# Astrostatistics Tools (astropy.stats)¶

## Introduction¶

The astropy.stats package holds statistical functions or algorithms used in astronomy. While the scipy.stats and statsmodel packages contains a wide range of statistical tools, they are general-purpose packages and are missing some tools that are particularly useful or specific to astronomy. This package is intended to provide such functionality, but not to replace scipy.stats if its implementation satisfies astronomers’ needs.

## Getting Started¶

A number of different tools are contained in the stats package, and they can be accessed by importing them:

>>> from astropy import stats


A full list of the different tools are provided below. Please see the documentation for their different usage. For example, sigma clipping, which is common way to estimate the background of an image, can be performed with the sigma_clip() function. By default, the function returns a masked array where outliers are masked:

>>> data = [1, 5, 6, 8, 100, 5, 3, 2]
>>> stats.sigma_clip(data, sigma=2, maxiters=5)
masked_array(data=[1, 5, 6, 8, --, 5, 3, 2],
mask=[False, False, False, False,  True, False, False, False],
fill_value=999999)


Alternatively, the SigmaClip class provides an object-oriented interface to sigma clipping, which also returns a masked array by default:

>>> sigclip = stats.SigmaClip(sigma=2, maxiters=5)
>>> sigclip(data)
masked_array(data=[1, 5, 6, 8, --, 5, 3, 2],
mask=[False, False, False, False,  True, False, False, False],
fill_value=999999)


In addition, there are also several convenience functions for making the calculation of statistics even easier. For example, sigma_clipped_stats() will return the mean, median, and standard deviation of a sigma-clipped array:

>>> stats.sigma_clipped_stats(data, sigma=2, maxiters=5)
(4.2857142857142856, 5.0, 2.2497165354319457)


There are also tools for calculating robust statistics, sampling the data, circular statistics, confidence limits, spatial statistics, and adaptive histograms.

Most tools are fairly self-contained, and include relevant examples in their docstrings.

## Using astropy.stats¶

More detailed information on using the package is provided on separate pages, listed below.

## Constants¶

The astropy.stats package defines two constants useful for converting between Gaussian sigma and full width at half maximum (FWHM):

gaussian_sigma_to_fwhm

Factor with which to multiply Gaussian 1-sigma standard deviation to convert it to full width at half maximum (FWHM).

>>> from astropy.stats import gaussian_sigma_to_fwhm
>>> gaussian_sigma_to_fwhm
2.3548200450309493

gaussian_fwhm_to_sigma

Factor with which to multiply Gaussian full width at half maximum (FWHM) to convert it to 1-sigma standard deviation.

>>> from astropy.stats import gaussian_fwhm_to_sigma
>>> gaussian_fwhm_to_sigma
0.42466090014400953


## Performance Tips¶

If you are finding sigma-clipping to be slow, and if you haven’t already done so, consider installing the bottleneck package, which will speed up some of the internal computations. In addition, if you are using standard functions for cenfunc and/or stdfunc, make sure you specify these as strings rather than passing a Numpy function - that is, use:

>>> sigma_clip(array, cenfunc='median')


>>> sigma_clip(array, cenfunc=np.nanmedian)


Using strings will allow the sigma clipping algorithm to pick the fastest implementation available for finding the median.

## Reference/API¶

### astropy.stats Package¶

This subpackage contains statistical tools provided for or used by Astropy.

While the scipy.stats package contains a wide range of statistical tools, it is a general-purpose package, and is missing some that are particularly useful to astronomy or are used in an atypical way in astronomy. This package is intended to provide such functionality, but not to replace scipy.stats if its implementation satisfies astronomers’ needs.

