Module « scipy.stats.qmc »

Fonction discrepancy - module scipy.stats.qmc

Signature de la fonction discrepancy

def discrepancy(sample: 'npt.ArrayLike', *, iterative: 'bool' = False, method: "Literal['CD', 'WD', 'MD', 'L2-star']" = 'CD', workers: 'IntNumber' = 1) -> 'float' 

Description

discrepancy.__doc__

Discrepancy of a given sample.

    Parameters
    ----------
    sample : array_like (n, d)
        The sample to compute the discrepancy from.
    iterative : bool, optional
        Must be False if not using it for updating the discrepancy.
        Default is False. Refer to the notes for more details.
    method : str, optional
        Type of discrepancy, can be ``CD``, ``WD``, ``MD`` or ``L2-star``.
        Refer to the notes for more details. Default is ``CD``.
    workers : int, optional
        Number of workers to use for parallel processing. If -1 is given all
        CPU threads are used. Default is 1.

    Returns
    -------
    discrepancy : float
        Discrepancy.

    Notes
    -----
    The discrepancy is a uniformity criterion used to assess the space filling
    of a number of samples in a hypercube. A discrepancy quantifies the
    distance between the continuous uniform distribution on a hypercube and the
    discrete uniform distribution on :math:`n` distinct sample points.

    The lower the value is, the better the coverage of the parameter space is.

    For a collection of subsets of the hypercube, the discrepancy is the
    difference between the fraction of sample points in one of those
    subsets and the volume of that subset. There are different definitions of
    discrepancy corresponding to different collections of subsets. Some
    versions take a root mean square difference over subsets instead of
    a maximum.

    A measure of uniformity is reasonable if it satisfies the following
    criteria [1]_:

    1. It is invariant under permuting factors and/or runs.
    2. It is invariant under rotation of the coordinates.
    3. It can measure not only uniformity of the sample over the hypercube,
       but also the projection uniformity of the sample over non-empty
       subset of lower dimension hypercubes.
    4. There is some reasonable geometric meaning.
    5. It is easy to compute.
    6. It satisfies the Koksma-Hlawka-like inequality.
    7. It is consistent with other criteria in experimental design.

    Four methods are available:

    * ``CD``: Centered Discrepancy - subspace involves a corner of the
      hypercube
    * ``WD``: Wrap-around Discrepancy - subspace can wrap around bounds
    * ``MD``: Mixture Discrepancy - mix between CD/WD covering more criteria
    * ``L2-star``: L2-star discrepancy - like CD BUT variant to rotation

    See [2]_ for precise definitions of each method.

    Lastly, using ``iterative=True``, it is possible to compute the
    discrepancy as if we had :math:`n+1` samples. This is useful if we want
    to add a point to a sampling and check the candidate which would give the
    lowest discrepancy. Then you could just update the discrepancy with
    each candidate using `update_discrepancy`. This method is faster than
    computing the discrepancy for a large number of candidates.

    References
    ----------
    .. [1] Fang et al. "Design and modeling for computer experiments".
       Computer Science and Data Analysis Series, 2006.
    .. [2] Zhou Y.-D. et al. Mixture discrepancy for quasi-random point sets.
       Journal of Complexity, 29 (3-4) , pp. 283-301, 2013.
    .. [3] T. T. Warnock. "Computational investigations of low discrepancy
       point sets". Applications of Number Theory to Numerical
       Analysis, Academic Press, pp. 319-343, 1972.

    Examples
    --------
    Calculate the quality of the sample using the discrepancy:

    >>> from scipy.stats import qmc
    >>> space = np.array([[1, 3], [2, 6], [3, 2], [4, 5], [5, 1], [6, 4]])
    >>> l_bounds = [0.5, 0.5]
    >>> u_bounds = [6.5, 6.5]
    >>> space = qmc.scale(space, l_bounds, u_bounds, reverse=True)
    >>> space
    array([[0.08333333, 0.41666667],
           [0.25      , 0.91666667],
           [0.41666667, 0.25      ],
           [0.58333333, 0.75      ],
           [0.75      , 0.08333333],
           [0.91666667, 0.58333333]])
    >>> qmc.discrepancy(space)
    0.008142039609053464

    We can also compute iteratively the ``CD`` discrepancy by using
    ``iterative=True``.

    >>> disc_init = qmc.discrepancy(space[:-1], iterative=True)
    >>> disc_init
    0.04769081147119336
    >>> qmc.update_discrepancy(space[-1], space[:-1], disc_init)
    0.008142039609053513