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There are many uses for linear fitting; we consider here the interpolation and denoising of data, as when the goal is to fit a smooth, flexible function to a set of noisy data points. Investigators often choose a polynomial basis, or a Fourier basis, or wavelets, or something equally general. They also choose an order, or number of basis functions to fit, and (often) some kind of regularization. We discuss how this basis-function fitting is done, with ordinary least squares and extensions thereof. We emphasize that it can be valuable to choose far more parameters than data points, despite folk rules to the contrary: Suitably regularized models with enormous numbers of parameters generalize well and make good predictions for held-out data; over-fitting is not (mainly) a problem of having too many parameters. It is even possible to take the limit of infinite parameters, at which, if the basis and regularization are chosen correctly, the least-squares fit becomes the mean of a Gaussian process, or a kernel regression. We recommend cross-validation as a good empirical method for model selection (for example, setting the number of parameters and the form of the regularization), and jackknife resampling as a good empirical method for estimating the uncertainties of the predictions made by the model. We also give advice for building stable computational implementations.

In this work, we propose a proper plasma analysis practice (PPAP), an updated procedure of plasma diagnostics in the era of spatially resolved spectroscopy. In particular, we emphasize the importance of performing both of the extinction correction and the direct method of plasma diagnostics simultaneously as an integrated process. This approach is motivated by the reciprocal dependence between critical parameters in these analyses, which can be resolved by iteratively seeking a converged solution. The use of PPAP allows us to eliminate unnecessary assumptions that prevent us from obtaining an exact solution at each element of the spectral imaging data. Using a suite of Hubble Space Telescope/WFC3 narrowband images of the planetary nebula, NGC 6720, we validate PPAP by (1) simultaneously and self-consistently deriving the extinction, c(Hβ), and electron density/temperature distribution, (nₑ([S II]), Tₑ([N II])), maps that are consistent with each other, and (2) obtaining identical metal abundance distribution maps, (n(N+)/n(H⁺), n(S⁺)/n(H⁺)), from multiple emission line maps at different wavelengths/transition energies. We also determine that the derived c(Hβ) consists both of the interstellar medium and circumsource components and that the ionized gas-to-dust mass ratio in the main ring is at least 437 and as high as about 1600. We find that, unless we deliberately seek self-consistency, uncertainties at tens of % can easily arise in outcomes, making it impossible to discern actual spatial variations that occurs at the same level, defeating the purpose of conducting spatially resolved spectroscopic observations.

Astrometry at centimeter wavelengths using Very Long Baseline Interferometry is approaching accuracies of ∼1 μas for the angle between a target and a calibrator source separated by ≲1° on the sky. The BeSSeL Survey and the Japanese VERA project are using this to map the spiral structure of the Milky Way by measuring trigonometric parallaxes of hundreds of maser sources associated with massive, young stars. This paper outlines how μas astrometry is done, including details regarding the scheduling of observations, calibration of data, and measuring positions.

The partition function, U, the number of available states in an atom or molecules, is crucial for understanding the physical state of any astrophysical system in thermodynamic equilibrium. There are surprisingly few useful discussions of the partition function’s numerical value. Textbooks often define U; some give tables of representative values, while others do a deep dive into the theory of dense plasma. Most say that it depends on temperature, atomic structure, density, and that it diverges, that is, it goes to infinity, at high temperatures, but few give practical examples. We aim to rectify this. We show that there are two limits, one- and two-electron (or closed-shell) systems like H or He, and species with a complicated electronic structure like C, N, O, and Fe. The high-temperature divergence does not occur for one- and two-electron systems in practical situations because, at high temperatures, species are collisionally ionized to higher-ionization stages and are not abundant. The partition function is then close to the statistical weight of the ground state. There is no such simplification for many-electron species. U is temperature sensitive across the range of temperatures where an ion is abundant but remains finite at even the highest practical temperatures. The actual value depends on highly uncertain truncation theories in highdensity plasmas. We show that there are various theories for continuum lowering but that they are not in good agreement. This remains a long-standing unsolved problem.

The Jupiter Observing Velocity Experiment (JOVE) is a solar-powered technology demonstration of rapid flight to outer solar system targets, performing a flyby of Jupiter 30 days after launch. This is achieved using a magnetic drag device to accelerate with the solar wind plasma. This “Wind Rider” propulsion system can potentially also decelerate against the Jovian magnetosphere dawn eddy, to enable Jupiter orbital insertion in future missions. The 16U cubesat bus contains scientific instruments to record the plasma parameters from the vicinity of the spacecraft, with principal measurements coming from a SPAN-I ion velocity sensor. This paper includes a description of the propulsive mechanisms and supporting subsystems and trajectory simulation results derived from solar wind measurements over the past two solar cycles. The objectives of the JOVE technology demonstrator design include: (1) verify Wind Rider stability and control; (2) characterize loss mechanisms in the solar wind, such as resistive losses in the plasma, as well as the magnetic field transient interaction time; (3) operate onboard instruments to measure the velocity and direction of the solar wind (SPAN-Ai) and speed of the spacecraft relative to the Earth (radio Doppler shift), to enable precision navigation on future science missions; and (4) characterize the Lift-to-Drag ratio of the plasma magnetic field. (The lift force enables lateral course control and maneuvering within the solar wind.) Applying existing scientific data from Voyagers and other deep space probes into new engineering models was important for enabling new insights about Wind Rider propulsion. It enables more science to be performed in a shorter amount of time, across the Jovian system.