#### Functions¶

 akaike_info_criterion(log_likelihood, …) Computes the Akaike Information Criterion (AIC). akaike_info_criterion_lsq(ssr, n_params, …) Computes the Akaike Information Criterion assuming that the observations are Gaussian distributed. bayesian_blocks(t[, x, sigma, fitness]) Compute optimal segmentation of data with Scargle’s Bayesian Blocks bayesian_info_criterion(log_likelihood, …) Computes the Bayesian Information Criterion (BIC) given the log of the likelihood function evaluated at the estimated (or analytically derived) parameters, the number of parameters, and the number of samples. bayesian_info_criterion_lsq(ssr, n_params, …) Computes the Bayesian Information Criterion (BIC) assuming that the observations come from a Gaussian distribution. binned_binom_proportion(x, success[, bins, …]) Binomial proportion and confidence interval in bins of a continuous variable x. binom_conf_interval(k, n[, conf, interval]) Binomial proportion confidence interval given k successes, n trials. biweight_location(data[, c, M, axis]) Compute the biweight location. biweight_midcorrelation(x, y[, c, M, …]) Compute the biweight midcorrelation between two variables. biweight_midcovariance(data[, c, M, …]) Compute the biweight midcovariance between pairs of multiple variables. biweight_midvariance(data[, c, M, axis, …]) Compute the biweight midvariance. biweight_scale(data[, c, M, axis, …]) Compute the biweight scale. bootstrap(data[, bootnum, samples, bootfunc]) Performs bootstrap resampling on numpy arrays. calculate_bin_edges(a[, bins, range, weights]) Calculate histogram bin edges like numpy.histogram_bin_edges. cdf_from_intervals(breaks, totals) Construct a callable piecewise-linear CDF from a pair of arrays. circcorrcoef(alpha, beta[, axis, …]) Computes the circular correlation coefficient between two array of circular data. circmean(data[, axis, weights]) Computes the circular mean angle of an array of circular data. circmoment(data[, p, centered, axis, weights]) Computes the p-th trigonometric circular moment for an array of circular data. circvar(data[, axis, weights]) Computes the circular variance of an array of circular data. fold_intervals(intervals) Fold the weighted intervals to the interval (0,1). freedman_bin_width(data[, return_bins]) Return the optimal histogram bin width using the Freedman-Diaconis rule histogram(a[, bins, range, weights]) Enhanced histogram function, providing adaptive binnings histogram_intervals(n, breaks, totals) Histogram of a piecewise-constant weight function. Compute the length of overlap of two intervals. Performs jackknife resampling on numpy arrays. jackknife_stats(data, statistic[, conf_lvl]) Performs jackknife estimation on the basis of jackknife resamples. knuth_bin_width(data[, return_bins, quiet]) Return the optimal histogram bin width using Knuth’s rule. kuiper(data[, cdf, args]) Compute the Kuiper statistic. Compute the false positive probability for the Kuiper statistic. kuiper_two(data1, data2) Compute the Kuiper statistic to compare two samples. mad_std(data[, axis, func, ignore_nan]) Calculate a robust standard deviation using the median absolute deviation (MAD). median_absolute_deviation(data[, axis, …]) Calculate the median absolute deviation (MAD). poisson_conf_interval(n[, interval, sigma, …]) Poisson parameter confidence interval given observed counts rayleightest(data[, axis, weights]) Performs the Rayleigh test of uniformity. scott_bin_width(data[, return_bins]) Return the optimal histogram bin width using Scott’s rule sigma_clip(data[, sigma, sigma_lower, …]) Perform sigma-clipping on the provided data. sigma_clipped_stats(data[, mask, …]) Calculate sigma-clipped statistics on the provided data. signal_to_noise_oir_ccd(t, source_eps, …) Computes the signal to noise ratio for source being observed in the optical/IR using a CCD. vonmisesmle(data[, axis]) Computes the Maximum Likelihood Estimator (MLE) for the parameters of the von Mises distribution. vtest(data[, mu, axis, weights]) Performs the Rayleigh test of uniformity where the alternative hypothesis H1 is assumed to have a known mean angle mu.

#### Classes¶

 BoxLeastSquares(*args, **kwargs) Compute the box least squares periodogram. BoxLeastSquaresResults(*args, **kwargs) The results of a BoxLeastSquares search. Events([p0, gamma, ncp_prior]) Bayesian blocks fitness for binned or unbinned events FitnessFunc([p0, gamma, ncp_prior]) Base class for bayesian blocks fitness functions LombScargle(*args, **kwargs) Compute the Lomb-Scargle Periodogram. PointMeasures([p0, gamma, ncp_prior]) Bayesian blocks fitness for point measures RegularEvents(dt[, p0, gamma, ncp_prior]) Bayesian blocks fitness for regular events RipleysKEstimator(area[, x_max, y_max, …]) Estimators for Ripley’s K function for two-dimensional spatial data. SigmaClip([sigma, sigma_lower, sigma_upper, …]) Class to perform sigma clipping.