State Variance Research Articles

Dynamically selected steady states and criticality in non-reciprocal networks

We consider a simple neural network model, evolving via non-linear coupled stochastic differential equations, where neural couplings are random Gaussian variables with non-zero mean and arbitrary degree of reciprocity. Using a path-integral approach, we analyse the dynamics, averaged over the network ensemble, in the thermodynamic limit. Our results show that for any degree of reciprocity in the couplings, two types of criticality emerge, corresponding to ferromagnetic and spin-glass order, respectively. The critical lines separating the disordered from the ordered phases is consistent with spectral properties of the coupling matrix, as derived from random matrix theory. As the non-reciprocity (or asymmetry) in the couplings increases, both ordered phases diminish in size, ultimately resulting in the disappearance of the spin-glass phase when the couplings become anti-symmetric. We investigate non-fixed point steady-state solutions for uncorrelated interactions. For such solutions the time-lagged correlation function evolves according to a gradient-descent dynamics on a potential, which depends on the stationary variance. Our analysis shows that in the spin-glass region, the variance dynamically selected by the system leads the correlation function to evolve on the separatrix curve, limiting different realizable steady states, whereas in the ferromagnetic region, a fixed point solution is selected as the only realizable steady state. In the spin-glass region, stationary solutions are unstable against perturbations that break time-translation invariance, indicating chaotic behaviour in large single network instances. Numerical analysis of Lyapunov exponents confirms that chaotic behaviour emerges throughout the spin-glass region, for any value of the coupling correlations. While negative correlations increase the strength of chaos, positive ones reduce it, with chaos disappearing for reciprocal (i.e. symmetric) couplings, where marginal stability is attained. On the other hand, in finite size non-reciprocal networks, fixed points and limit cycles can arise in the spin-glass region, especially close to the critical line. Finally, we show that chaos is suppressed when the strength of external noise exceeds a certain threshold. Intriguing analogies between chaotic phases in non-equilibrium systems and spin-glass phases in equilibrium are put forward.

Prediction of distributions of unit prices for real estate properties on the basis of the characteristics of PSI-processes

Real estate market price forecasting is always in the focus of interests of scientists-economists, market analysts, market participants (sellers and buyers), marketing services of building complex enterprises, analysts working for banks and insurance companies and investors. Under present day conditions, the price behavior of properties on real estate markets takes especially important meaning subject to the influence of such factors as changes in the structure of household incomes, changes in mortgage rates and their availability, dynamic changes in the macroeconomic and other external socio-economic and political type factors. However, unlike the financial and securities markets, the real estate market is always characterized by a delayed reaction to external perturbations, often up to half a year, which allows us to hope for an appropriate construction of forecasts, at least in time for the delayed reaction. Traditional autoregressive forecasting methods are characterized by rapidly increasing forecast variance, because they assume a factor of stochastic volatility. This paper proposes a model and method of forecast construction based on stochastic processes of the “Poisson random index” having a short time for reaching a stationary stable variance. The model is based on the “principle of replacements” of current prices with new ones. We analyze in detail an example of the application of the “principle of replacements” for construction of price forecasts on secondary residential real estate in St. Petersburg which is based on data of four-year observations of offer prices.

Stationary Jackknife

Variance estimation is an important aspect in statistical inference, especially in the dependent data situations. Resampling methods are ideal for solving this problem since these do not require restrictive distributional assumptions. In this paper, we develop a novel resampling method in the Jackknife family called the stationary jackknife. It can be used to estimate the variance of a statistic in the cases where observations are from a general stationary sequence. Unlike the moving block jackknife, the stationary jackknife computes the jackknife replication by deleting a variable length block and the length has a truncated geometric distribution. Under appropriate assumptions, we can show the stationary jackknife variance estimator is a consistent estimator for the case of the sample mean and, more generally, for a class of nonlinear statistics. Further, the stationary jackknife is shown to provide reasonable variance estimation for a wider range of expected block lengths when compared with the moving block jackknife by simulation.

Density regulation amplifies environmentally induced population fluctuations.

