Random Multiplicative Processes Research Articles

Functional generalized canonical correlation analysis for studying multiple longitudinal variables.

In this paper, we introduce functional generalized canonical correlation analysis, a new framework for exploring associations between multiple random processes observed jointly. The framework is based on the multiblock regularized generalized canonical correlation analysis framework. It is robust to sparsely and irregularly observed data, making it applicable in many settings. We establish the monotonic property of the solving procedure and introduce a Bayesian approach for estimating canonical components. We propose an extension of the framework that allows the integration of a univariate or multivariate response into the analysis, paving the way for predictive applications. We evaluate the method's efficiency in simulation studies and present a use case on a longitudinal dataset.

A generalization of the Lindley Distribution

The Lindley distribution is useful in a wide variety of fields, such as biology and astronomy. Many generalizations of the Lindley distribution have been introduced in the literature, with various motivations. Inspired by the concept of superstatistics in nonequilibrium statistical mechanics, we introduce here a novel generalization of the Lindley distribution, by regarding its shape parameter as a stochastic variable. This procedure is particularly well motivated in astronomy applications, since this parameter may fluctuate over a large spatiotemporal scale. By appealing to the central limit theorem, two families of distributions are constructed, associated with additive and multiplicative random processes. We discuss in detail these distributions, we study their asymptotic behavior, and compute their moments. We briefly discuss their possible use in astronomy, biology, and elsewhere.

An evaluation of common stock assessment diagnostic tools for choosing among state-space models with multiple random effects processes

State-space models have received increasing attention in fisheries stock assessments given their flexibility to incorporate multiple sources of process errors. Identifying which process errors to include is important because incorrectly including some process errors can induce bias in management quantities. Existing model selection tools commonly applied in traditional statistical catch-at-age models may not perform as well for state-space models. We evaluated the efficacy of common diagnostic tools for correctly identifying the presence, absence, and magnitude of three process errors (survival, selectivity, and natural mortality) in a simulation–estimation experiment. No model diagnostic tools could consistently identify the correct process error structure in all situations. Incorrectly attributing the process error from natural mortality to other processes, or vice versa, led to relatively large bias in management quantities. Furthermore, incorrectly including an additional source of process error in the assessment models exhibited similar performance to the correct model and generally showed unbiased estimates of management quantities; incorrectly excluding a source of process error, however, generated large biases. Thus, despite not having generally reliable model diagnostic tools for state-space assessments, practitioners should err on the side of using overly complex models, except for natural mortality unless there is external corroborating evidence of changing natural mortality.

A Reliability Evaluation Method for Gamma Processes with Multiple Random Effects

The multi-random-effects gamma process has a better characterization effect for degraded data with individual differences. In this paper, a reliability evaluation method for gamma stochastic processes with multiple random effects is studied. The mathematical model of multiple random effects gamma process was established. The model parameters estimation method was established based on the Bayesian approach. The prior distribution acquisition method was discussed, and the parameters of the multiple randomeffects gamma process were estimated by the MCMC-Gibbs method. The correctness of the model and method was verified by numerical simulation, the influence of algorithm parameters on the algorithm solving process was studied. In the fourth part, the reliability of aviation hydraulic rotary joints was studied by using multiple random effects gamma processes.

Bandwidth abstraction with the end-to-end latency-bounded reliability provisioning based on martingale theory

The provisioning of reliability regard to end-to-end (E2E) latency is crux for Ultra-Reliable Low Latency Communications (URLLCs). Based on martingale theory and stochastic network calculus, an E2E latency-bounded reliability analysis framework is proposed for the multi-hop system, where the arrival traffic possesses burstiness and the services provided by nodes are heterogeneous. The Wald martingales of arrival and service processes are constructed respectively, which contribute to reveal the influence of multiple random processes entangled with each other on the E2E latency. Leveraging the Doob maximum inequality of martingales and the features of moment generation functions, a tight upper bound of the unreliability regard to E2E latency is captured. A martingale parameter, named as tandem service descriptor, is proposed to embody the features of the tandem service mode provided for the specifically bursty traffic. The bandwidth abstraction algorithm is designed, which decouples the E2E latency-bounded reliability requirement as the demanded bandwidth of each node. An instantiation scheme of service rates is proposed to conduct the dynamical transmission power allocation according to the access states and channel characteristics in the wireless access network.

