Theory Of Random Point Processes Research Articles

A closure for the master equation starting from the dynamic cavity method

We consider classical spin systems evolving in continuous time with interactions given by a locally tree-like graph. Several approximate analysis methods have earlier been reported based on the idea of Belief Propagation / cavity method. We introduce a new such method which can be derived in a more systematic manner using the theory of Random Point Processes. Within this approach, the master equation governing the system’s dynamics is closed via a set of differential equations for the auxiliary cavity probabilities. The numerical results improve on the earlier versions of the closure on several important classes of problems. We re-visit here the cases of the Ising ferromagnet and the Viana–Bray spin-glass model.

Statistical Energy Analysis (SEA) based approaches for the mid- and high-frequency vibro-acoustic analysis of complex systems

This paper will present methods for predicting the statistics of the frequency response functions of complex random vibro-acoustic systems. The statistical properties include the ensemble mean, the ensemble variance, level crossing rates, extreme values, and quefrencies. The approach is based on a combination of statistical energy analysis (SEA), the finite element method, random point process theory, and random matrix theory. In the mid frequency range the finite element method is used to model components that have a low modal density, while other components are modelled as SEA subsystems. The coupling between the finite element components and the SEA subsystems is effected using the diffuse field reciprocity principle, which was developed some years ago. The blocked modes of the SEA subsystems are assumed to have natural frequency statistics that are governed by the Gaussian Orthogonal Ensemble, and this enables the statistical properties of the response to be found without any need for Monte Carlo simulations. The use of SEA leads to a relatively low number of degrees of freedom used in the model, and thus the approach is numerically efficient well suited to the design stage, where many design options may be explored.

Frequency response function statistics of diffuse systems

Vibro-acoustic analysis at high frequencies is challenging due to a large sensitivity to spatial variations and short acoustical wavelengths, rendering deterministic methods expensive. At these frequencies, the wave field is usually assumed diffuse. A realization of a diffuse field can be obtained considering that the mode shapes of the system are Gaussian random fields, and that its squared eigenfrequency spacings distribution conforms to the Gaussian orthogonal ensemble (GOE) eigenvalue spacings distribution. These diffuse field properties were for example used to extend statistical energy analysis (SEA) towards energetic variance prediction. However, energetic methods such as SEA lack the propagation of phase information, which means that, e.g., time-domain reconstruction is not possible. In this contribution, instead of total energies, expressions for the ensemble average and the cross-frequency covariance of frequency response functions are presented. These expressions are obtained from the generalized definition of a diffuse field, treating the eigenfrequencies of a diffuse subsystem as a collection of points randomly located on the frequency axis, making the analysis amenable to random point process theory. These expressions are numerically validated by comparison with computationally costly detailed models where random wave scatterers are modeled explicitly.

From random point processes to hierarchical cavity master equations for stochastic dynamics of disordered systems in random graphs: Ising models and epidemics.

We start from the theory of random point processes to derive n-point coupled master equations describing the continuous dynamics of discrete variables in random graphs. These equations constitute a hierarchical set of approximations that generalize and improve the cavity master equation (CME), a recently obtained closure for the usual master equation representing the dynamics. Our derivation clarifies some of the hypotheses and approximations that originally led to the CME, considered now as the first order of a more general technique. We tested the scheme in the dynamics of three models defined over diluted graphs: the Ising ferromagnet, the Viana-Bray spin-glass, and the susceptible-infectious-susceptible model for epidemics. In the first two, the equations perform similarly to the best-known approaches in literature. In the latter, they outperform the well-known pair quenched mean-field approximation.

Accident risk of road and weather conditions on different road types

This study was designed to investigate the relative accident risk of different road weather conditions and combinations of conditions. The study applied a recently developed method which is based on the notion of Palm probability, originating in the theory of random point processes, which in this case corresponds to picking a random vehicle from the traffic. The method consists of calculating the Palm distribution of different conditions and comparing it with the distribution of the same conditions as seen by the accidents. The condition affects the accident risk statistically, when these two distributions differ. The study included all police reported single- and multi-vehicle accidents (N = 10,646) occurring on 43 main roads in Finland during the years 2014–2016. A major contribution of this paper is the demonstration of the method on national scale by using estimated hourly traffic volumes on road segments instead of measured ones, which would have been available for few roads only. Accident risks are commonly examined in relation to traffic volume. This paper includes the speed of the traffic and thus, the paper examines accident risk in relation to the time spent on the road segment in certain conditions. The hour-level weather and road condition data per segment were obtained from nearby road weather stations. The relative accident risks were increased for poor road weather conditions; however, they were highest for icy rain and slippery and very slippery road conditions. When comparing the relative accident risk based on road type, the results showed that the risk in poor weather and road conditions was higher on motorways compared to two-lane and multiple-lane roads even though the overall risk was lower on motorways. Furthermore, the corresponding relative accident risks were generally higher for single-vehicle accidents compared to multi-vehicle accidents.

