import math from scipy import stats from scipy._lib._array_api import (xp_capabilities, array_namespace, xp_promote, xp_result_type) from scipy.stats._stats_py import _SimpleNormal, SignificanceResult, _get_pvalue from scipy.stats._axis_nan_policy import _axis_nan_policy_factory __all__ = ['chatterjeexi', 'spearmanrho'] def _xi_statistic(x, y, y_continuous, xp): # 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 = xp.argsort(x, axis=-1) j, y = xp.broadcast_arrays(j, y) y = xp.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) r, l = xp.astype(r, x.dtype), xp.astype(l, x.dtype) num = xp.sum(xp.abs(xp.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 * xp.sum((n - l) * l, axis=-1) statistic = 1 - n * num / den return statistic, r, l def _xi_std(r, l, y_continuous, xp): # Compute asymptotic standard deviation of xi under null hypothesis of independence # `axis=-1` is guaranteed by _axis_nan_policy decorator n = 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 xp.asarray(math.sqrt(2 / 5) / math.sqrt(n), dtype=r.dtype) # "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 = xp.arange(1, n + 1, dtype=r.dtype) u = xp.sort(r, axis=-1) v = xp.cumulative_sum(u, axis=-1) an = 1 / n**4 * xp.sum((2*n - 2*i + 1) * u**2, axis=-1) bn = 1 / n**5 * xp.sum((v + (n - i)*u)**2, axis=-1) cn = 1 / n**3 * xp.sum((2*n - 2*i + 1) * u, axis=-1) dn = 1 / n**3 * xp.sum((l * (n - l)), axis=-1) tau2 = (an - 2*bn + cn**2) / dn**2 return xp.sqrt(tau2) / math.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 @xp_capabilities(skip_backends=[('dask.array', 'no take_along_axis'), ('cupy', 'no rankdata (xp.repeats limitation)')]) @_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]_ recommends: "if there are ties among the Xi's, then choose an increasing rearrangement as above by breaking ties uniformly at random." This is easily accomplished by adding a small amount of random noise to `x`; see examples. [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]) Consider a case in which there are ties in `x`. >>> x = rng.integers(10, size=1000) >>> y = rng.integers(10, size=1000) [1]_ recommends breaking the ties uniformly at random. >>> d = rng.uniform(1e-5, size=x.size) >>> res = stats.chatterjeexi(x + d, y) >>> res.statistic -0.029919991638798438 Since this gives a randomized estimate of the statistic, [1]_ also suggests considering the average over all possibilities of breaking ties. This is computationally infeasible when there are many ties, but a randomized estimate of *this* quantity can be obtained by considering many random possibilities of breaking ties. >>> d = rng.uniform(1e-5, size=(9999, x.size)) >>> res = stats.chatterjeexi(x + d, y, axis=1) >>> np.mean(res.statistic) 0.001186895213756626 """ xp = array_namespace(x, y) # 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) x, y = xp_promote(x, y, force_floating=True, xp=xp) # 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, xp=xp) std = _xi_std(r, l, y_continuous, xp=xp) norm = _SimpleNormal() pvalue = _get_pvalue(xi / std, norm, alternative=alternative, xp=xp) 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, xp=xp)[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 xi = xi[()] if xi.ndim == 0 else xi pvalue = pvalue[()] if pvalue.ndim == 0 else pvalue return SignificanceResult(xi, pvalue) @xp_capabilities(cpu_only=True, exceptions=['jax.numpy'], skip_backends=[('dask.array', 'not supported by rankdata (take_along_axis)')] ) @_axis_nan_policy_factory(SignificanceResult, paired=True, n_samples=2, result_to_tuple=_unpack, n_outputs=2, too_small=1) def spearmanrho(x, y, /, *, alternative='two-sided', method=None, axis=0): r"""Calculate a Spearman rho correlation coefficient with associated p-value. The Spearman rank-order correlation coefficient is a nonparametric measure of the monotonicity of the relationship between two datasets. Like other correlation coefficients, it varies between -1 and +1 with 0 implying no correlation. Coefficients of -1 or +1 are associated with an exact monotonic relationship. Positive correlations indicate that as `x` increases, so does `y`; negative correlations indicate that as `x` increases, `y` decreases. The p-value is the probability of an