# Source code for astropy.stats.biweight

"""
This module contains functions for computing robust statistics using
Tukey's biweight function.
"""

import numpy as np

from .funcs import _expand_dims, median_absolute_deviation

__all__ = ['biweight_location', 'biweight_scale', 'biweight_midvariance',
'biweight_midcovariance', 'biweight_midcorrelation']

def _stat_functions(data, ignore_nan=False):
median_func = np.ma.median
sum_func = np.ma.sum
elif ignore_nan:
median_func = np.nanmedian
sum_func = np.nansum
else:
median_func = np.median
sum_func = np.sum

return median_func, sum_func

[docs]def biweight_location(data, c=6.0, M=None, axis=None, *, ignore_nan=False): r""" Compute the biweight location. The biweight location is a robust statistic for determining the central location of a distribution. It is given by: .. math:: \zeta_{biloc}= M + \frac{\sum_{|u_i|<1} \ (x_i - M) (1 - u_i^2)^2} {\sum_{|u_i|<1} \ (1 - u_i^2)^2} where :math:x is the input data, :math:M is the sample median (or the input initial location guess) and :math:u_i is given by: .. math:: u_{i} = \frac{(x_i - M)}{c * MAD} where :math:c is the tuning constant and :math:MAD is the median absolute deviation <https://en.wikipedia.org/wiki/Median_absolute_deviation>_. The biweight location tuning constant c is typically 6.0 (the default). Parameters ---------- data : array_like Input array or object that can be converted to an array. data can be a ~numpy.ma.MaskedArray. c : float, optional Tuning constant for the biweight estimator (default = 6.0). M : float or array_like, optional Initial guess for the location. If M is a scalar value, then its value will be used for the entire array (or along each axis, if specified). If M is an array, then its must be an array containing the initial location estimate along each axis of the input array. If None (default), then the median of the input array will be used (or along each axis, if specified). axis : None, int, or tuple of ints, optional The axis or axes along which the biweight locations are computed. If None (default), then the biweight location of the flattened input array will be computed. ignore_nan : bool, optional Whether to ignore NaN values in the input data. Returns ------- biweight_location : float or ~numpy.ndarray The biweight location of the input data. If axis is None then a scalar will be returned, otherwise a ~numpy.ndarray will be returned. See Also -------- biweight_scale, biweight_midvariance, biweight_midcovariance References ---------- .. [1] Beers, Flynn, and Gebhardt (1990; AJ 100, 32) (http://adsabs.harvard.edu/abs/1990AJ....100...32B) .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/biwloc.htm Examples -------- Generate random variates from a Gaussian distribution and return the biweight location of the distribution: >>> import numpy as np >>> from astropy.stats import biweight_location >>> rand = np.random.RandomState(12345) >>> biloc = biweight_location(rand.randn(1000)) >>> print(biloc) # doctest: +FLOAT_CMP -0.0175741540445 """ median_func, sum_func = _stat_functions(data, ignore_nan=ignore_nan) if isinstance(data, np.ma.MaskedArray) and ignore_nan: data = np.ma.masked_where(np.isnan(data), data, copy=True) data = np.asanyarray(data).astype(np.float64) if M is None: M = median_func(data, axis=axis) if axis is not None: M = _expand_dims(M, axis=axis) # NUMPY_LT_1_18 # set up the differences d = data - M # set up the weighting mad = median_absolute_deviation(data, axis=axis, ignore_nan=ignore_nan) if axis is None and mad == 0.: return M # return median if data is a constant array if axis is not None: mad = _expand_dims(mad, axis=axis) # NUMPY_LT_1_18 mad[mad == 0] = 1. # prevent divide by zero u = d / (c * mad) # now remove the outlier points # ignore RuntimeWarnings for comparisons with NaN data values with np.errstate(invalid='ignore'): mask = np.abs(u) >= 1 u = (1 - u ** 2) ** 2 u[mask] = 0 # Along the input axis if data is constant, d will be zero, thus # the median value will be returned along that axis. # Ignore RuntimeWarnings for divide by zero if all NaN along an axis with np.errstate(divide='ignore', invalid='ignore'): return M.squeeze() + (sum_func(d * u, axis=axis) / sum_func(u, axis=axis))
