Continuous-time Autoregressive Processes Research Articles

Multivariate continuous-time autoregressive moving-average processes on cones

In this article we study multivariate continuous-time autoregressive moving-average (MCARMA) processes with values in convex cones. More specifically, we introduce matrix-valued MCARMA processes with Lévy noise and present necessary and sufficient conditions for processes from this class to be cone valued. We derive specific hands-on conditions in the following two cases: First, for classical MCARMA on Rd with values in the positive orthant Rd+. Second, for MCARMA processes on real square matrices assuming values in the cone of symmetric and positive semi-definite matrices. Both cases are relevant for applications and we give several examples of positivity ensuring parameter specifications. In addition to the above, we discuss the capability of positive semi-definite MCARMA processes to model the spot covariance process in multivariate stochastic volatility models. We justify the relevance of MCARMA based stochastic volatility models by an exemplary analysis of the second order moment structure of positive semi-definite well-balanced Ornstein–Uhlenbeck based models.

Examining AGN UV/Optical Variability beyond the Simple Damped Random Walk

We present damped harmonic oscillator (DHO) light-curve modeling for a sample of 12,714 spectroscopically confirmed quasars in the Sloan Digital Sky Survey Stripe 82 region. DHO is a second-order continuous-time autoregressive moving-average process, which can be fully described using four independent parameters: a natural oscillation frequency (ω 0), a damping ratio (ξ), a characteristic perturbation timescale (τ perturb), and an amplitude for the perturbing white noise (σ ϵ ). The asymptotic variability amplitude of a DHO process is quantified by σ DHO—a function of ω 0, ξ, τ perturb, and σ ϵ . We find that both τ perturb and σ ϵ follow different dependencies with rest-frame wavelength (λ RF) on either side of 2500 Å, whereas σ DHO follows a single power-law relation with λ RF. After correcting for wavelength dependence, σ DHO exhibits anticorrelations with both the Eddington ratio and the black hole mass, while τ perturb—with a typical value of days in the rest frame—shows an anticorrelation with the bolometric luminosity. Modeling active galactic nuclei (AGN) variability as a DHO offers more insight into the workings of accretion disks close to the supermassive black holes at the center of AGN. The newly discovered short-term variability (characterized by τ perturb and σ ϵ ) and its correlation with bolometric luminosity pave the way for new algorithms that will derive fundamental properties (e.g., Eddington ratio) of AGN using photometric data alone.

Optimal Sparse Sampling for Detection of a Known Signal in Nonwhite Gaussian Noise

We address the problem of sparse sampling pattern design to maximize detection for a deterministic signal in colored noise. We model a colored noise as a continuous-time autoregressive process, which is obtained by passing a white noise through a causal linear-time invariant filter. This noise modeling is crucial to the development of the optimal sampling pattern design for a given number of sensors. We obtain a closed form expression for the whitening filter and consequently, for the Kullback-Leibler divergence at the whitening filter output, which is the detection index. The optimum sampling pattern is obtained by evaluating the detection index at Nyquist sampling rate, rank ordering the samples, and selecting the maximum values. We present some examples to illustrate the proposed procedure. We also extend our method to two-dimensional sampling. The advantage of our approach is its low computational complexity for both one-dimensional and two-dimensional cases and that optimality is guaranteed.

Pricing Weather Derivatives Index Based on Temperature: The Case of Bahir Dar, Ethiopia

In this paper we present a stochastic model for daily average temperature to calculate the temperature indices upon which temperature-based derivatives are written. We propose a seasonal mean and volatility model that describes the daily average temperature behavior using the mean-reverting Ornstein-Uhlenbeck process. We also use higher order continuous-time autoregressive process with lag 3 for modeling the time evolution of the temperatures after removing trend and seasonality. Our model is fitted to 11 years of data recorded, in the period 1 January 2005 to 31 December 2015, Bahir Dar, Ethiopia, obtained from Ethiopia National Meteorological Services Agency. The analytical approximation formulas are used to price heating degree days (HDD) and cooling degree days (CDD) futures. The suggested model is analytically tractable for derivation of explicit prices for CDD and HDD futures and option. The price of the CDD future is calculated, using analytical approximation formulas. Numerical examples are presented to indicate the accuracy of the method. The results show that our model performs better to predict CDD indices.

