There are three equivalent ways of representing two jointly observed real-valued signals: as a bivariate vector signal, as a single complex-valued signal, or as two analytic signals known as the rotary components. Each representation has unique advantages depending on the system of interest and the application goals. In this paper we provide a joint framework for all three representations in the context of frequency-domain stochastic modeling. This framework allows us to extend many established statistical procedures for bivariate vector time series to complex-valued and rotary representations. These include procedures for parametrically modeling signal coherence, estimating model parameters using the Whittle likelihood, performing semi-parametric modeling, and choosing between classes of nested models using model choice. We also provide a new method of testing for impropriety in complex-valued signals, which tests for noncircular or anisotropic second-order statistical structure when the signal is represented in the complex plane. Finally, we demonstrate the usefulness of our methodology in capturing the anisotropic structure of signals observed from fluid dynamic simulations of turbulence.
We study causality between bivariate curve time series using the Granger causality generalized measures of correlation. With this measure, we can investigate which curve time series Granger-causes the other; in turn, it helps determine the predictability of any two curve time series. Illustrated by a climatology example, we find that the sea surface temperature Granger-causes the sea-level atmospheric pressure. Motivated by a portfolio management application in finance, we single out those stocks that lead or lag behind Dow-Jones industrial averages. Given a close relationship between S&P 500 …
Highly accurate interval forecasting of electricity demand is fundamental to the success of reducing the risk when making power system planning and operational decisions by providing a range rather than point estimation. In this study, a novel modeling framework integrating bivariate empirical mode decomposition (BEMD) and support vector regression (SVR), extended from the well-established empirical mode decomposition (EMD) based time series modeling framework in the energy demand forecasting literature, is proposed for interval forecasting of electricity demand. The novelty of this study aris…
Aiming to generate realistic synthetic times series of the bivariate process of daily mean temperature and precipitations, we introduce a non-homogeneous hidden Markov model. The non-homogeneity lies in periodic transition probabilities between the hidden states, and time-dependent emission distributions. This enables the model to account for the non-stationary behaviour of weather variables. By carefully choosing the emission distributions, it is also possible to model the dependance structure between the two variables. The model is applied to several weather stations in Europe with various c…
Gaussian fields (GFs) are frequently used in spatial statistics for their versatility. The associated computational cost can be a bottleneck, especially in realistic applications. It has been shown that computational efficiency can be gained by doing the computations using Gaussian Markov random fields (GMRFs) as the GFs can be seen as weak solutions to corresponding stochastic partial differential equations (SPDEs) using piecewise linear finite elements. We introduce a new class of representations of GFs with bivariate splines instead of finite elements. This allows an easier implementation o…
In various situations in the insurance industry, in finance, in epidemiology, etc., one needs to represent the joint evolution of the number of occurrences of an event. In this paper, we present a multivariate integer-valued autoregressive (MINAR) model, derive its properties and apply the model to earthquake occurrences across various pairs of tectonic plates. The model is an extension of Pedelis & Karlis (2011) where cross autocorrelation (spatial contagion in a seismic context) is considered. We fit various bivariate count models and find that for many contiguous tectonic plates, spatial co…
In several disciplines it is common to find time series measured at irregular observational times. In particular, in astronomy there are a large number of surveys that gather information over irregular time gaps and in more than one passband. Some examples are Pan-STARRS, ZTF and also the LSST. However, current commonly used time series models that estimate the time dependency in astronomical light curves consider the information of each band separately (e.g, CIAR, IAR and CARMA models) disregarding the dependency that might exist between different passbands. In this paper we propose a novel b…
This paper deals with the comparison of several stationary processes with unequal sample sizes. We provide a detailed theoretical framework on the testing problem for equality of spectral densities in the bivariate case, after which the generalization of our approach to the m dimensional case and to other statistical applications (like testing for zero correlation or clustering of time series data with different length) is straightforward. We prove asymptotic normality of an appropriately standardized version of the test statistic both under the null and the alternative and investigate the fin…
A novel approach towards the spectral analysis of stationary random bivariate signals is proposed. Using the Quaternion Fourier Transform, we introduce a quaternion-valued spectral representation of random bivariate signals seen as complex-valued sequences. This makes possible the definition of a scalar quaternion-valued spectral density for bivariate signals. This spectral density can be meaningfully interpreted in terms of frequency-dependent polarization attributes. A natural decomposition of any random bivariate signal in terms of unpolarized and polarized components is introduced. Nonpara…
A bivariate integer-valued autoregressive process of order 1 (BINAR(1)) with copula-joint innovations is studied. Different parameter estimation methods are analyzed and compared via Monte Carlo simulations with emphasis on estimation of the copula dependence parameter. An empirical application on defaulted and non-defaulted loan data is carried out using different combinations of copula functions and marginal distribution functions covering the cases where both marginal distributions are from the same family, as well as the case where they are from different distribution families.
