Nonparametric Methods in Astronomy: Think, Regress, Observe -- Pick Any Three

Telescopes are much more expensive than astronomers, so it is essential to minimize required sample sizes by using the most data-efficient statistical methods possible. However, the most commonly used model-independent techniques for finding the relationship between two variables in astronomy are flawed. In the worst case they can lead without warning to subtly yet catastrophically wrong results, and even in the best case they require more data than necessary. Unfortunately, there is no single best technique for nonparametric regression. Instead, we provide a guide for how astronomers can choose the best method for their specific problem and provide a python library with both wrappers for the most useful existing algorithms and implementations of two new algorithms developed here.Comment: 19 pages, PAS

Spitzer Secondary Eclipses of the Dense, Modestly-irradiated, Giant Exoplanet HAT-P-20b Using Pixel-Level Decorrelation

HAT-P-20b is a giant exoplanet that orbits a metal-rich star. The planet itself has a high total density, suggesting that it may also have a high metallicity in its atmosphere. We analyze two eclipses of the planet in each of the 3.6- and 4.5 micron bands of Warm Spitzer. These data exhibit intra-pixel detector sensitivity fluctuations that were resistant to traditional decorrelation methods. We have developed a simple, powerful, and radically different method to correct the intra-pixel effect for Warm Spitzer data, which we call pixel-level decorrelation (PLD). PLD corrects the intra-pixel effect very effectively, but without explicitly using - or even measuring - the fluctuations in the apparent position of the stellar image. We illustrate and validate PLD using synthetic and real data, and comparing the results to previous analyses. PLD can significantly reduce or eliminate red noise in Spitzer secondary eclipse photometry, even for eclipses that have proven to be intractable using other methods. Our successful PLD analysis of four HAT-P-20b eclipses shows a best-fit blackbody temperature of 1134 +/-29K, indicating inefficient longitudinal transfer of heat, but lacking evidence for strong molecular absorption. We find sufficient evidence for variability in the 4.5 micron band that the eclipses should be monitored at that wavelength by Spitzer, and this planet should be a high priority for JWST spectroscopy. All four eclipses occur about 35 minutes after orbital phase 0.5, indicating a slightly eccentric orbit. A joint fit of the eclipse and transit times with extant RV data yields e(cos{omega}) = 0.01352 (+0.00054, -0.00057), and establishes the small eccentricity of the orbit to high statistical confidence. Given the existence of a bound stellar companion, HAT-P-20b is another excellent candidate for orbital evolution via Kozai migration or other three-body mechanism.Comment: version published in ApJ, minor text and figure revision

Robust nonparametric estimation via wavelet median regression

In this paper we develop a nonparametric regression method that is simultaneously adaptive over a wide range of function classes for the regression function and robust over a large collection of error distributions, including those that are heavy-tailed, and may not even possess variances or means. Our approach is to first use local medians to turn the problem of nonparametric regression with unknown noise distribution into a standard Gaussian regression problem and then apply a wavelet block thresholding procedure to construct an estimator of the regression function. It is shown that the estimator simultaneously attains the optimal rate of convergence over a wide range of the Besov classes, without prior knowledge of the smoothness of the underlying functions or prior knowledge of the error distribution. The estimator also automatically adapts to the local smoothness of the underlying function, and attains the local adaptive minimax rate for estimating functions at a point. A key technical result in our development is a quantile coupling theorem which gives a tight bound for the quantile coupling between the sample medians and a normal variable. This median coupling inequality may be of independent interest.Comment: Published in at http://dx.doi.org/10.1214/07-AOS513 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Brightness of Solar Magnetic Elements as a Function of Magnetic Flux at High Spatial Resolution

We investigate the relationship between the photospheric magnetic field of small-scale magnetic elements in the quiet Sun (QS) at disc centre, and the brightness at 214 nm, 300 nm, 313 nm, 388 nm, 397 nm, and at 525.02 nm. To this end we analysed spectropolarimetric and imaging time series acquired simultaneously by the IMaX magnetograph and the SuFI filter imager on-board the balloon-borne observatory Sunrise during its first science flight in 2009, with high spatial and temporal resolution. We find a clear dependence of the contrast in the near ultraviolet (NUV) and the visible on the line-of-sight component of the magnetic field, $B_{\rm LOS}$, which is best described by a logarithmic model. This function represents well the relationship between the Ca II H-line emission and $B_{\rm LOS}$, and works better than a power-law fit adopted by previous studies. This, along with the high contrast reached at these wavelengths, will help with determining the contribution of small-scale elements in the QS to the irradiance changes for wavelengths below 388 nm. At all wavelengths including the continuum at 525.40 nm the intensity contrast does not decrease with increasing $B_{\rm LOS}$. This result also strongly supports that Sunrise has resolved small strong magnetic field elements in the internetwork, resulting in constant contrasts for large magnetic fields in our continuum contrast at 525.40 nm vs. $B_{\rm LOS}$ scatterplot, unlike the turnover obtained in previous observational studies. This turnover is due to the intermixing of the bright magnetic features with the dark intergranular lanes surrounding them