Method for cycle detection in sparse, irregularly sampled, long-term neuro-behavioral timeseries: Basis pursuit denoising with polynomial detrending of long-term, inter-ictal epileptiform activity
Fig 2
Description of method objectives and signal assumptions.
(A) The core assumption of the model is that the underlying signal (ii) is the linear sum of (i) oscillations, a polynomial trend, and noise. Part (B) describes the overall workflow including (i) data input to the model, (ii) outputs, and (iii) estimated signal reconstructions. (Ci) Equation representing the core signal assumption that the observations come from a combination of oscillations, a polynomial trend, and noise or error. Notation includes y (m x 1 vector of observed data), Ψ (n x n discrete cosine transform (DCT) basis), Φ (m x n binary row subsampling matrix), x (n x 1 DCT coefficients), Τ (n x p Vandermonde matrix), z (p x 1 polynomial coefficients), (m x 1 error terms). (Cii) Expression for basis pursuit denoising containing x and z as unknowns, yielding a 2D minimization problem that is reduced to a 1D minimization problem (Ciii) by variable projection. (D) Schematic representation of the equations and sampling approach in (C).