Continuous Autoregressive Moving Average Research Articles

Modeling Survey Time Series Data with Flow-Observed CARMA Processes

Published survey data often are delivered as estimates computed over an epoch of time. Customers may desire to obtain survey estimates corresponding to epochs, or time points, that differ from the published estimates. This “change of support” problem can be addressed through the use of continuous-time models of the underlying population process, while taking into account the sampling error that survey data is subject to. The application of a Continuous AutoRegressive Moving Average (CARMA) model is investigated as a tool to provide change of support applications, thereby allowing interpolation for published survey estimates. A simulation study provides comparisons of competing estimation methods, and a synthetically constructed data set is developed in order to elucidate real data applications. The proposed method can be successful for change of support problems, despite modeling challenges with the CARMA framework.

Rapid optical flare in the extreme teraelectronvolt blazar 1ES 0229+200 on intraday timescales with the Transiting Exoplanet Survey Satellite

Context. The extreme teraelectronvolt (TeV) blazar 1ES 0229+200 is a high-frequency-peaked BL Lacertae object. It has not shown intraday variability in extensive optical and X-ray observations, nor has it shown any significant variability on any measurable timescale in the 1–100 GeV energy range over a 14-year span; however, variations in the source flux around its average are present in the energy range above 200 GeV. Aims. We aim to search for intraday optical variability in 1ES 0229+200 as part of an ongoing project to search for variability and quasi-periodic oscillations in the high-cadence (2 min), nearly uniformly sampled optical light curves of blazars provided by the Transiting Exoplanet Survey Satellite (TESS). Methods. 1ES 0229+200 was monitored by TESS in its Sectors 42, 43, and 44. We analysed the data of all these three sectors both with the TESS-provided lightkurve software and the eleanor reduction pipeline. We detected a strong, essentially symmetric flare that lasted about 6 h in Sector 42. We fitted the flare’s rising and declining phases to exponential functions. We also analysed the light curve of Sector 42 using the Lomb-Scargle periodogram (LSP) and continuous auto-regressive moving average (CARMA) methods. Results. The optical light curve of Sector 42 of the TESS observations displayed in the present work provides the first evidence of a strong, rapid, short-lived optical flare on the intraday timescale in the TeV blazar 1ES 0229+200. The variability timescale of the flare provides the upper limit for the size of the emission region to be within (3.3 ± 0.2–8.3 ± 0.5)×1015 cm. Away from the flare, the slope of the periodogram’s power spectrum is fairly typical of many blazars (α < 2), but the nominal slopes for the flaring regions are very steep (α ∼ 4.3), which may indicate that the electron distribution undergoes a sudden change. We discuss possible emission mechanisms that could explain this substantial and rapid flare.

TD-CARMA: Painless, Accurate, and Scalable Estimates of Gravitational Lens Time Delays with Flexible CARMA Processes

Cosmological parameters encoding our understanding of the expansion history of the universe can be constrained by the accurate estimation of time delays arising in gravitationally lensed systems. We propose TD-CARMA, a Bayesian method to estimate cosmological time delays by modeling observed and irregularly sampled light curves as realizations of a continuous auto-regressive moving average (CARMA) process. Our model accounts for heteroskedastic measurement errors and microlensing, an additional source of independent extrinsic long-term variability in the source brightness. The semiseparable structure of the CARMA covariance matrix allows for fast and scalable likelihood computation using Gaussian process modeling. We obtain a sample from the joint posterior distribution of the model parameters using a nested sampling approach. This allows for “painless” Bayesian computation, dealing with the expected multimodality of the posterior distribution in a straightforward manner and not requiring the specification of starting values or an initial guess for the time delay, unlike existing methods. In addition, the proposed sampling procedure automatically evaluates the Bayesian evidence, allowing us to perform principled Bayesian model selection. TD-CARMA is parsimonious, and typically includes no more than a dozen unknown parameters. We apply TD-CARMA to six doubly lensed quasars HS2209+1914, SDSS J1001+5027, SDSS J1206+4332, SDSS J1515+1511, SDSS J1455+1447, and SDSS J1349+1227, estimating their time delays as −21.96 ± 1.448, 120.93 ± 1.015, 111.51 ± 1.452, 210.80 ± 2.18, 45.36 ± 1.93, and 432.05 ± 1.950, respectively. These estimates are consistent with those derived in the relevant literature, but are typically two to four times more precise.

