Repository: scipy/scipy
File: scipy/stats/_stats_py.py (Python)

Code before:

    Notes
    -----
    For compatibility with older versions of SciPy, the return value acts
    like a ``namedtuple`` of length 5, with fields ``slope``, ``intercept``,
    ``rvalue``, ``pvalue`` and ``stderr``, so one can continue to write::

        slope, intercept, r, p, se = linregress(x, y)

    With that style, however, the standard error of the intercept is not
    available.  To have access to all the computed values, including the
    standard error of the intercept, use the return value as an object
    with attributes, e.g.::

        result = linregress(x, y)
        print(result.intercept, result.intercept_stderr)

    Examples
    --------
    >>> import numpy as np
    >>> import matplotlib.pyplot as plt
    >>> from scipy import stats
    >>> rng = np.random.default_rng()

    Generate some data:

    >>> x = rng.random(10)
    >>> y = 1.6*x + rng.random(10)

    Perform the linear regression:

    >>> res = stats.linregress(x, y)

    Coefficient of determination (R-squared):

    >>> print(f"R-squared: {res.rvalue**2:.6f}")
    R-squared: 0.717533

    Plot the data along with the fitted line:

    >>> plt.plot(x, y, 'o', label='original data')
    >>> plt.plot(x, res.intercept + res.slope*x, 'r', label='fitted line')
    >>> plt.legend()
    >>> plt.show()

    Calculate 95% confidence interval on slope and intercept:

    >>> # Two-sided inverse Students t-distribution
    >>> # p - probability, df - degrees of freedom
    >>> from scipy.stats import t
    >>> tinv = lambda p, df: abs(t.ppf(p/2, df))

Review comment (suggestion):
We could use `xp_cov` now that it's vectorized, but this is such a special case (just variances and covariance) that it's not so bad as written now.

Code after:
    >>> # Two-sided inverse Students t-distribution
    >>> # p - probability, df - degrees of freedom
    >>> from scipy.stats import t
    >>> tinv = lambda p, df: abs(t.ppf(p/2, df))

    >>> ts = tinv(0.05, len(x)-2)
    >>> print(f"slope (95%): {res.slope:.6f} +/- {ts*res.stderr:.6f}")
    slope (95%): 1.453392 +/- 0.743465
    >>> print(f"intercept (95%): {res.intercept:.6f}"
    ...       f" +/- {ts*res.intercept_stderr:.6f}")
    intercept (95%): 0.616950 +/- 0.544475

    """
    xp = array_namespace(x, y)
    x, y = xp_promote(x, y, force_floating=True, xp=xp)

    TINY = 1.0e-20

    # _axis_nan_policy decorator ensures that `axis=-1`
    n = x.shape[-1]
    xmean = xp.mean(x, axis=-1, keepdims=True)
    ymean = xp.mean(y, axis=-1, keepdims=True)

    # Average sums of square differences from the mean
    #   ssxm = mean( (x-mean(x))^2 )
    #   ssxym = mean( (x-mean(x)) * (y-mean(y)) )
    x_ = _demean(x, xmean, axis=-1, xp=xp)
    y_ = _demean(y, ymean, axis=-1, xp=xp, precision_warning=False)
    xmean = xp.squeeze(xmean, axis=-1)
    ymean = xp.squeeze(ymean, axis=-1)

    ssxm = xp.vecdot(x_, x_, axis=-1) / n
    ssym = xp.vecdot(y_, y_, axis=-1) / n
    ssxym = xp.vecdot(x_, y_, axis=-1) / n

    # R-value
    #   r = ssxym / sqrt( ssxm * ssym )
    degenerate = (ssxm == 0.0) | (ssym == 0.0)
    NaN = xp.asarray(xp.nan, dtype=ssxym.dtype)
    r = xpx.apply_where(
        ~degenerate,
        (ssxym, ssxm, ssym),
        lambda ssxym, ssxm, ssym: xp.clip(ssxym / xp.sqrt(ssxm * ssym), -1.0, 1.0),
        lambda ssxym, ssxm, ssym: xp.where(ssxym==0, NaN, 0.0)
    )

    slope = ssxym / ssxm
    intercept = ymean - slope*xmean
    if n == 2:
        # handle case when only two points are passed in
        one = xp.asarray(1.0, dtype=r.dtype)