Continuous Random Functions Research Articles

Cramer-Rao bounds in functional form: theory and application to passive optical ranging.

A functional approach to the multivariate statistical model of a generalized incoherent passive optical ranging and imaging system with a CCD sensor is proposed. This approach implies that a large number of discrete, statistically independent, random data (pixel readouts) can be approximated by a continuous random function. Thus, the joint probability density function (PDF) takes a functional form; the statistical averages of the infinite-variate PDF and the Fisher information become functional integrals that can be treated analytically in the Gaussian approximation. The Cramer-Rao bounds on estimator-error variances are obtained for the scalar and functional deterministic parameters of the model. An approximate expression is derived for the PDF of the sum of independent Gaussian and Poisson random variables using the steepest-descent method, and the resulting PDF is shown to be asymptotically Gaussian. As an illustration, we apply the developed approach to a passive optical rangefinder with chiral wavefront coding. Numerical and experimental examples are presented.

Stochastic transfer of polarized radiation in finite cloudy atmospheric media with reflective boundaries

The problem of monoenergetic radiative transfer in a finite planar stochastic atmospheric medium with polarized (vector) Rayleigh scattering is proposed. The solution is presented for an arbitrary absorption and scattering cross sections. The extinction function of the medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The joint probability distribution function of these Gaussian random variables is used to calculate the ensemble-averaged quantities, such as reflectivity and transmissivity, for an arbitrary correlation function. A modified Gaussian probability distribution function is also used to average the solution in order to exclude the probable negative values of the optical variable. Pomraning-Eddington approximation is used, at first, to obtain the deterministic analytical solution for both the total intensity and the difference function used to describe the polarized radiation. The problem is treated with specular reflecting boundaries and angular-dependent externally incident flux upon the medium from one side and with no flux from the other side. For the sake of comparison, two different forms of the weight function, which introduced to force the boundary conditions to be fulfilled, are used. Numerical results of the average reflectivity and average transmissivity are obtained for both Gaussian and modified Gaussian probability density functions at the different degrees of polarization.

Mid-frequency vibrations of a double-leaf plate with random inhomogeneities

Predicting vibrations of composite structures such as double-leaf plates is difficult because of the large number of components and random inhomogeneities in the components. In the low and high frequency ranges, the components may be homogenized, and consequently the model of a structure becomes simple enough to be mathematically and computationally tractable. However the vibrations in the mid-frequency range cannot be predicted using such methods because the wavelengths are comparable to the size of the components and junctions between components. Simply adding more details, e.g., higher resolution in finite element mesh, will not result in more accurate predictions. In this paper a double-leaf plate is modelled using the Kirchhoff plate and Euler beam theories. The elastic moduli and junctions are allowed to be inhomogeneous over the plates and beams. These inhomogeneities are simulated as continuous smooth random functions rather than series of discrete random numbers. The random functions are incorporated into the model using the variational formulation and the Fourier expansion of the vibration field. Various probability density functions are tested for the inhomogeneities. Then the distribution of resonant frequencies and the vibration field are studied and compared with other models.

Gradient Infinite-Dimensional Random Dynamical Systems

In this paper we introduce the concept of a gradient random dynamical system as a random semiflow possessing a continuous random Lyapunov function which describes the asymptotic regime of the system. Thus, we are able to analyze the dynamical properties on a random attractor described by its Morse decomposition for infinite-dimensional random dynamical systems. In particular, if a random attractor is characterized by a family of invariant random compact sets, we show the equivalence among the asymptotic stability of this family, the Morse decomposition of the random attractor, and the existence of a random Lyapunov function.

Random recursive triangulations of the disk via fragmentation theory

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension β* + 1, where β* = (√17 − 3)/2, and that it can be described as the geodesic lamination coded by a random continuous function which is Hölder continuous with exponent β* − ε, for every ε > 0. We also discuss recursive constructions of triangulations of the n-gon that give rise to the same continuous limit when n tends to infinity.

Robust, Adaptive Functional Regression in Functional Mixed Model Framework

Functional data are increasingly encountered in scientific studies, and their high dimensionality and complexity lead to many analytical challenges. Various methods for functional data analysis have been developed, including functional response regression methods that involve regression of a functional response on univariate/multivariate predictors with nonparametrically represented functional coefficients. In existing methods, however, the functional regression can be sensitive to outlying curves and outlying regions of curves, so is not robust. In this article, we introduce a new Bayesian method, robust functional mixed models (R-FMM), for performing robust functional regression within the general functional mixed model framework, which includes multiple continuous or categorical predictors and random effect functions accommodating potential between-function correlation induced by the experimental design. The underlying model involves a hierarchical scale mixture model for the fixed effects, random effect, and residual error functions. These modeling assumptions across curves result in robust nonparametric estimators of the fixed and random effect functions which down-weight outlying curves and regions of curves, and produce statistics that can be used to flag global and local outliers. These assumptions also lead to distributions across wavelet coefficients that have outstanding sparsity and adaptive shrinkage properties, with great flexibility for the data to determine the sparsity and the heaviness of the tails. Together with the down-weighting of outliers, these within-curve properties lead to fixed and random effect function estimates that appear in our simulations to be remarkably adaptive in their ability to remove spurious features yet retain true features of the functions. We have developed general code to implement this fully Bayesian method that is automatic, requiring the user to only provide the functional data and design matrices. It is efficient enough to handle large datasets, and yields posterior samples of all model parameters that can be used to perform desired Bayesian estimation and inference. Although we present details for a specific implementation of the R-FMM using specific distributional choices in the hierarchical model, 1D functions, and wavelet transforms, the method can be applied more generally using other heavy-tailed distributions, higher dimensional functions (e.g., images), and using other invertible transformations as alternatives to wavelets. Supplementary materials for this article are available online.

