Adaptive multiresolution analysis based on synchronization

We propose an adaptive multiscale approach to data analysis based on synchronization. The approach is nonlinear, data driven in the sense that it does not rely on a priori chosen basis, and automatically determines the data scale. Numerical results for one- and two-dimensional cases illustrate that the method works effectively for the usual modulated signals such as chirps, etc., as well as for more complicated data with multiple scales. The method extends straightforwardly to functions defined on weighted graphs and grids in high dimensions. Connections with some other recent approaches to multiscale analysis are briefly discussed.

Figure 1

Synchronization and Haar wavelets [axes: for the 1D case the oscillators are ordered in space and $j$ is the spatial coordinate of oscillators; $Y={\Omega }^0\left(V_k\right)$]. Various stages $n$ of synchronization-based MRA: $Y={\overline{\Omega }}^{10}\left(V_k\right)$ ($\times$ symbols); $Y={\overline{\Omega }}^{100}\left(V_k\right)$ (diamonds); $Y={\overline{\Omega }}^{500}\left(V_k\right)$ (circles); $c=0.25$. Top inset: Weighted averaging of the individual frequencies for a linear distribution of intrinsic frequencies—the signal (triangles), the Haar averaged signal $A2$ ($\times$ symbols), synchronized frequencies (circles). Bottom inset: The phases uniformly grow in time (axes: phases vs time).

Figure 2

A function with an abrupt frequency change [axes: $j$ is the spatial coordinate; $Y={\Omega }^0\left(V_k\right)$]. (a) ${\Omega }^0\left(V_k\right)$ [triangles (left)]; the averaged Haar signal $A4$ [triangles (up)]; the synchronization plateaus (circles). (b) The synchronized scales (the edge effects are neglected). (c) The continuous Morlet wavelet transform of ${\Omega }^0\left(V_k\right)$. Synchronization gives a sharper overall value of the scale, comparing to the continuous Morlet wavelet transform.

Figure 3

Two chirps and a harmonic signal [axes: $j$ is the spatial coordinate; $Y$ = ${\Omega }^0\left(V_k\right)$]. (a) Chirp No. 1 [triangles (up)]; chirp No. 2 (circles); harmonic signal ($\times$ symbols). (b) The corresponding synchronized frequencies. Note that the length and the height of the plateaus corresponding to the harmonic signal are constant, while the scale and the height of the plateaus corresponding to the chirps [triangles (up), circles] reflect the frequency changes of the chirps.

Figure 4

Multiscale analysis based on synchronization [axes: $j$ is the spatial coordinate; $Y={\Omega }^0\left(V_k\right)$]. Main figure: data with multiple scales ($\times$ symbols); various stages $n$ of synchronization-based MRA: $Y={\overline{\Omega }}^1\left(V_k\right)$ [triangles (up)]; $Y={\overline{\Omega }}^{40}\left(V_k\right)$ [triangles (right)], $Y={\overline{\Omega }}^{80}\left(V_k\right)$ [triangles (left)]. Inset: A zoomed in version of the main figure.

Figure 5

A function ${\Omega }^0\left(V_k\right)$ defined on a 2D grid (sampled from harmonics with three different wave numbers). (a) The data [axes: $i,j$ are the spatial coordinates, $z={\Omega }^0\left(V_k\right)$]. (b) The scales in the data revealed by synchronization [axes: $i,j$ are the spatial coordinates, $z={\overline{\Omega }}^6\left(V_k\right)$]; colors are proportional to the function values, with darker colors corresponding to extrema.