Density-dependent regulation is ubiquitous in population dynamics, and its potential interaction with environmental stochasticity complicates the characterization of the random component of population dynamics. Yet, this issue has not received attention commensurate with its relevance for descriptive and predictive modeling of population dynamics. Here we use a Bayesian modeling approach to investigate the contribution of density regulation to population variability in stochastic environments. We analytically derive a formula linking the stationary variance of population abundance/density under Gompertz regulation in a stochastic environment with constant variance to the environmental variance and the strength of density feedback, to investigate whether and how density regulation affects the stationary variance. We examine through simulations whether the relationship between stationary variance and density regulation inferred analytically under the Gompertz model carries over to the Ricker model, widely used in population dynamics modeling. The analytical decomposition of the stationary variance under stochastic Gompertz dynamics implies higher variability for strongly regulated populations. Simulation results demonstrate that the pattern of increasing population variability with increasing density feedback found under the Gompertz model holds for the Ricker model as well, and is expected to be a general phenomenon with stochastic population models. We also analytically established and empirically validated that the square of the autoregressive parameter of the Gompertz model in AR(1) form represents the proportion of stationary variance due to density dependence. Our results suggest that neither environmental stochasticity nor density regulation can alone explain the patterns of population variability in stochastic environments, as these two components of temporal variation interact, with a tendency for density regulation to amplify the magnitude of environmentally induced population fluctuations. This finding has far-reaching implications for population viability. It implies that intense intra-specific resource competition increases the risk of environment-driven population collapse at high density, making opportune harvesting a sensible practice for improving the resistance of managed populations such as fish stocks to environmental perturbations. The separation of density-dependent and density-independent processes will help improve population dynamics modeling, while providing a basis for evaluating the relative importance of these two categories of processes that remains a topic of long-standing controversy among ecologists.

Stochastic analysis of small-scale beams with internal and external damping

Random flexural vibrations of small-scale Bernoulli–Euler beams with internal and external damping are investigated in the paper. Such a kind of problem occurs in design and optimization of structural components of smart miniaturized electromechanical systems. Internal damping is related to the viscoelastic stress–strain law describing the beam time-dependent behavior while the external damping represents the interaction between the structure and the surrounding viscous environment. In the proposed formulation, Boltzmann superposition integral and fractional-order viscoelasticity are exploited to capture the above mentioned non-conventional phenomena. Size effects caused by long range interactions arising at small-scale are also taken into account and modeled by means of an integral nonlocal formulation based on the stress-driven approach. Moreover, in order to reproduce stochastic vibrations of small-size devices, such as resonators and sensors, the external loading are modeled by taking into account their random nature. The relevant dynamic problem is thus described by a fractional-order stochastic partial differential equation in space and time equipped with standard and non-classical boundary conditions. A proper differential eigenanalysis for the nonlocal viscoelastic bending problem is performed to find the modal shapes and closed-form expressions of power spectral density and stationary variance of displacements are provided. The proposed methodology, accounting for size-effects, hereditariness behaviors, viscous properties of surrounding environment and randomness of external loadings, is suitable to model and capture the effective behavior of miniaturized devices. The presented theoretical findings and numerical outcomes show the influence of size-effects and viscoelastic parameters on the structural response.

Forecasting Indonesian Oil, Non-Oil and Gas Import Export with Fuzzy Time Series

Indonesia is active in export and import activities. Some of the commodities traded are oil and gas, as well as food and other industrial materials. Export and import activities play a role in determining the stability of the country's economy seen from its trade balance. According to the Central Statistics Agency, Indonesia experienced a deficit of USD 864 million due to a decline in exports at the beginning of 2020. Based on the state of the trade balance, the government needs to make policies in order to maintain the stability of the Indonesian economy. The right decision-making must be supported by accurate information, therefore, through this research, the value of Indonesia's exports and imports will be forecasted in the oil and gas and non-oil and gas sectors for the next period using the Fuzzy Time Series (FTS). FTS was chosen as the forecasting method because it is able to predict free real time data with arbitrary patterns. The data used is data on the value of exports and imports of oil and gas and non-oil and gas sectors for 2011-2020. To overcome the problem of stationary data variance and reduce the error value, a Box Cox transformation will be applied. The research stages include data transformation with Box Cox, forming universe and linguistic sets, determining interval length, fuzzification, forming FLR and FLR, defuzzification and forecasting. The final forecast results estimate that exports and imports in the oil and gas sector in 2021 will decline, while for the non-oil and gas sector will fluctuate and increase from the previous year. Forecasting with Box Cox transform data is more accurate with MAPE 19.56% and RMSE 121.52 compared to forecasting with original data with MAPE 74.89% and RMSE 132.09.

