biweight_midvariance¶
- astropy.stats.biweight.biweight_midvariance(data, c=9.0, M=None, axis=None, modify_sample_size=False, *, ignore_nan=False)[source]¶
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:
\[\zeta_{bivar} = n \ \frac{\sum_{|u_i| < 1} \ (x_i - M)^2 (1 - u_i^2)^4} {(\sum_{|u_i| < 1} \ (1 - u_i^2) (1 - 5u_i^2))^2}\]where \(x\) 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 midvariance, \(n\) 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 variance for small sample sizes or for a large number of rejected values.
- Parameters:
- dataarray_like
Input array or object that can be converted to an array.
data
can be aMaskedArray
.- 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 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
None
,int
, ortuple
ofint
, 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_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 midvariance. 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 variance 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:
See also
References
[2]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 midvariance of the distribution:
>>> import numpy as np >>> from astropy.stats import biweight_midvariance >>> rand = np.random.default_rng(12345) >>> bivar = biweight_midvariance(rand.standard_normal(1000)) >>> print(bivar) 1.0484350639638342