lenspack.stats module¶

STATS MODULE

This module contains some common statistical measures useful in weak-lensing studies. For example, the higher-order moments of filtered convergence maps can be used to constrain cosmological parameters.

lenspack.stats.mad(x)[source]¶

Compute median absolute deviation (MAD) of an array.

Parameters: x (array_like) – Input array.
Returns: The MAD of x.
Return type: float

lenspack.stats.skew(x)[source]¶

Compute the skewness of an array as the third standardized moment.

Parameters: x (array_like) – Input array. If x is multidimensional, skewness is automatically computed over all elements in the array, not just along one axis.
Returns: Skewness of x.
Return type: float

Notes

For array x, the calculation is carried out as E[((x - mu) / sigma)^3], where E() is the expectation value, mu is the mean of x, and sigma is the uncorrected sample standard deviation.

This is equivalent to

>>> n = len(x)
>>> numer = np.power(x - mu, 3).sum() / n
>>> denom = np.power(np.power(x - mu, 2).sum() / (n - 1), 3 / 2)
>>> skew = numer / denom

and to the normalized 3rd-order central moment

>>> mu_n(x, 3, normed=True)

This function’s output matches that of scipy.stats.skew exactly, but the latter is typically faster.

lenspack.stats.kurt(x, fisher=True)[source]¶

Compute the kurtosis of an array as the fourth standardized moment.

Parameters

x (array_like) – Input array. If x is multidimensional, kurtosis is automatically computed over all elements in the array, not just along one axis.
fisher (bool, optional) – If True, use Fisher’s normalization, i.e. subtract 3 from the result.

Returns

Kurtosis of x.

Return type

float

Notes

For array x, the calculation is carried out as E[((x - mu) / sigma)^4], where E() is the expectation value, mu is the mean of x, and sigma is the uncorrected sample standard deviation.

This is equivalent to the normalized 4th-order central moment

>>> mu_n(x, 4, normed=True) - 3

This function’s output matches that of scipy.stats.kurtosis very well, but the latter is typically faster.

lenspack.stats.mu_n(x, order, normed=False)[source]¶

Compute the (normalized) nth-order central moment of a distribution.

Parameters

x (array_like) – Input array. If x is multidimensional, the moment is automatically computed over all elements in the array, not just along one axis.
order (int (positive)) – Order of the moment.
normed (bool, optional) – If True, normalize the result by sigma^order, where sigma is the corrected sample standard deviation of x. Default is False.

Returns

Nth-order moment of x.

Return type

float

Notes

This function’s output matches that of scipy.stats.moment very well, but the latter is typically faster.

lenspack.stats.kappa_n(x, order)[source]¶

Compute the nth-order cumulant of a distribution.

Parameters

x (array_like) – Input array. If x is multidimensional, the cumulant is automatically computed over all elements in the array, not just along one axis.
order (int between 2 and 6, inclusive) – Order of the cumulant.

Returns

Nth-order cumulant of x.

Return type

float

Notes

This function’s output matches that of scipy.stats.kstat very well, but the latter is typically faster.

lenspack.stats.fdr(x, tail, alpha=0.05, kde=False, n_samples=10000, debug=False)[source]¶

Compute the false discovery rate (FDR) threshold of a distribution.

Parameters

x (array_like) – Samples of the distribution. If x is multidimentional, it will first be flattened.
tail ({'left', 'right'}) – Side of the distribution for which to compute the threshold.
alpha (float, optional) – Maximum average false discovery rate. Default is 0.05.
kde (bool, optional) – If True, compute p-values from the distribution smoothed by kernel density estimation. Not recommended for large x. Default is False.
n_samples (int, optional) – Number of samples to draw if using KDE. Default is 10000.
debug (bool, optional) – If True, print the number of detections and the final p-value. Default is False.

Returns

FDR threshold.

Return type

float

Examples

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lenspack.stats.hc(x, kind=1)[source]¶

Compute a higher criticism statistic of a distribution.

Parameters

x (array_like) – Samples of the distribution. If x is multidimentional, it will first be flattened.
kind (int, optional) – Either 1 (HC^*) or 2 (HC^+). Default is 1.

Returns

Higher criticism of the first (*) or second (+) kind.

Return type

float

References

Donoho & Jin, The Annals of Statistics 32, 962 (2004)
Pires, Starck, Amara, et al., A&A 505, 969 (2009)

Examples

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