In this paper we examine the Direct Method for measuring electron temperatures in H II regions, and the extent to which such measurements can provide meaningful information on the physical conditions in these regions. We discuss the limits to what can be inferred about electron temperatures from nebular emission line fluxes. We provide a new simplified method for estimating electron temperatures, including parameters that can be used to determine this from UV [O III] and [O II] oxygen lines observable in high-redshift objects using ground-based telescopes. We test this method on published UV high redshift observations and compare the results with reported electron temperatures.

This tutorial focuses on trials that assess outcomes by counting events that can occur zero, one, or more than one time in each participant. Trials and meta‐analyses can estimate treatment effects for count outcomes using rate differences or rate ratios. We explain why it may be appropriate to meta‐analyze count data to estimate rate ratios rather than odds ratios, risk ratios, or risk differences. We explain what count data are, how trials may estimate treatment effects, how to interpret such estimates, and how to extract data from trials that use count outcomes for meta‐analysis. Finally, we discuss some common misunderstandings and subtleties. Supplementary materials include an Excel file for performing calculations, mathematical background, and additional advice. This tutorial focuses on trials that assess outcomes by counting events that can occur zero, one, or more than one time in each participant. Trials and meta‐analyses can estimate treatment effects for count outcomes using rate differences or rate ratios. We explain why it may be appropriate to meta‐analyze count data to estimate rate ratios rather than odds ratios, risk ratios, or risk differences. We explain what count data are, how trials may estimate treatment effects, how to interpret such estimates, and how to extract data from trials that use count outcomes for meta‐analysis. Finally, we discuss some common misunderstandings and subtleties. Supplementary materials include an Excel file for performing calculations, mathematical background, and additional advice. Count data and rates micro learning module

The aim of this article is to present the most common statistical errors in meta‐analyses included in systematic reviews; these are confusing standard deviation and standard error, using heterogeneity estimators for choosing between a common‐effect and random‐effects model, improper handling of multiarm trials, and unnecessary and misinterpreted subgroup analyses. We introduce some useful terminology and explain what authors can do to avoid these errors and how peer reviewers can spot them. We have also developed a micro‐learning module to provide practical hands‐on tutorial. The aim of this article is to present the most common statistical errors in meta‐analyses included in systematic reviews; these are confusing standard deviation and standard error, using heterogeneity estimators for choosing between a common‐effect and random‐effects model, improper handling of multiarm trials, and unnecessary and misinterpreted subgroup analyses. We introduce some useful terminology and explain what authors can do to avoid these errors and how peer reviewers can spot them. We have also developed a micro‐learning module to provide practical hands‐on tutorial. Common Statistical Errors micro learning module

This work develops application techniques for stochastic modeling of Active Galactic Nuclei (AGNs) variability as a probe of accretion disk physics. Stochastic models, specifically Continuous Auto-Regressive Moving Average (CARMA) models, characterize light curves with a perturbation spectrum and an Impulse-Response function, which crucially provides an interpretation for variability timescales. CARMA timescales are not physical but rather, they describe correlation structure and ordered information in stochastic processes. We begin this tutorial by reviewing discrete auto-regressive and moving-average processes, we bridge these components to their continuous analogs, and lastly we investigate the significance of CARMA timescales, obtained by modeling a light curve in the time domain, in relation to the shape of the power spectrum (PSD) and structure function. We determine that higher order CARMA models, for example the Damped Harmonic Oscillator (DHO or CARMA(2, 1)) are more sensitive to deviations from a single-slope power-law description of AGN variability; unlike Damped Random Walks (DRW or CAR(1)) where the PSD slope is fixed, the DHO slope is not. Higher complexity stochastic models than the DRW capture additional covariance in data and output additional characteristic timescales that probe the driving mechanisms of variability. We provide code using Kali software to generate simulations of diverse complexity stochastic light curves. We also provide a heuristic discussion of aliasing effects in ground-based cadences and the importance of light curve length in regards to uncertainty and limitations in timescale estimation.

This tutorial provides guidance on creating clear and informative summary of findings tables for systematic reviews of interventions. This tutorial provides guidance on creating clear and informative summary of findings tables for systematic reviews of interventions. We will explain what information is required in the different sections of the table. In the accompanying micro‐learning module, you can test your knowledge of the key components of a summary of findings table and practice calculating absolute risks. Summary of findings table micro learning module