Second-order Markov Process Research Articles

Modelling output gaps in the Euro Area with structural breaks: The COVID-19 recession

This paper proposes a novel unobserved component model with a COVID-19 structural break in the trend growth rate to model output gaps. Using historical real GDP data for the Euro Area between 1995Q1 and 2022Q1, we test our framework against a battery of competing models, including a standard unobserved components model, a correlated model with a second-order Markov process, a Hodrick–Prescott filter and an augmented version of it. To examine the impact on the fitting performance, we test the inclusion and exclusion of pandemic quarters and we also extend the estimation to a country-level detail. We find that: (i) our suggested model outperforms the competing ones; (ii) when excluding pandemic quarters, the standard unobserved component model outperforms their counterparts; (iii) our model yields the best fitting performance for most of the Euro Area countries and (iv) the Hodrick–Prescott filter model has the poorest fitting performance.

Comparative Analysis of Pitching Prediction Algorithms

Two algorithms are described in the paper; one of them is the Kalman filter, which is based on the use of a pitching mathematical model, and the second uses a neural network in which the model is considered unknown. The results of the algorithms sensitivity analysis to the parameters of the model and its influence on the potential accuracy of prediction are presented. A stationary narrow-band second-order Markov process is used as a model of the ship pitching, which was used to form the input signal of the algorithms. Also, the results of the algorithms simulation in predicting real data are presented.

Accommodating the role of site memory in dynamic species distribution models.

First-order dynamic occupancy models (FODOMs) are a class of state-space model in which the true state (occurrence) is observed imperfectly. An important assumption of FODOMs is that site dynamics only depend on the current state and that variations in dynamic processes are adequately captured with covariates or random effects. However, it is often difficult to understand and/or measure the covariates that generate ecological data, which are typically spatiotemporally correlated. Consequently, the non-independent error structure of correlated data causes underestimation of parameter uncertainty and poor ecological inference. Here, we extend the FODOM framework with a second-order Markov process to accommodate site memory when covariates are not available. Our modeling framework can be used to make reliable inference about site occupancy, colonization, extinction, turnover, and detection probabilities. We present a series of simulations to illustrate the data requirements and model performance. We then applied our modeling framework to 13yr of data from an amphibian community in southern Arizona, USA. In this analysis, we found residual temporal autocorrelation of population processes for most species, even after accounting for long-term drought dynamics. Our approach represents a valuable advance in obtaining inference on population dynamics, especially as they relate to metapopulations.

Gaussian Process Trajectory Learning and Synthesis of Individualized Gait Motions.

This paper proposes a Gaussian process-based method for trajectory learning and generation of individualized gait motions at arbitrary user-designated walking speeds, intended to be used in generating reference motions for robotic gait rehabilitation systems. We utilize a nonlinear dimension reduction technique based on Gaussian process dynamical models (GPDMs), in which the internal dynamics is modeled as a second-order Markov process evolving in a lower-dimensional latent space. After the GPDM parameters are identified with training data obtained from gait motions of healthy subjects walking at different speeds, our method then employs Gaussian process regression (GPR) to predict the initial two states of the latent space dynamics from any arbitrary desired walking speed and the anthropometric parameters of the test subject. Motions are then generated by directly mapping the latent space dynamics to joint trajectories. Experimental studies involving more than 100 subjects indicate that our method generates gait patterns with 30% less mean square prediction errors compared to recent state-of-the-art methods, while also allowing for arbitrary user-specified walking speeds.

A nonlinear second-order filtering strategy for state estimation of uncertain systems

In this paper, a new strategy referred to as the nonlinear second-order (NSO) filter is presented and used for estimation of linear and nonlinear systems in the presence of uncertainties. Similar to the popular Kalman filter estimation strategy, the proposed strategy is model-based and formulated as a predictor-corrector. The NSO filter is based on variable structure theory that utilizes a switching term and gain that ensures some level of estimation stability. It offers improvements in terms of robustness to modeling uncertainties and errors. The proof of stability is derived based on Lyapunov that demonstrates convergence of estimates towards the true state values. The proposed filtering strategy is based on a second-order Markov process that utilizes information from the current and past two time steps. An experimental system was setup and characterized in order to demonstrate the proposed filtering strategy's performance. The strategy was compared with the popular Kalman filter (and its nonlinear form) and the smooth variable structure filter (SVSF). Experimental results demonstrate that the proposed nonlinear second-order filter provides improvements in terms of state estimation accuracy and robustness to modeling uncertainties and external disturbances.