[docs]def biweight_scale(data, c=9.0, M=None, axis=None, modify_sample_size=False, *, ignore_nan=False): r""" Compute the biweight scale. The biweight scale is a robust statistic for determining the standard deviation of a distribution. It is the square root of the biweight midvariance <https://en.wikipedia.org/wiki/Robust_measures_of_scale#The_biweight_midvariance>_. It is given by: .. math:: \zeta_{biscl} = \sqrt{n} \ \frac{\sqrt{\sum_{|u_i| < 1} \ (x_i - M)^2 (1 - u_i^2)^4}} {|(\sum_{|u_i| < 1} \ (1 - u_i^2) (1 - 5u_i^2))|} where :math:x is the input data, :math:M is the sample median (or the input location) and :math:u_i is given by: .. math:: u_{i} = \frac{(x_i - M)}{c * MAD} where :math:c is the tuning constant and :math:MAD is the median absolute deviation <https://en.wikipedia.org/wiki/Median_absolute_deviation>_. The biweight midvariance tuning constant c is typically 9.0 (the default). For the standard definition of biweight scale, :math:n is the total number of points in the array (or along the input axis, if specified). That definition is used if modify_sample_size is False, which is the default. However, if modify_sample_size = True, then :math:n is the number of points for which :math:|u_i| < 1 (i.e. the total number of non-rejected values), i.e. .. math:: n = \sum_{|u_i| < 1} \ 1 which results in a value closer to the true standard deviation for small sample sizes or for a large number of rejected values. Parameters ---------- data : array_like Input array or object that can be converted to an array. data can be a ~numpy.ma.MaskedArray. c : float, optional Tuning constant for the biweight estimator (default = 9.0). M : float or array_like, optional The location estimate. If M is a scalar value, then its value will be used for the entire array (or along each axis, if specified). If M is an array, then its must be an array containing the location estimate along each axis of the input array. If None (default), then the median of the input array will be used (or along each axis, if specified). axis : None, int, or tuple of ints, optional The axis or axes along which the biweight scales are computed. If None (default), then the biweight scale of the flattened input array will be computed. modify_sample_size : bool, optional If False (default), then the sample size used is the total number of elements in the array (or along the input axis, if specified), which follows the standard definition of biweight scale. If True, then the sample size is reduced to correct for any rejected values (i.e. the sample size used includes only the non-rejected values), which results in a value closer to the true standard deviation for small sample sizes or for a large number of rejected values. ignore_nan : bool, optional Whether to ignore NaN values in the input data. Returns ------- biweight_scale : float or ~numpy.ndarray The biweight scale of the input data. If axis is None then a scalar will be returned, otherwise a ~numpy.ndarray will be returned. See Also -------- biweight_midvariance, biweight_midcovariance, biweight_location, astropy.stats.mad_std, astropy.stats.median_absolute_deviation References ---------- .. [1] Beers, Flynn, and Gebhardt (1990; AJ 100, 32) (http://adsabs.harvard.edu/abs/1990AJ....100...32B) .. [2] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/biwscale.htm Examples -------- Generate random variates from a Gaussian distribution and return the biweight scale of the distribution: >>> import numpy as np >>> from astropy.stats import biweight_scale >>> rand = np.random.RandomState(12345) >>> biscl = biweight_scale(rand.randn(1000)) >>> print(biscl) # doctest: +FLOAT_CMP 0.986726249291 """ return np.sqrt( biweight_midvariance(data, c=c, M=M, axis=axis, modify_sample_size=modify_sample_size, ignore_nan=ignore_nan))
[docs]def biweight_midvariance(data, c=9.0, M=None, axis=None, modify_sample_size=False, *, ignore_nan=False): r""" Compute the biweight midvariance. The biweight midvariance is a robust statistic for determining the variance of a distribution. Its square root is a robust estimator of scale (i.e. standard deviation). It is given by: .. math:: \zeta_{bivar} = n \ \frac{\sum_{|u_i| < 1} \ (x_i - M)^2 (1 - u_i^2)^4} {(\sum_{|u_i| < 1} \ (1 - u_i^2) (1 - 5u_i^2))^2} where :math:x is the input data, :math:M is the sample median (or the input location) and :math:u_i is given by: .. math:: u_{i} = \frac{(x_i - M)}{c * MAD} where :math:c is the tuning constant and :math:MAD is the median absolute deviation <https://en.wikipedia.org/wiki/Median_absolute_deviation>_. The biweight midvariance tuning constant c is typically 9.0 (the default). For the standard definition of biweight midvariance <https://en.wikipedia.org/wiki/Robust_measures_of_scale#The_biweight_midvariance>_, :math:n