COVID-19 cases and deaths in the United States follow Taylor’s law for heavy-tailed distributions with infinite variance

The spatial and temporal patterns of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) cases and COVID-19 deaths in the United States are poorly understood. We show that variations in the cumulative reported cases and deaths by county, state, and date exemplify Taylor's law of fluctuation scaling. Specifically, on day 1 of each month from April 2020 through June 2021, each state's variance (across its counties) of cases is nearly proportional to its squared mean of cases. COVID-19 deaths behave similarly. The lower 99% of counts of cases and deaths across all counties are approximately lognormally distributed. Unexpectedly, the largest 1% of counts are approximately Pareto distributed, with a tail index that implies a finite mean and an infinite variance. We explain why the counts across the entire distribution conform to Taylor's law with exponent two using models and mathematics. The finding of infinite variance has practical consequences. Local jurisdictions (counties, states, and countries) that are planning for prevention and care of largely unvaccinated populations should anticipate the rare but extremely high counts of cases and deaths that occur in distributions with infinite variance. Jurisdictions should prepare collaborative responses across boundaries, because extremely high local counts of cases and deaths may vary beyond the resources of any local jurisdiction.

On Irreversible Metropolis Sampling Related to Langevin Dynamics

There has been considerable interest in designing Markov chain Monte Carlo algorithms by exploiting numerical methods for Langevin dynamics, which includes Hamiltonian dynamics as a deterministic case. A prominent approach is Hamiltonian Monte Carlo (HMC), where a leapfrog discretization of Hamiltonian dynamics is employed. We investigate a recently proposed class of irreversible sampling algorithms, called Hamiltonian assisted Metropolis sampling (HAMS), which uses an augmented target density similarly as in HMC but involves a flexible proposal scheme and a carefully formulated acceptance-rejection scheme to achieve generalized reversibility. We show that as the step size tends to 0, the HAMS proposal satisfies a class of stochastic differential equations including Langevin dynamics as a special case. We provide theoretical results for HAMS, including algebraic properties of the acceptance probability, the stationary variance from the HAMS proposal, and the expected acceptance rate under a product Gaussian target distribution and the convergence rate under standard Gaussian. From these results, we derive default choices of tuning parameters for HAMS such that only the step size needs to be tuned in applications. Various relatively recent algorithms for Langevin dynamics are also shown to fall in the class of HAMS proposals up to negligible differences. Our numerical experiments on sampling high-dimensional latent variables confirm that the HAMS algorithms consistently achieve superior performance compared with several Metropolis-adjusted algorithms based on popular integrators of Langevin dynamics.

Two New Mathematical Equalities in the Life Table

There is a rich body of literature on equalities in the period life table, which also can be interpreted as a stationary population, and a smaller, but no less rich, body on inequalities. The latter is important because it provides information on health disparities and, like the equality literature, serves as a foundation for formal mortality analysis. We straddle both of these bodies by reconciling a known inequality such that a new mathematical equality emerges in the period life table. We then show that this new equality links life expectancy at birth (mean age at death) directly to the average lifespan of the living members of a stationary population. This linkage represents a second newly identified equality in this paper that has the potential to yield useful insights because it links the average lifespan of the living members of a stationary population directly to elements that are central to the dynamics and structure of a stationary population, life expectancy, and variance in age at death.