A random multiplicative model of Piéron’s law and choice reaction times

We present a random multiplicative model with additive noise of human reaction/response times based on the power-law function, Piéron’s law. We study the role of weak additive noise in two different scenarios: in the first case, the multiplicative model describes the differences between simple, and two-choice reaction times in Piéron’s law. In the second case, we investigate how choice reaction times depend on the transfer of information in neurons. A transition is found at 0.5 bits due to weak additive noise. Reaction times follow an U-shaped function that lead to both anti-Hick’s and Hick’s effects. We discuss the implications of random multiplicative processes, and minimum transfer of information in decision making, and neural control.

Measuring and Testing Mutual Dependence of Multivariate Functional Data

Abstract This paper considers new measures of mutual dependence between multiple multivariate random processes representing multidimensional functional data. In the case of two processes, the extension of functional distance correlation is used by selecting appropriate weight function in the weighted distance between characteristic functions of joint and marginal distributions. For multiple random processes, two measures are sums of squared measures for pairwise dependence. The dependence measures are zero if and only if the random processes are mutually independent. This property is used to construct permutation tests for mutual independence of random processes. The finite sample properties of these tests are investigated in simulation studies. The use of the tests and the results of simulation studies are illustrated with an example based on real data.

Blunt extension of discrete universal multifractal cascades: development and application to downscaling

ABSTRACT Scale issues are ubiquitous in geosciences. Because of their simplicity and intuitiveness, and despite strong limitations, notably its non-stationarity features, discrete random multiplicative cascade processes are very often used to address these scale issues. A novel approach based on the parsimonious framework of Universal Multifractals (UM) is introduced to tackle this issue while preserving the simple structure of discrete cascades. It basically consists in smoothing at each cascade step the random multiplicative increments with the help of a geometric interpolation over a moving window. The window size enables to introduce non-conservativeness in the simulated fields. It is established theoretically,] and numerically confirmed, that the simulated fields also exhibit a multifractal behaviour with expected features. It is shown that such an approach remains valid over a limited range of UM parameters. Finally, we test downscaling of rainfall fields with the help of this blunt discrete cascade process, and we discuss challenges for future developments.

Kernel density approach to error estimation of MF-DFA measures on time series

MF-DFA is a widely used method for the extraction of multifractal patterns that may appear in time series. Current techniques for determining the range of scales employed in the MF-DFA procedure are mainly empirical and in some cases require the researcher to visually select such range, so multifractal measures often differ among experiments, and the estimation of errors in such measures is often disregarded. This problem also represents a serious obstacle for incorporating MF-DFA in automated processes. In this paper, we present a Kernel Density based method that provides metrics for errors on the estimations of the multifractal spectrum obtained by MF-DFA, and gives the probabilities of errors being inside a certain range for each moment considered in the spectrum. Our approach makes use of Kernel Density Estimation in combination with a variant of the Theil–Sen estimator to compute MF-DFA exponents. We tested our technique on simulated and real-world time series. Comparison between this approach and the conventional method applied to deterministic and random multiplicative processes demonstrates robustness to spurious estimations of generalized fractal dimensions in MF-DFA.

Resource allocation for hybrid energy powered cloud radio access network with battery leakage

This study proposes a resource allocation policy for hybrid energy powered cloud radio access network with battery leakage. To optimise the network utility, the authors first formulate a network utility maximisation problem while jointly considering multiple random processes, which include energy harvesting, data arrival, wireless channel condition, and grid energy prices. To tackle this problem, they exploit the Lyapunov optimisation technique to develop an online dynamic resource allocation framework, which contains four subproblems, i.e. data admission, hybrid energy management, power allocation and route scheduling. Based on the solutions of these subproblems, a network utility optimisation resource allocation (ORA) algorithm is proposed. Specifically, the ORA algorithm only needs to track the current system states without requiring a prior knowledge about channel and energy conditions. Theoretical performance analyses and simulation results verify that the proposed algorithm can achieve close-to-optimal utility with bounded data buffer and battery capacity.

