@@ -73,7 +73,8 @@ def chatterjee_xi(x: np.ndarray, y: np.ndarray) -> float:
7373 underlying distributions of the variable.
7474
7575 It ranges from 0 (variables are completely independent) to 1
76- (one is a measurable function of the other).
76+ (one is a measurable function of the other). But a lot of the times the maximum
77+ value of the coefficient is lower than 1.
7778
7879 This implementation does not break ties at random, instead
7980 it break ties depending on order. This makes it dependent on
@@ -212,10 +213,10 @@ def concordance_rate(x: np.ndarray, y: np.ndarray) -> float:
212213 sem_y = np .std (y , ddof = 0 ) / n ** 0.5
213214 return float (
214215 (
215- np .sum ((x > mean_x + sem_x ) & (y > mean_y + sem_y ))
216- + np .sum ((x < mean_x - sem_x ) & (y > mean_y + sem_y ))
217- - np .sum ((x < mean_x - sem_x ) & (y < mean_y - sem_y ))
218- - np .sum ((x > mean_x + sem_x ) & (y < mean_y - sem_y ))
216+ np .sum ((x >= mean_x + sem_x ) & (y >= mean_y + sem_y ))
217+ - np .sum ((x <= mean_x - sem_x ) & (y >= mean_y + sem_y ))
218+ + np .sum ((x <= mean_x - sem_x ) & (y <= mean_y - sem_y ))
219+ - np .sum ((x >= mean_x + sem_x ) & (y <= mean_y - sem_y ))
219220 )
220221 / n
221222 )
@@ -228,7 +229,8 @@ def symmetric_chatterjee_xi(x: np.ndarray, y: np.ndarray) -> float:
228229 underlying distributions of the variable.
229230
230231 It ranges from 0 (variables are completely independent) to 1
231- (one is a measurable function of the other).
232+ (one is a measurable function of the other). But a lot of the times the maximum
233+ value of the coefficient is lower than 1.
232234
233235 This implementation does not break ties at random, instead
234236 it break ties depending on order. This makes it dependent on
@@ -311,28 +313,27 @@ def zhang_i(x: np.ndarray, y: np.ndarray) -> float:
311313
312314 References
313315 ----------
314- Zhang, Q. (2023).
315- On relationships between Chatterjee's and Spearman's correlation coefficients.
316- arXiv preprint arXiv:2302.10131.
317-
318- Notes
319- -----
320- This measure is assymetric: (x, y) != (y, x).
316+ Zhang, Q. (2025).
317+ On the extensions of the Chatterjee-Spearman test.
318+ Journal of Nonparametric Statistics, 1-30.
321319
322320 See Also
323321 --------
324322 scipy.stats.spearmanr - Spearman R coefficient.
325- obscure_stats.associaton.chatterjee_xi - Chatterjee Xi coefficient.
323+ obscure_stats.associaton.symmetric_chatterjee_xi - Chatterjee Xi coefficient.
326324 """
327325 if _check_arrays (x , y ):
328326 return np .nan
329327 x , y = _prep_arrays (x , y )
330328 if _check_arrays (x , y ):
331329 return np .nan
332330 return float (
333- max (
334- abs (stats .spearmanr (x , y , nan_policy = "omit" )[0 ]),
335- 2.5 ** 0.5 * chatterjee_xi (x , y ),
331+ min (
332+ 1.0 ,
333+ max (
334+ abs (stats .spearmanr (x , y , nan_policy = "omit" )[0 ]),
335+ 2.5 ** 0.5 * symmetric_chatterjee_xi (x , y ),
336+ ),
336337 )
337338 )
338339
@@ -596,13 +597,11 @@ def tukey_correlation(x: np.ndarray, y: np.ndarray) -> float:
596597 s_y = gini_mean_difference (y )
597598 x_norm = x / s_x
598599 y_norm = y / s_y
599- return float (
600- 0.25
601- * (
602- gini_mean_difference (x_norm + y_norm ) ** 2
603- - gini_mean_difference (x_norm - y_norm ) ** 2
604- )
600+ coef = 0.25 * (
601+ gini_mean_difference (x_norm + y_norm ) ** 2
602+ - gini_mean_difference (x_norm - y_norm ) ** 2
605603 )
604+ return float (max (min (coef , 1.0 ), - 1.0 ))
606605
607606
