Numerical Stochastic Model of Non-stationary Time Series of the Wind Chill Index

A numerical stochastic model of the high-resolution time series of the wind chill index is considered. The model is constructed under the assumption that time series of the wind chill index are non-stationary non-Gaussian random processes with time-dependent one-dimensional distributions. This assumption makes possible to take into account both daily and seasonal variations of real meteorological processes. Data of the long-term real observations at weather stations were used for estimating the model parameters and for the verification of the model. Based on the simulated trajectories, some statistical properties of rare and adverse weather events, like long periods of time with a low wind chill index, are studied. The model is also used to study the dependence of the statistical properties of the wind chill index time series on a climate change.

     

Stochastic Model of Conditional Non-stationary Time Series of the Wind Chill Index in West Siberia

In this paper, we propose a stochastic model of the conditional time series of the wind chill index. The model is based on the inverse distribution function method and on the normalization method for simulation of the non-Gaussian non-stationary random processes as well as on the method of conditional distributions for simulation of the conditional Gaussian processes. In the framework of the approach considered, two types of conditions (point conditions and interval conditions) are imposed on the time series. The model in question was verified using the real data collected at the weather stations located in West Siberia (Russia). It is shown that the simulated trajectories are close in their statistical properties to the real time series. The model proposed was used for stochastic forecasting of the wind chill index and the results of the numerical experiments have shown that the accuracy of the short-term forecasts is high enough.

     

OU models based on positive and negative subordinate processes applying in SHIBOR time series analysis and derivative pricing – through discrete differential method

ABSTRACT

On account of that the OU models based on Gaussian process cannot describe the characteristics of peak, bias and asymmetric thick tail in SHIBOR time series, this paper replaces the Gaussian process in OU model with Levy process which can be decomposed into positive and negative subordinate processes, constructs OU model based on positive and negative subordinate processes. Methods parameter estimation and stochastic simulation were carried out by making discrete the stochastic differential equations into stochastic difference equations. The result shows that non-Gaussian OU process based on positive and negative subordinate processes not only fits the time series but also has better economic interpretation. The innovation of our research is to build a model of Non-Gaussian OU process based on positive and negative subordinate processes with less stochastic terms, and it provides an efficient tool for forecasting SHIBOR time series.     

Efficient and robust estimation for autoregressive regression models using shape mixtures of skewt normal distribution

Multiple linear regression model based on normally distributed and uncorrelated errors is a popular statistical tool with application in various fields. But these assumptions of normality and no serial correlation are hardly met in real life. Hence, this study considers the linear regression time series model for series with outliers and autocorrelated errors. These autocorrelated errors are represented by a covariance-stationary autoregressive process where the independent innovations are driven by shape mixture of skew-t normal distribution. The shape mixture of skew-t normal distribution is a flexible extension of the skew-t normal with an additional shape parameter that controls skewness and kurtosis. With this error model, stochastic modeling of multiple outliers is possible with an adaptive robust maximum likelihood estimation of all the parameters. An Expectation Conditional Maximization Either algorithm is developed to carryout the maximum likelihood estimation. We derive asymptotic standard errors of the estimators through an information-based approximation. The performance of the estimation procedure developed is evaluated through Monte Carlo simulations and real life data analysis.

     

Multivariate polynomial regression for identification of chaotic time series

Multivariate polynomial regression was used to generate polynomial iterators for time series exhibiting autocorrelations. A stepwise technique was used to add and remove polynomial terms to ensure the model contained only those terms that produce a statistically significant contribution to the fit. An approach is described in which datasets are divided into three subsets for identification, estimation, and validation. This produces a parsimonious global model that is can greatly reduce the tendency towards undesirable behaviours such as overfitting or instability. The technique was found to be able to identify the nonlinear dynamic behaviour of simulated time series, as reflected in the geometry of the attractor and calculation of multiple Lyapunov exponents, even in noisy systems.

The technique was applied to times series data obtained from simulations of the Lorenz and Mackey – Glass equations with and without measurement noise. The model was also used to determine the embedding dimension of the Mackey – Glass equation.     

Visibility graphs of fractional Wu–Baleanu time series

ABSTRACT

We study time series generated by the parametric family of fractional discrete maps introduced by Wu and Baleanu, presenting an alternative way of introducing these maps. For the values of the parameters that yield chaotic time series, we have studied the Shannon entropy of the degree distribution of the natural and horizontal visibility graphs associated to these series. In these cases, the degree distribution can be fitted with a power law. We have also compared the Shannon entropy and the exponent of the power law fitting for the different values of the fractionary exponent and the scaling factor of the model. Our results illustrate a connection between the fractionary exponent and the scaling factor of the maps, with the respect to the onset of the chaos.     