Legacy Survey of Space and Time cadence strategy evaluations for active galactic nucleus time-series data in Wide-Fast-Deep field

ABSTRACT Machine learning is a promising tool to reconstruct time-series phenomena, such as variability of active galactic nuclei (AGNs), from sparsely sampled data. Here, we use three Continuous Autoregressive Moving Average (CARMA) representations of AGN variability – the Damped Random Walk (DRW) and (over/under)Damped Harmonic Oscillator – to simulate 10-yr AGN light curves as they would appear in the upcoming Vera Rubin Observatory Legacy Survey of Space and Time (LSST), and provide a public tool to generate these for any survey cadence. We investigate the impact on AGN science of five proposed cadence strategies for LSST’s primary Wide-Fast-Deep (WFD) survey. We apply for the first time in astronomy a novel Stochastic Recurrent Neural Network (SRNN) algorithm to reconstruct input light curves from the simulated LSST data, and provide a metric to evaluate how well SRNN can help recover the underlying CARMA parameters. We find that the light-curve reconstruction is most sensitive to the duration of gaps between observing season, and that of the proposed cadences, those that change the balance between filters, or avoid having long gaps in the g band perform better. Overall, SRNN is a promising means to reconstruct densely sampled AGN light curves and recover the long-term structure function of the DRW process (SF∞) reasonably well. However, we find that for all cadences, CARMA/SRNN models struggle to recover the decorrelation time-scale (τ) due to the long gaps in survey observations. This may indicate a major limitation in using LSST WFD data for AGN variability science.

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 1: Frequency analysis

Abstract. We develop a general framework for the frequency analysis of irregularly sampled time series. It is based on the Lomb–Scargle periodogram, but extended to algebraic operators accounting for the presence of a polynomial trend in the model for the data, in addition to a periodic component and a background noise. Special care is devoted to the correlation between the trend and the periodic component. This new periodogram is then cast into the Welch overlapping segment averaging (WOSA) method in order to reduce its variance. We also design a test of significance for the WOSA periodogram, against the background noise. The model for the background noise is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, more general than the classical Gaussian white or red noise processes. CARMA parameters are estimated following a Bayesian framework. We provide algorithms that compute the confidence levels for the WOSA periodogram and fully take into account the uncertainty in the CARMA noise parameters. Alternatively, a theory using point estimates of CARMA parameters provides analytical confidence levels for the WOSA periodogram, which are more accurate than Markov chain Monte Carlo (MCMC) confidence levels and, below some threshold for the number of data points, less costly in computing time. We then estimate the amplitude of the periodic component with least-squares methods, and derive an approximate proportionality between the squared amplitude and the periodogram. This proportionality leads to a new extension for the periodogram: the weighted WOSA periodogram, which we recommend for most frequency analyses with irregularly sampled data. The estimated signal amplitude also permits filtering in a frequency band. Our results generalise and unify methods developed in the fields of geosciences, engineering, astronomy and astrophysics. They also constitute the starting point for an extension to the continuous wavelet transform developed in a companion article (Lenoir and Crucifix, 2018). All the methods presented in this paper are available to the reader in the Python package WAVEPAL.

A general theory on frequency and time–frequency analysis of irregularly sampled time series based on projection methods – Part 2: Extension to time–frequency analysis

Abstract. Geophysical time series are sometimes sampled irregularly along the time axis. The situation is particularly frequent in palaeoclimatology. Yet, there is so far no general framework for handling the continuous wavelet transform when the time sampling is irregular. Here we provide such a framework. To this end, we define the scalogram as the continuous-wavelet-transform equivalent of the extended Lomb–Scargle periodogram defined in Part 1 of this study (Lenoir and Crucifix, 2018). The signal being analysed is modelled as the sum of a locally periodic component in the time–frequency plane, a polynomial trend, and a background noise. The mother wavelet adopted here is the Morlet wavelet classically used in geophysical applications. The background noise model is a stationary Gaussian continuous autoregressive-moving-average (CARMA) process, which is more general than the traditional Gaussian white and red noise processes. The scalogram is smoothed by averaging over neighbouring times in order to reduce its variance. The Shannon–Nyquist exclusion zone is however defined as the area corrupted by local aliasing issues. The local amplitude in the time–frequency plane is then estimated with least-squares methods. We also derive an approximate formula linking the squared amplitude and the scalogram. Based on this property, we define a new analysis tool: the weighted smoothed scalogram, which we recommend for most analyses. The estimated signal amplitude also gives access to band and ridge filtering. Finally, we design a test of significance for the weighted smoothed scalogram against the stationary Gaussian CARMA background noise, and provide algorithms for computing confidence levels, either analytically or with Monte Carlo Markov chain methods. All the analysis tools presented in this article are available to the reader in the Python package WAVEPAL.

