Section: Research Program

Tools for characterizing and measuring regularity

Fractional Dimensions

Although the main focus of our team is on characterizing local regularity, on occasions, it is interesting to use a global index of regularity. Fractional dimensions provide such an index. In particular, the regularization dimension, that was defined in [31] , is well adapted to the study stochastic processes, as its definition allows to build robust estimators in an easy way. Since its introduction, regularization dimension has been used by various teams worldwide in many different applications including the characterization of certain stochastic processes, statistical estimation, the study of mammographies or galactograms for breast carcinomas detection, ECG analysis for the study of ventricular arrhythmia, encephalitis diagnosis from EEG, human skin analysis, discrimination between the nature of radioactive contaminations, analysis of porous media textures, well-logs data analysis, agro-alimentary image analysis, road profile analysis, remote sensing, mechanical systems assessment, analysis of video games, ...(see http://regularity.saclay.inria.fr/theory/localregularity/biblioregdim for a list of works using the regularization dimension).

Hölder exponents

The simplest and most popular measures of local regularity are the pointwise and local Hölder exponents. For a stochastic process ${\left\{X\left(t\right)\right\}}_{t\in ℝ}$ whose trajectories are continuous and nowhere differentiable, these are defined, at a point $t_0$, as the random variables:

and

Although these quantities are in general random, we will omit as is customary the dependency in $\omega$ and $X$ and write $\alpha \left(t_0\right)$ and $\tilde{\alpha }\left(t_0\right)$ instead of ${\alpha }_X(t_0,\omega )$ and ${\tilde{\alpha }}_X(t_0,\omega )$.

The random functions $t\mapsto {\alpha }_X(t_0,\omega )$ and $t\mapsto {\tilde{\alpha }}_X(t_0,\omega )$ are called respectively the pointwise and local Hölder functions of the process $X$.

The pointwise Hölder exponent is a very versatile tool, in the sense that the set of pointwise Hölder functions of continuous functions is quite large (it coincides with the set of lower limits of sequences of continuous functions [6] ). In this sense, the pointwise exponent is often a more precise tool (i.e. it varies in a more rapid way) than the local one, since local Hölder functions are always lower semi-continuous. This is why, in particular, it is the exponent that is used as a basis ingredient in multifractal analysis (see section 3.2 ). For certain classes of stochastic processes, and most notably Gaussian processes, it has the remarkable property that, at each point, it assumes an almost sure value [18] . SRP, mBm, and processes of this kind (see sections 3.3 and 3.3 ) rely on the sole use of the pointwise Hölder exponent for prescribing the regularity.

However, ${\alpha }_X$ obviously does not give a complete description of local regularity, even for continuous processes. It is for instance insensitive to “oscillations”, contrarily to the local exponent. A simple example in the deterministic frame is provided by the function $x^{\gamma }sin\left(x^{-\beta }\right)$, where $\gamma ,\beta$ are positive real numbers. This so-called “chirp function” exhibits two kinds of irregularities: the first one, due to the term $x^{\gamma }$ is measured by the pointwise Hölder exponent. Indeed, $\alpha \left(0\right)=\gamma$. The second one is due to the wild oscillations around 0, to which $\alpha$ is blind. In contrast, the local Hölder exponent at 0 is equal to $\frac{\gamma }{1+\beta }$, and is thus influenced by the oscillatory behaviour.

Another, related, drawback of the pointwise exponent is that it is not stable under integro-differentiation, which sometimes makes its use complicated in applications. Again, the local exponent provides here a useful complement to $\alpha$, since $\tilde{\alpha }$ is stable under integro-differentiation.

Both exponents have proved useful in various applications, ranging from image denoising and segmentation to TCP traffic characterization. Applications require precise estimation of these exponents.

Stochastic 2-microlocal analysis

Neither the pointwise nor the local exponents give a complete characterization of the local regularity, and, although their joint use somewhat improves the situation, it is far from yielding the complete picture.

A fuller description of local regularity is provided by the so-called 2-microlocal analysis, introduced by J.M. Bony [53] . In this frame, regularity at each point is now specified by two indices, which makes the analysis and estimation tasks more difficult. More precisely, a function $f$ is said to belong to the 2-microlocal space $C_{x_0}^{s,s^{\text{'}}}$, where $s+s^{\text{'}}>0,s^{\text{'}}<0$, if and only if its $m=[s+s^{\text{'}}]-$th order derivative exists around $x_0$, and if there exists $\delta >0$, a polynomial $P$ with degree lower than $\left[s\right]-m$, and a constant $C$, such that

for all $x,y$ such that $0<|x-x_0|<\delta$, $0<|y-x_0|<\delta$. This characterization was obtained in [25] , [32] . See [64] , [65] for other characterizations and results. These spaces are stable through integro-differentiation, i.e. $f\in C_x^{s,s^{\text{'}}}$ if and only if $f^{\text{'}}\in C_x^{s-1,s^{\text{'}}}$. Knowing to which space $f$ belongs thus allows to predict the evolution of its regularity after derivation, a useful feature if one uses models based on some kind differential equations. A lot of work remains to be done in this area, in order to obtain more general characterizations, to develop robust estimation methods, and to extend the “2-microlocal formalism” : this is a tool allowing to detect which space a function belongs to, from the computation of the Legendre transform of an auxiliary function known as its 2-microlocal spectrum. This spectrum provide a wealth of information on the local regularity.