Principal Components in Time Series Forecast

Abstract

This article describes a time series forecast method based on the principal component analysis applied to the data matrix derived from the initial time series.     

Invariants of Multidimensional Time Series Based on Their Iterated-Integral Signature

We introduce a novel class of features for multidimensional time series that are invariant with respect to transformations of the ambient space. The general linear group, the group of rotations and the group of permutations of the axes are considered. The starting point for their construction is Chen’s iterated-integral signature.

     

A Multinomial Approximation Approach for the Finite Time Survival Probability Under the Markov-modulated Risk Model

In this paper, we consider the problem of computing different types of finite time survival probabilities for a Markov-Modulated risk model and a Markov-Modulated risk model with reinsurance, both with varying premium rates. We use the multinomial approximation scheme to derive an efficient recursive algorithm to compute finite time survival probabilities and finite time draw-down survival probabilities. Numerical results show that by comparing with MCMC approximation, discretize approximation and diffusion approximation methods, the proposed scheme performs accurate results in all the considered cases and with better computation efficiency.

     

A local discontinuous Galerkin method for pattern formation dynamical model in polymerizing action flocks

In this paper, we apply local discontinuous Galerkin methods to the pattern formation dynamical model in polymerizing action flocks. Optimal error estimates for the density and filament polarization in different norms are established. We use a semi-implicit spectral deferred correction time method for time discretization, which allows a relative large time step and avoids computation of a Jacobian matrix. Numerical experiments are presented to verify the theoretical analysis and to show the capability for simulations of action wave formation.

     

A ecessary and Sufficient Condition for The Stability of General molecular Systems

We study here the binding of atoms and molecules and the stability of general molecular systems including molecular ions. This is the first paper of a series devoted to the study of these general problems. We obtain here a general necessary and sufficient condition for the stability of general molecular ststem in the context of thomasz-Fermi-Von Weiasäcker, Thomas-Fermi-Dirac-Von Weizsaäcker, Hartree or Hartree-Fock theories

SUMARY OF PART 1

1.Introduction.

II.Presentation of the models

III.Diatomic molecular systems and hartree-Fock theory

IV.Diatomic molecular systems and Hartree or Thomas-Fermi theories

V.General molecular systems

Appendix 1: Hartree-Fock models when Z > N ― 1

Appendix 2: Dichotomy yields equal Lagrange multipliers

Appendix 3: The problem at infinty for the TRDW model     

Forecasting financial time series with Boltzmann entropy through neural networks

Neural networks have recently been established as state-of-the-art in forecasting financial time series. However, many studies show how one architecture, the Long-Short Term Memory, is the most widespread in financial sectors due to its high performance over time series. Considering some stocks traded in financial markets and a crypto ticker, this paper tries to study the effectiveness of the Boltzmann entropy as a financial indicator to improve forecasting, comparing it with financial analysts’ most commonly used indicators. The results show how Boltzmann’s entropy, born from an Agent-Based Model, is an efficient indicator that can also be applied to stocks and cryptocurrencies alone and in combination with some classic indicators. This critical fact allows obtaining good results in prediction ability using Network architecture that is not excessively complex.

     

An asymptotic expansion for the normalizing constant of the Conway–Maxwell–Poisson distribution

The Conway–Maxwell–Poisson distribution is a two-parameter generalization of the Poisson distribution that can be used to model data that are under- or over-dispersed relative to the Poisson distribution. The normalizing constant \(Z(\lambda ,\nu )\) is given by an infinite series that in general has no closed form, although several papers have derived approximations for this sum. In this work, we start by using probabilistic argument to obtain the leading term in the asymptotic expansion of \(Z(\lambda ,\nu )\) in the limit \(\lambda \rightarrow \infty \) that holds for all \(\nu >0\). We then use an integral representation to obtain the entire asymptotic series and give explicit formulas for the first eight coefficients. We apply this asymptotic series to obtain approximations for the mean, variance, cumulants, skewness, excess kurtosis and raw moments of CMP random variables. Numerical results confirm that these correction terms yield more accurate estimates than those obtained using just the leading-order term.

