Multi-channel/multi-trial time-frequency analysis Time-Frequency analysis for multi-channel and/or multi-trial signals B. Torr´ esani Aix-Marseille Univ. Laboratoire d’Analyse, Topologie et Probabilit´ es ESI, December 2012 B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 1 / 45
Multi-channel/multi-trial time-frequency analysis Outline Introduction 1 Multi-channel signals, multi-trial signals Time-frequency analysis Multi-channel signals and time-frequency 2 The need for structures A regression model Introducing time dependencies : a detection model 3 Time dependencies via Markov chain A case study : alpha waves based characterization of multiple sclerosis Conclusions 4 B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 2 / 45
Multi-channel/multi-trial time-frequency analysis Motivation : Multi-sensor biosignals, such as EEG, MEG,... contain information that shows up differently in various channels, and may be difficult to extract from single channel. In this context, one often looks for features that are localized in some joint space-time-frequency domain. To detect weak signals, experiments are often repeated several times : multi-trial signals Problem : tackle inter-trial variability... which may sometimes be modelled as time-frequency jitter and amplitude variability... B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 3 / 45
Multi-channel/multi-trial time-frequency analysis Outline Introduction 1 Multi-channel signals, multi-trial signals Time-frequency analysis Multi-channel signals and time-frequency 2 The need for structures A regression model Introducing time dependencies : a detection model 3 Time dependencies via Markov chain A case study : alpha waves based characterization of multiple sclerosis Conclusions 4 B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 4 / 45
Multi-channel/multi-trial time-frequency analysis Time-frequency analysis : Time-frequency transforms are inherently single channel techniques. Can be trivially extended to multi-channel signals, by individually transforming each channel ; multi-channel cooperation is enforced by post-processing. Synthesis-based frameworks allow one to enforce multi-channel information sharing already in the first stage. The multi-trial situation is much more complex... need of time-frequency registration techniques prior to trial averaging. B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 5 / 45
Multi-channel/multi-trial time-frequency analysis Notations : Gabor atoms Modulated and translated copies of a reference window g kn [ t ] = e 2 i π n ν 0 ( t − kb 0 ) / L g [ t − kb 0 ] , k ∈ Z K , n ∈ Z N where ν 0 and b 0 are divisors of L , K = L / b 0 et N = L / ν 0 . Given f ∈ C L , the family of coefficients L − 1 f [ t ] g [ t − kb 0 ] e − 2 i π n ν 0 ( t − kb 0 ) / L ∑ V g f [ k , n ] = � f , g kn � = t = 0 form a short time Fourier transform (if b 0 = ν 0 = 1) or a Gabor transform of f . B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 6 / 45
Multi-channel/multi-trial time-frequency analysis Examples of Gabor atoms : B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 7 / 45
Multi-channel/multi-trial time-frequency analysis Notations : MDCT atoms In C L , let M ∈ Z + be a un divisor of L . Z L is segmented into K = L / N intervals of length N For all k = 0 ,... K − 1, let w k ∈ C L be such that w k [ t ] = 0 for t < ( k − 1 / 2 ) N and t > ( k + 3 / 2 ) N . w k [ kN + τ ] = w k + 1 [ kN − τ ] for all τ = 1 − N / 2 ,... N / 2 − 1 w k [ kN + τ ] 2 + w k + 1 [ kN + τ ] 2 = 1 for all τ = 1 − N / 2 ,... N / 2 − 1 Denote by u kn ∈ C L the vectors defined by � � � � � 2 n + 1 u kn [ t ] = N w k [ t ] cos π ( t − kN ) 2 The collection { u kn } is an orthonormal basis of C L . B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 8 / 45
Multi-channel/multi-trial time-frequency analysis Examples of MDCT atoms : Being a basis has a price : the time-frequency localization of MDCT atoms is more difficult to control. B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 9 / 45
Multi-channel/multi-trial time-frequency analysis Frames and bases Given a time-frequency basis Ψ = { ψ tf } : the transform x ∈ C L → {� x , ψ tf �} is unitary. Any x has a unique expansion x = ∑ α tf ψ tf . tf Given a time-frequency frame Ψ = { ψ tf } (which is not a basis). Any x has infinitely many expansions of the form x = ∑ α tf ψ tf , tf finding the most relevant one requires extra information,... and is application dependent. B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 10 / 45