Global Mapping of the Nonstationary Semidiurnal Internal Tide Using Argo Data

AbstractData from Argo floats equipped with Iridium communications are used to obtain a global map of the total amplitude (or variance) of the semidiurnal internal tide at 1,000 dbar. The results are confirmed by a comparison with data from an historical collection of moored instruments. The obtained amplitude is in turn compared with the High‐Resolution Empirical Tide (HRET) model, based on satellite altimetry. While HRET only contains the stationary component, with a fixed phase difference to the astronomical tide, the present results capture the total amplitude, including the nonstationary component. We estimate the global average ratio of total (Argo) to stationary (HRET) semidiurnal internal tide variance to be 6.5, and the amplitude ratio to be 3.6. Our estimate of the stationary fraction of the semidiurnal internal tide is subject to significant uncertainties. In particular, HRET is thought to mainly represent baroclinic mode‐1 waves, while Argo data contain contributions from all modes.

Short-term mine scheduling targeting stationary grades

In short-term mine planning, mining scheduling is generally defined by designing dig-lines, allocated on benches. The mined ore will be sent to stockpiles, homogenization piles, or a concentration plant. The process to design dig-lines is usually done manually, whereby multiple simultaneous mining fronts are time-consuming and labour-intensive. The manual design of dig-lines tends to produce high variability of the grades throughout certain periods. Due to the limited time to manually multiple test dig-line design alternatives in short term planning, it is impossible to ensure production under stationary mean grades and variance. This article proposes an alternative to design short-term dig-lines, through an optimization process that joins and sequences the blocks in the block model over weeks or months, ensuring low variability of grades among periods. The methodology proposed generates multiple random paths starting at seed-points representing the locations and numbers of shovels previously selected by the mine planner. It tests multiple polygons representing a set of first dig-lines, comparing them with others, and keeping the dig-lines of low variability closer to a specific ore grade probability distribution, discarding the rest of the iterations. The process is repeated for the next dig-line. The block grades' probability distribution of all iterations is compared to a reference-grade histogram, and the iterations with the grade histogram more adherent are selected. Union-find and genetic algorithms were used to optimize the dig-lines aiming at the possible stationary grade distribution. The mean and variance of the reference model are 2.13% and 0.64%2, respectively. The mean for the automated draw dig-lines is closer to these values than the ones manually drawn. The method ensures more constant quality and quantity of ore production along a period planned, matching a target grade probability distribution. The methodology is illustrated using SiO2 values at a major iron ore mine.

Generalized Exact Scheduling: A Minimal-Variance Distributed Deadline Scheduler

In this paper, we adapt tools from optimization and control theory to characterize the optimal distributed policies in a broad range of settings without any approximation. We show that Exact Scheduling minimizes both the stationary mean and variance of the service capacity subject to strict demand and deadline requirements. For more general settings, we characterize the minimal-variance distributed policies with soft demand requirements, soft deadline requirements, or both. Moreover, we derive the Pareto-optimality condition for distributed policies that balance the variance and mean square of the service capacity. The performance of the optimal distributed policies is compared with that of the optimal centralized policy by deriving closed-form bounds. Finally, we discuss a scalable partially centralized algorithm that uses centralized information to boost performance and a method to deal with missing information on service requirements. Our finding can ultimately lead to more efficient power distribution networks and cloud computing services, which optimally match the service capacity to changing demands.

Continuous‐time multi‐type Ehrenfest model and related Ornstein–Uhlenbeck diffusion on a star graph

We deal with a continuous‐time Ehrenfest model defined over an extended star graph, defined as a lattice formed by the integers of d semiaxis joined at the origin. The dynamics on each ray are regulated by linear transition rates, whereas the switching among rays at the origin occurs according to a general stochastic matrix. We perform a detailed investigation of the transient and asymptotic behavior of this process. We also obtain a diffusive approximation of the considered model, which leads to an Ornstein–Uhlenbeck diffusion process over a domain formed by semiaxis joined at the origin, named spider. We show that the approximating process possesses a truncated Gaussian stationary density. Finally, the goodness of the approximation is discussed through comparison of stationary distributions, means, and variances.