is the total number of points in the array (or along the input axis, if specified). That definition is used if modify_sample_size is False, which is the default. However, if modify_sample_size = True, then :math:n is the number of points for which :math:|u_i| < 1 (i.e. the total number of non-rejected values), i.e. .. math:: n = \sum_{|u_i| < 1} \ 1 which results in a value closer to the true variance for small sample sizes or for a large number of rejected values. Parameters ---------- data : array_like Input array or object that can be converted to an array. data can be a ~numpy.ma.MaskedArray. c : float, optional Tuning constant for the biweight estimator (default = 9.0). M : float or array_like, optional The location estimate. If M is a scalar value, then its value will be used for the entire array (or along each axis, if specified). If M is an array, then its must be an array containing the location estimate along each axis of the input array. If None (default), then the median of the input array will be used (or along each axis, if specified). axis : None, int, or tuple of ints, optional The axis or axes along which the biweight midvariances are computed. If None (default), then the biweight midvariance of the flattened input array will be computed. modify_sample_size : bool, optional If False (default), then the sample size used is the total number of elements in the array (or along the input axis, if specified), which follows the standard definition of biweight midvariance. If True, then the sample size is reduced to correct for any rejected values (i.e. the sample size used includes only the non-rejected values), which results in a value closer to the true variance for small sample sizes or for a large number of rejected values. ignore_nan : bool, optional Whether to ignore NaN values in the input data. Returns ------- biweight_midvariance : float or ~numpy.ndarray The biweight midvariance of the input data. If axis is None then a scalar will be returned, otherwise a ~numpy.ndarray will be returned. See Also -------- biweight_midcovariance, biweight_midcorrelation, astropy.stats.mad_std, astropy.stats.median_absolute_deviation References ---------- .. [1] https://en.wikipedia.org/wiki/Robust_measures_of_scale#The_biweight_midvariance .. [2] Beers, Flynn, and Gebhardt (1990; AJ 100, 32) (http://adsabs.harvard.edu/abs/1990AJ....100...32B) Examples -------- Generate random variates from a Gaussian distribution and return the biweight midvariance of the distribution: >>> import numpy as np >>> from astropy.stats import biweight_midvariance >>> rand = np.random.RandomState(12345) >>> bivar = biweight_midvariance(rand.randn(1000)) >>> print(bivar) # doctest: +FLOAT_CMP 0.97362869104 """ median_func, sum_func = _stat_functions(data, ignore_nan=ignore_nan) if isinstance(data, np.ma.MaskedArray) and ignore_nan: data = np.ma.masked_where(np.isnan(data), data, copy=True) data = np.asanyarray(data).astype(np.float64) if M is None: M = median_func(data, axis=axis) if axis is not None: M = _expand_dims(M, axis=axis) # NUMPY_LT_1_18 # set up the differences d = data - M # set up the weighting mad = median_absolute_deviation(data, axis=axis, ignore_nan=ignore_nan) if axis is None and mad == 0.: return 0. # return zero if data is a constant array if axis is not None: mad = _expand_dims(mad, axis=axis) # NUMPY_LT_1_18 mad[mad == 0] = 1. # prevent divide by zero u = d / (c * mad) # now remove the outlier points # ignore RuntimeWarnings for comparisons with NaN data values with np.errstate(invalid='ignore'): mask = np.abs(u) < 1 if isinstance(mask, np.ma.MaskedArray): mask = mask.filled(fill_value=False) # exclude masked data values u = u ** 2 if modify_sample_size: n = sum_func(mask, axis=axis) else: # set good values to 1, bad values to 0 include_mask = np.ones(data.shape) if isinstance(data, np.ma.MaskedArray): include_mask[data.mask] = 0 if ignore_nan: include_mask[np.isnan(data)] = 0 n = np.sum(include_mask, axis=axis) f1 = d * d * (1. - u)**4 f1[~mask] = 0. f1 = sum_func(f1, axis=axis) f2 = (1. - u) * (1. - 5.*u) f2[~mask] = 0. f2 = np.abs(np.sum(f2, axis=axis))**2 with np.errstate(divide='ignore', invalid='ignore'): return n * f1 / f2
