Editing Dynamic time warping (section)

== Alternative approaches ==

In [[functional data analysis]], time series are regarded as discretizations of smooth (differentiable) functions of time. By viewing the observed samples at smooth functions, one can utilize continuous mathematics for analyzing data.<ref>{{Cite journal|title = On the Registration of Time and the Patterning of Speech Movements|last1 = Lucero|first1 = J. C.|last2 = Munhall|first2 = K. G.|last3 = Gracco|first3 = V. G.|last4 = Ramsay|first4 = J. O.|journal = Journal of Speech, Language, and Hearing Research|volume = 40|issue = 5|pages = 1111–1117|year = 1997|doi=10.1044/jslhr.4005.1111|pmid = 9328881}}</ref> Smoothness and monotonicity of time warp functions may be obtained for instance by integrating a time-varying [[radial basis function]], thus being a one-dimensional [[diffeomorphism]].<ref>{{cite journal |last1=Durrleman |first1=S |last2= Pennec |first2=X.|last3=Trouvé|first3=A.|last4=Braga|first4=J.|last5=Gerig|first5=G.|last6=Ayache|first6=N.|name-list-style=amp |date=2013 |title=Toward a Comprehensive Framework for the Spatiotemporal Statistical Analysis of Longitudinal Shape Data|url=https://hal.inria.fr/hal-00813825/document |journal=International Journal of Computer Vision |volume=103|issue=1 |pages=22–59 |doi=10.1007/s11263-012-0592-x|pmid=23956495 |pmc=3744347}}</ref> Optimal nonlinear time warping functions are computed by minimizing a measure of distance of the set of functions to their warped average. Roughness penalty terms for the warping functions may be added, e.g., by constraining the size of their curvature. The resultant warping functions are smooth, which facilitates further processing. This approach has been successfully applied to analyze patterns and variability of speech movements.<ref>{{Cite book|title = Speech Motor Control: New Developments in Basic and Applied Research|last1 = Howell|first1 = P.|publisher = Oxford University Press|year = 2010|isbn = 978-0199235797|pages = 215–225|last2 = Anderson|first2 = A.|last3 = Lucero|first3 = J. C.|chapter = Speech motor timing and fluency|editor-last = Maassen|editor-first = B.|editor-last2 = van Lieshout|editor-first2 = P.}}</ref><ref>{{Cite journal|title = Speech production variability in fricatives of children and adults: Results of functional data analysis|journal = The Journal of the Acoustical Society of America|date = 2008|issn = 0001-4966|pmc = 2677351|pmid = 19045800|pages = 3158–3170|volume = 124|issue = 5|doi = 10.1121/1.2981639|first1 = Laura L.|last1 = Koenig|first2 = Jorge C.|last2 = Lucero|first3 = Elizabeth|last3 = Perlman|bibcode = 2008ASAJ..124.3158K}}</ref>

Another related approach are [[hidden Markov model]]s (HMM) and it has been shown that the [[Viterbi algorithm]] used to search for the most likely path through the HMM is equivalent to stochastic DTW.<ref>{{Cite journal|last1=Nakagawa|first1=Seiichi|last2=Nakanishi|first2=Hirobumi|date=1988-01-01|title=Speaker-Independent English Consonant and Japanese Word Recognition by a Stochastic Dynamic Time Warping Method|journal=IETE Journal of Research|volume=34|issue=1|pages=87–95|doi=10.1080/03772063.1988.11436710|issn=0377-2063}}</ref><ref>{{Cite web|url=http://eecs.ceas.uc.edu/~fangcg/course/FromDTWtoHMM_ChunshengFang.pdf|title=From Dynamic Time Warping (DTW) to Hidden Markov Model (HMM)|last=Fang|first=Chunsheng}}</ref><ref>{{Cite journal|last=Juang|first=B. H.|date=September 1984|title=On the hidden Markov model and dynamic time warping for speech recognition #x2014; A unified view|journal=AT&T Bell Laboratories Technical Journal|volume=63|issue=7|pages=1213–1243|doi=10.1002/j.1538-7305.1984.tb00034.x|s2cid=8461145|issn=0748-612X}}</ref>

DTW and related warping methods are typically used as pre- or post-processing steps in data analyses. If the observed sequences contain both random variation in both their values, shape of observed sequences and random temporal misalignment, the warping may overfit to noise leading to biased results. A simultaneous model formulation with random variation in both values (vertical) and time-parametrization (horizontal) is an example of a [[nonlinear mixed-effects model]].<ref name="Raket_et_al_2014">{{cite journal |vauthors=Raket LL, Sommer S, Markussen B |year=2014 |title=A nonlinear mixed-effects model for simultaneous smoothing and registration of functional data |journal=Pattern Recognition Letters |volume=38|pages=1–7 |doi=10.1016/j.patrec.2013.10.018|bibcode=2014PaReL..38....1R }}</ref> In human movement analysis, simultaneous nonlinear mixed-effects modeling has been shown to produce superior results compared to DTW.<ref name="Raket_et_al_2016">{{cite journal |vauthors=Raket LL, Grimme B, Schöner G, Igel C, Markussen B |year=2016 |title=Separating timing, movement conditions and individual differences in the analysis of human movement |journal=PLOS Computational Biology |volume=12|issue=9|pages=e1005092 |doi=10.1371/journal.pcbi.1005092|pmid=27657545 |pmc=5033575 |bibcode=2016PLSCB..12E5092R |doi-access=free |arxiv=1601.02775 }}</ref>