Step detection

<h2 id="algorithms">Algorithms</h2>
When the step detection must be performed as and when the data arrives, then <a href="/facts/Online_algorithm/64TsU7Yz">online algorithms</a> are usually used,<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> 
and it becomes a special case of <a href="/facts/Sequential_analysis/K8nla6cd">sequential analysis</a>.
Such algorithms include the classical <a href="/facts/CUSUM/CmhqdV1U">CUSUM</a> method applied to changes in mean.
<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
By contrast, offline algorithms are applied to the data potentially long after it has been received. Most offline algorithms for step detection in digital data can be categorised as top-down, bottom-up, sliding window, or global methods.

<h3>Top-down</h3>
These algorithms start with the assumption that there are no steps and introduce possible candidate steps one at a time, testing each candidate to find the one that minimizes some criteria (such as the least-squares fit of the estimated, underlying piecewise constant signal). An example is the stepwise jump placement algorithm, first studied in geophysical problems,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> that has found recent uses in modern biophysics.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a>

<h3>Bottom-up</h3>
Bottom-up algorithms take the "opposite" approach to top-down methods, first assuming that there is a step in between every sample in the digital signal, and then successively merging steps based on some criteria tested for every candidate merge.

<h3>Sliding window</h3>
By considering a small "window" of the signal, these algorithms look for evidence of a step occurring within the window. The window "slides" across the time series, one time step at a time. The evidence for a step is tested by statistical procedures, for example, by use of the two-sample <a href="/facts/Student%27s_t-test/dMH8uC28">Student's t-test</a>. Alternatively, a <a href="/facts/Nonlinear_filter/sgt9wmvb">nonlinear filter</a> such as the <a href="/facts/Median_filter/YU4NdjKT">median filter</a> is applied to the signal. Filters such as these attempt to remove the noise whilst preserving the abrupt steps.

<h3>Global</h3>
Global algorithms consider the entire signal in one go, and attempt to find the steps in the signal by some kind of optimization procedure. Algorithms include <a href="/facts/Wavelet/3XcX4uIP">wavelet</a> methods,<a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> and <a href="/facts/Total_variation_denoising/rKa5n2wI">total variation denoising</a> which uses methods from <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a>. Where the steps can be modelled as a <a href="/facts/Markov_chain/5oP5hntk">Markov chain</a>, then <a href="/facts/Hidden_Markov_Models/ur1zTAhP">Hidden Markov Models</a> are also often used (a popular approach in the biophysics community<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>). When there are only a few unique values of the mean, then <a href="/facts/K-means_clustering/vxIcqHwN">k-means clustering</a> can also be used.

<h2 id="linear-versus-nonlinear-signal-processing-methods-for-step-detection">Linear versus nonlinear signal processing methods for step detection</h2>
Because steps and (independent) noise have theoretically infinite <a href="/facts/Bandwidth_(signal_processing)/7eMDzfJI">bandwidth</a> and so overlap in the <a href="/facts/Fourier_transform/XXhKJtc8">Fourier basis</a>, <a href="/facts/Signal_processing/Npaja6zb">signal processing</a> approaches to step detection generally do not use classical smoothing techniques such as the <a href="/facts/Low_pass_filter/QFsUjw7h">low pass filter</a>. Instead, most algorithms are explicitly nonlinear or time-varying.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>

<h2 id="step-detection-and-piecewise-constant-signals">Step detection and piecewise constant signals</h2>
Because the aim of step detection is to find a series of instantaneous jumps in the mean of a signal, the wanted, underlying, mean signal is <a href="/facts/Piecewise_constant/VnrEjo4B">piecewise constant</a>. For this reason, step detection can be profitably viewed as the problem of recovering a piecewise constant signal corrupted by noise. There are two complementary models for piecewise constant signals: as <a href="/facts/Spline_(mathematics)/yafkQtqp">0-degree splines</a> with a few knots, or as <a href="/facts/Level_set/JUmNbdNh">level sets</a> with a few unique levels. Many algorithms for step detection are therefore best understood as either 0-degree spline fitting, or level set recovery, methods.

<h3>Step detection as level set recovery</h3>
When there are only a few unique values of the mean, clustering techniques such as <a href="/facts/K-means_clustering/vxIcqHwN">k-means clustering</a> or <a href="/facts/Mean-shift/5eRTqvTa">mean-shift</a> are appropriate. These techniques are best understood as methods for finding a level set description of the underlying piecewise constant signal.

