Least-Squares Fitting of Multidimensional Spectra to Kubo Lineshape Models

We report a comprehensive study of the efficacy of least-squares fitting of multidimensional spectra to generalized Kubo lineshape models and introduce a novel least-squares fitting metric, termed the Scale Invariant Gradient Norm (SIGN), that enables a highly reliable and versatile algorithm. The precision of dephasing parameters is between 8× to 50× better for nonlinear model fitting compared to the CLS method, which effectively increases data acquisition efficiency by one to two orders of magnitude. Whereas the center-line-slope (CLS) method requires sequential fitting of both the nonlinear and linear spectra, our model fitting algorithm only requires nonlinear spectra, but accurately predicts the linear spectrum. We show an experimental example in which the CLS time constants differ by 60% for independent measurements of the same system, while the Kubo time constants differ by only 10% for model fitting. This suggests that model fitting is a far more robust method of measuring spectral diffusion than the CLS method, which is more susceptible to structured residual signals that are not removable by pure solvent subtraction. Statistical analysis of the CLS method reveals a fundamental oversight in accounting for the propagation of uncertainty by Kubo time constants in the process of fitting to the linear absorption spectrum. A standalone desktop app and source code for the least-squares fitting algorithm are freely available with example lineshape models and data. We have written the MATLAB source code in a generic framework where users may supply custom lineshape models. Using this application, a standard desktop fits a 12-parameter generalized Kubo model to a 106 data-point spectrum in a few minutes.

A Short Guide to Using Python For Data Analysis In Experimental Physics

Common signal processing tasks in the numerical handling of experimental data include interpolation, smoothing, and propagation of uncertainty. A comparison of experimental results to a theoretical model further requires curve fitting, the plotting of functions and data, and a determination of the goodness of fit. These tasks often typically require an interactive, exploratory approach to the data, yet for the results to be reliable, the original data needs to be freely available and resulting analysis readily reproducible. In this article, we provide examples of how to use the Numerical Python (Numpy) and Scientific Python (SciPy) packages and interactive Jupyter Notebooks to accomplish these goals for data stored in a common plain text spreadsheet format. Sample Jupyter notebooks containing the Python code used to carry out these tasks are included and can be used as templates for the analysis of new data.

Getting started with uncertainty evaluation using the Monte Carlo method in R

The evaluation of measurement uncertainty is often perceived by laboratory staff as complex and quite distant from daily practice. Nevertheless, standards such as ISO/IEC 17025, ISO 15189 and ISO 17034 that specify requirements for laboratories to enable them to demonstrate they operate competently, and are able to generate valid results, require that measurement uncertainty is evaluated and reported. In response to this need, a European project entitled “Advancing measurement uncertainty—comprehensive examples for key international standards” started in July 2018 that aims at developing examples that contribute to a better understanding of what is required and aid in implementing such evaluations in calibration, testing and research. The principle applied in the project is “learning by example”. Past experience with guidance documents such as EA 4/02 and the Eurachem/CITAC guide on measurement uncertainty has shown that for practitioners it is often easier to rework and adapt an existing example than to try to develop something from scratch. This introductory paper describes how the Monte Carlo method of GUM (Guide to the expression of Uncertainty in Measurement) Supplement 1 can be implemented in R, an environment for mathematical and statistical computing. An implementation of the law of propagation of uncertainty is also presented in the same environment, taking advantage of the possibility of evaluating the partial derivatives numerically, so that these do not need to be derived by analytic differentiation. The implementations are shown for the computation of the molar mass of phenol from standard atomic masses and the well-known mass calibration example from EA 4/02.

Hydrostatic Determination of Test Weight Density and Uncertainty Evaluation by Monte Carlo Method

This paper presents the characterization of the hydrostatic weighing facility of the National Metrology Laboratory (NML) of the Philippines. The study aimed to evaluate its suitability for determination of solid density. It was used to hydrostatically measure the density of a stainless steel (OIML Class F1) test weight weighing 200 g. The measurement result obtained was 7.5827 g cm-3 ± 0.0041 g cm-3 at an approximately 95 % level of confidence. The uncertainty evaluated by the Law of Propagation of Uncertainty (LPU) according to JCGM 100:2008 (GUM) was verified by the Monte Carlo method (MCM), which gave a result of 0.0040 g cm-3. The value determined for the solid density of the sample with its associated expanded uncertainty was found to be within the tolerance interval between 7.39 g cm-3 and 8.73 g cm-3 as required in OIML R111-1 for Class F1 test weights.

Interpretation and use of standard atomic weights (IUPAC Technical Report)

Many calculations for science or trade require the evaluation and propagation of measurement uncertainty. Although relative atomic masses (standard atomic weights) of elements in normal terrestrial materials and chemicals are widely used in science, the uncertainties associated with these values are not well understood. In this technical report, guidelines for the use of standard atomic weights are given. This use involves the derivation of a value and a standard uncertainty from a standard atomic weight, which is explained in accordance with the requirements of the Guide to the Expression of Uncertainty in Measurement. Both the use of standard atomic weights with the law of propagation of uncertainty and the Monte Carlo method are described. Furthermore, methods are provided for calculating uncertainties of relative molecular masses of substances and their mixtures. Methods are also outlined to compute material-specific atomic weights whose associated uncertainty may be smaller than the uncertainty associated with the standard atomic weights.

ANALYTICAL EXPRESSION FOR UNCERTAINTY PROPAGATION OF AERIAL COOPERATIVE NAVIGATION

In this paper, the propagation of uncertainty in a cooperative navigation algorithm (CNA) for a group of flying robots (FRs) is investigated. Each FR is equipped with an inertial measurement unit (IMU) and range-bearing sensors to measure the relative distance and bearing angles between the agents. In this regard, an extended Kalman filter (EKF) is implemented to estimate the position and rotation angles of all the agents. For further studies, a relaxed analytical performance index through a closed-form solution is derived. Moreover, the effects of the sensors noise covariance and the number of FRs on the growth rate of the position error covariance is investigated. Analytically, it is shown that the covariance of position error in the vehicles equipped with the IMU is proportional to the cube of time. However, the growth rate of the navigation error is, considerably more rapid compared to a mobile robot group. Furthermore, the covariance of position error is independent of the path and noise resulting from the relative position measurements. Further, it merely depends on both the size of the group and noise characteristics of the accelerometers. Lastly, the analytical results are validated through comprehensive Guidance, Navigation, and Control (GNC) in-the-loop simulations.