Quantile regression for functional partially linear model in ultra-high dimensions

Quantile regression for functional partially linear model in ultra-high dimensions is proposed and studied. By focusing on the conditional quantiles, where conditioning is on both multiple random processes and high-dimensional scalar covariates, the proposed model can lead to a comprehensive description of the scalar response. To select and estimate important variables, a double penalized functional quantile objective function with two nonconvex penalties is developed, and the optimal tuning parameters involved can be chosen by a two-step technique. Based on the difference convex analysis (DCA), the asymptotic properties of the resulting estimators are established, and the convergence rate of the prediction of the conditional quantile function can be obtained. Simulation studies demonstrate a competitive performance against the existing approach. A real application to Alzheimer’s Disease Neuroimaging Initiative (ADNI) data is used to illustrate the practicality of the proposed model.

Power-law Exponent in Multiplicative Langevin Equation with Temporally Correlated Noise

Power-law distributions are ubiquitous in nature. Random multiplicative processes are a basic model for the generation of power-law distributions. It is known that, for discrete-time systems, the power-law exponent decreases as the autocorrelation time of the multiplier increases. However, for continuous-time ystems, it has not yet been elucidated as to how the temporal correlation affects the power-law behavior. Herein, we have analytically investigated a multiplicative Langevin equation with colored noise. We show that the power-law exponent depends on the details of the multiplicative noise, in contrast to the case of discrete-time systems.

Power-Law Distributions from Sigma-Pi Structure of Sums of Random Multiplicative Processes

We introduce a simple growth model in which the sizes of entities evolve as multiplicative random processes that start at different times. A novel aspect we examine is the dependence among entities. For this, we consider three classes of dependence between growth factors governing the evolution of sizes: independence, Kesten dependence and mixed dependence. We take the sum X of the sizes of the entities as the representative quantity of the system, which has the structure of a sum of product terms (Sigma-Pi), whose asymptotic distribution function has a power-law tail behavior. We present evidence that the dependence type does not alter the asymptotic power-law tail behavior, nor the value of the tail exponent. However, the structure of the large values of the sum X is found to vary with the dependence between the growth factors (and thus the entities). In particular, for the independence case, we find that the large values of X are contributed by a single maximum size entity: the asymptotic power-law tail is the result of such single contribution to the sum, with this maximum contributing entity changing stochastically with time and with realizations.

Non-stationary random vibration analysis of structures under multiple correlated normal random excitations

An algorithm that integrates Karhunen-Loeve expansion (KLE) and the finite element method (FEM) is proposed to perform non-stationary random vibration analysis of structures under excitations, represented by multiple random processes that are correlated in both time and spatial domains. In KLE, the auto-covariance functions of random excitations are discretized using orthogonal basis functions. The KLE for multiple correlated random excitations relies on expansions in terms of correlated sets of random variables reflecting the cross-covariance of the random processes. During the response calculations, the eigenfunctions of KLE used to represent excitations are applied as forcing functions to the structure. The proposed algorithm is applied to a 2DOF system, a 2D cantilever beam and a 3D aircraft wing under both stationary and non-stationary correlated random excitations. Two methods are adopted to obtain the structural responses: a) the modal method and b) the direct method. Both the methods provide the statistics of the dynamic response with sufficient accuracy. The structural responses under the same type of correlated random excitations are bounded by the response obtained by perfectly correlated and uncorrelated random excitations. The structural response increases with a decrease in the correlation length and with an increase in the correlation magnitude. The proposed methodology can be applied for the analysis of any complex structure under any type of random excitation.

Recurrent hierarchical patterns and the fractal distribution of fossil localities

Abstract Understanding the spatial structure of fossil localities is critical for interpreting Earth system processes based on their geographic distribution. Coordinates of marine and terrestrial sites in the conterminous United States for 17 time bins were analyzed using point pattern statistics. Lacunarity analysis shows that the spatial distributions of sites are fractal for almost every studied interval, indicating that clumping of localities occurs at multiple scales. Random hierarchical multiplicative processes provide a theoretical null model for the distribution of collecting sites, consistent with their occurrence being a complex product of numerous biological, geological, and anthropogenic processes acting at many spatial and temporal scales. Mechanistic models for the formation, preservation, and exposure of fossil localities and other geologic entities can be tested using point pattern and related spatial statistics.

