org 1404.1214. To appear.
www.stochastik.math.uni-goettingen.de/munk
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TOP DATA, Porquerolles October 22, 2015 Axel Munk 80