In [18] , we have laid some foundations for a stochastic version of 2-microlocal analysis. We believe this will provide a fine analysis of the local regularity of random processes in a direction different from the one detailed for instance in [69] .We have defined random versions of the 2-microlocal spaces, and given almost sure conditions for continuous processes to belong to such spaces. More precise results have also been obtained for Gaussian processes. A preliminary investigation of the 2-microlocal behaviour of Wiener integrals has been performed.

Multifractal analysis of stochastic processes

A direct use of the local regularity is often fruitful in applications. This is for instance the case in RR analysis or terrain modeling. However, in some situations, it is interesting to supplement or replace it by a more global approach known as multifractal analysis (MA). The idea behind MA is to group together all points with same regularity (as measured by the pointwise Hölder exponent) and to measure the “size” of the sets thus obtained [28] , [54] , [60] . There are mainly two ways to do so, a geometrical and a statistical one.

In the geometrical approach, one defines the Hausdorff multifractal spectrum of a process or function $X$ as the function: $\alpha \mapsto f_h\left(\alpha \right)=dim\{t:{\alpha }_X\left(t\right)=\alpha \}$, where $dimE$ denotes the Hausdorff dimension of the set $E$. This gives a fine measure-theoretic information, but is often difficult to compute theoretically, and almost impossible to estimate on numerical data.

The statistical path to MA is based on the so-called large deviation multifractal spectrum:

where:

and ${\alpha }_n^k$ is the “coarse grained exponent” corresponding to the interval $I_n^k=\left[\frac{k}{n},\frac{k+1}{n}\right]$, i.e.:

Here, $Y_n^k$ is some quantity that measures the variation of $X$ in the interval $I_n^k$, such as the increment, the oscillation or a wavelet coefficient.

The large deviation spectrum is typically easier to compute and to estimate than the Hausdorff one. In addition, it often gives more relevant information in applications.

Under very mild conditions (e.g. for instance, if the support of $f_g$ is bounded, [27] ) the concave envelope of $f_g$ can be computed easily from an auxiliary function, called the Legendre multifractal spectrum. To do so, one basically interprets the spectrum $f_g$ as a rate function in a large deviation principle (LDP): define, for $q\in ℝ$,

with the convention $0^q:=0$ for all $q\in ℝ$. Let:

$\tau \left(q\right)=\underset{n\to \infty }{lim\; inf}\frac{log{S}_n\left(q\right)}{-log\left(n\right)}.$

The Legendre multifractal spectrum of $X$ is defined as the Legendre transform ${\tau }^{*}$ of $\tau$:

$f_l\left(\alpha \right):={\tau }^{*}\left(\alpha \right):=\underset{q\in ℝ}{inf}(q\alpha -\tau \left(q\right)).$

To see the relation between $f_g$ and $f_l$, define the sequence of random variables $Z_n:=log\left|Y_n^k\right|$ where the randomness is through a choice of $k$ uniformly in $\{0,...,n-1\}$. Consider the corresponding moment generating functions:

$c_n\left(q\right):=-\frac{log{E}_n[exp\left(q{Z}_n\right)]}{log\left(n\right)}$

where $E_n$ denotes expectation with respect to $P_n$, the uniform distribution on $\{0,...,n-1\}$. A version of Gärtner-Ellis theorem ensures that if $lim{c}_n\left(q\right)$ exists (in which case it equals $1+\tau \left(q\right)$), and is differentiable, then $c^{*}=f_g-1$. In this case, one says that the weak multifractal formalism holds, i.e. $f_g=f_l$. In favorable cases, this also coincides with $f_h$, a situation referred to as the strong multifractal formalism.

Multifractal spectra subsume a lot of information about the distribution of the regularity, that has proved useful in various situations. A most notable example is the strong correlation reported recently in several works between the narrowing of the multifractal spectrum of ECG and certain pathologies of the heart [61] , [63] . Let us also mention the multifractality of TCP traffic, that has been both observed experimentally and proved on simplified models of TCP [2] , [49] .

Another colour in local regularity: jumps

As noted above, apart from Hölder exponents and their generalizations, at least another type of irregularity may sometimes be observed on certain real phenomena: discontinuities, which occur for instance on financial logs and certain biomedical signals. In this frame, it is of interest to supplement Hölder exponents and their extensions with (at least) an additional index that measures the local intensity and size of jumps. This is a topic we intend to pursue in full generality in the near future. So far, we have developed an approach in the particular frame of multistable processes. We refer to section 3.3 for more details.