Compute the number of significant digits based on the paper Confidence Intervals for Stochastic Arithmetic. This package is also inspired by the Jupyter Notebook included with the publication.
This synthetic example illustrates how to compute significant digits of a results sample with a given known reference:
>>> import significantdigits as sd
>>> import numpy as np
>>> from numpy.random import uniform as U
>>> np.random.seed(0)
>>> eps = 2**-52
>>> # simulates results with epsilon differences
>>> X = [1+U(-1,1)*eps for _ in range(10)]
>>> sd.significant_digits(X, reference=1)
>>> 51.02329058847853
or with the CLI interface assuming X
is in test.txt
:
> significantdigits --metric significant -i "$(cat test.txt)" --input-format stdin --reference 1
> (51.02329058847853,)
If the reference is unknown, one can use the sample average:
...
>>> sd.significant_digits(X, reference=np.mean(X))
>>> 51.02329058847853
python3 -m pip install -U significantdigits
or if you want the latest version of the code, you can install it from the repository directly
python3 -m pip install -U git+https://github.com/verificarlo/significantdigits.git
# or if you don't have 'git' installed
python3 -m pip install -U https://github.com/verificarlo/significantdigits/zipball/master
Functions accept the following types of inputs:
InputType: ArrayLike
Those types are accessible with the numpy.typing.ArrayLike
type.
Metrics are computed using Z, the distance between the samples and the reference. There are four possible cases depending on the distance and the nature of the reference that are summarized in this table:
constant reference (x) | random variable reference (Y) | |
---|---|---|
Absolute precision | Z = X - x | Z = X - Y |
Relative precision | Z = X/x - 1 | Z = X/Y - 1 |
_compute_z(array: InternalArrayType,
reference: InternalArrayType | None,
error: Error | str,
axis: int,
shuffle_samples: bool = False) -> InternalArrayType
Compute Z, the distance between the random variable and the reference
Compute Z, the distance between the random variable and the reference
with three cases depending on the dimensions of array and reference:
X = array
Y = reference
Three cases:
- Y is none
The case when X = Y
We split X in two and set one group to X and the other to Y
- X.ndim == Y.ndim
X and Y have the same dimension
It it the case when Y is a random variable
- X.ndim - 1 == Y.ndim or Y.ndim == 0
Y is a scalar value
Parameters
----------
array : InternalArrayType
The random variable
reference : Optional[InternalArrayType]
The reference to compare against
error : Error | str
The error function to compute Z
axis : int, default=0
The axis or axes along which compute Z
shuflle_samples : bool, default=False
If True, shuffles the groups when the reference is None
Returns
-------
array : InternalArrayType
The result of Z following the error method choose
Two methods exist for computing both significant and contributing digits depending on whether the sample follows a Centered Normal distribution or not.
You can pass the method to the function by using the Method
enum provided by the package.
The functions also accept the name as a string
"cnh"
for Method.CNH
and "general"
for Method.General
.
class Method(AutoName):
"""
CNH: Centered Normality Hypothesis
X follows a Gaussian law centered around the reference or
Z follows a Gaussian law centered around 0
General: No assumption about the distribution of X or Z
"""
CNH = auto()
General = auto()
significant_digits(array: InputType,
reference: ReferenceType | None = None,
axis: int = 0,
basis: int = 2,
error: Error | str,
method: Method | str,
probability: float = 0.95,
confidence: float = 0.95,
shuffle_samples: bool = False,
dtype: DTypeLike | None = None
) -> ArrayLike
Compute significant digits
This function computes with a certain probability
the number of bits that are significant.
Parameters
----------
array: InputType
Element to compute
reference: ReferenceType | None, optional=None
Reference for comparing the array
axis: int, optional=0
Axis or axes along which the significant digits are computed
basis: int, optional=2
Basis in which represent the significant digits
error : Error | str, optional=Error.Relative
Error function to use to compute error between array and reference.
method : Method | str, optional=Method.CNH
Method to use for the underlying distribution hypothesis
probability : float, default=0.95
Probability for the significant digits result
confidence : float, default=0.95
Confidence level for the significant digits result
shuffle_samples : bool, optional=False
If reference is None, the array is split in two and \
comparison is done between both pieces. \
If shuffle_samples is True, it shuffles pieces.
dtype : dtype_like | None, default=None
Numerical type used for computing significant digits
Widest format between array and reference is taken if no supplied.
Returns
-------
ndarray
array_like containing significant digits
contributing_digits(array: InputType,
reference: ReferenceType | None = None,
axis: int = 0,
basis: int = 2,
error: Error | str,
method: Method | str,
probability: float = 0.51,
confidence: float = 0.95,
shuffle_samples: bool = False,
dtype: DTypeLike | None = None
) -> ArrayLike
Compute contributing digits
This function computes with a certain probability the number of bits
of the mantissa that will round the result towards the correct reference
value[1]_
Parameters
----------
array: InputArray
Element to compute
reference: ReferenceArray | None, default=None
Reference for comparing the array
axis: int, default=0
Axis or axes along which the contributing digits are computed
default: None
basis: int, optional=2
basis in which represent the contributing digits
error : Error | str, default=Error.Relative
Error function to use to compute error between array and reference.
method : Method | str, default=Method.CNH
Method to use for the underlying distribution hypothesis
probability : float, default=0.51
Probability for the contributing digits result
confidence : float, default=0.95
Confidence level for the contributing digits result
shuffle_samples : bool, default=False
If reference is None, the array is split in two and
comparison is done between both pieces.
If shuffle_samples is True, it shuffles pieces.
dtype : dtype_like | None, default=None
Numerical type used for computing contributing digits
Widest format between array and reference is taken if no supplied.
Returns
-------
ndarray
array_like containing contributing digits
These are utility functions for the general case.
Estimates the lower bound probability given the sample size.
probability_estimation_general(success: int, trials: int, confidence: float) -> float
Computes probability lower bound for Bernouilli process
This function computes the probability associated with metrics
computed in the general case (without assumption on the underlying
distribution). Indeed, in that case the probability is given by the
sample size with a certain confidence level.
Parameters
----------
success : int
Number of success for a Bernoulli experiment
trials : int
Number of trials for a Bernoulli experiment
confidence : float
Confidence level for the probability lower bound estimation
Returns
-------
float
The lower bound probability with `confidence` level to have `success` successes for `trials` trials
Returns the minimal sample size required to reach the requested probability
and confidence
.
minimum_number_of_trials(probability: float, confidence: float) -> int
Computes the minimum number of trials to have probability and confidence
This function computes the minimal sample size required to have
metrics with a certain probability and confidence for the general case
(without assumption on the underlying distribution).
For example, if one wants significant digits with proabability p = 99%
and confidence (1 - alpha) = 95%, it requires at least 299 observations.
Parameters
----------
probability : float
Probability
confidence : float
Confidence
Returns
-------
int
Minimal sample size to have given probability and confidence
This file is part of the Verificarlo project, under the Apache License v2.0 with LLVM Exceptions. SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception. See https://llvm.org/LICENSE.txt for license information.
Copyright (c) 2020-2023 Verificarlo Contributors