608607def gaussain_rank_correlation (x : np .ndarray , y : np .ndarray ) -> float :
@@ -640,10 +639,10 @@ def gaussain_rank_correlation(x: np.ndarray, y: np.ndarray) -> float:
640639 norm_factor = 1 / (n + 1 )
641640 x_ranks_norm = (np .argsort (x ) + 1 ) * norm_factor
642641 y_ranks_norm = (np .argsort (y ) + 1 ) * norm_factor
643- return float (
644- np .sum (stats .norm .ppf (x_ranks_norm ) * stats .norm .ppf (y_ranks_norm ))
645- / np .sum (stats .norm .ppf (np .arange (1 , n + 1 ) * norm_factor ) ** 2 )
642+ coef = np .sum (stats .norm .ppf (x_ranks_norm ) * stats .norm .ppf (y_ranks_norm )) / np .sum (
643+ stats .norm .ppf (np .arange (1 , n + 1 ) * norm_factor ) ** 2
646644 )
645+ return float ((coef - 0.5 ) * 2 )
647646
648647
649648def quantile_correlation (x : np .ndarray , y : np .ndarray , q : float = 0.5 ) -> float :
@@ -689,3 +688,82 @@ def quantile_correlation(x: np.ndarray, y: np.ndarray, q: float = 0.5) -> float:
689688 np .mean ((q - (y < np .quantile (y , q = q ))) * (x - np .mean (x )))
690689 / (((q - q ** 2 ) * np .var (x )) ** 0.5 )
691690 )
691+
692+
693+ def normalized_chatterjee_xi (x : np .ndarray , y : np .ndarray ) -> float :
694+ """Calculate normalizd Xi correlation coefficient.
695+
696+ Another variation of rank correlation which does not make any assumptions about
697+ underlying distributions of the variable.
698+
699+ It ranges from 0 (variables are completely independent) to 1
700+ (one is a measurable function of the other). This variant normalizes Chatterjee Xi,
701+ so it's maximum will always be 1.0.
702+
703+ This implementation does not break ties at random, instead
704+ it break ties depending on order. This makes it dependent on
705+ data sorting, which could be useful in application like time
706+ series.
707+
708+ The arrays will be flatten before any calculations.
709+
710+ Parameters
711+ ----------
712+ x : array_like
713+ Input array.
714+ y : array_like
715+ Input array.
716+
717+ Returns
718+ -------
719+ nxi : float.
720+ The value of the normalized xi correlation coefficient.
721+
722+ References
723+ ----------
724+ Dalitz, C.; Arning, J.; Goebbels, S. (2024).
725+ A Simple Bias Reduction for Chatterjee's Correlation.
726+ J Stat Theory Pract 18, 51.
727+
728+ Notes
729+ -----
730+ This measure is assymetric: (x, y) != (y, x).
731+ """
732+ if _check_arrays (x , y ):
733+ return np .nan
734+ x , y = _prep_arrays (x , y )
735+ if _check_arrays (x , y ):
736+ return np .nan
737+ n = len (x )
738+ # y ~ f(x)
739+ y_forward_ordered = y [np .argsort (x )]
740+ _ , y_unique_indexes , y_counts = np .unique (
741+ y_forward_ordered , return_inverse = True , return_counts = True
742+ )
743+ right_xy = np .cumsum (y_counts )[y_unique_indexes ]
744+ left_xy = np .cumsum (y_counts [::- 1 ])[len (y_counts ) - y_unique_indexes - 1 ]
745+ # y ~ f(y)
746+ y_ordered = y [np .argsort (y )]
747+ _ , y_unique_indexes , y_counts = np .unique (
748+ y_ordered , return_inverse = True , return_counts = True
749+ )
750+ right_yy = np .cumsum (y_counts )[y_unique_indexes ]
751+ left_yy = np .cumsum (y_counts [::- 1 ])[len (y_counts ) - y_unique_indexes - 1 ]
752+ # divide one by another
753+ return float (
754+ max (
755+ - 1 ,
756+ (
757+ 1
758+ - 0.5
759+ * np .sum (np .abs (np .diff (right_xy )))
760+ / np .mean (left_xy * (n - left_xy ))
761+ )
762+ / (
763+ 1
764+ - 0.5
765+ * np .sum (np .abs (np .diff (right_yy )))
766+ / np .mean (left_yy * (n - left_yy )),
767+ ),
768+ )
769+ )
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