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"&&!a%12&...C# % a?c$|jd}|rC|tjdtj|z |jS|d|dz|j}||d}||d}d|dzz |d|zd|zz dz|dzzdz}d|dzz ||||z |zzdzdz} d|d zz |d|zd|zz dz|zdz} d|d zz ||||z zdz} |d| zz | dzz| dzz } || tj|z S) Nrg?)rrrrr) rasarraymathsqrtrarangesortcumulative_sumr) r'r(r#r$r%iuvanbncndntau2s r,_xi_stdr@)s  AJzz$)E**TYq\\9zIII !QU!' **A A !"%%A QTBFFAaC!A#IMQT1F;; ;B QTBFFAQ MA-BF77 7B QTBFFAaC!A#IMQ.RF88 8B QTBFFAQKrF22 2B 2IA Q &D 774==49Q<< ''r.c|dvrtdt|tjs+|}d}|dkrt|||fS)N>FTz`y_continuous` must be boolean.z@`method` must be 'asymptotic' or a `PermutationMethod` instance. asymptotic) ValueError isinstancerPermutationMethodlower)r#rmessages r,_chatterjeexi_ivrHCsi=((:;;; fe5 6 6&T \ ! !W%% %  r.c|j|jfS)N)r*pvalue)res_s r,_unpackrMSs =#* $$r.) dask.arrayzno take_along_axis)cupyz#no rankdata (xp.repeats limitation)) skip_backendsTrr)paired n_samplesresult_to_tuple n_outputs too_smallFrB)rr#rcP t| t|\}t|d \}d}|dkrOt| \}}}t || } t } t || z | | } nZt|tj r@tj d |f fd|dd | d d i} | j | j } }|jd kr|d n|}| jd kr| d n| } t|| S)aCompute 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 Tforce_floatingr$greaterrBr$) alternativer$c6t|dS)NrZr)r-)r"rr!r$r#s r,zchatterjeexi..smAq,2&N&N&Nq&Qr.pairings)datar*r[permutation_typerrr)rrHrr-r@rr rDrrEpermutation_test_asdictr*rJndimr )r!r"rr#rr[xir'r(stdnormrJrKr$s` ` @r,r r Wsv~ A  B,L&AAL& a4B 7 7 7DAq K  A|;;;AqaLR000R#XtLLL FE3 4 4/$QQQQQQ#j EKNNDTDT   ]CJF7a<<BRB!;!++VBZZF b& ) ))r.z jax.numpy)rNz+not supported by rankdata (take_along_axis))cpu_only exceptionsrPz two-sided)r[rrcft||}t||d|}tj||}tj||}|||d}|||d}tj|||||} t | j| jS)aCalculate 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. ]]) TrWrF)copy)rr[r) rrrrrpearsonrr r*rJ) r!r"r[rrr$rrxryrKs r,r r sH A  B 1a < < rts 333333333333PPPPPPPPPPAAAAAA = ):(((4    %%% D O QRRR,TQ*1Q!MMM u\X*X*X*X*MMRRX*v$K=PQ,TQ*1Q!MMM(3DqF9F9F9F9MM F9F9F9r.