     

On Data-Driven Computation of Information Transfer for Causal Inference in Discrete-Time Dynamical Systems

In this paper, we provide a novel approach to capture causal interaction in a dynamical system from time series data. In Sinha and Vaidya (in: IEEE conference on decision and control, pp 7329–7334, 2016), we have shown that the existing measures of information transfer, namely directed information, Granger causality and transfer entropy, fail to capture the causal interaction in a dynamical system and proposed a new definition of information transfer that captures direct causal interactions. The novelty of the information transfer definition used in this paper is the fact that it can differentiate between direct and indirect influences Sinha and Vaidya (2016). The main contribution of this paper is to show that the proposed definition of information transfers in Sinha and Vaidya (2016) and Sinha and Vaidya (in: Indian control conference, pp 303–308, 2017) can be computed from time series data, and thus, the direct influences in a dynamical system can be identified from time series data. We use transfer operator theoretic framework, involving Perron–Frobenius and Koopman operators for the data-driven approximation of the system dynamics and computation of information transfer. Several examples, involving linear and nonlinear system dynamics, are presented to verify the efficiency of the developed algorithm.

     

Far Casting Cross-Validation

Cross-validation has long been used for choosing tuning parameters and other model selection tasks. It generally performs well provided the data are independent, or nearly so. Improvements have been suggested which address ordinary cross-validation’s (OCV) shortcomings in correlated data. Whereas these techniques have merit, they can still lead to poor model selection in correlated data or are not readily generalizable to high-dimensional data.

The proposed solution, far casting cross-validation (FCCV), addresses these problems. FCCV withholds correlated neighbors in every aspect of the cross-validation procedure. The result is a technique that stresses a fitted model’s ability to extrapolate rather than interpolate. This generally leads to better model selection in correlated datasets.

Whereas FCCV is less than optimal in the independence case, our improvement of OCV applies more generally to higher dimensional error processes and to both parametric and nonparametric model selection problems. To facilitate introduction, we consider only one application, namely estimating global bandwidths for curve estimation with local linear regression. We provide theoretical motivation and report some comparative results from a simulation experiment and on a time series of annual global temperature deviations. For such data, FCCV generally has lower average squared error when disturbances are correlated.

Supplementary materials are available online.     

A Kendall correlation coefficient between functional data

Measuring dependence is a very important tool to analyze pairs of functional data. The coefficients currently available to quantify association between two sets of curves show a non robust behavior under the presence of outliers. We propose a new robust numerical measure of association for bivariate functional data. We extend in this paper Kendall coefficient for finite dimensional observations to the functional setting. We also study its statistical properties. An extensive simulation study shows the good behavior of this new measure for different types of functional data. Moreover, we apply it to establish association for real data, including microarrays time series in genetics.

     

q-ENGEL SERIES EXPANSIONS AND SLATER'S IDENTITIES

Abstract

Dedicated to the memory of John Knopfmacher (1937–1999)

We describe the q-Engel series expansion for Laurent series discovered by John Knopfmacher and use this algorithm to shed new light on partition identities related to two entries from Slater's list. In our study Al-Salam/Ismail and Santos polynomials play a crucial r?ole.     

Some new q-series identities through an extended Bailey transform

In this note, a variety of new formulas for basic hypergeometric series will be derived by an application of a single Bailey-type transform to some q-series formulas which appeared recently in the literature.

     

Stochastic analysis of a n-unit repairable system with slow switch

This paper deals with the costn–benefit analysis of a cold standby system composed of n identical repairable units, subject to slow switch. Two models of system functioning are studied in this paper. In model 1, the repair time of a unit is assumed to follow exponential distribution and the other time distributions as arbitrary, while in model 2, the repair time of a unit is assumed to be arbitrarily distributed and the other time distributions follow exponential law. For both the models, the system characteristics, namely

(i) the expected upn–time of the system during the period (O,t]

(ii) the expected busyn–period of the repair facility during the period (0,t] and

(iii) the expected time the units spend in the switchover/installation state during the period (O,t]

are studied by identifying the system a t suitable regeneration epochs. The cost-benefit analysis is carried out using these characteristics     

Efficient Simulation of Ruin Probabilities When Claims are Mixtures of Heavy and Light Tails

We consider the classical Cramér-Lundberg risk model with claim sizes that are mixtures of phase-type and subexponential variables. Exploiting a specific geometric compound representation, we propose control variate techniques to efficiently simulate the ruin probability in this situation. The resulting estimators perform well for both small and large initial capital. We quantify the variance reduction as well as the efficiency gain of our method over another fast standard technique based on the classical Pollaczek-Khinchine formula. We provide a numerical example to illustrate the performance, and show that for more time-consuming conditional Monte Carlo techniques, the new series representation also does not compare unfavorably to the one based on the Pollaczek-Khinchine formula.