Multi-channel/multi-trial time-frequency analysis Outline Introduction 1 Multi-channel signals, multi-trial signals Time-frequency analysis Multi-channel signals and time-frequency 2 The need for structures A regression model Introducing time dependencies : a detection model 3 Time dependencies via Markov chain A case study : alpha waves based characterization of multiple sclerosis Conclusions 4 B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 11 / 45
Multi-channel/multi-trial time-frequency analysis Multichannel signals : x = { x c , c = 1 ,... N c } signals from different channels are often dependent the dependence structure is often complex, and not necessarily known in advance Example (Propagation from two sources) Signals, originating from two inner “sources”, propagating to the boundary of some region where they are measured. Quasi-static approximation : time-locked signals B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 12 / 45
Multi-channel/multi-trial time-frequency analysis Example : EEG signals B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 13 / 45
Multi-channel/multi-trial time-frequency analysis Multichannel time-frequency expansions In the framework of quasi-static type approximations (no time delay) : use the same time-frequency dictionary for all channels : Ψ = { ψ tf } . Transform + post-processing : example Compute time-frequency transform coefficients α = { α c α c tf = � x c , ψ tf � tf } , Describe the data cube α via space-time-frequency modes, using factor decomposition (PARAFAC, Kruskal,...) α = ∑ tf = ∑ α c C c C k ⊗ T k ⊗ F k + res. , k T kt F kf + res. k k B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 14 / 45
Multi-channel/multi-trial time-frequency analysis Outline Introduction 1 Multi-channel signals, multi-trial signals Time-frequency analysis Multi-channel signals and time-frequency 2 The need for structures A regression model Introducing time dependencies : a detection model 3 Time dependencies via Markov chain A case study : alpha waves based characterization of multiple sclerosis Conclusions 4 B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 15 / 45
Multi-channel/multi-trial time-frequency analysis Synthesis models estimate multichannel time-frequency expansions of the form x c = ∑ α c tf ψ tf + noise t , f Elementary models : channel, time and frequency are independent variables ; e.g. bridge regression type approaches : for 1 ≤ p ≤ 2, solve � � 2 � � 1 + µ � � � x c − ∑ p � α � p α c 2 ∑ min � tf ψ tf � p � α c t , f B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 16 / 45
Multi-channel/multi-trial time-frequency analysis More complex models : introduce correlation structures, via either function space models (involving sophisticated mixed norms) or probabilistic models. Gaussian and Gaussian mixture models Gaussian prior model : p ( α ) ∼ N ( 0 , Σ ) Gaussian mixture prior model : for example p ( α ) ∼ ∑ K k = 1 p k N ( 0 , Σ k ) Generalizations... In most cases, Σ and/or Σ k are large matrices : difficult to estimate... and to exploit. B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 17 / 45
Multi-channel/multi-trial time-frequency analysis Gaussian mixture prior � � � � 2 1 � � � x − ∑ � α tf ψ tf � − ln ( p ( α )) min 2 σ 2 � 0 t , f with p a Gaussian or Gaussian mixture prior. Gaussian prior : the (explicit) solution requires the inversion of a large matrix involving the inverse covariance matrix Σ − 1 and the Gram matrix of the frame. Gaussian mixture priors : MM numerical strategies require at each iteration the inversion of matrices of the same size. Typical size : N c ≈ 20 channels, time-frequency blocks of dimension MN ≈ 1000... yields matrices of size ≈ 20000 × 20000. B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 18 / 45
Multi-channel/multi-trial time-frequency analysis Extra structure has to be assumed for the covariance model : coefficient cube α with independent slices B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 19 / 45
Multi-channel/multi-trial time-frequency analysis Extra structure has to be assumed for the covariance model : coefficient cube α with independent slices If the time-frequency frame and the covariance structure are compatible, corresponding estimation algorithms can be designed. B. Torr´ esani (LATP , Aix-Marseille Univ.) ESI, December 2012 19 / 45
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