import numpy as np from scipy import stats from scipy.stats._stats_py import _SimpleNormal, SignificanceResult, _get_pvalue from scipy.stats._axis_nan_policy import _axis_nan_policy_factory __all__ = ['chatterjeexi'] # TODO: # - Adjust to respect dtype def _xi_statistic(x, y, y_continuous): # Compute xi correlation statistic # `axis=-1` is guaranteed by _axis_nan_policy decorator n = x.shape[-1] # "Rearrange the data as (X(1), Y(1)), . . . ,(X(n), Y(n)) # such that X(1) ≤ ··· ≤ X(n)" j = np.argsort(x, axis=-1) j, y = np.broadcast_arrays(j, y) y = np.take_along_axis(y, j, axis=-1) # "Let ri be the rank of Y(i), that is, the number of j such that Y(j) ≤ Y(i)" r = stats.rankdata(y, method='max', axis=-1) # " additionally define li to be the number of j such that Y(j) ≥ Y(i)" # Could probably compute this from r, but that can be an enhancement l = stats.rankdata(-y, method='max', axis=-1) num = np.sum(np.abs(np.diff(r, axis=-1)), axis=-1) if y_continuous: # [1] Eq. 1.1 statistic = 1 - 3 * num / (n ** 2 - 1) else: # [1] Eq. 1.2 den = 2 * np.sum((n - l) * l, axis=-1) statistic = 1 - n * num / den return statistic, r, l def _xi_std(r, l, y_continuous): # Compute asymptotic standard deviation of xi under null hypothesis of independence # `axis=-1` is guaranteed by _axis_nan_policy decorator n = np.float64(r.shape[-1]) # "Suppose that X and Y are independent and Y is continuous. Then # √n·ξn(X, Y) → N(0, 2/5) in distribution as n → ∞" if y_continuous: # [1] Theorem 2.1 return np.sqrt(2 / 5) / np.sqrt(n) # "Suppose that X and Y are independent. Then √n·ξn(X, Y) # converges to N(0, τ²) in distribution as n → ∞ # [1] Eq. 2.2 and surrounding math i = np.arange(1, n + 1) u = np.sort(r, axis=-1) v = np.cumsum(u, axis=-1) an = 1 / n**4 * np.sum((2*n - 2*i + 1) * u**2, axis=-1) bn = 1 / n**5 * np.sum((v + (n - i)*u)**2, axis=-1) cn = 1 / n**3 * np.sum((2*n - 2*i + 1) * u, axis=-1) dn = 1 / n**3 * np.sum((l * (n - l)), axis=-1) tau2 = (an - 2*bn + cn**2) / dn**2 return np.sqrt(tau2) / np.sqrt(n) def _chatterjeexi_iv(y_continuous, method): # Input validation for `chatterjeexi` # x, y, `axis` input validation taken care of by decorator if y_continuous not in {True, False}: raise ValueError('`y_continuous` must be boolean.') if not isinstance(method, stats.PermutationMethod): method = method.lower() message = "`method` must be 'asymptotic' or a `PermutationMethod` instance." if method != 'asymptotic': raise ValueError(message) return y_continuous, method def _unpack(res): return res.statistic, res.pvalue @_axis_nan_policy_factory(SignificanceResult, paired=True, n_samples=2, result_to_tuple=_unpack, n_outputs=2, too_small=1) def chatterjeexi(x, y, *, axis=0, y_continuous=False, method='asymptotic'): r"""Compute the xi correlation and perform a test of independence The xi correlation coefficient is a measure of association between two variables; the value tends to be close to zero when the variables are independent and close to 1 when there is a strong association. Unlike other correlation coefficients, the xi correlation is effective even when the association is not monotonic. Parameters ---------- x, y : array-like The samples: corresponding observations of the independent and dependent variable. The (N-d) arrays must be broadcastable. axis : int, default: 0 Axis along which to perform the test. method : 'asymptotic' or `PermutationMethod` instance, optional Selects the method used to calculate the *p*-value. Default is 'asymptotic'. The following options are available. * ``'asymptotic'``: compares the standardized test statistic against the normal distribution. * `PermutationMethod` instance. In this case, the p-value is computed using `permutation_test` with the provided configuration options and other appropriate settings. y_continuous : bool, default: False Whether `y` is assumed to be drawn from a continuous distribution. If `y` is drawn from a continuous distribution, results are valid whether this is assumed or not, but enabling this assumption will result in faster computation and typically produce similar results. Returns ------- res : SignificanceResult An object containing attributes: statistic : float The xi correlation statistic. pvalue : float The associated *p*-value: the probability of a statistic at least as high as the observed value under the null hypothesis of independence. See Also -------- scipy.stats.pearsonr, scipy.stats.spearmanr, scipy.stats.kendalltau Notes ----- There is currently no special handling of ties in `x`; they are broken arbitrarily by the implementation. [1]_ notes that the statistic is not symmetric in `x` and `y` *by design*: "...we may want to understand if :math:`Y` is a function :math:`X`, and not just if one of the variables is a function of the other." See [1]_ Remark 1. References ---------- .. [1] Chatterjee, Sourav. "A new coefficient of correlation." Journal of the American Statistical Association 116.536 (2021): 2009-2022. :doi:`10.1080/01621459.2020.1758115`. Examples -------- Generate perfectly correlated data, and observe that the xi correlation is nearly 1.0. >>> import numpy as np >>> from scipy import stats >>> rng = np.random.default_rng(348932549825235) >>> x = rng.uniform(0, 10, size=100) >>> y = np.sin(x) >>> res = stats.chatterjeexi(x, y) >>> res.statistic np.float64(0.9012901290129013) The probability of observing such a high value of the statistic under the null hypothesis of independence is very low. >>> res.pvalue np.float64(2.2206974648177804e-46) As noise is introduced, the correlation coefficient decreases. >>> noise = rng.normal(scale=[[0.1], [0.5], [1]], size=(3, 100)) >>> res = stats.chatterjeexi(x, y + noise, axis=-1) >>> res.statistic array([0.79507951, 0.41824182, 0.16651665]) Because the distribution of `y` is continuous, it is valid to pass ``y_continuous=True``. The statistic is identical, and the p-value (not shown) is only slightly different. >>> stats.chatterjeexi(x, y + noise, y_continuous=True, axis=-1).statistic array([0.79507951, 0.41824182, 0.16651665]) """ # x, y, `axis` input validation taken care of by decorator # In fact, `axis` is guaranteed to be -1 y_continuous, method = _chatterjeexi_iv(y_continuous, method) # A highly negative statistic is possible, e.g. # x = np.arange(100.), y = (x % 2 == 0) # Unclear whether we should expose `alternative`, though. alternative = 'greater' if method == 'asymptotic': xi, r, l = _xi_statistic(x, y, y_continuous) std = _xi_std(r, l, y_continuous) norm = _SimpleNormal() pvalue = _get_pvalue(xi / std, norm, alternative=alternative) elif isinstance(method, stats.PermutationMethod): res = stats.permutation_test( # Could be faster if we just permuted the ranks; for now, keep it simple. data=(y,), statistic=lambda y, axis: _xi_statistic(x, y, y_continuous)[0], alternative=alternative, permutation_type='pairings', **method._asdict(), axis=-1) # `axis=-1` is guaranteed by _axis_nan_policy decorator xi, pvalue = res.statistic, res.pvalue return SignificanceResult(xi, pvalue)