biweight_scale¶
- astropy.stats.biweight.biweight_scale(data, c=9.0, M=None, axis=None, modify_sample_size=False, *, ignore_nan=False)[source]¶
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. It is given by:
\[\zeta_{biscl} = \sqrt{n} \ \frac{\sqrt{\sum_{|u_i| < 1} \ (x_i - M)^2 (1 - u_i^2)^4}} {|(\sum_{|u_i| < 1} \ (1 - u_i^2) (1 - 5u_i^2))|}\]where \(x\) is the input data, \(M\) is the sample median (or the input location) and \(u_i\) is given by:
\[u_{i} = \frac{(x_i - M)}{c * MAD}\]where \(c\) is the tuning constant and \(MAD\) is the median absolute deviation. The biweight midvariance tuning constant
c
is typically 9.0 (the default).If \(MAD\) is zero, then zero will be returned.
For the standard definition of biweight scale, \(n\) is the total number of points in the array (or along the input
axis
, if specified). That definition is used ifmodify_sample_size
isFalse
, which is the default.However, if
modify_sample_size = True
, then \(n\) is the number of points for which \(|u_i| < 1\) (i.e. the total number of non-rejected values), i.e.\[n = \sum_{|u_i| < 1} \ 1\]which results in a value closer to the true standard deviation for small sample sizes or for a large number of rejected values.
- Parameters:
- datanumpy:array_like
Input array or object that can be converted to an array.
data
can be aMaskedArray
.- c
python:float
, optional Tuning constant for the biweight estimator (default = 9.0).
- M
python:float
or numpy:array_like, optional The location estimate. If
M
is a scalar value, then its value will be used for the entire array (or along eachaxis
, if specified). IfM
is an array, then its must be an array containing the location estimate along eachaxis
of the input array. IfNone
(default), then the median of the input array will be used (or along eachaxis
, if specified).- axis
python:None
,python:int
, orpython:tuple
ofpython:int
, 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_sizebool, optional
If
False
(default), then the sample size used is the total number of elements in the array (or along the inputaxis
, if specified), which follows the standard definition of biweight scale. IfTrue
, 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_nanbool, optional
Whether to ignore NaN values in the input
data
.
- Returns:
- biweight_scale
python:float
orndarray
The biweight scale of the input data. If
axis
isNone
then a scalar will be returned, otherwise andarray
will be returned.
- biweight_scale
See also
References
[1]Beers, Flynn, and Gebhardt (1990; AJ 100, 32) (https://ui.adsabs.harvard.edu/abs/1990AJ….100…32B)
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.default_rng(12345) >>> biscl = biweight_scale(rand.standard_normal(1000)) >>> print(biscl) 1.0239311812635818