[docs]def biweight_midcovariance(data, c=9.0, M=None, modify_sample_size=False): r""" Compute the biweight midcovariance between pairs of multiple variables. The biweight midcovariance is a robust and resistant estimator of the covariance between two variables. This function computes the biweight midcovariance between all pairs of the input variables (rows) in the input data. The output array will have a shape of (N_variables, N_variables). The diagonal elements will be the biweight midvariances of each input variable (see :func:biweight_midvariance). The off-diagonal elements will be the biweight midcovariances between each pair of input variables. For example, if the input array data contains three variables (rows) x, y, and z, the output ~numpy.ndarray midcovariance matrix will be: .. math:: \begin{pmatrix} \zeta_{xx} & \zeta_{xy} & \zeta_{xz} \\ \zeta_{yx} & \zeta_{yy} & \zeta_{yz} \\ \zeta_{zx} & \zeta_{zy} & \zeta_{zz} \end{pmatrix} where :math:\zeta_{xx}, :math:\zeta_{yy}, and :math:\zeta_{zz} are the biweight midvariances of each variable. The biweight midcovariance between :math:x and :math:y is :math:\zeta_{xy} (:math:= \zeta_{yx}). The biweight midcovariance between :math:x and :math:z is :math:\zeta_{xz} (:math:= \zeta_{zx}). The biweight midcovariance between :math:y and :math:z is :math:\zeta_{yz} (:math:= \zeta_{zy}). The biweight midcovariance between two variables :math:x and :math:y is given by: .. math:: \zeta_{xy} = n_{xy} \ \frac{\sum_{|u_i| < 1, \ |v_i| < 1} \ (x_i - M_x) (1 - u_i^2)^2 (y_i - M_y) (1 - v_i^2)^2} {(\sum_{|u_i| < 1} \ (1 - u_i^2) (1 - 5u_i^2)) (\sum_{|v_i| < 1} \ (1 - v_i^2) (1 - 5v_i^2))} where :math:M_x and :math:M_y are the medians (or the input locations) of the two variables and :math:u_i and :math:v_i are given by: .. math:: u_{i} = \frac{(x_i - M_x)}{c * MAD_x} v_{i} = \frac{(y_i - M_y)}{c * MAD_y} where :math:c is the biweight tuning constant and :math:MAD_x and :math:MAD_y are the median absolute deviation <https://en.wikipedia.org/wiki/Median_absolute_deviation>_ of the :math:x and :math:y variables. The biweight midvariance tuning constant c is typically 9.0 (the default). For the standard definition of biweight midcovariance, :math:n_{xy} is the total number of observations of each variable. That definition is used if modify_sample_size is False, which is the default. However, if modify_sample_size = True, then :math:n_{xy} is the number of observations for which :math:|u_i| < 1 and/or :math:|v_i| < 1, i.e. .. math:: n_{xx} = \sum_{|u_i| < 1} \ 1 .. math:: n_{xy} = n_{yx} = \sum_{|u_i| < 1, \ |v_i| < 1} \ 1 .. math:: n_{yy} = \sum_{|v_i| < 1} \ 1 which results in a value closer to the true variance for small sample sizes or for a large number of rejected values. Parameters ---------- data : 2D or 1D array_like Input data either as a 2D or 1D array. For a 2D array, it should have a shape (N_variables, N_observations). A 1D array may be input for observations of a single variable, in which case the biweight midvariance will be calculated (no covariance). Each row of data represents a variable, and each column a single observation of all those variables (same as the numpy.cov convention). c : float, optional Tuning constant for the biweight estimator (default = 9.0). M : float or 1D array_like, optional The location estimate of each variable, either as a scalar or array. If M is an array, then its must be a 1D array containing the location estimate of each row (i.e. a.ndim elements). If M is a scalar value, then its value will be used for each variable (row). If None (default), then the median of each variable (row) will be used. modify_sample_size : bool, optional If False (default), then the sample size used is the total number of observations of each variable, which follows the standard definition of biweight midcovariance. If True, then the sample size is reduced to correct for any rejected values (see formula above), which results in a value closer to the true covariance for small sample sizes or for a large number of rejected values. Returns ------- biweight_midcovariance : ~numpy.ndarray A 2D array representing the biweight midcovariances between each pair of the variables (rows) in the input array. The output array will have a shape of (N_variables, N_variables). The diagonal elements will be the biweight midvariances of each input variable. The off-diagonal elements will be the biweight midcovariances between each pair of input variables. See Also -------- biweight_midvariance, biweight_midcorrelation, biweight_scale, biweight_location References ---------- .. [1] https://www.itl.nist.gov/div898/software/dataplot/refman2/auxillar/biwmidc.htm Examples -------- Compute the biweight