.. _stats-robust: .. testsetup:: >>> import numpy as np >>> np.random.seed(0) ***************************** Robust Statistical Estimators ***************************** Robust statistics provides reliable estimates of basic statistics for complex distributions. The statistics package includes several robust statistical functions that are commonly used in astronomy. This includes methods for rejecting outliers as well as statistical description of the underlying distributions. In addition to the functions mentioned here, models can be fit with outlier rejection using :func:`~astropy.modeling.fitting.FittingWithOutlierRemoval`. Sigma Clipping ============== Sigma clipping provides a fast method for identifying outliers in a distribution. For a distribution of points, a center and a standard deviation are calculated. Values which are less or more than a specified number of standard deviations from a center value are rejected. The process can be iterated to further reject outliers. The `astropy.stats` package provides both a functional and object-oriented interface for sigma clipping. The function is called :func:`~astropy.stats.sigma_clip` and the class is called :class:`~astropy.stats.SigmaClip`. By default, they both return a masked array where the rejected points are masked. Examples -------- .. EXAMPLE START Functional Sigma Clipping with astropy.stats.sigma_clip We can start by generating some data that has a mean of 0 and standard deviation of 0.2, but with outliers: .. doctest-requires:: scipy >>> import numpy as np >>> import scipy.stats as stats >>> rng = np.random.default_rng(0) >>> x = np.arange(200) >>> y = np.zeros(200) >>> c = stats.bernoulli.rvs(0.35, size=x.shape) >>> y += (rng.normal(0., 0.2, x.shape) + ... c * rng.normal(3.0, 5.0, x.shape)) Now we can use :func:`~astropy.stats.sigma_clip` to perform sigma clipping on the data: .. doctest-requires:: scipy >>> from astropy.stats import sigma_clip >>> filtered_data = sigma_clip(y, sigma=3, maxiters=10) The output masked array then can be used to calculate statistics on the data, fit models to the data, or otherwise explore the data. .. EXAMPLE END .. EXAMPLE START Object-Oriented Sigma Clipping with the astropy.stats.SigmaClip Class To perform the same sigma clipping with the :class:`~astropy.stats.SigmaClip` class: .. doctest-requires:: scipy >>> from astropy.stats import SigmaClip >>> sigclip = SigmaClip(sigma=3, maxiters=10) >>> print(sigclip) # doctest: +SKIP sigma: 3 sigma_lower: None sigma_upper: None maxiters: 10 cenfunc: stdfunc: >>> filtered_data = sigclip(y) Note that once the ``sigclip`` instance is defined above, it can be applied to other data using the same already defined sigma-clipping parameters. .. EXAMPLE END For basic statistics, :func:`~astropy.stats.sigma_clipped_stats` is a convenience function to calculate the sigma-clipped mean, median, and standard deviation of an array. As can be seen, rejecting the outliers returns accurate values for the underlying distribution. .. EXAMPLE START Calculating the Sigma-Clipped Mean, Median, and Standard Deviation of an Array To use :func:`~astropy.stats.sigma_clipped_stats` for sigma-clipped statistics calculation: .. doctest-requires:: scipy >>> from astropy.stats import sigma_clipped_stats >>> y.mean(), np.median(y), y.std() # doctest: +FLOAT_CMP (0.7068938765410144, 0.013567387681385379, 3.599605215851649) >>> sigma_clipped_stats(y, sigma=3, maxiters=10) # doctest: +FLOAT_CMP (-0.0228473012826993, -0.02356858871405204, 0.2079616996908159) :func:`~astropy.stats.sigma_clip` and :class:`~astropy.stats.SigmaClip` can be combined with other robust statistics to provide improved outlier rejection as well. .. plot:: :include-source: import numpy as np import scipy.stats as stats from matplotlib import pyplot as plt from astropy.stats import sigma_clip, mad_std # Generate fake data that has a mean of 0 and standard deviation of 0.2 with outliers rng = np.random.default_rng(0) x = np.arange(200) y = np.zeros(200) c = stats.bernoulli.rvs(0.35, size=x.shape) y += (rng.normal(0., 0.2, x.shape) + c * rng.normal(3.0, 5.0, x.shape)) filtered_data = sigma_clip(y, sigma=3, maxiters=1, stdfunc=mad_std) # plot the original and rejected data plt.figure(figsize=(8,5)) plt.plot(x, y, '+', color='#1f77b4', label="original data") plt.plot(x[filtered_data.mask], y[filtered_data.mask], 'x', color='#d62728', label="rejected data") plt.xlabel('x') plt.ylabel('y') plt.legend(loc=2, numpoints=1) .. automodapi:: astropy.stats.sigma_clipping .. EXAMPLE END Median Absolute Deviation ========================= The median absolute deviation (MAD) is a measure of the spread of a distribution and is defined as ``median(abs(a - median(a)))``. The MAD can be calculated using `~astropy.stats.median_absolute_deviation`. For a normal distribution, the MAD is related to the standard deviation by a factor of 1.4826, and a convenience function, `~astropy.stats.mad_std`, is available to apply the conversion. .. note:: A function can be supplied to the `~astropy.stats.median_absolute_deviation` to specify the median function to be used in the calculation. Depending on the version of NumPy and whether the array is masked or contains irregular values, significant performance increases can be had by preselecting the median function. If the median function is not specified, `~astropy.stats.median_absolute_deviation` will attempt to select the most relevant function according to the input data. Biweight Estimators =================== A set of functions are included in the `astropy.stats` package that use the biweight formalism. These functions have long been used in astronomy, particularly to calculate the velocity dispersion of galaxy clusters [1]_. The following set of tasks are available for biweight measurements: .. automodapi:: astropy.stats.biweight References ---------- .. [1] Beers, Flynn, and Gebhardt (1990; AJ 100, 32) (https://ui.adsabs.harvard.edu/abs/1990AJ....100...32B)