The effect of generalised force correlations on the response statistics of a harmonically driven random system

If the physical properties of a structural component are sufficiently random then the statistical distribution of the natural frequencies and mode shapes tends to a universal distribution associated with the Gaussian Orthogonal Ensemble (GOE) of random matrices. Previous work has exploited this result to yield expressions for the relative variance of the energy of the response of a random system to harmonic excitation. The derivation of these expressions employed random point process theory, and in the theoretical development it was assumed that the modal generalised forces were uncorrelated. Although this assumption is often valid, there are cases in which correlations between the generalised forces can significantly affect the response variance, and in the present work the existing theory is extended to include correlations of this type. The extended theory is applicable to both single frequency responses and to band average responses, and the developed closed form expressions are validated by comparison with direct simulations for a random plate structure.

Cavity master equation for the continuous time dynamics of discrete-spin models.

We present an alternate method to close the master equation representing the continuous time dynamics of interacting Ising spins. The method makes use of the theory of random point processes to derive a master equation for local conditional probabilities. We analytically test our solution studying two known cases, the dynamics of the mean-field ferromagnet and the dynamics of the one-dimensional Ising system. We present numerical results comparing our predictions with Monte Carlo simulations in three different models on random graphs with finite connectivity: the Ising ferromagnet, the random field Ising model, and the Viana-Bray spin-glass model.

INCREASE IN INFORMATIVE VALUE OF RAILWAY TRACK MAINTENANCE ASSESSMENT

[For the English abstract and full text of the article please see the attached PDF-File (English version follows Russian version)].ABSTRACT Using the methods of statistical analysis based on the theory of random process emissions, the theory of random point processes, the method of least squares, the method of least modules, filtering algorithms, the choice of a type of a theoretical function and its parameters that allow reducing the error of the measured characteristics of defects and deviations of the track from the design position, a mathematical model is formulated that provides an increase in the informative value of the assessment of the actual state of the railway track. High information value is needed primarily for formation of adequate control actions on the track in order to ensure traffic safety. Keywords: railway track, measured parameters, information-measuring system, defect detection, statistical analysis, theory of random processes, mathematical model.

Transient response of structures with uncertain properties to nonlinear shock loading

A method is presented to predict the transient response of a structure at the driving point following an impact or a shock loading. The displacement and the contact force are calculated solving the discrete convolution between the impulse response and the contact force itself, expressed in terms of a nonlinear Hertzian contact stiffness. Application of random point process theory allows the calculation of the impulse response function from knowledge of the modal density and the geometric characteristics of the structure only. The theory is applied to a wide range of structures and results are experimentally verified for the case of a rigid object hitting a beam, a plate, a thin and a thick cylinder and for the impact between two cylinders. The modal density of the flexural modes for a thick slender cylinder is derived analytically. Good agreement is found between experimental, simulated and published results, showing the reliability of the method for a wide range of situations including impacts and pyroshock applications.

Shock dynamics of random structures

Predicting the response of a structure following an impact is of interest in situations where parts of a complex assembly may come into contact. Standard approaches are based on the knowledge of the impulse response function, requiring the knowledge of the modes and the natural frequencies of the structure. In real engineering structures the statistics of higher natural frequencies follows those of the Gaussian Orthogonal Ensemble, this allows the application of random point process theory to get a mean impulse response function by the knowledge of the modal density of the structure. An ensemble averaged time history for both the response and the impact force can be predicted. Once the impact characteristics are known in the time domain, a simple Fourier Transform allows the frequency range of the impact excitation to be calculated. Experimental and numerical results for beams, plates, and cylinders are presented to confirm the validity of the method.

Scaling for the ensemble statistics prediction of a random system subjected to harmonic excitations

This paper is concerned with the ensemble statistics of the dynamic responses of a random system subjected to harmonic excitations. Random point process theory is employed to derive general scaling laws with the Gaussian orthogonal ensemble assumption about the system natural frequencies. A scaled model is built to simulate the high-frequency vibrations of the original system. Specific forms of the scaling laws are presented for a mass-loaded plate regarding the scaling factors for the structural parameters. The ensemble statistics predicted from the scaled model are compared favorably with those obtained from the original system.