<h3>Step detection as 0-degree spline fitting</h3>
Many algorithms explicitly fit 0-degree splines to the noisy signal in order to detect steps (including stepwise jump placement methods<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>), but there are other popular algorithms that can also be seen to be spline fitting methods after some transformation, for example <a href="/facts/Total_variation_denoising/rKa5n2wI">total variation denoising</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15">15</a>

<h3>Generalized step detection by piecewise constant denoising</h3>
All the algorithms mentioned above have certain advantages and disadvantages in particular circumstances, yet, a surprisingly large number of these step detection algorithms are special cases of a more general algorithm.<a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> This algorithm involves the minimization of a global functional:<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a>

<table><tbody><tr><td> H [ m ] = ∑ i = 1 N ∑ j = 1 N Λ ( x i − m j , m i − m j , x i − x j , i − j ) {\displaystyle H[m]=\sum _{i=1}^{N}\sum _{j=1}^{N}\Lambda (x_{i}-m_{j},m_{i}-m_{j},x_{i}-x_{j},i-j)} </td> <td></td> <td>1</td></tr></tbody></table>
Here, xi for i = 1, ...., N is the discrete-time input signal of length N, and mi is the signal output from the algorithm. The goal is to minimize H[m] with respect to the output signal m. The form of the function 
 
 
 
 
 Λ
 
 
 
 {\displaystyle \scriptstyle \Lambda }
 
 determines the particular algorithm. For example, choosing:

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    {\displaystyle \Lambda ={\frac {1}{2}}\left|x_{i}-m_{j}\right|^{2}I(i-j=0)+\gamma \left|m_{i}-m_{j}\right|I(i-j=1)}

where I(S) = 0 if the condition S is false, and one otherwise, obtains the <a href="/facts/Total_variation_denoising/rKa5n2wI">total variation denoising</a> algorithm with regularization parameter 
 
 
 
 γ
 
 
 {\displaystyle \gamma }
 
. Similarly:

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    {\displaystyle \Lambda =\min \left\{{\frac {1}{2}}\left|m_{i}-m_{j}\right|^{2},W\right\}}

leads to the <a href="/facts/Mean_shift/5eRTqvTa">mean shift</a> algorithm, when using an adaptive step size Euler integrator initialized with the input signal x.<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a> Here W > 0 is a parameter that determines the support of the mean shift kernel. Another example is:

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    {\displaystyle \Lambda ={\frac {1-\exp(-\beta |m_{i}-m_{j}|^{2}/2)}{\beta }}\cdot I(|i-j|\leq W)}

leading to the <a href="/facts/Bilateral_filter/1z7dHYaJ">bilateral filter</a>, where 
 
 
 
 
 β
 >
 0
 
 
 
 {\displaystyle \scriptstyle \beta >0}
 
 is the tonal kernel parameter, and W is the spatial kernel support. Yet another special case is:

Λ
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    {\displaystyle \Lambda ={\frac {1}{2}}\left|x_{i}-m_{j}\right|^{2}I(i-j=0)+\gamma \left|m_{i}-m_{j}\right|^{0}I(i-j=1)}

specifying a group of algorithms that attempt to greedily fit 0-degree splines to the signal.<a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a> Here, 
 
 
 
 
 
 
 |
 x
 |
 
 
 0
 
 
 
 
 
 {\displaystyle \scriptstyle \left|x\right|^{0}}
 
 is defined as zero if x = 0, and one otherwise.
Many of the functionals in equation (1) defined by the particular choice of 
 
 
 
 
 Λ
 
 
 
 {\displaystyle \scriptstyle \Lambda }
 
 are <a href="/facts/Convex_function/IbzG5SLF">convex</a>: they can be minimized using methods from <a href="/facts/Convex_optimization/7D6U4MTN">convex optimization</a>. Still others are non-convex but a range of algorithms for minimizing these functionals have been devised.<a class="footnote-ref" id="fnref:21" href="#fn:21">21</a>

<h3>Step detection using the Potts model</h3>
A classical variational method for step detection is the Potts model. It is given by the non-convex optimization problem

u
          
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    {\displaystyle u^{*}=\arg \min _{u\in \mathbb {R} ^{N}}\gamma \|\nabla u\|_{0}+\|u-x\|_{p}^{p}}

The term 
 
 
 
