Period, epoch and prediction errors of ephemeris from continuous sets of timing measurements

Space missions such as Kepler and CoRoT have led to large numbers of eclipse or transit measurements in nearly continuous time series. This paper shows how to obtain the period error in such measurements from a basic linear least-squares fit, and how to correctly derive the timing error in the prediction of future transit or eclipse events. Assuming strict periodicity, a formula for the period error of such time series is derived: sigma_P = sigma_T (12/( N^3-N))^0.5, where sigma_P is the period error; sigma_T the timing error of a single measurement and N the number of measurements. Relative to the iterative method for period error estimation by Mighell & Plavchan (2013), this much simpler formula leads to smaller period errors, whose correctness has been verified through simulations. For the prediction of times of future periodic events, the usual linear ephemeris where epoch errors are quoted for the first time measurement, are prone to overestimation of the error of that prediction. This may be avoided by a correction for the duration of the time series. An alternative is the derivation of ephemerides whose reference epoch and epoch error are given for the centre of the time series. For long continuous or near-continuous time series whose acquisition is completed, such central epochs should be the preferred way for the quotation of linear ephemerides. While this work was motivated from the analysis of eclipse timing measures in space-based light curves, it should be applicable to any other problem with an uninterrupted sequence of discrete timings for which the determination of a zero point, of a constant period and of the associated errors is needed.Comment: Astronomy and Astrophysics, accepte

Coherent Predictions of Low Count Time Series

The application of traditional forecasting methods to discrete count data yields forecasts that are non-coherent. That is, such methods produce non-integer point and interval predictions which violate the restrictions on the sample space of the integer variable. This paper presents a methodology for producing coherent forecasts of low count time series. The forecasts are based on estimates of the p-step ahead predictive mass functions for a family of distributions nested in the integer-valued first-order autoregressive (INAR(1)) class. The predictive mass functions are constructed from convolutions of the unobserved components of the model, with uncertainty associated with both parameter values and model specifcation fully incorporated. The methodology is used to analyse two sets of Canadian wage loss claims data.Forecasting; Discrete Time Series; INAR(1); Bayesian Prediction; Bayesian Model Averaging.

Learning Linear Dynamical Systems via Spectral Filtering

We present an efficient and practical algorithm for the online prediction of discrete-time linear dynamical systems with a symmetric transition matrix. We circumvent the non-convex optimization problem using improper learning: carefully overparameterize the class of LDSs by a polylogarithmic factor, in exchange for convexity of the loss functions. From this arises a polynomial-time algorithm with a near-optimal regret guarantee, with an analogous sample complexity bound for agnostic learning. Our algorithm is based on a novel filtering technique, which may be of independent interest: we convolve the time series with the eigenvectors of a certain Hankel matrix.Comment: Published as a conference paper at NIPS 201

The Trace Problem for Toeplitz Matrices and Operators and its Impact in Probability

The trace approximation problem for Toeplitz matrices and its applications to stationary processes dates back to the classic book by Grenander and Szeg\"o, "Toeplitz forms and their applications". It has then been extensively studied in the literature. In this paper we provide a survey and unified treatment of the trace approximation problem both for Toeplitz matrices and for operators and describe applications to discrete- and continuous-time stationary processes. The trace approximation problem serves indeed as a tool to study many probabilistic and statistical topics for stationary models. These include central and non-central limit theorems and large deviations of Toeplitz type random quadratic functionals, parametric and nonparametric estimation, prediction of the future value based on the observed past of the process, etc. We review and summarize the known results concerning the trace approximation problem, prove some new results, and provide a number of applications to discrete- and continuous-time stationary time series models with various types of memory structures, such as long memory, anti-persistent and short memory

AN APPROACH OF TRAFFIC FLOW PREDICTION USING ARIMA MODEL WITH FUZZY WAVELET TRANSFORM

It is essential for intelligent transportation systems to be capable of producing an accurate forecast of traffic flow in both the short and long terms. However, the counting datasets of traffic volume are non-stationary time series, which are integrally noisy. As a result, the accuracy of traffic prediction carried out on such unrefined data is reduced by the arbitrary components. A prior study shows that Box-Jenkins’ Autoregressive Integrated Moving Average (ARIMA) models convey demand of noise-free dataset for model construction. Therefore, this study proposes to overcome the noise issue by using a hybrid approach that combines the ARIMA model with fuzzy wavelet transform. In this approach, fuzzy rules are developed to categorize traffic datasets according to influencing factors such as the time of a day, the season of a year, and weather conditions. As the input of linear data series for ARIMA model needs to be converted into linear time series for traffic flow prediction, the discrete wavelet transform is applied to help separating the nonlinear and linear part of the time series along with denoised time series traffic data

Recursive estimation for continuous time stochastic volatility models

AbstractVolatility plays an important role in portfolio management and option pricing. Recently, there has been a growing interest in modeling volatility of the observed process by nonlinear stochastic process [S.J. Taylor, Asset Price Dynamics, Volatility, and Prediction, Princeton University Press, 2005; H. Kawakatsu, Specification and estimation of discrete time quadratic stochastic volatility models, Journal of Empirical Finance 14 (2007) 424–442]. In [H. Gong, A. Thavaneswaran, J. Singh, Filtering for some time series models by using transformation, Math Scientist 33 (2008) 141–147], we have studied the recursive estimates for discrete time stochastic volatility models driven by normal errors. In this paper, we study the recursive estimates for various classes of continuous time nonlinear non-Gaussian stochastic volatility models used for option pricing in finance