Statistically simulated time series of wave parameters are required for many coastal and offshore engineering applications, often at the resolution of approximately one hour. Various studies have relied on autoregressive moving-average (ARMA) processes to simulate synthetic series of wave parameters in a Monte Carlo sense. However, accurately representing inter-series dependencies has remained a challenge. In particular, the relationship between wave height and period statistics is complex, due to the limiting steepness condition. Here, we present a new simulation method for joint time series …
Spatiotemporal simulation of minimum and maximum temperature is a fundamental requirement for climate impact studies and hydrological or agricultural models. Particularly over regions with variable orography, these simulations are difficult to produce due to terrain driven nonstationarity. We develop a bivariate stochastic model for the spatiotemporal field of minimum and maximum temperature. The proposed framework splits the bivariate field into two components of "local climate" and "weather." The local climate component is a linear model with spatially varying process coefficients capturing …
Let {X(t)} be a stationary time series with a.e. positive spectrum. Two consequences of that the bispectrum of {X(t)} is real-valued but nonzero: 1) if {X(t)} is also linear, then it is reversible; 2) {X(t),} can not be causal linear. A corollary of the first statement: if {X(t)} is linear, and the skewness of X(0) is nonzero, then third order reversibility implies reversibility. In this paper the notion of bispectrum is of a broader scope.
In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. …
We investigate shrinkage priors on power spectral densities for complex-valued circular-symmetric autoregressive processes. We construct shrinkage predictive power spectral densities, which asymptotically dominate (i) the Bayesian predictive power spectral density based on the Jeffreys prior and (ii) the estimative power spectral density with the maximal likelihood estimator, where the Kullback-Leibler divergence from the true power spectral density to a predictive power spectral density is adopted as a risk. Furthermore, we propose general constructions of objective priors for Kähler paramete…
We study distributional properties of a quadratic form of a stationary functional time series under mild moment conditions. As an important application, we obtain consistency rates of estimators of spectral density operators and prove joint weak convergence to a vector of complex Gaussian random operators. Weak convergence is established based on an approximation of the form via transforms of Hilbert-valued martingale difference sequences. As a side-result, the distributional properties of the long-run covariance operator are established.
The generalizations of instantaneous frequency and instantaneous bandwidth to a bivariate signal are derived. These are uniquely defined whether the signal is represented as a pair of real-valued signals, or as one analytic and one anti-analytic signal. A nonstationary but oscillatory bivariate signal has a natural representation as an ellipse whose properties evolve in time, and this representation provides a simple geometric interpretation for the bivariate instantaneous moments. The bivariate bandwidth is shown to consist of three terms measuring the degree of instability of the time-varyin…
We construct a procedure to test the stochastic order of two samples of interval-valued data. We propose a test statistic which belongs to U-statistic and derive its asymptotic distribution under the null hypothesis. We compare the performance of the newly proposed method with the existing one-sided bivariate Kolmogorov-Smirnov test using real data and simulated data.
This paper provides an algorithm for simulating improper (or noncircular) complex-valued stationary Gaussian processes. The technique utilizes recently developed methods for multivariate Gaussian processes from the circulant embedding literature. The method can be performed in $\mathcal{O}(n\log_2 n)$ operations, where $n$ is the length of the desired sequence. The method is exact, except when eigenvalues of prescribed circulant matrices are negative. We evaluate the performance of the algorithm empirically, and provide a practical example where the method is guaranteed to be exact for all $n$…
We introduce a flexible and scalable class of Bayesian geostatistical models for discrete data, based on the class of nearest neighbor mixture transition distribution processes (NNMP), referred to as discrete NNMP. The proposed class characterizes spatial variability by a weighted combination of first-order conditional probability mass functions (pmfs) for each one of a given number of neighbors. The approach supports flexible modeling for multivariate dependence through specification of general bivariate discrete distributions that define the conditional pmfs. Moreover, the discrete NNMP allo…
Based on a novel dynamic Whittle likelihood approximation for locally stationary processes, a Bayesian nonparametric approach to estimating the time-varying spectral density is proposed. This dynamic frequency-domain based likelihood approximation is able to depict the time-frequency evolution of the process by utilizing the moving periodogram previously introduced in the bootstrap literature. The posterior distribution is obtained by updating a bivariate extension of the Bernstein-Dirichlet process prior with the dynamic Whittle likelihood. Asymptotic properties such as sup-norm posterior con…
We propose a simple stochastic process for modeling improper or noncircular complex-valued signals. The process is a natural extension of a complex-valued autoregressive process, extended to include a widely linear autoregressive term. This process can then capture elliptical, as opposed to circular, stochastic oscillations in a bivariate signal. The process is order one and is more parsimonious than alternative stochastic modeling approaches in the literature. We provide conditions for stationarity, and derive the form of the covariance and relation sequence of this model. We describe how par…