Full Information H2 Control of Borel-Measurable Markov Jump Systems with Multiplicative Noises

This paper addresses an H2 optimal control problem for a class of discrete-time stochastic systems with Markov jump parameter and multiplicative noises. The involved Markov jump parameter is a uniform ergodic Markov chain taking values in a Borel-measurable set. In the presence of exogenous white noise disturbance, Gramian characterization is derived for the H2 norm, which quantifies the stationary variance of output response for the considered systems. Moreover, under the condition that full information of the system state is accessible to measurement, an H2 dynamic optimal control problem is shown to be solved by a zero-order stabilizing feedback controller, which can be represented in terms of the stabilizing solution to a set of coupled stochastic algebraic Riccati equations. Finally, an iterative algorithm is provided to get the approximate solution of the obtained Riccati equations, and a numerical example illustrates the effectiveness of the proposed algorithm.

First Exit-Time Analysis for an Approximate Barndorff-Nielsen and Shephard Model with Stationary Self-Decomposable Variance Process

In this paper, an approximate version of the Barndorff-Nielsen and Shephard model, driven by a Brownian motion and a L\'evy subordinator, is formulated. The first-exit time of the log-return process for this model is analyzed. It is shown that with certain probability, the first-exit time process of the log-return is decomposable into the sum of the first exit time of the Brownian motion with drift, and the first exit time of a L\'evy subordinator with drift. Subsequently, the probability density functions of the first exit time of some specific L\'evy subordinators, connected to stationary, self-decomposable variance processes, are studied. Analytical expressions of the probability density function of the first-exit time of three such L\'evy subordinators are obtained in terms of various special functions. The results are implemented to empirical S&P 500 dataset.

A Nonstationary Uncertainty Model and Bayesian Calibration of Strain-Life Models

Abstract The Coffin–Manson–Basquin–Haford (CMBH) model is a well-accepted strain-life relationship to model fatigue life as a function of applied strain. In this paper, we propose a nonstationary uncertainty model for the CMBH model, alongside a Bayesian framework for model calibration and estimation of confidence and prediction intervals. Using Inconel 617 coupon test data, we compared our approach to traditional stationary variance models. The proposed uncertainty model successfully captures the fact that the variance of strain amplitude decreases with increasing fatigue life. Additionally, a discussion on how to use the proposed Bayesian framework to compensate for the lack of data by using prior information coming from similar alloys is also presented considering Hastealloy-X and Inconel 617 coupon data.

A note on stationary bootstrap variance estimator under long-range dependence

The stationary bootstrap method is popularly used to compute the standard errors or confidence regions of estimators, generated from time processes exhibiting weakly dependent stationarity. Most previous stationary bootstrap methods have focused on studying large-sample properties of stationary bootstrap inference about a sample mean under short-range dependence. For long-range dependence, recent studies have investigated the properties of block bootstrap methods using overlapping and non-overlapping blocking techniques with fixed block lengths. However, the characteristics of a stationary bootstrap with random block lengths are less well-known under long-range dependence. We investigate the asymptotic property of a stationary bootstrap variance estimator for a sample mean under long-range dependence. Our theoretical and simulation results indicate that the stationary bootstrap method does not have n−consistency for stationary and long-range dependent time processes.