Temperature modelling and pricing of temperature index insurance

In this paper, we use continuous-time autoregressive process to explain evolution of the temperature dynamics. We take data of New York daily average temperatures as our empirical study. Our analysis indicates continuous-time autoregressive of order 3 fits data very well. The model is employed in the pricing of index based temperature insurance. In the context of agricultural industry, our study revealed that the price of insurance obtained from temperature dynamical modelling is far more expensive than the price obtained from classical approaches of burn analysis and index modelling.

Correlation properties of continuous-time autoregressive processes delayed by the inverse of the stable subordinator

We define the delayed Lévy-driven continuous-time autoregressive process via the inverse of the stable subordinator. We derive correlation structure for the observed non-stationary delayed Lévy-driven continuous-time autoregressive processes of order p, emphasizing low orders, and we show they exhibit long-range dependence property. Distributional properties are discussed as well.

Autoregressive continuous logit: Formulation and application to time-of-day choice modeling

In this paper, we formulate the autoregressive continuous logit model as a novel continuous choice model capable of representing correlations across alternatives in the continuous spectrum. We formulate this model by considering two approaches: combining a discrete-time autoregressive process of order one (i.e., a linear stochastic difference equation) with the continuous logit model, which leads to the discrete-time autoregressive continuous logit; and combining a continuous-time autoregressive process (i.e., a linear stochastic differential equation), known in the stochastic process literature as Ornstein-Uhlenbeck process with the continuous logit model, which leads to the continuous-time autoregressive continuous logit. The autoregressive nature of the model is in its error structure, allowing correlations in unobserved heterogeneity. For both approaches, we study their properties numerically. We also compare both approaches to highlight their relation to each other.

Parameters estimation for continuous-time heavy-tailed signals modeled by α-stable autoregressive processes

In this paper, we focus on the heavy-tailed stochastic signals generated through continuous-time autoregressive (CAR) models excited by infinite-variance α-stable processes. Our goal is to estimate the parameters of the continuous-time model, such as the autoregressive coefficients and the distribution parameters related to the excitation process for the α-stable CAR process with 0<α<2 based on the state-space representation. Likewise, we investigate the closed form expressions for the parameters of equivalent model in the discrete-time setting via regular samples of the process. We analyze the estimator based on the Monte Carlo simulations and illustrate the estimator consistency to the desired values when sampling frequency and sample size tend to infinity. We also apply the proposed method to the two types of real-world data, financial and ground magnetometer data, to evaluate its performance in real environments.

Stable continuous-time autoregressive process driven by stable subordinator

In this paper we examine the continuous-time autoregressive moving average process driven by α-stable Lévy motion delayed by inverse stable subordinator. This process can be applied to high-frequency data with visible jumps and so-called “trapping-events”. Those properties are often visible in financial time series but also in amorphous semiconductors, technical data describing the rotational speed of a machine working under various load regimes or data related to indoor air quality. We concentrate on the main characteristics of the examined subordinated process expressed in the language of the measures of dependence which are main tools used in statistical investigation of real data. However, because the analyzed system is based on the α-stable distribution therefore we cannot consider here the correlation (or covariance) as a main measure which indicates at the dependence inside the process. In the paper we examine the codifference, the more general measure of dependence defined for wide class of processes. Moreover we present the simulation procedure of the considered system and indicate how to estimate its parameters. The theoretical results we illustrate by the simulated data analysis.