Influence of membrane properties on spike synchronization in neurons: theory and experiments

In the brain spike synchronization in neurons is involved in information transfer and certain forms of dysfunction. The theory of random point processes was used to relate the statistical properties of input point processes to synchronization between firing in neurons viewed as threshold devices. Derived analytical relations describe normalized synchronization in the case of shared input with balanced excitation and inhibition. For neuronal models with unbalanced shared input and spike generating Hodgkin-Huxley type conductances, the theory satisfactorily describes the temporal dependence of spike synchronization on the delay between spikes. Computer generated stochastic stimulus current was used to stimulate motoneurons in turtle spinal cord slices. Theory was able to approximate the temporal dependence of spike synchronization on the delay between spikes when the membrane time constant and the relative spike threshold measured were used in calculations. In agreement with the theoretical prediction, normalized spike synchrony was reduced when the threshold for spike generation was lowered by injection of steady depolarizing bias current. In spinal motoneurons the relative spike threshold can be lowered by a persistent inward current facilitated by activation of certain metabotropic transmitter receptors. After induction of this inward current spike synchronization was reduced several times. It is suggested that downregulation of the persistent inward current in motoneurons by disruption of brainstem modulatory systems, as in Parkinson disease, can facilitate tremor due to the increased synchrony between motoneurons.

The ensemble statistics of the energy of a random system subjected to harmonic excitation

This paper is concerned with the ensemble statistics of the response to harmonic excitation of a single dynamic system such as a plate or an acoustic volume. Random point process theory is employed, and various statistical assumptions regarding the system natural frequencies are compared, namely: (i) Poisson natural frequency spacings, (ii) statistically independent Rayleigh natural frequency spacings, and (iii) natural frequency spacings conforming to the Gaussian orthogonal ensemble (GOE). The GOE is found to be the most realistic assumption, and simple formulae are derived for the variance of the energy of the system under either point loading or rain-on-the-roof excitation. The theoretical results are compared favourably with numerical simulations and experimental data for the case of a mass loaded plate.

Influence of membrane properties on spike synchronization in neurons: theory and experiments

In the brain spike synchronization in neurons is involved in information transfer and certain forms of dysfunction. The theory of random point processes was used to relate the statistical properties of input point processes to synchronization between firing in neurons viewed as threshold devices. Derived analytical relations describe normalized synchronization in the case of shared input with balanced excitation and inhibition. For neuronal models with unbalanced shared input and spike generating Hodgkin–Huxley type conductances, the theory satisfactorily describes the temporal dependence of spike synchronization on the delay between spikes.Computer generated stochastic stimulus current was used to stimulate motoneurons in turtle spinal cord slices. Theory was able to approximate the temporal dependence of spike synchronization on the delay between spikes when the membrane time constant and the relative spike threshold measured were used in calculations. In agreement with the theoretical prediction, normalized spike synchrony was reduced when the threshold for spike generation was lowered by injection of steady depolarizing bias current.In spinal motoneurons the relative spike threshold can be lowered by a persistent inward current facilitated by activation of certain metabotropic transmitter receptors. After induction of this inward current spike synchronization was reduced several times. It is suggested that downregulation of the persistent inward current in motoneurons by disruption of brainstem modulatory systems, as in Parkinson disease, can facilitate tremor due to the increased synchrony between motoneurons.

Random matrix theory, random point process theory, and natural frequency statistics

There has been much recent interest in the application of random matrix theory to the dynamics of uncertain engineering structures. In particular, it has been found experimentally and numerically that the results arising from the Gaussian orthogonal ensemble (GOE) are applicable to natural frequency spacing statistics. The occurrence of natural frequencies can also be viewed as a random point process, and although there is much in common between random point process theory and random matrix theory, the two subjects appear to have been developed independently. It is shown here that the two approaches employ very similar mathematical functions, albeit under a different name: For example, the distribution function (point process theory) corresponds to the n-point correlation function (random matrix theory), and the cumulant function corresponds to the n-level cluster function. By recognizing this similarity, it is possible to apply a number of established results in random point process theory to the statistics of the eigenvalues of a random matrix. This leads to new insights into the statistics of natural frequency spacings, and helps to explain why the Wigner surmise (which states that the spacings have a Rayleigh distribution) is applicable to a much wider class of matrix than the GOE.

Levy diffusion in a force field, huber relaxation kinetics, and nonequilibrium thermodynamics: H theorem for enhanced diffusion with Levy white noise