A note on uniform exponential convergence of sample average approximation of random functions

Shapiro and Xu (2008) [17] investigated uniform large deviation of a class of Hölder continuous random functions. It is shown under some standard moment conditions that with probability approaching one at exponential rate with the increase of sample size, the sample average approximation of the random function converges to its expected value uniformly over a compact set. This note extends the result to a class of discontinuous functions whose expected values are continuous and the Hölder continuity may be violated for some negligible random realizations. The extension entails the application of the exponential convergence result to a substantially larger class of practically interesting functions in stochastic optimization.

A developed solution of the stochastic Milne problem using probabilistic transformations

This paper considers the solution of the Milne problem of radiative transfer with isotropic scattering in a continuous stochastic medium. Properties of the medium are assumed to be continuous random functions of the spatial dimensions. The available solutions – in literature – for this stochastic integro-differential equation (SIDE) are represented only by the ensemble average of the radiant energy density. In this paper, a developed algorithm, based on the implementation of the random variable transformation technique together with an integral transformation to the stochastic properties, is introduced. A complete stochastic solution represented by the probability-density function (p.d.f) of the radiant energy density is obtained. Using the closed form of the p.d.f, the nth moment of the stochastic solution is evaluated. In realization of this work, Exponential and Gaussian statistics for the medium properties are assumed. Results are physically acceptable and found to be compatible with those in the literature.

Sample Functions of Stochastic Measures and Besov Spaces

This paper considers stochastic measures, i.e., sets of functions given on the Borel sigma-algebra in $[0,1]^d$ sigma-additive with respect to probability. It is shown that realizations of continuous random functions generated by stochastic measures belong to the Besov spaces under some general sufficiently assumptions.

A general real-time formulation for multi-rate mass transfer problems

Abstract. Many flow and transport phenomena, ranging from delayed storage in pumping tests to tailing in river or aquifer tracer breakthrough curves or slow kinetics in reactive transport, display non-equilibrium (NE) behavior. These phenomena are usually modeled by non-local in time formulations, such as multi-porosity, multiple processes non equilibrium, continuous time random walk, memory functions, integro-differential equations, fractional derivatives or multi-rate mass transfer (MRMT), among others. We present a MRMT formulation that can be used to represent all these models of non equilibrium. The formulation can be extended to non-linear phenomena. Here, we develop an algorithm for linear mass transfer, which is accurate, computationally inexpensive and easy to implement in existing groundwater or river flow and transport codes. We illustrate this approach by application to published data involving NE groundwater flow and solute transport in rivers and aquifers.

Hausdorff measure of arcs and Brownian motion on Brownian spatial trees

A Brownian spatial tree is defined to be a pair $(\mathcal{T},\phi)$, where $\mathcal{T}$ is the rooted real tree naturally associated with a Brownian excursion and φ is a random continuous function from $\mathcal{T}$ into ℝd such that, conditional on $\mathcal{T}$, φ maps each arc of $\mathcal{T}$ to the image of a Brownian motion path in ℝd run for a time equal to the arc length. It is shown that, in high dimensions, the Hausdorff measure of arcs can be used to define an intrinsic metric $d_{\mathcal{S}}$ on the set $\mathcal{S}:=\phi(\mathcal{T})$. Applications of this result include the recovery of the spatial tree $(\mathcal{T},\phi)$ from the set $\mathcal{S}$ alone, which implies in turn that a Dawson–Watanabe super-process can be recovered from its range. Furthermore, $d_{\mathcal{S}}$ can be used to construct a Brownian motion on $\mathcal{S}$, which is proved to be the scaling limit of simple random walks on related discrete structures. In particular, a limiting result for the simple random walk on the branching random walk is obtained.

On the integral with respect to the tensor product of two random measures

A Fubini-type formula for the integral with respect to the tensor product of two random measures is established in an intrinsic way. This permits one to consider a convolution product. The results are applied to a stationary continuous random function (which may be multiplicatively written with two stationary components) and to principal component analysis in the frequency domain.

Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique

Much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation (SDE). The Random Variable Transformation technique (RVT) is a powerful technique to get the complete solution for the SDE represented by the probability-density function of the solution process. In this paper, the RVT technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one-speed transport equation for neutral particles in a semi-infinite stochastic medium with linear anisotropic scattering. The extinction function of the medium (input stochastic process) is assumed to be a continuous random function of position. The probability-density function and hence, the higher order statistical moments of the solution process are presented. Numerical results are given for different distributions of the input stochastic process.