Theory of transcription bursting: stochasticity in the transcription rates

Transcription bursting creates variation among the individuals of a given population. Bursting emerges as the consequence of turning on and off the transcription process randomly. There are at least three sub-processes involved in the bursting phenomenon with different timescale regimes viz. flipping across the on–off state channels, microscopic transcription elongation events and the mesoscopic transcription dynamics along with the mRNA recycling. We demonstrate that when the flipping dynamics is coupled with the microscopic elongation events, then the distribution of the resultant transcription rates will be over-dispersed. This in turn reflects as the transcription bursting with over-dispersed non-Poisson type distribution of mRNA numbers. We further show that there exist optimum flipping rates (αC, βC) at which the stationary state Fano factor and variance associated with the mRNA numbers attain maxima. These optimum points are connected via $$ \alpha_{C} = \sqrt {\beta_{C} \left( {\beta_{C} + \gamma_{r} } \right)} $$ . Here α is the rate of flipping from the on-state to the off-state, β is the rate of flipping from the off-state to the on-state and γr is the decay rate of mRNA. When α = β = χ with zero rate in the off-state channel, then there exist optimum flipping rates at which the non-stationary Fano factor and variance attain maxima. Here $$ \chi_{C,v} \simeq {{3k_{r}^{ + } } \mathord{\left/ {\vphantom {{3k_{r}^{ + } } {2\left( {1 + k_{r}^{ + } t} \right)}}} \right. \kern-0pt} {2\left( {1 + k_{r}^{ + } t} \right)}} $$ (here $$ k_{r}^{ + } $$ is the rate of transcription purely through the on-state elongation channel) is the optimum flipping rate at which the variance of mRNA attains a maximum and $$ \chi_{C,\kappa } \simeq {{1.72} \mathord{\left/ {\vphantom {{1.72} t}} \right. \kern-0pt} t} $$ is the optimum flipping rate at which the Fano factor attains a maximum. Close observation of the transcription mechanism reveals that the RNA polymerase performs several rounds of stall-continue type dynamics before generating a complete mRNA. Based on this observation, we model the transcription event as a stochastic trajectory of the transcription machinery across these on–off state elongation channels. Each mRNA transcript follows different trajectory. The total time taken by a given trajectory is the first passage time (FPT). Inverse of this FPT is the resultant transcription rate associated with the particular mRNA. Therefore, the time required to generate a given mRNA transcript will be a random variable. For a stall-continue type dynamics of RNA polymerase, we show that the overall average transcription rate can be expressed as $$ k_{r} \simeq h_{\infty }^{ + } k_{r}^{ + } $$ where $$ k_{r}^{ + } \simeq {{\lambda_{r}^{ + } } \mathord{\left/ {\vphantom {{\lambda_{r}^{ + } } L}} \right. \kern-0pt} L} $$ , λ+ is the microscopic transcription elongation rate in the on-state channel and L is the length of a complete mRNA transcript and h∞+ = [β/(α + β)] is the stationary state probability of finding the transcription machinery in the on-state channel.

Random vibrations of stress-driven nonlocal beams with external damping

Stochastic flexural vibrations of small-scale Bernoulli-Euler beams with external damping are investigated by stress-driven nonlocal mechanics. Damping effects are simulated considering viscous interactions between beam and surrounding environment. Loadings are modeled by accounting for their random nature. Such a dynamic problem is characterized by a stochastic partial differential equation in space and time governing time-evolution of the relevant displacement field. Differential eigenanalyses are performed to evaluate modal time coordinates and mode shapes, providing a complete stochastic description of response solutions. Closed-form expressions of power spectral density, correlation function, stationary and non-stationary variances of displacement fields are analytically detected. Size-dependent dynamic behaviour is assessed in terms of stiffness, variance and power spectral density of displacements. The outcomes can be useful for design and optimization of structural components of modern small-scale devices, such as Micro- and Nano-Electro-Mechanical-Systems (MEMS and NEMS).

Estimation of a non‐stationary prior covariance from seismic data

ABSTRACTNon‐stationarity in statistical properties of the subsurface is often ignored. In a classical linear Bayesian inversion setting of seismic data, the prior distribution of physical parameters is often assumed to be stationary. Here we propose a new method of handling non‐stationarity in the variance of physical parameters in seismic data. We propose to infer the model variance prior to inversion using maximum likelihood estimators in a sliding window approach. A traditional, and a localized shrinkage estimator is defined for inferring the prior model variance. The estimators are assessed in a synthetic base case with heterogeneous variance of the acoustic impedance in a zero‐offset seismic cross section. Subsequently, this data is inverted for acoustic impedance using a non‐stationary model set up with the inferred variances. Results indicate that prediction as well as posterior resolution is greatly improved using the non‐stationary model compared with a common prior model with stationary variance. The localized shrinkage predictor is shown to be slightly more robust than the traditional estimator in terms of amplitude differences in the variance of acoustic impedance and size of local neighbourhood. Finally, we apply the methodology to a real data set from the North Sea basin. Inversion results show a more realistic posterior model than using a conventional approach with stationary variance.