Time series models for credit default swap premiums

We present statistical models for the continuous-time Dynamics of credit Default swap (CDS) premium within an intensity-based credit risk modeling Framework. Based on historical daily CDS Premiums for a large set of different corporate refence entities from several developed countries, we fit continuous-time autoregressive moving-average processes of an appropriate order driven by a Levy process. We recover the driving noise process, which only Shows a stochastic volatility effect for particular branches. On a distributional Level, the increments of the noise process are, as a rule, best modeled by a normal inverse Gaussian distribution.

Stochastic dynamical modelling of spot freight rates

Based on empirical analysis of the Capesize and Panamax indices, we propose different continuous-time stochastic processes to model their dynamics. The models go beyond the standard geometric Brownian motion, and incorporate observed effects like heavy-tailed returns, stochastic volatility and memory. In particular, we suggest stochastic dynamics based on exponential Levy processes with normal inverse Gaussian distributed logarithmic returns. The Barndorff-Nielsen and Shephard stochastic volatility model is shown to capture time-varying volatility in the data. Finally, continuous-time autoregressive processes provide a class of models sufficiently rich to incorporate short-term persistence of freight rates. Our proposed dynamical models are fitted to market data. Finally, a numerical Value-at-Risk exercise is provided that illustrate the significance of using more sophisticated models than geometric Brownian motion.

Bootstrapping continuous-time autoregressive processes

We develop a bootstrap procedure for Levy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regularly sampled CAR processes of higher order satisfy ARMA equations with uncorrelated (but in general dependent) noise, a more general bootstrap procedure is needed for such processes. We consider statistics depending on observations of the CAR process at the uniformly-spaced times, together with auxiliary observations on a finer grid, which give approximations to the derivatives of the continuous time process. This enables us to approximate the state-vector of the CAR process which is a vector-valued CAR(1) process, and whose sampled version, on the uniformly-spaced grid, is a multivariate AR(1) process with i.i.d. noise. This leads to a valid residual-based bootstrap which allows replication of CAR $$(p)$$ processes on the underlying discrete time grid. We show that this approach is consistent for empirical autocovariances and autocorrelations.

Is Uniqueness Lost for Under-Sampled Continuous-Time Auto-Regressive Processes?

We consider the problem of sampling continuous-time auto-regressive processes on a uniform grid. We investigate whether a given sampled process originates from a single continuous-time model, and address this uniqueness problem by introducing an alternative description of poles in the complex plane. We then utilize Kronecker's approximation theorem and prove that the set of non-unique continuous-time AR(2) models has Lebesgue measure zero in this plane. This is a key aspect in current estimation algorithms that use sampled data, as it allows one to remove the sampling rate constraint that is imposed currently.

Optimal sampling regime for detecting significant differences in peak mass of chicks: a case study with the wandering albatross Diomedea exulans