A characteristic functional approach is suggested for Levy diffusion in disordered systems with external force fields. We study the overdamped motion of an ensemble of independent particles and assume that the force acting upon one particle is made up of two additive components: a linear term generated by a harmonic potential and a second term generated by the interaction with the disordered system. The stochastic properties of the second term are evaluated by using Huber's approach to complex relaxation [Phys. Rev. B 31, 6070 (1985)]. We assume that the interaction between a moving particle and the environment can be expressed by the contribution of a large number of relaxation channels, each channel having a very small probability of being open and obeying Poisson statistics. Two types of processes are investigated: (a) Levy diffusion with static disorder for which the fluctuations of the random force are frozen and last forever and (b) diffusion with strong dynamic disorder and independent Levy fluctuations (Levy white noise). In both cases we show that the probability distribution of the position of a diffusing particle tends towards a stationary nonequilibrium form. The characteristic functional of concentration fluctuations is evaluated in both cases by using the theory of random point processes. For large times the fluctuations of the concentration field are stationary and the corresponding probability density functional can be evaluated analytically. In this limit the fluctuations depend on the distribution of the total number of particles but are independent of the initial positions of the particles. We show that the logarithm of the stationary probability functional plays the role of a nonequilibrium thermodynamic potential, which has a structure similar to the Helmholtz free energy in equilibrium thermodynamics: it is made up of the sum of an energetic component, depending on the external mechanical potential, and of an entropic component, depending on the concentration field. We show that the conditions for the existence and stability of the nonequilibrium steady state, which emerges for large times, can be expressed in terms of the stochastic potential. For Levy white noise the average concentration field can be expressed as the solution of a fractional Fokker-Planck equation. We show that the stochastic potential is a Lyapunov function of the fractional Fokker-Planck equation, which ensures that all transient solutions for the average concentration field tend towards a unique stationary form.

Influence of Temporal Correlation of Synaptic Input on the Rate and Variability of Firing in Neurons

The spike trains that transmit information between neurons are stochastic. We used the theory of random point processes and simulation methods to investigate the influence of temporal correlation of synaptic input current on firing statistics. The theory accounts for two sources for temporal correlation: synchrony between spikes in presynaptic input trains and the unitary synaptic current time course. Simulations show that slow temporal correlation of synaptic input leads to high variability in firing. In a leaky integrate-and-fire neuron model with spike afterhyperpolarization the theory accurately predicts the firing rate when the spike threshold is higher than two standard deviations of the membrane potential fluctuations. For lower thresholds the spike afterhyperpolarization reduces the firing rate below the theory’s predicted level when the synaptic correlation decays rapidly. If the synaptic correlation decays slower than the spike afterhyperpolarization, spike bursts can occur during single broad peaks of input fluctuations, increasing the firing rate over the prediction. Spike bursts lead to a coefficient of variation for the interspike intervals that can exceed one, suggesting an explanation of high coefficient of variation for interspike intervals observed in vivo.

Fractional diffusion equation on fractals: Self-similar stationary solutions in a force field derived from a logarithmic potential

The fractional diffusion equation of Giona and Roman is extended for diffusion processes in force fields. For Euclidean diffusion self-similar stationary concentration profiles may exist provided that the force fields derive from logarithmic potentials. The physically significant potentials obey a non-linear differential equation which can be integrated analytically. By using these potentials a fractional Euclidean diffusion equation is derived. By using the heuristic method of Giona and Roman this equation is extended for diffusion on fractals. The self-similar stationary concentration profile is computed in the fractal case and a relationship among the different scaling exponents is derived. If the stationary concentration profile corresponds to thermodynamic equilibrium then a generalized Einstein equation may be derived: it establishes a connection between the fractal diffusion coefficient and the generalized mobility which depends on the anomalous diffusion exponent but is independent of the fractal dimension of the system. The fluctuations of the concentration profile for finite systems of non-interacting particles are analysed by using the theory of random point processes. A statistical correlation of the positions of different particles exists which is due to the finite number of particles; it vanishes as the total number of particles tends to infinity. In this context a fractal analogue of the thermodynamic limit is introduced.

Maximum likelihood identification of neural point process systems.

Using the theory of random point processes, a method is presented whereby functional relationships between neurons can be detected and modeled. The method is based on a point process characterization involving stochastic intensities and an additive rate function model. Estimates are based on the maximum likelihood (ML) principle and asymptotic properties are examined in the absence of a stationarity assumption. An iterative algorithm that computes the ML estimates is presented. It is based on the expectation/maximization (EM) procedure of Dempster et al. (1977) and makes ML identification accessible to models requiring many parameters. Examples illustrating the use of the method are also presented. These examples are derived from simulations of simple neural systems that cannot be identified using correlation techniques. It is shown that the ML method correctly identifies each of these systems.

Use of time-delay correlators and wave-shaping techniques in the statistical analysis of Barkhausen pulses

A new class of digital correlators and other related instruments have been developed recently for photoelectron statistics applications. In this work we explore the possibility of using these low cost devices for analyzing trains of Barkhausen pulses. We also outline the applications of the theory of random point processes (specifically, self-exciting point processes) to Barkhausen noise. Finally, we present a new technique for exposing ferromagnetic samples to a linearly variable induction field. Our technique is fully digital and offers several advantages in terms of accuracy and reproducibility of control over the conventional tape-recorder method.