Real-valued Process Research Articles

Regression estimation for continuous-time functional data processes with missing at random response

In this paper, we are interested in nonparametric kernel estimation of a generalised regression function based on an incomplete sample ( X t , Y t , ζ t ) t ∈ [ 0 , T ] copies of a continuous-time stationary and ergodic process ( X , Y , ζ ) . The predictor X is valued in some infinite-dimensional space, whereas the real-valued process Y is observed when the Bernoulli process ζ = 1 and missing whenever ζ = 0 . Uniform almost sure consistency rate as well as the evaluation of the conditional bias and asymptotic mean square error are established. The asymptotic distribution of the estimator is provided with a discussion on its use in building asymptotic confidence intervals. To illustrate the performance of the proposed estimator, a first simulation is performed to compare the efficiency of discrete-time and continuous-time estimators. A second simulation is conducted to discuss the selection of the optimal sampling mesh in the continuous-time case. Then, a third simulation is considered to build asymptotic confidence intervals. An application to financial time series is used to study the performance of the proposed estimator in terms of point and interval prediction of the IBM asset price log-returns. Finally, a second application is introduced to discuss the usage of the initial estimator to impute missing household-level peak electricity demand.

A novel Bayesian image despeckling method using 2D CGARCH-M model in 2D dost framework

Image denoising is a crucial task in the field of image processing, and Bayesian estimation has emerged as a prominent approach. To effectively employ Bayesian estimation, it is essential to assume a priori distribution corresponding to the transform coefficients of the noise-free image. In this paper, we propose a novel Bayesian image despeckling method that leverages the 2D Complex Generalized Autoregressive Conditional Heteroscedasticity Mixture (CGARCH-M) model within the framework of the 2D Discrete Orthonormal Stockwell Transform (DOST). Prior to introducing this method, we present a novel statistical analysis of the 2D DOST coefficients of log-transformed images. Speckle noise, commonly found in synthetic aperture radar (SAR) images, is modeled as multiplicative noise. To effectively suppress speckle noise, our proposed method utilizes a novel adaptive Bayes risk estimator known as compound maximum a posteriori (CMAP). By employing CMAP, we estimate the noise-free 2D DOST coefficients from the noisy ones, which are modeled using the 2D CGARCH-M. Notably, this paper introduces the 2D CGARCH-M model for 2D complex stochastic processes for the first time, extending the capabilities of the GARCH model and its subsequent extensions that were limited to real-valued processes. The proposed model incorporates location-dependent conditional variances to capture the non-Gaussian statistics of 2D DOST coefficients of log-transformed images and the dependencies between them. Through our statistical analysis, we establish the compatibility between the 2D CGARCH-M model and these coefficients. Our proposed method takes into account both magnitude and phase by utilizing the real and imaginary components of the DOST coefficients concurrently. It provides an optimal and closed-form solution, significantly reducing memory and computational requirements. Moreover, unlike state-of-the-art approaches in this domain, our method exhibits robustness to initial parameter settings. To evaluate the effectiveness of our approach, we conduct comparisons with other denoising methods on artificially speckled aerial images and actual SAR images. The results demonstrate the superior performance of our method.

Permutation group entropy: A new route to complexity for real-valued processes.

This is a review of group entropy and its application to permutation complexity. Specifically, we revisit a new approach to the notion of complexity in the time series analysis based on both permutation entropy and group entropy. As a result, the permutation entropy rate can be extended from deterministic dynamics to random processes. More generally, our approach provides a unified framework to discuss chaotic and random behaviors.

Complexity-based permutation entropies: From deterministic time series to white noise

This is a paper in the intersection of time series analysis and complexity theory that presents new results on permutation complexity in general and permutation entropy in particular. In this context, permutation complexity refers to the characterization of time series by means of ordinal patterns (permutations), entropic measures, decay rates of missing ordinal patterns, and more. Since the inception of this “ordinal” methodology, its practical application to any type of scalar time series and real-valued processes have proven to be simple and useful. However, the theoretical aspects have remained limited to noiseless deterministic series and dynamical systems, the main obstacle being the super-exponential growth of allowed permutations with length when randomness (also in form of observational noise) is present in the data. To overcome this difficulty, we take a new approach through complexity classes, which are precisely defined by the growth of allowed permutations with length, regardless of the deterministic or noisy nature of the data. We consider three major classes: exponential, sub-factorial and factorial. The next step is to adapt the concept of Z-entropy to each of those classes, which we call permutation entropy because it coincides with the conventional permutation entropy on the exponential class. Z-entropies are a family of group entropies, each of them extensive on a given complexity class. The result is a unified approach to the ordinal analysis of deterministic and random processes, from dynamical systems to white noise, with new concepts and tools. Numerical simulations show that permutation entropy discriminates time series from all complexity classes.

