Pulsar timing analysis in the presence of correlated noise

Pulsar timing observations are usually analysed with least-square-fitting procedures under the assumption that the timing residuals are uncorrelated (statistically "white"). Pulsar observers are well aware that this assumption often breaks down and causes severe errors in estimating the parameters of the timing model and their uncertainties. Ad hoc methods for minimizing these errors have been developed, but we show that they are far from optimal. Compensation for temporal correlation can be done optimally if the covariance matrix of the residuals is known using a linear transformation that whitens both the residuals and the timing model. We adopt a transformation based on the Cholesky decomposition of the covariance matrix, but the transformation is not unique. We show how to estimate the covariance matrix with sufficient accuracy to optimize the pulsar timing analysis. We also show how to apply this procedure to estimate the spectrum of any time series with a steep red power-law spectrum, including those with irregular sampling and variable error bars, which are otherwise very difficult to analyse.Comment: Accepted by MNRA

Functional Regression

Functional data analysis (FDA) involves the analysis of data whose ideal units of observation are functions defined on some continuous domain, and the observed data consist of a sample of functions taken from some population, sampled on a discrete grid. Ramsay and Silverman's 1997 textbook sparked the development of this field, which has accelerated in the past 10 years to become one of the fastest growing areas of statistics, fueled by the growing number of applications yielding this type of data. One unique characteristic of FDA is the need to combine information both across and within functions, which Ramsay and Silverman called replication and regularization, respectively. This article will focus on functional regression, the area of FDA that has received the most attention in applications and methodological development. First will be an introduction to basis functions, key building blocks for regularization in functional regression methods, followed by an overview of functional regression methods, split into three types: [1] functional predictor regression (scalar-on-function), [2] functional response regression (function-on-scalar) and [3] function-on-function regression. For each, the role of replication and regularization will be discussed and the methodological development described in a roughly chronological manner, at times deviating from the historical timeline to group together similar methods. The primary focus is on modeling and methodology, highlighting the modeling structures that have been developed and the various regularization approaches employed. At the end is a brief discussion describing potential areas of future development in this field

Millisecond and Binary Pulsars as Nature's Frequency Standards. II. Effects of Low-Frequency Timing Noise on Residuals and Measured Parameters

Pulsars are the most stable natural frequency standards. They can be applied to a number of principal problems of modern astronomy and time-keeping metrology. The full exploration of pulsar properties requires obtaining unbiased estimates of the spin and orbital parameters. These estimates depend essentially on the random noise component being revealed in the residuals of time of arrivals (TOA). In the present paper, the influence of low-frequency ("red") timing noise with spectral indices from 1 to 6 on TOA residuals, variances, and covariances of estimates of measured parameters of single and binary pulsars are studied. In order to determine their functional dependence on time, an analytic technique of processing of observational data in time domain is developed which takes into account both stationary and non-stationary components of noise. Our analysis includes a simplified timing model of a binary pulsar in a circular orbit and procedure of estimation of pulsar parameters and residuals under the influence of red noise. We reconfirm that uncorrelated white noise of errors of measurements of TOA brings on gradually decreasing residuals, variances and covariances of all parameters. On the other hand, we show that any red noise causes the residuals, variances, and covariances of certain parameters to increase with time. Hence, the low frequency noise corrupts our observations and reduces experimental possibilities for better tests of General Relativity Theory. We also treat in detail the influence of a polynomial drift of noise on the residuals and fitting parameters. Results of the analitic analysis are used for discussion of a statistic describing stabilities of kinematic and dynamic pulsar time scales.Comment: 40 pages, 1 postscript figure, 1 picture, uses mn.sty, accepted to Mon. Not. Roy. Astron. So

Interpolation of nonstationary high frequency spatial-temporal temperature data

The Atmospheric Radiation Measurement program is a U.S. Department of Energy project that collects meteorological observations at several locations around the world in order to study how weather processes affect global climate change. As one of its initiatives, it operates a set of fixed but irregularly-spaced monitoring facilities in the Southern Great Plains region of the U.S. We describe methods for interpolating temperature records from these fixed facilities to locations at which no observations were made, which can be useful when values are required on a spatial grid. We interpolate by conditionally simulating from a fitted nonstationary Gaussian process model that accounts for the time-varying statistical characteristics of the temperatures, as well as the dependence on solar radiation. The model is fit by maximizing an approximate likelihood, and the conditional simulations result in well-calibrated confidence intervals for the predicted temperatures. We also describe methods for handling spatial-temporal jumps in the data to interpolate a slow-moving cold front.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS633 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

Robustness of ToF estimators - an Empirical Evaluation

The performance of ToF estimators for acoustic tone bursts is empirically evaluated. In indoor applications, the observed waveform is likely to be disrupted by multiple echoes. These echoes can cause complex interference patterns. The paper presents the results of a comparison study of the robustness of various ToF estimators against such type of disruptions

Optimal Estimation of Several Linear Parameters in the Presence of Lorentzian Thermal Noise

In a previous article we developed an approach to the optimal (minimum variance, unbiased) statistical estimation technique for the equilibrium displacement of a damped, harmonic oscillator in the presence of thermal noise. Here, we expand that work to include the optimal estimation of several linear parameters from a continuous time series. We show that working in the basis of the thermal driving force both simplifies the calculations and provides additional insight to why various approximate (not optimal) estimation techniques perform as they do. To illustrate this point, we compare the variance in the optimal estimator that we derive for thermal noise with those of two approximate methods which, like the optimal estimator, suppress the contribution to the variance that would come from the irrelevant, resonant motion of the oscillator. We discuss how these methods fare when the dominant noise process is either white displacement noise or noise with power spectral density that is inversely proportional to the frequency ($1/f$ noise). We also construct, in the basis of the driving force, an estimator that performs well for a mixture of white noise and thermal noise. To find the optimal multi-parameter estimators for thermal noise, we derive and illustrate a generalization of traditional matrix methods for parameter estimation that can accommodate continuous data. We discuss how this approach may help refine the design of experiments as they allow an exact, quantitative comparison of the precision of estimated parameters under various data acquisition and data analysis strategies.Comment: 16 pages, 10 figures. Accepted for publication in Classical and Quantum Gravit