An Improved Differential Space-Time Block Coding Scheme Based on Viterbi Algorithm

This letter proposes a novel improved differential space-time block code (DSTBC) scheme. By introducing a specific redundancy mechanism, the proposed DSTBC encoder behaves as a second-order discrete Markov process that we model by a multilevel trellis diagram. At the receiver side, the well-known Viterbi algorithm is applied without channel state information (CSI). The new scheme completely compensates for the 3-dB penalty, over wide SNR range, noticed in conventional DSTBC scheme and shows a closer performance to coherent STBC.

Exploring sensitivity of a multistate occupancy model to inform management decisions

1. Dynamic occupancy models are often used to investigate questions regarding the processes that influence patch occupancy and are prominent in the fields of population and community ecology and conservation biology. Recently, multistate occupancy models have been developed to investigate dynamic systems involving more than one occupied state, including reproductive states, relative abundance states and joint habitat-occupancy states. Here we investigate the sensitivities of the equilibrium-state distribution of multistate occupancy models to changes in transition rates. 2. We develop equilibrium occupancy expressions and their associated sensitivity metrics for dynamic multistate occupancy models. To illustrate our approach, we use two examples that represent common multistate occupancy systems. The first example involves a three-state dynamic model involving occupied states with and without successful reproduction (California spotted owl Strix occidentalis occidentalis), and the second involves a novel way of using a multistate occupancy approach to accommodate second-order Markov processes (wood frog Lithobates sylvatica breeding and metamorphosis). 3. In many ways, multistate sensitivity metrics behave in similar ways as standard occupancy sensitivities. When equilibrium occupancy rates are low, sensitivity to parameters related to colonisation is high, while sensitivity to persistence parameters is greater when equilibrium occupancy rates are high. Sensitivities can also provide guidance for managers when estimates of transition probabilities are not available. 4. Synthesis and applications. Multistate models provide practitioners a flexible framework to define multiple, distinct occupied states and the ability to choose which state, or combination of states, is most relevant to questions and decisions about their own systems. In addition to standard multistate occupancy models, we provide an example of how a second-order Markov process can be modified to fit a multistate framework. Assuming the system is near equilibrium, our sensitivity analyses illustrate how to investigate the sensitivity of the system-specific equilibrium state(s) to changes in transition rates. Because management will typically act on these transition rates, sensitivity analyses can provide valuable information about the potential influence of different actions and when it may be prudent to shift the focus of management among the various transition rates.

Diffusion via space discretization method to study the concentration dependence of self-diffusivity under confinement

The concentration dependence of self-diffusivity is investigated by means of a novel method, extending our previously developed second-order Markov process model to periodic media. Introducing the concept of minimum-crossing surface, we obtain a unique decomposition of the self-diffusion coefficient into two parameters with specific physical meanings. Two case studies showing a maximum in self-diffusivity as a function of concentration are investigated, along with two cases where such a maximum cannot be present. Subsequently, the method is applied to the large cavity pore network of the ITQ-1 (Mobil tWenty tWo, MWW) zeolite for methane (displaying a maximum in self-diffusivity) and carbon dioxide (no maximum), explaining the diffusivity trend on the basis of the evolution of the model parameters as a function of concentration.

A second-order Markov process for modeling diffusive motion through spatial discretization

A new "mesoscopic" stochastic model has been developed to describe the diffusive behavior of a system of particles at equilibrium. The model is based on discretizing space into slabs by drawing equispaced parallel planes along a coordinate direction. A central role is played by the probability that a particle exits a slab via the face opposite to the one through which it entered (transmission probability), as opposed to exiting via the same face through which it entered (reflection probability). A simple second-order Markov process invoking this probability is developed, leading to an expression for the self-diffusivity, applicable for large slab widths, consistent with a continuous formulation of diffusional motion. This model is validated via molecular dynamics simulations in a bulk system of soft spheres across a wide range of densities.

