2023-08-21

2023-08-21

Iterative Filtering

The critical question is how do we choose the moving average operator L(X) to replace the mean of the envelopes. Ideally this choice should be data adaptive, easy to implement and analyze, and the sifting algorithm should converge. The simplest choice for the moving average is an adaptive local weighted average. We consider the moving average Y = L(X) given by Y (n) = ∑m j=−m aj(n)X(n + j), where m = m(n). a(n) = (aj(n))jm=−m is called the mask (or filter coefficients) for L at n. We say L has uniform mask if a(n) = a are independent of n, and use La to denote this operator. In other words, La(X) = Y has Y (n) = ∑m j=−m aj X(n + j). The fact that IFs use uniform mask (chosen adaptively) for the operator L in the process of extracting each IMF may be a limitation when the data to be processed are nonstaionary. As we shall see, for stationary data IFs work very well. The filter should be: convergence and (optionally) symmetric.

We have primarily focused on using the double averaging filter to perform the alternative algorithm of EMD using IFs. This is partly because of its simplicity as well as its effectiveness in our experiments. For effective sifting we need to choose both the right window size for the double averaging mask and the right stopping criteria. In our approach we choose the window size of our filter primarily according to the frequency of oscillations in the signal, which is very much related to how the envelopes are obtained.

For nonstationary data, especially extremely nonstationary data, it will be wise to use a more adaptive moving average with nonuniform mask. More sophisticated method for choosing the mask will be needed. It is rather natural to choose a moving average operator whose mask at time t depends on the local density of the extrema. We shall describe in detail how such nonuniform masks were chosen in [7] in one of the examples later. The convergence of sifting algorithm with nonuniform mask in the continuous time setting is established in [7] for certain class of signals.

To perform EMD using nonuniform double average filter mask one needs to find a good mechanism for specifying the length of the mask at a given k. This is done as follows in [7]. The mask length w(k) for any other k is given by the interpolated value using the spline. Using this non-uniform mask the EMD yields essentially a perfect decomposition with two IMF’s, see the right figure in Figure 8. In comparison, the EMD using uniform mask is given on the left of Figure 8.

When the data are nonstationary moving averages with nonuniform masks may need to be used to perform the decomposition effectively. There are still some mathematical questions that need to be further studied. The IFs have one advantage over the classical EMD: It is more stable under perturbations.

(linIterativeFilteringAlternative2009, link, DOI, zolib)

The convergence for the inner loop is guaranteed for periodic signals [21] and it has been studied for l∞ functions in [33]. However, the convergence of IF applied to general signals using uniform or non–uniform filter hasn’t been explored yet. With this in mind, in the following we provide an explicit formula for each IMF and we show sufficient conditions on the filter w(t) that ensure the convergence of the IF inner loop. In Section 4 we present a class of filters that fulfill these sufficient conditions. So every filter given by the convolution of a symmetric, nonnegative and finitely supported L2 filter with itself satisfies the sufficient conditions of Theorem 1. In Section 4 we present a class of this kind of filters.

An important aspect in ALIF and IF methods is the choice of the low pass filters used in the inner loops. To handle non–linear and non–stationary signals, we want to design smooth filters with compact support. We produce such filters by means of a diffusion process. The idea is very natural: when applying the diffusion to the data, the oscillations will eventually be eliminated and a curve will be generated as the average. It is well known that diffusion processes are associated with PDEs. For a given diffusion equation, we can get its fundamental solution and treat it as the filter in the iterative filtering algorithm. Here we note that the well known heat equation may not be a good choice, because its solution leads to a filter with infinite support which is not desirable. To have compact support and smoothness for the filters, we select Fokker–Planck equations to construct what we call FP filters.

(ciconeAdaptiveLocalIterative2015, link, DOI, zolib)

If we consider filters like the Fokker-Planck filters [4] or any filter with smooth finite support properties. So for a filter w with smooth finite support properties the set Iw,γ,N is made up of a finite number of disjoint compact intervals, containing zeros of p w, together with the intervals p ́8, ́Ξ0s and rΞ0, 8q.

As an example of a compactly supported filter we can consider the triangular filter function.

Definition 3. A vector w P Rn, n odd number, is called a filter if its values are symmetric with respect to the middle, nonnegative, and řn p“1 wp “ 1. We assume that a filter shape has been selected a priori, like one of the Fokker-Planck filters described in [4], and that some invertible and monotone scaling function gm has been chosen so that wmpξq can be computed as described in (8).

f we consider the case of linear scaling, making use of (9), the matrix Wm becomes... We point out that the previous formula represent an ideal Wm, however we need to take into account the quadrature formula we use to compute the numerical convolution in order to build the appropriate Wm to be used in the DIF algorithm.

If we take into account all the aforementioned details we can reproduce a matrix Wm which is row stochastic. We observe that in the implemented code we simply scale each row of Wm by its sum so that the matrix becomes row stochastic.

Since Wm P Rnˆn represents the discrete convolution operator, it can be a circulant matrix, Toeplitz matrix or it can have a more complex structure. Its structure depends on the way we extend the signal outside its boundaries.

(ciconeNumericalAnalysisIterative2018, link, DOI, zolib)