Stochastic Mortality Modelling: Some Extensions Based on LLvy CARMA Models

Force of mortality is defined using an exponential function of Legendre polynomials, as in Renshaw et al. (1996), plus an extra term which captures mortality shocks. For the extra term Ballotta Haberman (2006) and Ahmadi et al. (2015) consider an Ornstein-Uhlenbeck while we suggest using Levy Continuous Autoregressive Moving Average (CARMA) models. The proposed model encompasses as special cases the existing approaches. We present some extensions in the theory of Levy CARMA models that enter in the construction of the model estimation procedure. Males life tables for USA, Taiwan and Japan for the time period 1998-2013 are used for the presentation of fitting and projection results.

SDSS J14584479+3720215: A BENCHMARKJHKSBLAZAR LIGHT CURVE FROM THE 2MASS CALIBRATION SCANS

Active galactic nuclei (AGNs) are well-known to exhibit flux variability across a wide range of wavelength regimes, but the precise origin of the variability at different wavelengths remains unclear. To investigate the relatively unexplored near-IR variability of the most luminous AGNs, we conduct a search for variability using well sampled JHKs-band light curves from the 2MASS survey calibration fields. Our sample includes 27 known quasars with an average of 924 epochs of observation over three years, as well as one spectroscopically confirmed blazar (SDSSJ14584479+3720215) with 1972 epochs of data. This is the best-sampled NIR photometric blazar light curve to date, and it exhibits correlated, stochastic variability that we characterize with continuous auto-regressive moving average (CARMA) models. None of the other 26 known quasars had detectable variability in the 2MASS bands above the photometric uncertainty. A blind search of the 2MASS calibration field light curves for AGN candidates based on fitting CARMA(1,0) models (damped-random walk) uncovered only 7 candidates. All 7 were young stellar objects within the {\rho} Ophiuchus star forming region, five with previous X-ray detections. A significant {\gamma}-ray detection (5{\sigma}) for the known blazar using 4.5 years of Fermi photon data is also found. We suggest that strong NIR variability of blazars, such as seen for SDSSJ14584479+3720215, can be used as an efficient method of identifying previously-unidentified {\gamma}-ray blazars, with low contamination from other AGN.

Implementation of Lévy CARMA model in Yuima package

The paper shows how to use the R package yuima available on CRAN for the simulation and the estimation of a general Levy Continuous Autoregressive Moving Average (CARMA) model. The flexibility of the package is due to the fact that the user is allowed to choose several parametric Levy distribution for the increments. Some numerical examples are given in order to explain the main classes and the corresponding methods implemented in yuima package for the CARMA model.

A Note on Non-Negative Continuous Time Processes

SummaryRecently there has been much work on developing models that are suitable for analysing the volatility of a continuous time process. One general approach is to define a volatility process as the convolution of a kernel with a non-decreasing Lévy process, which is non-negative if the kernel is non-negative. Within the framework of time continuous autoregressive moving average (CARMA) processes, we derive a necessary and sufficient condition for the kernel to be non-negative. This condition is in terms of the Laplace transform of the CARMA kernel, which has a simple form. We discuss some useful consequences of this result and delineate the parametric region of stationarity and non-negative kernel for some lower order CARMA models.

Fitting Continuous ARMA Models to Unequally Spaced Spatial Data

Abstract Methods for fitting continuous spatial autoregressive moving average (ARMA) models to unequally spaced observations in two dimensions are reviewed and extended. These are models with rational two-dimensional spectra. Assuming Gaussian input noise and observational errors, maximum likelihood methods are used to estimate the ARMA parameters and the regression coefficients of the deterministic trend. When the number of observations is too large for exact maximum likelihood estimation, approximate maximum likelihood estimation is used based on nearest neighbors. Comparisons of nearest-neighbor methods with exact likelihood methods are presented. Predictions of the height of the field at unobserved points can be calculated with confidence intervals.