A new weather generator based on spectral properties of surface air temperatures

We present a new weather generator, which simulates daily precipitation occurrence and daily maximum and minimum air temperature. Precipitation occurrence is simulated with a two-state, second-order Markov process, while maximum and minimum daily temperatures are simulated using spectral methods. Data generated at nine stations in the Southeastern USA are then compared to: (1) observed station data, and (2) data generated by a variant of the commonly used autoregressive weather generator. Precipitation occurrence simulation is found to be excellent for our study region. Maximum and minimum temperatures generated using the spectral weather generator are found to be superior to those generated by the autoregressive weather generator in terms of means and variances across multiple timescales as well as the number of freeze events, skewness of temperature distributions, and the relationship between daily minimum and maximum air temperatures. Both weather generators are found to slightly underestimate interannual variability in all three generated variables.

Geometrically Nonlinear Cantilever under Stochastic Loading Vectors

This paper deals with the static and dynamic response of acantilever beam affected by Gaussian and non-Gaussian vectorprocesses. The load is modeled by three different correlationstructures based on the second-order Markov process, and thenon-Gaussian random features are described by translationprocesses. Both the linear and geometrically nonlinear behaviorare described using the finite element approach, and the beammechanical characteristics (natural frequencies and damping) arevaried to investigate their influence on the response. Finally,the Monte Carlo approach is used to estimate the responsestatistics. The transverse displacement and the bending moment atthe beam joints are used to summarize the obtained results interms of skewness and kurtosis coefficients, and probabilitydensity functions. The study presented in this paper providesinformation on the influence of the load nature and the structuralbehavior on the response random features.

Numerical Analysis of Structural Systems Subjected to Non-Gaussian Random Fields

This paper deals with the stochastic response of structures loaded by non-Gaussian random fields. A finite element model is used to describe a cantilever beam assuming both linear and non-linear behavior. The cross-correlated stochastic field is generated by a numerical procedure based on the translation processes theory. The marginal distribution of the load is assumed to be lognormal and the correlation structure is based on the second-order Markov process. The statistical analysis of the results highlights the effects of the involved non-linearity and non-Gaussianity properties on the structure’s response.

Stochastic models for sunshine duration and solar irradiation

Harmonic analysis of sunshine duration and solar irradiation measured at Sebele, Botswana is carried out. The data used consists of the monthly averages and the Julian-days averages of sunshine duration and solar irradiation sequences. This study involves splitting the time series into deterministic and stochastic components, and determining the proportion of the variance explained by each component. The stochastic component is analyzed for persistence using the Box and Jenkins technique. It is found that the stochastic component for monthly averages solar radiation series is best described by the second-order autoregressive Markov process, while that for Julian-days averages series has no memory.

Modelling of gear stochastic vibration considering non-white noise errors

A model to simulate the non-Gaussian non-white-noise random errors of gears is developed with the influence of rotational speed considered. The gear transmission errors is first decomposed into harmonic and random components based on the spectral analysis. Then, a second-order Markov process with time-variant parameters is used to simulate the non-Gaussian non-white-noise random component of gears. The second-order Markov process is used as a shaping filter. The input of the filter is a white Gaussian noise process, and the output is the random erros of gears. Furthermore, a new dynamic and stochastic simulation model for the rotational bivration of gear transmission system is proposed. The influences of the random errors on vibration acceleration are examined by use of the developed simulation system.

