astro py:docs

- Index
- Modules

### Navigation

- Astropy v5.2.1 »
- Module code »

# Source code for astropy.stats.jackknife

    # Licensed under a 3-clause BSD style license - see LICENSE.rst

    import numpy as np

    __all__ = ["jackknife_resampling", "jackknife_stats"]
    __doctest_requires__ = {"jackknife_stats": ["scipy"]}

    def jackknife_resampling(data):
        """Performs jackknife resampling on numpy arrays.

        Jackknife resampling is a technique to generate 'n' deterministic samples
        of size 'n-1' from a measured sample of size 'n'. Basically, the i-th
        sample, (1<=i<=n), is generated by means of removing the i-th measurement
        of the original sample. Like the bootstrap resampling, this statistical
        technique finds applications in estimating variance, bias, and confidence
        intervals.

        Parameters
        ----------
        data : ndarray
            Original sample (1-D array) from which the jackknife resamples will be
            generated.

        Returns
        -------
        resamples : ndarray
            The i-th row is the i-th jackknife sample, i.e., the original sample
            with the i-th measurement deleted.

        References
        ----------
        .. [1] McIntosh, Avery. "The Jackknife Estimation Method".
            <https://arxiv.org/abs/1606.00497>

        .. [2] Efron, Bradley. "The Jackknife, the Bootstrap, and other
            Resampling Plans". Technical Report No. 63, Division of Biostatistics,
            Stanford University, December, 1980.

        .. [3] Jackknife resampling <https://en.wikipedia.org/wiki/Jackknife_resampling>
        """

        n = data.shape[0]
        if n <= 0:
            raise ValueError("data must contain at least one measurement.")

        resamples = np.empty([n, n - 1])

        for i in range(n):
            resamples[i] = np.delete(data, i)

        return resamples

    def jackknife_stats(data, statistic, confidence_level=0.95):
        """Performs jackknife estimation on the basis of jackknife resamples.

        This function requires `SciPy <https://www.scipy.org/>`_ to be installed.

        Parameters
        ----------
        data : ndarray
            Original sample (1-D array).
        statistic : function
            Any function (or vector of functions) on the basis of the measured
            data, e.g, sample mean, sample variance, etc. The jackknife estimate of
            this statistic will be returned.
        confidence_level : float, optional
            Confidence level for the confidence interval of the Jackknife estimate.
            Must be a real-valued number in (0,1). Default value is 0.95.

        Returns
        -------
        estimate : float or `~numpy.ndarray`
            The i-th element is the bias-corrected "jackknifed" estimate.

        bias : float or `~numpy.ndarray`
            The i-th element is the jackknife bias.

        std_err : float or `~numpy.ndarray`
            The i-th element is the jackknife standard error.

        conf_interval : ndarray
            If ``statistic`` is single-valued, the first and second elements are
            the lower and upper bounds, respectively. If ``statistic`` is
            vector-valued, each column corresponds to the confidence interval for
            each component of ``statistic``. The first and second rows contain the
            lower and upper bounds, respectively.

        Examples
        --------
        1. Obtain Jackknife resamples:

        >>> import numpy as np
        >>> from astropy.stats import jackknife_resampling
        >>> from astropy.stats import jackknife_stats
        >>> data = np.array([1,2,3,4,5,6,7,8,9,0])
        >>> resamples = jackknife_resampling(data)
        >>> resamples
        array([[2., 3., 4., 5., 6., 7., 8., 9., 0.],
               [1., 3., 4., 5., 6., 7., 8., 9., 0.],
               [1., 2., 4., 5., 6., 7., 8., 9., 0.],
               [1., 2., 3., 5., 6., 7., 8., 9., 0.],
               [1., 2., 3., 4., 6., 7., 8., 9., 0.],
               [1., 2., 3., 4., 5., 7., 8., 9., 0.],
               [1., 2., 3., 4., 5., 6., 8., 9., 0.],
               [1., 2., 3., 4., 5., 6., 7., 9., 0.],
               [1., 2., 3., 4., 5., 6., 7., 8., 0.],
               [1., 2., 3., 4., 5., 6., 7., 8., 9.]])
        >>> resamples.shape
        (10, 9)

        2. Obtain Jackknife estimate for the mean, its bias, its standard error,
        and its 95% confidence interval:

        >>> test_statistic = np.mean
        >>> estimate, bias, stderr, conf_interval = jackknife_stats(
        ...     data, test_statistic, 0.95)
        >>> estimate
        4.5
        >>> bias
        0.0
        >>> stderr  # doctest: +FLOAT_CMP
        0.95742710775633832
        >>> conf_interval
        array([2.62347735,  6.37652265])

        3. Example for two estimates

        >>> test_statistic = lambda x: (np.mean(x), np.var(x))
        >>> estimate, bias, stderr, conf_interval = jackknife_stats(
        ...     data, test_statistic, 0.95)
        >>> estimate
        array([4.5       ,  9.16666667])
        >>> bias
        array([ 0.        , -0.91666667])
        >>> stderr
        array([0.95742711,  2.69124476])
        >>> conf_interval
        array([[ 2.62347735,   3.89192387],
               [ 6.37652265,  14.44140947]])

        IMPORTANT: Note that confidence intervals are given as columns
        """
        # jackknife confidence interval
        if not (0 < confidence_level < 1):
            raise ValueError("confidence level must be in (0, 1).")

        # make sure original data is proper
        n = data.shape[0]
        if n <= 0:
            raise ValueError("data must contain at least one measurement.")

        # Only import scipy if inputs are valid
        from scipy.special import erfinv

        resamples = jackknife_resampling(data)

        stat_data = statistic(data)
        jack_stat = np.apply_along_axis(statistic, 1, resamples)
        mean_jack_stat = np.mean(jack_stat, axis=0)

        # jackknife bias
        bias = (n - 1) * (mean_jack_stat - stat_data)

        # jackknife standard error
        std_err = np.sqrt(
            (n - 1)
            * np.mean((jack_stat - mean_jack_stat) * (jack_stat - mean_jack_stat), axis=0)
        )

        # bias-corrected "jackknifed estimate"
        estimate = stat_data - bias

        z_score = np.sqrt(2.0) * erfinv(confidence_level)
        conf_interval = estimate + z_score * np.array((-std_err, std_err))

        return estimate, bias, std_err, conf_interval