SPHARMA approximations for stationary functional time series on the sphere

In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in Caponera and Marinucci (Ann Stat 49(1):346–369, 2021) and Caponera et al. (Stoch Process Appl 137:167–199, 2021); more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.

Universality for Persistence Exponents of Local Times of Self-Similar Processes with Stationary Increments

AbstractWe show that $$\mathbb {P}( \ell _X(0,T] \le 1)=(c_X+o(1))T^{-(1-H)}$$ P ( ℓ X ( 0 , T ] ≤ 1 ) = ( c X + o ( 1 ) ) T - ( 1 - H ) , where $$\ell _X$$ ℓ X is the local time measure at 0 of any recurrent H-self-similar real-valued process X with stationary increments that admits a sufficiently regular local time and $$c_X$$ c X is some constant depending only on X. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound $$1-H$$ 1 - H on the decay exponent of $$\mathbb {P}( \ell _X(0,T] \le 1)$$ P ( ℓ X ( 0 , T ] ≤ 1 ) . Our approach establishes a new connection between persistence probabilities and Palm theory for self-similar random measures, thereby providing a general framework which extends far beyond the Gaussian case.

On universal algorithms for classifying and predicting stationary processes

This is a survey of results on universal algorithms for classification and prediction of stationary processes. The classification problems include discovering the order of a k-step Markov chain, determining memory words in finitarily Markovian processes and estimating the entropy of an unknown process. The prediction problems cover both discrete and real valued processes in a variety of situations. Both the forward and the backward prediction problems are discussed with the emphasis being on pointwise results. This survey is just a teaser. The purpose is merely to call attention to results on classification and prediction. We will refer the interested reader to the sources. Throughout the paper we will give illuminating examples.

Remark on pathwise uniqueness of stochastic differential equations driven by Lévy processes

Consider real-valued processes determined by stochastic differential equations driven by Lévy processes. The jump parts of the driving Lévy process are not always α-stable ones, nor symmetric ones. In the present article, we shall study the pathwise uniqueness of the solutions to the stochastic differential equations under the conditions on the coefficients that the diffusion and the jump terms are Hölder continuous, while the drift one is monotonic. Our approach is based on Gronwall’s inequality.

Two-Time-Scale Hybrid Traffic Models for Pedestrian Crowds

This paper introduces new models to describe pedestrian crowd dynamics in a typical unidirectional environment, such as corridors, pathways, and railway platforms. Pedestrian movements are represented in a two-dimensional space that is further divided into narrow virtual lanes. Consequently, pedestrians either move in a lane following each other or change lanes, when it is desirable. Within this framework, the motions of pedestrians are modeled as a two-dimensional and two-time-scale hybrid system. A pedestrian’s movement along the crowd direction is labeled as the $x$ direction and modeled by a real-valued process, a solution of a differential equation in continuous time, the lane change is labeled as the $y$ direction. In contrast to the $x$ direction dynamics, the movements in the $y$ direction only happen at some time epoch. Although the movements are still on the same time horizon as the $x$ direction movements, with a slight abuse of notation and for simplicity and convenience, we use discrete time as the time indicator, and model the movements by a recursive equation taking values in a finite set. Under common assumptions of crowd movements, we prove that the crowd movements in the $x$ direction will converge to a uniform distance distribution and the convergence rate is exponential. Furthermore, by using a velocity-distance function to represent the common crowd and traffic congestion scenarios, we show that all pedestrians will asymptotically move with a uniform group speed. In the $y$ direction, when pedestrians naturally wish to change to faster lanes, we show that the numbers in each virtual lanes converge to a balanced distribution and hence achieves asymptotic consensus as shown typically in a crowd behavior. Stability and convergence analysis is carried out rigorously by using properties of circular matrices, stability of networked systems, and stochastic approximations. Simulation studies are used to demonstrate the main properties of our modeling approach and establish its usefulness in representing pedestrian dynamics.