Stochastic modelling of water temperatures in a small stream using air to water relations

Stream water temperature is a very important parameter when assessing aquatic ecosystem dynamics. For instance, cold-water fishes such as salmon can be adversely affected by maximum summer temperatures or by those exaggerated by land-use activities such as deforestation. The present study deals with the modelling of stream water temperatures using a stochastic approach to relate air and water temperatures in Catamaran Brook, a small stream in New Brunswick where long-term multidisciplinary habitat research is being carried out. The first step in the modelling approach was to establish the long-term annual component (pattern) in stream water temperatures. This was possible by fitting a Fourier series to stream water temperatures. The short-term residual temperatures (departure from the long-term annual component) were modelled using different air to water relations, namely a multiple regression analysis, a second-order Markov process, and a Box-Jenkins time-series model. The results indicated that it was possible to predict daily water temperatures for small streams using air temperatures and that the three models produced similar results in predicting stream temperatures. The root mean square error (RSME) varied between 0.59°C and 1.68°C on an annual basis from 1990 to 1995, with the warmest year (1994) showing the highest RMSE. Although 1992 was an exceptionally cold summer (coldest in 30 years), good predictions of stream water temperature were obtained, with an RMSE of approximately 1.24°C. Of the three models, the second-order Markov process was preferred based on its performance and its simplicity in development.Key words: small stream, water temperature, model, stochastic, root mean square error, Markov process.

Nonvalidity of the telegrapher's diffusion equation in two and three dimensionsfor crystalline solids

We use a classical analog of two-dimensional (2D) and 3D quantum S-matrix scattering theory to study classical mesoscopic diffusion in isotropic, crystalline, solids. The individual collisions include transmission, reflection, and lateral scattering probabilities. The resulting stochastic process is a second-order Markov process in phase space, which is known in the literature as 2D (3D) persistent random walk. In striking contrast with the 1D case, in the continuum limit, the 2D and 3D total densities do not satisfy the telegrapher's diffusion equation. We explain this fact deriving the anomalous Maxwell-Cattaneo equation in the case of discrete diffusion processes. We find that inertial memory, giving the forward scattering a preferential direction, breaks the x-y symmetry.

From the quantum random walk to classical mesoscopic diffusion in crystalline solids.

We use a discrete approximation of quantum mechanics called the quantum random walk to study diffusion in one-dimensional crystalline nanostructures. There, intense fluctuations in the density are a consequence of quantum interference. As the size of the crystal increase, relative quantum fluctuations decrease and, if a coarse-grained averaging is taken, interference may be neglected. When this happens, we describe a far-from-equilibrium, classical, dynamic mesoscopic diffusion regime. In this regime, diffusion equations are given by a persistent-random-walk process. Density becomes a second-order Markov process, and in the continuum limit, the current satisfies the non-Fickian Maxwell-Cattaneo relationship. The generalized diffusion equation in the telegraphist's equation.

Rate-aided multisignal time delay detection and tracking

Time delay detection and tracking problems have been investigated in previous papers. The approach used is applicable to many problems. In those papers, a first-order Markov process was assumed for the signal parameter (time delay) dynamic model and a constant signal parameter value was assumed within a block, or processing unit of time. A performance degradation is incurred when the signal parameter value is nearly constant-rate. Also, crossing signal parameter values from different signals cannot be tracked. The detector-trackers will switch signals. To combat these difficulties, the signal parameter dynamic model is assumed to be a second-order Markov process, and signal parameter rate of change information is computed within a block. A rate-aided multisignal detection and tracking system is developed with these assumptions. >

Performance analysis of a star topology of interconnected networks under 2nd-order Markov network output processes

The concept of approximating the output process of slotted multiuser random-access communication networks (i.e. the process of the successfully transmitted packets within the networks) by a second-order Markov process is introduced. A method is developed for analytically calculating the parameters of the approximating process for a class of random-access algorithms. The method is illustrated by considering a specific random-access algorithm from that class. The mean time that a packet spends in the central node of a star topology of interconnected networks is incorporated in the evaluation of the accuracy of the proposed approximation. This quantity is calculated under the proposed approximation on the output processes of the interconnected networks and is compared to simulation results from the actual system. Results showing the accuracy of the proposed approximation for networks operating under a specific random-access algorithm are presented. >

Modeling of perturbed covariance sequences

An algorithm for obtaining a state-space (Markovian) model of a random process from a finite number of estimated covariance lags is presented and compared to other SVD-based methods. The new algorithm performs well for small data sets in which the estimated covariance sequence is perturbed by estimation errors, and may not even be positive definite. Simulation results are presented for a second-order Markov process.