ESR Endangered Species Research Contact the journal Facebook Twitter RSS Mailing List Subscribe to our mailing list via Mailchimp HomeLatest VolumeAbout the JournalEditorsSpecials ESR 11:167-173 (2010) - DOI: https://doi.org/10.3354/esr00270 Optimal sampling regime for detecting significant differences in peak mass of chicks: a case study with the wandering albatross Diomedea exulans Steven G. Candy*, John van den Hoff Australian Antarctic Division, 203 Channel Highway, Kingston 7050, Tasmania, Australia *Email: steve.candy@aad.gov.au ABSTRACT: Special emphasis should be placed on determining optimal sampling regimes to detect statistically significant differences in key indicators between populations for species with high conservation values, especially when invasive handling techniques are the only method of gathering data. In procellariiform nestlings one such key indicator is peak mass, which serves as a measure of parental investment and is an important factor for future survival. Daily measurements of mass for 15 wandering albatross Diomedea exulans chicks ranging in age from 14 to 270 d were obtained in 1963 on Bird Island, South Georgia. Log-transformed mass was modelled using a linear mixed model (LMM) that included a linear term for age combined with smooth nonlinear departures estimated as random effects, random chick effects and a continuous-time autoregressive error process. The relative efficiency of different sample sizes of chicks versus frequency of measurement was investigated by fitting this LMM to sub-samples of the dataset of either 5, 10 or 15 chicks with the corresponding frequency manipulated to give the same total number of measurements for each sub-sample. Measures of precision of predictions from the LMM obtained for each dataset included (1) the width of single standard error bounds about predicted peak mass as a percentage of this mass (PSEPM), and (2) the probability of detecting a P- percentage difference in peak mass between 2 theoretical populations for the same sample size of chicks and same frequency of measurement in each (PRDPM). The most efficient of these sampling regimes for detecting a difference between populations or cohorts was 15 birds sampled every third day, which gave a PRDPM of 0.95 for a P of 15%. KEY WORDS: Optimal sampling · Albatross · Chick growth · Cubic splines · Autocorrelated data Full text in pdf format PreviousNextCite this article as: Candy SG, van den Hoff J (2010) Optimal sampling regime for detecting significant differences in peak mass of chicks: a case study with the wandering albatross Diomedea exulans. Endang Species Res 11:167-173. https://doi.org/10.3354/esr00270 Export citation RSS - Facebook - Tweet - linkedIn Cited by Published in ESR Vol. 11, No. 2. Online publication date: April 16, 2010 Print ISSN: 1863-5407; Online ISSN: 1613-4796 Copyright © 2010 Inter-Research.

High-Accuracy Instrumental Variable Identification of Continuous-Time Autoregressive Processes From Irregularly Sampled Noisy Data

A computationally efficient estimator of continuous-time autoregressive (AR) process parameters from irregularly sampled data affected by discrete-time white measurement noise is presented. It is described how an instrumental variable approach can be used for estimating the AR process parameters with high accuracy. Possible estimators of the incremental variance of the driving continuous-time white noise source and of the variance of the discrete-time white measurement noise are also discussed.

Variance Analysis of a Cross-Covariance Matching Method for Continuous-Time ARX Parameter Estimation

A method for estimating the parameters of a continuous-time autoregressive exogenous process from discrete-time data is analyzed. The method consists of fitting an expression for the cross-covariance function, parameterized by the unknown parameters, to sample cross-covariances. The main contribution of the note is the derivation of an approximate expression for the covariance matrix of the estimated parameter vector.

Long-memory continuous-time correlation models

This paper introduces a rather general class of stationary continuous-time processes with long memory by randomizing the time-scale of short-memory processes. In particular, by randomizing the time-scale of continuous-time autoregressive and moving-average processes, many power-law decay and slow decay correlation functions are obtained.

Long-memory continuous-time correlation models

This paper introduces a rather general class of stationary continuous-time processes with long memory by randomizing the time-scale of short-memory processes. In particular, by randomizing the time-scale of continuous-time autoregressive and moving-average processes, many power-law decay and slow decay correlation functions are obtained.

Finite sample performance of density estimators from unequally spaced data

For broad classes of deterministic and random sampling schemes { τ k }, exact mean integrated squared error (MISE) expressions for the kernel estimator of the marginal density of a first-order continuous-time autoregressive process are derived. The obtained expressions show that the effect on MISE due to both the sampling scheme and the sampling rate is significant for finite samples. The results are also extended to a case where the irregular observations are generated from a mixture of first-order continuous-time processes.

Performance evaluation of methods for identifying continuous-time autoregressive processes

Identification of continuous-time autoregressive processes from discrete-time data by replacing the differentiation operator by an approximation is considered. A linear regression model can then be formulated. The least-squares method and the instrumental variables method must be used with some care to get parameter estimates of good quality. The bias is studied explicitly in the paper together with the asymptotic distribution, and expressions are presented for the covariance matrix of the estimated parameters. It turns out that there are small differences in the dominating bias term for the different methods, whereas the statistical properties are comparable. Overall, the performance is similar to that of a prediction error method for short sampling intervals.