The Microcanonical Multiscale Formalism
3.2 The micro-canonical formalism
h|Γrs|pi=Aprτp (3.2) whereAp= h|Γls|pil−τp is a factor that is independent of the scalerand the expo-nent of this power law (τp) is called the Legendre spectrum. Kolomogrov intuition was that τp varies linearly with p: τp = δp. However, experimental observations show that the Legendre spectrum is not exactly a linear function ofpin form ofδp. It is actually diverging from this linear form for large values of pand is a curved convex function ofp[8].
The deviation of τp from the linear scaling with p is due to a common basic feature of complex signals, called intermittency. An intermittent signal displays ac-tivity only in a fraction of time and this fraction decreases with the scale under consideration. Indeed, while going toward finer scales, the reproduction of eddies becomes less space filling and hence the exponent of the power-law structure func-tion decreases [42].
As such, the mapp→τpis an important representation characterizing the inter-mittent character of complex signals. Indeed, Eq. (3.2) implies that having known τp, the p-th order statistics of the physical variable s at the fine scale r, can be deduced from that of the coarser scalel. Thus, τp can uniquely describe the inter-scale organization and the intermittent character of the complex signal. Other than complete characterization of multi-scale hierarchy from a statistical point of view, once this multiplicative relationship is validated, it might be used to obtain a more parsimonious representation, using the fact that if Eq. (3.1) holds for a set of scales, the information in finer scales might be deduced from those of coarser scales.
3.2 The micro-canonical formalism
The exponent of the canonical power-law in Eq. (3.2) describes the intermittent character of a complex signal, only from a global statistical point of view. Indeed, the Legendre spectrum is a global description and provides no information about local complexity. Therefore, it can be only used to recognize the existence of an underlying multi-scale structure, without any information about its geometric or-ganization 2. Besides, the evaluation of this power-law requires the calculation of p-th order moments of variables (the structure function) which imposes stationar-ity assumptions. Moreover, the computation of these structure functions is highly demanding in data, which prohibits their use for empirical data in most cases [130].
2 There exist however attempts in the canonical framework to geometrically access these complexities.
An example is the Wavelet Transform Modulus Maxima method [81,82,90] which is discussed in section3.3.1.2
tel-00821896, version 1 - 13 May 2013
The microcanonical framework (the MMF) provides computationally efficient tools for geometricalcharacterization of this inter-scale relationship. MMF provides access to local scaling parameters which provide valuable information about the local dynamics of a complex signal and can be used for precise detection of critical events inside the signal. As such, the micro-canonical formalism not only recog-nizes the global existence of complex multi-scale structures, but also it showslocally where the complexity appears and how it organizes itself. Indeed, as we will see shortly, rigid theoretical links has been made between this geometrical evaluation and the statistical viewpoint in section 3.1. These links also serve to provide theo-retical evidence regarding the meaningfulness of such local analysis [43].
MMF does not rely on statistical values for ensemble averages, but rather look at what is going on around any given point. It is based on the computation of a scaling exponenth(t) at every point in a signal domain and out of any stationarity assumption. These exponents are formally defined by the evaluation of the limiting behavior of a multi-scale functionalΓr(s(t))at each pointtover a set of fine scales r:
Γr(s(t)) = α(t)rh(t)+o rh(t)
r→0 (3.3) where Γr(s(t)) can be any multi-scale functional complying with this power-law and the multiplicative factor α(t) generally depends on the chosenΓr, but for sig-nals conforming to the multi-scale hierarchy explained in section3.1, the exponent h(t)is independent of it. The term o rh(t)
means that for very small scales the ad-ditive terms are negligible compared to the factor and thus h(t)dominantly quan-tifies the multi-scale behavior of the signal at the time instant t. Indeed, close to a critical point, the details on the microscopic dynamics of the system disappear and the macroscopic characteristics are purely determined by the value of this ex-ponent, called the Singularity Exponent (SE) [113]. A central concern in MMF is the proper choice of the multi-scale functional Γr(s(t)) so as to precisely estimate these exponents. We will address this subject in following sections, but for now let us assume the availability of precise estimates ofh(t)and develop the link between this geometric representation and the global one in the canonical formalism.