midcovariance between two random variables: >>> import numpy as np >>> from astropy.stats import biweight_midcovariance >>> # Generate two random variables x and y >>> rng = np.random.RandomState(1) >>> x = rng.normal(0, 1, 200) >>> y = rng.normal(0, 3, 200) >>> # Introduce an obvious outlier >>> x[0] = 30.0 >>> # Calculate the biweight midcovariances between x and y >>> bicov = biweight_midcovariance([x, y]) >>> print(bicov) # doctest: +FLOAT_CMP [[ 0.82483155 -0.18961219] [-0.18961219 9.80265764]] >>> # Print standard deviation estimates >>> print(np.sqrt(bicov.diagonal())) # doctest: +FLOAT_CMP [ 0.90820237 3.13091961] """ data = np.asanyarray(data).astype(np.float64) # ensure data is 2D if data.ndim == 1: data = data[np.newaxis, :] if data.ndim != 2: raise ValueError('The input array must be 2D or 1D.') # estimate location if not given if M is None: M = np.median(data, axis=1) M = np.asanyarray(M) if M.ndim > 1: raise ValueError('M must be a scalar or 1D array.') # set up the differences d = (data.T - M).T # set up the weighting mad = median_absolute_deviation(data, axis=1) mad[mad == 0] = 1. # prevent divide by zero u = (d.T / (c * mad)).T # now remove the outlier points mask = np.abs(u) < 1 u = u ** 2 if modify_sample_size: maskf = mask.astype(float) n = np.inner(maskf, maskf) else: n = data[0].size usub1 = (1. - u) usub5 = (1. - 5. * u) usub1[~mask] = 0. numerator = d * usub1 ** 2 denominator = (usub1 * usub5).sum(axis=1)[:, np.newaxis] numerator_matrix = np.dot(numerator, numerator.T) denominator_matrix = np.dot(denominator, denominator.T) return n * (numerator_matrix / denominator_matrix)
[docs]def biweight_midcorrelation(x, y, c=9.0, M=None, modify_sample_size=False): r""" Compute the biweight midcorrelation between two variables. The biweight midcorrelation <https://en.wikipedia.org/wiki/Biweight_midcorrelation>_ is a measure of similarity between samples. It is given by: .. math:: r_{bicorr} = \frac{\zeta_{xy}}{\sqrt{\zeta_{xx} \ \zeta_{yy}}} where :math:\zeta_{xx} is the biweight midvariance of :math:x, :math:\zeta_{yy} is the biweight midvariance of :math:y, and :math:\zeta_{xy} is the biweight midcovariance of :math:x and :math:y. Parameters ---------- x, y : 1D array_like Input arrays for the two variables. x and y must be 1D arrays and have the same number of elements. c : float, optional Tuning constant for the biweight estimator (default = 9.0). See biweight_midcovariance for more details. M : float or array_like, optional The location estimate. If M is a scalar value, then its value will be used for the entire array (or along each axis, if specified). If M is an array, then its must be an array containing the location estimate along each axis of the input array. If None (default), then the median of the input array will be used (or along each axis, if specified). See biweight_midcovariance for more details. modify_sample_size : bool, optional If False (default), then the sample size used is the total number of elements in the array (or along the input axis, if specified), which follows the standard definition of biweight midcovariance. If True, then the sample size is reduced to correct for any rejected values (i.e. the sample size used includes only the non-rejected values), which results in a value closer to the true midcovariance for small sample sizes or for a large number of rejected values. See biweight_midcovariance for more details. Returns ------- biweight_midcorrelation : float The biweight midcorrelation between x and y. See Also -------- biweight_scale, biweight_midvariance, biweight_midcovariance, biweight_location References ---------- .. [1] https://en.wikipedia.org/wiki/Biweight_midcorrelation Examples -------- Calculate the biweight midcorrelation between two variables: >>> import numpy as np >>> from astropy.stats import biweight_midcorrelation >>> rng = np.random.RandomState(12345) >>> x = rng.normal(0, 1, 200) >>> y = rng.normal(0, 3, 200) >>> # Introduce an obvious outlier >>> x[0] = 30.0 >>> bicorr = biweight_midcorrelation(x, y) >>> print(bicorr) # doctest: +FLOAT_CMP -0.0495780713907 """ x = np.asanyarray(x) y = np.asanyarray(y) if x.ndim != 1: raise ValueError('x must be a 1D array.') if y.ndim != 1: raise ValueError('y must be a 1D array.') if x.shape != y.shape: raise ValueError('x and y must have the same shape.') bicorr = biweight_midcovariance([x, y], c=c, M=M, modify_sample_size=modify_sample_size) return bicorr[0, 1] / (np.sqrt(bicorr[0, 0] * bicorr[1, 1]))