Long memory estimation for complex-valued time series

Long memory has been observed for time series across a multitude of fields, and the accurate estimation of such dependence, for example via the Hurst exponent, is crucial for the modelling and prediction of many dynamic systems of interest. Many physical processes (such as wind data) are more naturally expressed as a complex-valued time series to represent magnitude and phase information (wind speed and direction). With data collection ubiquitously unreliable, irregular sampling or missingness is also commonplace and can cause bias in a range of analysis tasks, including Hurst estimation. This article proposes a new Hurst exponent estimation technique for complex-valued persistent data sampled with potential irregularity. Our approach is justified through establishing attractive theoretical properties of a new complex-valued wavelet lifting transform, also introduced in this paper. We demonstrate the accuracy of the proposed estimation method through simulations across a range of sampling scenarios and complex- and real-valued persistent processes. For wind data, our method highlights that inclusion of the intrinsic correlations between the real and imaginary data, inherent in our complex-valued approach, can produce different persistence estimates than when using real-valued analysis. Such analysis could then support alternative modelling or policy decisions compared with conclusions based on real-valued estimation.

Goodness-of-fit Tests for Continuous-time Stationary Processes

The paper considers the following problem of hypotheses testing: based on a finite realization {X(t)}, 0 ≤ t ≤ T of a zero mean real-valued mean square continuous stationary Gaussian process X(t), t ϵ R, construct goodness-of-fit tests for testing a hypothesis H0 that the hypothetical spectral density of the process X(t) has the specified form. We show that in the case where the hypothetical spectral density of X(t) does not depend on unknown parameters (the hypothesis H0 is simple), then the suggested test statistic has a chi-square distribution. In the case where the hypothesis H0 is composite, that is, the hypothetical spectral density of X(t) depends on an unknown p–dimensional vector parameter, we choose an appropriate estimator for unknown parameter and describe the limiting distribution of the test statistic, which is similar to that of obtained by Chernov and Lehman in the case of independent observations. The testing procedure works both for short- and long-memory models.

An Approach to Stochastic Integration in General Separable Banach Spaces

We suggest a new approach to stochastic integration in infinite-dimensional spaces that is based on representing random variables on Banach spaces as real-valued processes on an interval. We prove stochastic integrability of operator-valued processes on general separable Banach spaces under the conditions that do not depend on the norm of the space and show how our methods can be applied to studying infinite-dimensional stochastic differential equations. In particular, our results provide a natural construction of the stochastic integral in abstract Wiener spaces.

A pathwise inference method for the parameters of diffusion terms

ABSTRACTWe consider inference of the parameters of the diffusion term for continuous time stochastic processes with a power-type dependence of the diffusion coefficient from the underlying process such as Cox–Ingersoll–Ross, CKLS, and similar processes. We suggest some original pathwise estimates for this coefficient and for the power index based on an analysis of an auxiliary continuous time complex-valued process generated by the underlying real-valued process. These estimates do not rely on the distribution of the underlying process and on a particular choice of the drift. Some numerical experiments are used to illustrate the feasibility of the suggested method.

Weak decreasing stochastic order

We introduce the notion of weak decreasing stochastic (WDS) ordering for real-valued processes with negative means, which, to our knowledge, has not been studied before. Thanks to Madan–Yor’s argument, it follows that the WDS ordering is a necessary and sufficient condition for a family of integrable probability measures with negative mean to be embeddable in a standard Brownian motion by the Cox and Hobson extension of the Azéma–Yor algorithm. The resulting process is a supermartingale and if, in addition, the measures have densities, this supermartingale is Markovian. Then the Cox–Hobson algorithm provides a special solution of Kellerer’s theorem relying on the stronger hypothesis of WDS order.

On wide sense stationary processes over finite non-abelian groups

Let X be a real-valued wide sense stationary process over a finite non-abelian group G. We provide results on optimal orthogonal decomposition of X into real-valued mutually orthogonal components and using this decomposition we develop a test for correlation of X over the group G. Applications of these results to the analysis of variance of the carry-over effects in the cross-over designs in clinical studies are given. Our focus will be on groups S3,S4, and A4.