 ‖
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 u
 
 ‖
 
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 =
 #
 {
 i
 :
 
 u
 
 i
 
 
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 u
 
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 }
 
 
 {\displaystyle \|\nabla u\|_{0}=\#\{i:u_{i}\neq u_{i+1}\}}
 
 penalizes the number of jumps and the term 
 
 
 
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 ‖
 
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 p
 
 
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 {\displaystyle \|u-x\|_{p}^{p}=\sum _{i=1}^{N}|u_{i}-x_{i}|^{p}}
 
 measures fidelity to data x. The parameter γ > 0 controls the tradeoff between regularity and <a href="/facts/Data_fidelity/g3XGbKIH">data fidelity</a>. Since the minimizer 
 
 
 
 
 u
 
 ∗
 
 
 
 
 {\displaystyle u^{*}}
 
 is piecewise constant the steps are given by the non-zero locations of the gradient 
 
 
 
 ∇
 
 u
 
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 {\displaystyle \nabla u^{*}}
 
.
For 
 
 
 
 p
 =
 2
 
 
 {\displaystyle p=2}
 
 and 
 
 
 
 p
 =
 1
 
 
 {\displaystyle p=1}
 
 there are fast algorithms which give an exact solution of the Potts problem in 
 
 
 
 O
 (
 
 N
 
 2
 
 
 )
 
 
 {\displaystyle O(N^{2})}
 
. 
<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a><a class="footnote-ref" id="fnref:25" href="#fn:25">25</a>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Time-series_segmentation/L6WP6Cjh">Time-series segmentation</a></li></ul>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://www.maxlittle.net/software/">PWCTools: Flexible Matlab and Python software for step detection by piecewise constant denoising</a></li>
<li><a href="http://pottslab.de/">Pottslab: Matlab toolbox for piecewise constant estimation based on the Potts model</a></li></ul>