uncorrelated system producing datasets with a Spearman correlation at least as extreme as the one computed from the observed dataset. Parameters ---------- x, y : array-like The samples: corresponding observations of the independent and dependent variable. The (N-d) arrays must be broadcastable. alternative : {'two-sided', 'less', 'greater'}, optional Defines the alternative hypothesis. Default is 'two-sided'. The following options are available: * 'two-sided': the correlation is nonzero * 'less': the correlation is negative (less than zero) * 'greater': the correlation is positive (greater than zero) method : ResamplingMethod, optional Defines the method used to compute the p-value. If `method` is an instance of `PermutationMethod`/`MonteCarloMethod`, the p-value is computed using `scipy.stats.permutation_test`/`scipy.stats.monte_carlo_test` with the provided configuration options and other appropriate settings. Otherwise, the p-value is computed using an asymptotic approximation of the null distribution. axis : int or None, optional If axis=0 (default), then each column represents a variable, with observations in the rows. If axis=1, the relationship is transposed: each row represents a variable, while the columns contain observations. If axis=None, then both arrays will be raveled. Like other `scipy.stats` functions, `axis` is interpreted after the arrays are broadcasted. Returns ------- res : SignificanceResult An object containing attributes: statistic : floating point array or NumPy scalar Spearman correlation coefficient pvalue : floating point array NumPy scalar The p-value - the probabilitiy of realizing such an extreme statistic value under the null hypothesis that two samples have no ordinal correlation. See `alternative` above for alternative hypotheses. Warns ----- `~scipy.stats.ConstantInputWarning` Raised if an input is a constant array. The correlation coefficient is not defined in this case, so ``np.nan`` is returned. Notes ----- `spearmanrho` was created to make improvements to SciPy's implementation of the Spearman correlation test without making backward-incompatible changes to `spearmanr`. Advantages of `spearmanrho` over `spearmanr` include: - `spearmanrho` follows standard array broadcasting rules. - `spearmanrho` is compatible with some non-NumPy arrays. - `spearmanrho` can compute exact p-values, even in the presence of ties, when an appropriate instance of `PermutationMethod` is provided via the `method` argument. References ---------- .. [1] Zwillinger, D. and Kokoska, S. (2000). CRC Standard Probability and Statistics Tables and Formulae. Chapman & Hall: New York. 2000. Section 14.7 .. [2] Kendall, M. G. and Stuart, A. (1973). The Advanced Theory of Statistics, Volume 2: Inference and Relationship. Griffin. 1973. Section 31.18 Examples -------- Univariate samples, approximate p-value. >>> import numpy as np >>> from scipy import stats >>> x = [1, 2, 3, 4, 5] >>> y = [5, 6, 7, 8, 7] >>> res = stats.spearmanrho(x, y) >>> res.statistic np.float64(0.8207826816681233) >>> res.pvalue np.float64(0.08858700531354405) Univariate samples, exact p-value. >>> res = stats.spearmanrho(x, y, method=stats.PermutationMethod()) >>> res.statistic np.float64(0.8207826816681233) >>> res.pvalue np.float64(0.13333333333333333) Batch of univariate samples, one vectorized call. >>> rng = np.random.default_rng(98145152315484) >>> x2 = rng.standard_normal((2, 100)) >>> y2 = rng.standard_normal((2, 100)) >>> res = stats.spearmanrho(x2, y2, axis=-1) >>> res.statistic array([ 0.16585659, -0.12151215]) >>> res.pvalue array([0.0991155 , 0.22846869]) Bivariate samples using standard broadcasting rules. >>> res = stats.spearmanrho(x2[np.newaxis, :], x2[:, np.newaxis], axis=-1) >>> res.statistic array([[ 1. , -0.14670267], [-0.14670267, 1. ]]) >>> res.pvalue array([[0. , 0.14526128], [0.14526128, 0. ]]) """ xp = array_namespace(x, y) dtype = xp_result_type(x, y, force_floating=True, xp=xp) rx = stats.rankdata(x, axis=axis) ry = stats.rankdata(y, axis=axis) rx = xp.astype(rx, dtype, copy=False) ry = xp.astype(ry, dtype, copy=False) res = stats.pearsonr(rx, ry, method=method, alternative=alternative, axis=axis) return SignificanceResult(res.statistic, res.pvalue)