Transport Of Neutral Particles In A Finite Continuous Stochastic Medium With Gaussian Statistics

The stationary solution of the one-speed transport equation in a finite stochastic medium with linear anisotropic scattering is considered. The solution is presented for an arbitrary absorption and scattering cross sections. The total cross section of medium is assumed to be a continuous random function of position, with fluctuations about the mean taken as Gaussian distributed. The Pomraning-Eddington technique is used at first to solve the problem in the deterministic case. Two correlated random variables appear in the solution; namely, the optical space variable and the optical thickness of the medium. The dual Gaussian-probability density function of these two random variables is derived from which the ensemble-averaged solution is calculated for an arbitrary correlation function. The first and the second statistical moments of some quantities of interest, such as radiant energy, net flux, reflectivity, and transmissivity, are calculated. The problem is treated with specular-refecting boundaries and an incident flux of particles on the medium from the left and with no flux from the right.

Neutron transport through semi-infinite continuous stochastic media using Gaussian statistics

The stationary solution of the one-speed neutron transport equation in a semi-infinite stochastic medium with linear anisotropic scattering is considered. The cross-section function of the medium is assumed to be a continuous random function of position with fluctuations about the mean taken as Gaussian distributed. The joint probability distribution function of these Gaussian random variables is used to calculate the ensemble-averaged quantities, such as radiant neutron energy and net neutron flux, for an arbitrary correlation function. The problem is solved at first in the deterministic case, then the solution is averaged using Gaussian joint probability distribution function. A modified Gaussian probability distribution function is also used to average the solution. Numerical results are given for the sake of comparison.

Functional data analysis for cash flow and transactions intensity continuous-time prediction using Hilbert-valued autoregressive processes

In this paper we use a functional autoregressive model as a robust predictor of the cash flow and intensity of transactions in a credit card payment systems. Intraday economic time series are treated as random continuous functions projected onto low dimensional subspace. Wavelet bases are considered for data smoothing. We compare two linear wavelet methods for the prediction problem of a continuous-time stochastic process on an entire time interval. Ex poste prediction is used to check the models.

Algorithmic randomness of continuous functions

We investigate notions of randomness in the space $${{\mathcal C}(2^{\mathbb N})}$$ of continuous functions on $${2^{\mathbb N}}$$ . A probability measure is given and a version of the Martin-Lof test for randomness is defined. Random $${\Delta^0_2}$$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any $${y \in 2^{\mathbb N}}$$ , there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set.

Groupe d'opérateurs unitaires déduit d'une mesure spectrale – une application

In this Note, given the stationary continuous random function ( X t ) t ∈ R and a pair ( a , b ) of reals we will define all the stationary processes ( X ( t 1 , t 2 ) ′ ) ( t 1 , t 2 ) ∈ R × R such that ( X ( b t , a t ) ′ ) t ∈ R = ( X t ) t ∈ R . For this purpose, we use the product of spectral measure convolution, as defined in [A. Boudou, Y. Romain, On spectral and random measures associated to discrete and continuous-time processes, Statist. Probab. Lett. 59 (2002) 145–157], and we develop other mathematical tools which offer a wider interest than our initial preoccupation. So, we use the concept of transposed homomorphism, which allows us to evenly define the concept of a group of unitary operators deduced from a spectral measure. To cite this article: A. Boudou, C. R. Acad. Sci. Paris, Ser. I 344 (2007).

Random Continuous Functions

We investigate notions of algorithmic randomness in the space C(2N) of continuous functions on 2N. A probability measure is given and a version of the Martin-Löf test for randomness is defined which allows us to define a class of (Martin-Löf) random continuous functions. We show that random Δ20 continuous functions exist, but no computable function can be random. We show that a random function maps any computable real to a random real and that the image of a random continuous function is always a perfect set and hence uncountable. We show that for any y∈2N, there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set.

Assessing the uncertainty associated with intermittent rainfall fields

In many practical situations where spatial rainfall estimates are needed, rainfall occurs as a spatially intermittent phenomenon. An efficient geostatistical method for rainfall estimation in the case of intermittency has previously been published and comprises the estimation of two independent components: a binary random function for modeling the intermittency and a continuous random function that models the rainfall inside the rainy areas. The final rainfall estimates are obtained as the product of the estimates of these two random functions. However the published approach does not contain a method for estimation of uncertainties. The contribution of this paper is the presentation of the indicator maximum likelihood estimator from which the local conditional distribution of the rainfall value at any location may be derived using an ensemble approach. From the conditional distribution, representations of uncertainty such as the estimation variance and confidence intervals can be obtained. An approximation to the variance can be calculated more simply by assuming rainfall intensity is independent of location within the rainy area. The methodology has been validated using simulated and real rainfall data sets. The results of these case studies show good agreement between predicted uncertainties and measured errors obtained from the validation data.