<h2 id="references">References</h2>

<ol>
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<li id="fn:2">Gill, D. (1970). "Application of a statistical zonation method to reservoir evaluation and digitized log analysis". American Association of Petroleum Geologists Bulletin. 54: 719–729. doi:10.1306/5d25ca35-16c1-11d7-8645000102c1865d. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Snijders, A.M.; et al. (2001). "Assembly of microarrays for genome-wide measurement of DNA copy number". Nature Genetics. 29 (3): 263–264. doi:10.1038/ng754. PMID 11687795. S2CID 19460203. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">Sowa, Y.; Rowe, A. D.; Leake, M. C.; Yakushi, T.; Homma, M.; Ishijima, A.; Berry, R. M. (2005). "Direct observation of steps in rotation of the bacterial flagellar motor". Nature. 437 (7060): 916–919. Bibcode:2005Natur.437..916S. doi:10.1038/nature04003. PMID 16208378. S2CID 262329024. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Serra, J.P. (1982). Image analysis and mathematical morphology. London; New York: Academic Press. <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Basseville, M.; I.V. Nikiforov (1993). Detection of Abrupt Changes: Theory and Application. Prentice Hall. <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Rodionov, S.N., 2005a: A brief overview of the regime shift detection methods. link to PDF In: Large-Scale Disturbances (Regime Shifts) and Recovery in Aquatic Ecosystems: Challenges for Management Toward Sustainability, V. Velikova and N. Chipev (Eds.), UNESCO-ROSTE/BAS Workshop on Regime Shifts, 14–16 June 2005, Varna, Bulgaria, 17-24. <a href="http://www.beringclimate.noaa.gov/regimes/rodionov_overview.pdf" target="_blank">http://www.beringclimate.noaa.gov/regimes/rodionov_overview.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Gill, D. (1970). "Application of a statistical zonation method to reservoir evaluation and digitized log analysis". American Association of Petroleum Geologists Bulletin. 54: 719–729. doi:10.1306/5d25ca35-16c1-11d7-8645000102c1865d. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Kerssemakers, J.W.J.; Munteanu, E.L.; Laan, L.; Noetzel, T.L.; Janson, M.E.; Dogterom, M. (2006). "Assembly dynamics of microtubules at molecular resolution". Nature. 442 (7103): 709–712. Bibcode:2006Natur.442..709K. doi:10.1038/nature04928. PMID 16799566. S2CID 4359681. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Mallat, S.; Hwang, W.L. (1992). "Singularity detection and processing with wavelets". IEEE Transactions on Information Theory. 38 (2): 617–643. CiteSeerX 10.1.1.36.5153. doi:10.1109/18.119727. <a href="/wiki/CiteSeerX_(identifier)" target="_blank">/wiki/CiteSeerX_(identifier)</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">McKinney, S. A.; Joo, C.; Ha, T. (2006). "Analysis of Single-Molecule FRET Trajectories Using Hidden Markov Modeling". Biophysical Journal. 91 (5): 1941–1951. Bibcode:2006BpJ....91.1941M. doi:10.1529/biophysj.106.082487. PMC 1544307. PMID 16766620. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1544307" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1544307</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Little, M.A.; Jones, N.S. (2011). "Generalized methods and solvers for noise removal from piecewise constant signals: Part I. Background theory". Proceedings of the Royal Society A. 467 (2135): 3088–3114. Bibcode:2011RSPSA.467.3088L. doi:10.1098/rspa.2010.0671. PMC 3191861. PMID 22003312. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3191861" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3191861</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gill, D. (1970). "Application of a statistical zonation method to reservoir evaluation and digitized log analysis". American Association of Petroleum Geologists Bulletin. 54: 719–729. doi:10.1306/5d25ca35-16c1-11d7-8645000102c1865d. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">Kerssemakers, J.W.J.; Munteanu, E.L.; Laan, L.; Noetzel, T.L.; Janson, M.E.; Dogterom, M. (2006). "Assembly dynamics of microtubules at molecular resolution". Nature. 442 (7103): 709–712. Bibcode:2006Natur.442..709K. doi:10.1038/nature04928. PMID 16799566. S2CID 4359681. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">Chan, D.; T. Chan (2003). "Edge-preserving and scale-dependent properties of total variation regularization". Inverse Problems. 19 (6): S165 – S187. Bibcode:2003InvPr..19S.165S. doi:10.1088/0266-5611/19/6/059. S2CID 30704800. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">Little, M.A.; Jones, N.S. (2011). "Generalized methods and solvers for noise removal from piecewise constant signals: Part I. Background theory". Proceedings of the Royal Society A. 467 (2135): 3088–3114. Bibcode:2011RSPSA.467.3088L. doi:10.1098/rspa.2010.0671. PMC 3191861. PMID 22003312. <a href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3191861" target="_blank">https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3191861</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Mrazek, P.; Weickert, J.; Bruhn, A. (2006). "On robust estimation and smoothing with spatial and tonal kernels". Geometric properties for incomplete data. Berlin, Germany: Springer. <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Mrazek, P.; Weickert, J.; Bruhn, A. (2006). "On robust estimation and smoothing with spatial and tonal kernels". Geometric properties for incomplete data. Berlin, Germany: Springer. <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Gill, D. (1970). "Application of a statistical zonation method to reservoir evaluation and digitized log analysis". American Association of Petroleum Geologists Bulletin. 54: 719–729. doi:10.1306/5d25ca35-16c1-11d7-8645000102c1865d. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Kerssemakers, J.W.J.; Munteanu, E.L.; Laan, L.; Noetzel, T.L.; Janson, M.E.; Dogterom, M. (2006). "Assembly dynamics of microtubules at molecular resolution". Nature. 442 (7103): 709–712. Bibcode:2006Natur.442..709K. doi:10.1038/nature04928. PMID 16799566. S2CID 4359681. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Mrazek, P.; Weickert, J.; Bruhn, A. (2006). "On robust estimation and smoothing with spatial and tonal kernels". Geometric properties for incomplete data. Berlin, Germany: Springer. <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on pure and applied mathematics, 42(5), 577-685. <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">Winkler, G.; Liebscher, V. (2002). "Smoothers for discontinuous signals". Journal of Nonparametric Statistics. 14 (1–2): 203–222. doi:10.1080/10485250211388. S2CID 119562495. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">Friedrich; et al. (2008). "Complexity penalized M-estimation: fast computation". Journal of Computational and Graphical Statistics. 17 (1): 201–224. doi:10.1198/106186008x285591. S2CID 117951377. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">A. Weinmann, M. Storath, L. Demaret. "The 
 
 
 
 
 L
 
 1
 
 
 
 
 {\displaystyle L^{1}}
 
-Potts functional for robust jump-sparse reconstruction."
SIAM Journal on Numerical Analysis, 53(1):644-673 (2015). <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
</ol>

Step detection open-in-new

Step detection