Power law in random multiplicative processes with spatio-temporal correlated multipliers

It is well known that random multiplicative processes generate power-law probability distributions. We study how the spatio-temporal correlation of the multipliers influences the power-law exponent. We investigate two sources of the time correlation: the local environment and the global environment. In addition, we introduce two simple models through which we analytically and numerically show that the local and global environments yield different trends in the power-law exponent.

Non-Gaussian signatures and collective effects in charge noise affecting a dynamically decoupled qubit

The effects of a collection of classical two-level charge fluctuators on the coherence of a dynamically-decoupled qubit are studied. Distinct dynamics are found at different qubit working positions. Exact analytical formulae are derived at pure dephasing and approximate solutions are found at the general working position, for weakly- and strongly-coupled fluctuators. Analysis of these solutions, combined with numerical simulations of the multiple random telegraph processes, reveal the scaling of the noise with the number of fluctuators and the number of control pulses, as well as dependence on other parameters of the qubit-fluctuators system. These results can be used to determine potential microscopic models for the charge environment by performing noise spectroscopy.

Reliability and maintenance models for a dependent competing-risk system with multiple time-scales

In this article, reliability and maintenance models are developed for systems subjected to dependent competing risks and multiple time-scales. Multiple degradation processes and random shocks involve in system failure. The random shocks cause sudden increment jumps in the degradation processes. We use copula method to fit the dependent relationship among various degradation processes. We also study the impact of multiple random usage processes (time-scales) on reliability measures of systems subjected to competing risks. A time-based maintenance policy is proposed to maximize the availability of such systems. The time interval between two successive preventive maintenances and maximum number of preventive maintenances before replacement is found so that the availability of the system is maximized.

Emergence of Heavy-Tailed Distributions in a Random Multiplicative Model Driven by a Gaussian Stochastic Process

We consider a random multiplicative stochastic process with multipliers given by the exponential of a Brownian motion. The positive integer moments of the distribution function can be computed exactly, and can be represented as the grand partition function of an equivalent lattice gas with attractive 2-body interactions. The numerical results for the positive integer moments display a sharp transition at a critical value of the model parameters, which corresponds to a phase transition in the equivalent lattice gas model. The shape of the terminal distribution changes suddenly at the critical point to a heavy-tailed distribution. The transition can be related to the position of the complex zeros of the grand partition function of the lattice gas, in analogy with the Lee, Yang picture of phase transitions in statistical mechanics. We study the properties of the equivalent lattice gas in the thermodynamical limit, which corresponds to the continuous time limit of the random multiplicative model, and derive the asymptotics of the approach to the continuous time limit. The results can be generalized to a wider class of random multiplicative processes, driven by the exponential of a Gaussian stochastic process.

Non-Gaussianity effects in petrophysical quantities

It has been proved that there are many indicators (petrophysical quantities) for the estimation of petroleum reservoirs. The value of information contained in each indicator is yet to be addressed. In this work, the most famous and applicable petrophysical quantities for a reservoir, which are the gamma emission (GR), sonic transient time (DT), neutron porosity (NPHI), bulk density (RHOB), and deep induced resistivity (ILD), have been analyzed in order to characterize a reservoir. The implemented technique is the well-logging method. Based on the log-normal model defined in random multiplicative processes, the probability distribution function (PDF) for the data sets is described. The shape of the PDF depends on the parameter λ2 which determines the efficiency of non-Gaussianity. When non-Gaussianity appears, it is a sign of uncertainty and phase transition in the critical regime. The large value and scale-invariant behavior of the non-Gaussian parameter λ2 is an indication of a new phase which proves adequate for the existence of petroleum reservoirs. Our results show that one of the indicators (GR) is more non-Gaussian than the other indicators, scale wise. This means that GR is a continuously critical indicator. But by moving windows with various scales, the estimated λ2 shows that the most appropriate indicator for distinguishing the critical regime is ILD, which shows an increase at the end of the measured region of the well.