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scipy.stats.sampling.FastGeneratorInversion.evaluate_error#

FastGeneratorInversion.evaluate_error(size=100000, random_state=None, x_error=False)[source]#

Evaluate the numerical accuracy of the inversion (u- and x-error).

Parameters:
sizeint, optional

The number of random points over which the error is estimated. Default is 100000.

random_state{None, int, numpy.random.Generator,

A NumPy random number generator or seed for the underlying NumPy random number generator used to generate the stream of uniform random numbers. If random_state is None, use self.random_state. If random_state is an int, np.random.default_rng(random_state) is used. If random_state is already a Generator or RandomState instance then that instance is used.

Returns:
u_error, x_errortuple of floats

A NumPy array of random variates.

Notes

The numerical precision of the inverse CDF ppf is controlled by the u-error. It is computed as follows: max |u - CDF(PPF(u))| where the max is taken size random points in the interval [0,1]. random_state determines the random sample. Note that if ppf was exact, the u-error would be zero.

The x-error measures the direct distance between the exact PPF and ppf. If x_error is set to True`, it is computed as the maximum of the minimum of the relative and absolute x-error: ``max(min(x_error_abs[i], x_error_rel[i])) where x_error_abs[i] = |PPF(u[i]) - PPF_fast(u[i])|, x_error_rel[i] = max |(PPF(u[i]) - PPF_fast(u[i])) / PPF(u[i])|. Note that it is important to consider the relative x-error in the case that PPF(u) is close to zero or very large.

By default, only the u-error is evaluated and the x-error is set to np.nan. Note that the evaluation of the x-error will be very slow if the implementation of the PPF is slow.

Further information about these error measures can be found in [1].

References

[1]

Derflinger, Gerhard, Wolfgang Hörmann, and Josef Leydold. “Random variate generation by numerical inversion when only the density is known.” ACM Transactions on Modeling and Computer Simulation (TOMACS) 20.4 (2010): 1-25.

Examples

>>> import numpy as np
>>> from scipy import stats
>>> from scipy.stats.sampling import FastGeneratorInversion

Create an object for the normal distribution:

>>> d_norm_frozen = stats.norm()
>>> d_norm = FastGeneratorInversion(d_norm_frozen)

To confirm that the numerical inversion is accurate, we evaluate the approximation error (u-error and x-error).

>>> u_error, x_error = d_norm.evaluate_error(x_error=True)

The u-error should be below 1e-10:

>>> u_error
8.785783212061915e-11  # may vary

Compare the PPF against approximation ppf:

>>> q = [0.001, 0.2, 0.4, 0.6, 0.8, 0.999]
>>> diff = np.abs(d_norm_frozen.ppf(q) - d_norm.ppf(q))
>>> x_error_abs = np.max(diff)
>>> x_error_abs
1.2937954707581412e-08

This is the absolute x-error evaluated at the points q. The relative error is given by

>>> x_error_rel = np.max(diff / np.abs(d_norm_frozen.ppf(q)))
>>> x_error_rel
4.186725600453555e-09

The x_error computed above is derived in a very similar way over a much larger set of random values q. At each value q[i], the minimum of the relative and absolute error is taken. The final value is then derived as the maximum of these values. In our example, we get the following value:

>>> x_error
4.507068014335139e-07  # may vary
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