When correctly defined and estimated, the values of singularity exponents h(t) define a hierarchy of sets having a multi-scale structure closely related to the cascad-ing properties of some random variables associated to the macroscopic description of the system under study, similar to the one observed in the canonical framework.
Formally, this hierarchy can be represented by the definition of singularity compo-nentsFhas the level-sets of the SEs:
tel-00821896, version 1 - 13 May 2013
3.2 t h e m i c r o-c a n o n i c a l f o r m a l i s m 23
Fh={t|h(t) =h} (3.4) These level sets, each highlight a set of irregularly spaced points having the same SE values. Consequently, they can be used to decompose the signal into a hierarchy of subsets, the "multi-scale hierarchy". Particularly, they can be used to detect the most informative subset of points called the Most Singular Manifold (MSM) and also, they can be used to provide a global statistical view of the complex system by the use of the so-called singularity spectrum.
3.2.1 The Most Singular Manifold
In the MMF, a particular set of interest is the level set comprising the points hav-ing the smallest SE values and provides indications in the acquired signal about the most critical transitions of the associated dynamics [131]. These are the points where sharp and sudden local variations take place and hence, they have the lowest predictability: the degree in which they can be predicted from their neighboring samples is minimal. MSM is formed as the collection of points having the smallest values of SE. In other words, the smaller the h(t) is for a given point, the higher the predictability is in the neighborhood of this point. It has been established that the critical transitions of the system occurs at these points. This property has been successfully used in several applications [127, 128, 134]. The formal definition of MSM reads:
F∞={t|h(t) =h∞}, h∞=min(h(t)) (3.5) In practice, once the signal is discretized, h∞ should be defined within a certain quantization range and hence MSM is formed as a set of points whereh(t)is below a certain threshold.
The significance of the MSM is particularly demonstrated in the framework of reconstructible systems: it has been shown that, for many natural signals, the whole signal can be reconstructed using only the information carried by the MSM [127, 131]. For example, a reconstruction operator is defined for natural images in [127] which allows very accurate reconstruction of the whole image when applied to its gradient information over the MSM. The reconstruction quality can be further improved, using theΓrmeasure defined in [126] which makes a local evaluation of the reconstruction operator to geometrically quantify the unpredictability of each point.
tel-00821896, version 1 - 13 May 2013
Although simple, the notion of MSM plays an important role in most of the ap-plications we have developed in this thesis. By those apap-plications, we show how the MSM corresponds to the subset of physically important points in the speech signal.
3.2.2 The singularity spectrum
As the points in each singularity component Fh are irregularly spaced and do not fill their topological space, a non-integer value D(h) can be assigned to each one of them as the Hausdorff dimension of these subsets:
D(h) =dimHFh (3.6) and the map h → D(h) is the so called singularity spectrum. D(h) gives an in-sight about the hierarchical arrangement and the probability of occurrence of each singularity component. It describes the statistics of change in scales, just like the exponent of canonical power-lawτpin Eq. (3.2) (the Legendre spectrum).D(h)can also be used to bound completely different physical systems through the concept of universality class. It is known that, different physical systems having similar dis-tributions of singular exponents share common multi-scale properties, even if they are completely different physical systems [131].
Parisi and Frich proved that under some assumptions on the shape ofD(h)[130], the Legendre spectrumτp can be computed from the singularity spectrum [43]:
τp=inf
h {ph+d−D(h)} (3.7)
wheredstands for the dimension of the embedding space. Hence, as in the canon-ical framework, the singularity spectrum (if well estimated) could analogously de-fine the multi-scale structure of real world intermittent signals (based on the cascade model of energy dissipations). In the canonical formalism, as there is no easy way to have an estimate of singularity exponents at each point, it is difficult to correctly es-timateD(h). Instead, the structure functions in Eq. (3.2) can be used to first estimate τpand thenD(h)is accessible through the inverse Legendre transform of Eq. (3.7).
As the inverse transform, computes the convex hull of the variables, this methods imposes a limiting constraint on D(h) to be a convex function. In Microcanonical framework however, as we have access to precise estimates ofh(t)(and hence singu-larity components can be formed), the frequency of occurrence of particular values of SEs can be used for the estimation ofD(h)by the histogram method. Indeed, the empirical histogram of SE (ρr(h)) at small scalerverifies [130]: