Fitting Effective Diffusion Models to Data Associated with a "Glassy Potential": Estimation, Classical Inference Procedures and Some Heuristics

A variety of researchers have successfully obtained the parameters of low dimensional diffusion models using the data that comes out of atomistic simulations. This naturally raises a variety of questions about efficient estimation, goodness-of-fit tests, and confidence interval estimation. The first part of this article uses maximum likelihood estimation to obtain the parameters of a diffusion model from a scalar time series. I address numerical issues associated with attempting to realize asymptotic statistics results with moderate sample sizes in the presence of exact and approximated transition densities. Approximate transition densities are used because the analytic solution of a transition density associated with a parametric diffusion model is often unknown.I am primarily interested in how well the deterministic transition density expansions of Ait-Sahalia capture the curvature of the transition density in (idealized) situations that occur when one carries out simulations in the presence of a "glassy" interaction potential. Accurate approximation of the curvature of the transition density is desirable because it can be used to quantify the goodness-of-fit of the model and to calculate asymptotic confidence intervals of the estimated parameters. The second part of this paper contributes a heuristic estimation technique for approximating a nonlinear diffusion model. A "global" nonlinear model is obtained by taking a batch of time series and applying simple local models to portions of the data. I demonstrate the technique on a diffusion model with a known transition density and on data generated by the Stochastic Simulation Algorithm.Comment: 30 pages 10 figures Submitted to SIAM MMS (typos removed and slightly shortened

SARAS 2: A Spectral Radiometer for probing Cosmic Dawn and the Epoch of Reionization through detection of the global 21 cm signal

The global 21 cm signal from Cosmic Dawn (CD) and the Epoch of Reionization (EoR), at redshifts $z \sim 6-30$, probes the nature of first sources of radiation as well as physics of the Inter-Galactic Medium (IGM). Given that the signal is predicted to be extremely weak, of wide fractional bandwidth, and lies in a frequency range that is dominated by Galactic and Extragalactic foregrounds as well as Radio Frequency Interference, detection of the signal is a daunting task. Critical to the experiment is the manner in which the sky signal is represented through the instrument. It is of utmost importance to design a system whose spectral bandpass and additive spurious can be well calibrated and any calibration residual does not mimic the signal. SARAS is an ongoing experiment that aims to detect the global 21 cm signal. Here we present the design philosophy of the SARAS 2 system and discuss its performance and limitations based on laboratory and field measurements. Laboratory tests with the antenna replaced with a variety of terminations, including a network model for the antenna impedance, show that the gain calibration and modeling of internal additives leave no residuals with Fourier amplitudes exceeding 2~mK, or residual Gaussians of 25 MHz width with amplitudes exceeding 2~mK. Thus, even accounting for reflection and radiation efficiency losses in the antenna, the SARAS~2 system is capable of detection of complex 21-cm profiles at the level predicted by currently favoured models for thermal baryon evolution.Comment: 44 pages, 17 figures; comments and suggestions are welcom

Nonlinear time-series analysis revisited

In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data---typically univariate---via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems

Time series analysis of private healthcare expenditures GDP: cointegration results with structural breaks

This paper analyses the time-series behaviour of private health expenditure and GDP to understand whether there is long-term equilibrium relationship between these two variables and estimate income elasticity of private health expenditure. The study uses cointegration analysis with structural breaks and estimates these relationships using FM OLS (fully modified ordinary least squares) method. The findings suggest that income elasticity of private health expenditures is 1.95 indicating that for every one per cent increase in per capita income the private health expenditure has gone up by 1.95 per cent. The private health expenditure was 2.4 per cent of GDP in 1960 and this has risen to 5.8 per cent in 2003. In nominal terms it has grown at the rate of 11.3 per cent since 1960 and during 1990’s the growth rate is 18 per cent per annum. The study discusses four reasons for this high growth experience. These are: (i) financing mechanisms including provider payment system, (ii) demographic trends and epidemiological transition, (iii) production function of private health services delivery system, and (iv) dwindling financing support to public health system. In developing countries where per se the need for spending on health is high, high levels of private health expenditures pose serious challenge to policy makers. The sheer size of these expenditures once it has risen to high levels can impede control of health expenditures itself. The high private health expenditures are also cause of concern because most of these expenditures are out-of-pocket, insurance mechanisms cover small segment of population, provider payment systems are primarily based on fee-for-services and the professional regulation and accountability systems are weak and non-functioning in many ways. It is not clear whether these expenditures are sustainable as it can have number of undesirable consequences making the health system high cost, unaffordable, and vulnerable to provider payment system.

Misleading Regressions with Constructed Variables.

It is common practice to examine empirical models in which one of the regressors is constructed as the weighted average or sum of a set of series that includes the dependent variable. Examples include models relating money and wealth, consumption and income and regional and national unemployment. In this paper we show that biased results are likely to be generated by such models and that the identified bias is distinct from the more familiar simultaneous equation bias. The theoretical arguments are illustrated with simulation experiments and as a practical example we consider the relationship between regional and national unemployment in Australia.REGRESSION ANALYSIS ; ESTIMATOR ; EMPLOYMENT ; REGIONAL DEVELOPMENT