def windowed_statistic(pos, values, statistic, size=None, start=None,
                       stop=None, step=None, windows=None, fill=np.nan):
    """Calculate a statistic from items in windows over a single
    chromosome/contig.

    Parameters
    ----------

    pos : array_like, int, shape (n_items,)
        The item positions in ascending order, using 1-based coordinates..
    values : array_like, int, shape (n_items,)
        The values to summarise. May also be a tuple of values arrays,
        in which case each array will be sliced and passed through to the
        statistic function as separate arguments.
    statistic : function
        The statistic to compute.
    size : int, optional
        The window size (number of bases).
    start : int, optional
        The position at which to start (1-based).
    stop : int, optional
        The position at which to stop (1-based).
    step : int, optional
        The distance between start positions of windows. If not given,
        defaults to the window size, i.e., non-overlapping windows.
    windows : array_like, int, shape (n_windows, 2), optional
        Manually specify the windows to use as a sequence of (window_start,
        window_stop) positions, using 1-based coordinates. Overrides the
        size/start/stop/step parameters.
    fill : object, optional
        The value to use where a window is empty, i.e., contains no items.

    Returns
    -------

    out : ndarray, shape (n_windows,)
        The value of the statistic for each window.
    windows : ndarray, int, shape (n_windows, 2)
        The windows used, as an array of (window_start, window_stop) positions,
        using 1-based coordinates.
    counts : ndarray, int, shape (n_windows,)
        The number of items in each window.

    Notes
    -----

    The window stop positions are included within a window.

    The final window will be truncated to the specified stop position,
    and so may be smaller than the other windows.

    Examples
    --------

    Count non-zero (i.e., True) items in non-overlapping windows::

        >>> import allel
        >>> pos = [1, 7, 12, 15, 28]
        >>> values = [True, False, True, False, False]
        >>> nnz, windows, counts = allel.windowed_statistic(
        ...     pos, values, statistic=np.count_nonzero, size=10
        ... )
        >>> nnz
        array([1, 1, 0])
        >>> windows
        array([[ 1, 10],
               [11, 20],
               [21, 28]])
        >>> counts
        array([2, 2, 1])

    Compute a sum over items in half-overlapping windows::

        >>> values = [3, 4, 2, 6, 9]
        >>> x, windows, counts = allel.windowed_statistic(
        ...     pos, values, statistic=np.sum, size=10, step=5, fill=0
        ... )
        >>> x
        array([ 7, 12,  8,  0,  9])
        >>> windows
        array([[ 1, 10],
               [ 6, 15],
               [11, 20],
               [16, 25],
               [21, 28]])
        >>> counts
        array([2, 3, 2, 0, 1])

    """

    # assume sorted positions
    if not isinstance(pos, SortedIndex):
        pos = SortedIndex(pos, copy=False)

    # check lengths are equal
    if isinstance(values, tuple):
        # assume multiple values arrays
        check_equal_length(pos, *values)
    else:
        # assume a single values array
        check_equal_length(pos, values)

    # setup windows
    if windows is None:
        windows = position_windows(pos, size, start, stop, step)
    else:
        windows = asarray_ndim(windows, 2)

    # find window locations
    locs = window_locations(pos, windows)

    # setup outputs
    out = []
    counts = []

    # iterate over windows
    for start_idx, stop_idx in locs:

        # calculate number of values in window
        n = stop_idx - start_idx

        if n == 0:
            # window is empty
            s = fill

        else:

            if isinstance(values, tuple):
                # assume multiple values arrays
                wv = [v[start_idx:stop_idx] for v in values]
                s = statistic(*wv)

            else:
                # assume a single values array
                wv = values[start_idx:stop_idx]
                s = statistic(wv)

        # store outputs
        out.append(s)
        counts.append(n)

    # convert to arrays for output
    return np.asarray(out), windows, np.asarray(counts)
Calculate a statistic from items in windows over a single
    chromosome/contig.

    Parameters
    ----------

    pos : array_like, int, shape (n_items,)
        The item positions in ascending order, using 1-based coordinates..
    values : array_like, int, shape (n_items,)
        The values to summarise. May also be a tuple of values arrays,
        in which case each array will be sliced and passed through to the
        statistic function as separate arguments.
    statistic : function
        The statistic to compute.
    size : int, optional
        The window size (number of bases).
    start : int, optional
        The position at which to start (1-based).
    stop : int, optional
        The position at which to stop (1-based).
    step : int, optional
        The distance between start positions of windows. If not given,
        defaults to the window size, i.e., non-overlapping windows.
    windows : array_like, int, shape (n_windows, 2), optional
        Manually specify the windows to use as a sequence of (window_start,
        window_stop) positions, using 1-based coordinates. Overrides the
        size/start/stop/step parameters.
    fill : object, optional
        The value to use where a window is empty, i.e., contains no items.

    Returns
    -------

    out : ndarray, shape (n_windows,)
        The value of the statistic for each window.
    windows : ndarray, int, shape (n_windows, 2)
        The windows used, as an array of (window_start, window_stop) positions,
        using 1-based coordinates.
    counts : ndarray, int, shape (n_windows,)
        The number of items in each window.

    Notes
    -----

    The window stop positions are included within a window.

    The final window will be truncated to the specified stop position,
    and so may be smaller than the other windows.

    Examples
    --------

    Count non-zero (i.e., True) items in non-overlapping windows::

        >>> import allel
        >>> pos = [1, 7, 12, 15, 28]
        >>> values = [True, False, True, False, False]
        >>> nnz, windows, counts = allel.windowed_statistic(
        ...     pos, values, statistic=np.count_nonzero, size=10
        ... )
        >>> nnz
        array([1, 1, 0])
        >>> windows
        array([[ 1, 10],
               [11, 20],
               [21, 28]])
        >>> counts
        array([2, 2, 1])

    Compute a sum over items in half-overlapping windows::

        >>> values = [3, 4, 2, 6, 9]
        >>> x, windows, counts = allel.windowed_statistic(
        ...     pos, values, statistic=np.sum, size=10, step=5, fill=0
        ... )
        >>> x
        array([ 7, 12,  8,  0,  9])
        >>> windows
        array([[ 1, 10],
               [ 6, 15],
               [11, 20],
               [16, 25],
               [21, 28]])
        >>> counts
        array([2, 3, 2, 0, 1])