Confidence interval with specified precision for the mean value in a sequence of Gaussian variables

Consider the class C of real-valued stochastic processes of the form Xt=m+mt+ξt, with discrete t=1,2,⋯, such that mt→0 and the ξt are random variables with normal distributions N(0,σt), where σt→0. Required is a sequence of estimators m^t for the parameter m, determined by X1,⋯,Xt and a stopping rule τ, such that for every e>0 and 0 e} 0 and the sequences {at}, {bt} are (1) known, and (2) their limiting behavior satisfies some detailed conditions. Furthermore, assume that k sample functions of Xt were observed: {X(i)t}, i=1,⋯,k, with k≥2. Then for the estimators m^t=∑ki=1X(i)t/k one can construct the required stopping rule.

Multifractal analysis for the occupation measure of stable-like processes

In this article, we investigate the local behavior of the occupation measure $\mu $ of a class of real-valued Markov processes $\mathcal{M} $, defined via a SDE. This (random) measure describes the time spent in each set $A\subset \mathbb{R} $ by the sample paths of $\mathcal{M} $. We compute the multifractal spectrum of $\mu $, which turns out to be random, depending on the trajectory. This remarkable property is in sharp contrast with the results previously obtained on occupation measures of other processes (such as Levy processes), where the multifractal spectrum is usually deterministic, almost surely. In addition, the shape of this multifractal spectrum is very original, reflecting the richness and variety of the local behavior. The proof is based on new methods, which lead for instance to fine estimates on Hausdorff dimensions of certain jump configurations in Poisson point processes.

On an approach to boundary crossing by stochastic processes

In this paper we provide an overview as well as new (definitive) results of an approach to boundary crossing. The first published results in this direction appeared in de la Peña and Giné (1999) book on decoupling. They include order of magnitude bounds for the first hitting time of the norm of continuous Banach-Space valued processes with independent increments. One of our main results is a sharp lower bound for the first hitting time of càdlàg real-valued processes X(t), where X(0)=0 with arbitrary dependence structure: ETrγ≥∫01{a−1(rα)}γdα, where Tr=inf{t>0:X(t)≥r},a(t)=E{sup0≤s≤tX(s)} and γ>0. Under certain extra conditions, we also obtain an upper bound for ETrγ. As the main text suggests, although Tr is defined as the hitting time of X(t) hitting a level boundary, the bounds developed can be extended to more general processes and boundaries. We shall illustrate applications of the bounds derived for additive processes, Gaussian Processes, Bessel Processes, Bessel bridges among others. By considering the non-random function a(t), we can show that in various situations, ETr≈a−1(r).

Affine LIBOR models driven by real-valued affine processes

ABSTRACTThe class of affine LIBOR models is appealing since it satisfies three central requirements of interest rate modeling. It is arbitrage-free, interest rates are nonnegative, and caplet and swaption prices can be calculated analytically. In order to guarantee nonnegative interest rates affine LIBOR models are driven by nonnegative affine processes, a restriction that makes it hard to produce volatility smiles. We modify the affine LIBOR models in such a way that real-valued affine processes can be used without destroying the nonnegativity of interest rates. Numerical examples show that in this class of models, pronounced volatility smiles are possible.

A new encoding of coalescent processes: applications to the additive and multiplicative cases

We revisit the discrete additive and multiplicative coalescents, starting with n particles with unit mass. These cases are known to be related to some “combinatorial coalescent processes”: a time reversal of a fragmentation of Cayley trees or a parking scheme in the additive case, and the random graph process $$(G(n,p))_p$$ in the multiplicative case. Time being fixed, encoding these combinatorial objects in real-valued processes indexed by the line is the key to describing the asymptotic behaviour of the masses as $$n\rightarrow +\infty $$ . We propose to use the Prim order on the vertices instead of the classical breadth-first (or depth-first) traversal to encode the combinatorial coalescent processes. In the additive case, this yields interesting connections between the different representations of the process. In the multiplicative case, it allows one to answer to a stronger version of an open question of Aldous (Ann Probab 25:812–854, 1997): we prove that not only the sequence of (rescaled) masses, seen as a process indexed by the time $$\lambda $$ , converges in distribution to the reordered sequence of lengths of the excursions above the current minimum of a Brownian motion with parabolic drift $$(B_t+\lambda t - t^2/2, t\ge 0)$$ , but we also construct a version of the standard augmented multiplicative coalescent of Bhamidi et al. (Probab Theory Relat, 2013) using an additional Poisson point process.