Adjust Pearson's $r$ to Measure Arbitrary Monotone Dependence
Authors: Xinbo Ai
Venue: NeurIPS 2024 poster (2024)
Keywords: Pearson's r
correlation coefficient
rearrangement inequality
nonlinear monotone dependence
Primary area: probabilistic_methods
TL;DR: The most used Pearson's r, typically regarded as a measure only for linear dependence, has been enhanced to accurately measure nonlinear monotone dependence,  with the aid of an inequality tighter than Cauchy-Schwartz Inequality.

Abstract: Pearson's $r$, the most widely-used correlation coefficient, is traditionally regarded as exclusively capturing linear dependence, leading to its discouragement in contexts involving nonlinear relationships. However, recent research challenges this notion, suggesting that Pearson's $r$ should not be ruled out a priori for measuring nonlinear monotone relationships. Pearson's $r$ is essentially a scaled covariance, rooted in the renowned Cauchy-Schwarz Inequality. Our findings reveal that different scaling bounds yield coefficients with different capture ranges, and interestingly, tighter bounds actually expand these ranges. We derive a tighter inequality than Cauchy-Schwarz Inequality, leverage it to refine Pearson's $r$, and propose a new correlation coefficient, i.e., rearrangement correlation. This coefficient is able to capture arbitrary monotone relationships, both linear and nonlinear ones. It reverts to Pearson's $r$ in linear scenarios. Simulation experiments and real-life investigations show that the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.

Adjust Pearson's r to Measure Arbitrary Monotone Dependence
Xinbo Ai
School of Intelligent Engineering and Automation Beijing University Posts and Telecommunications Beijing 100876, China axb@bupt. edu.. cn
Abstract
Pearson's r, the most widely-used correlation coefficient, is traditionally regarded as exclusively capturing linear dependence, leading to its discouragement in con- texts involving nonlinear relationships. However, recent research challenges this notion, suggesting that Pearson's 's should not be ruled out priori for measuring nonlinear monotone relationships. Pearson's r essentially a scaled covariance, Schwarz Inequality. Our findings reveal that differ-
rooted in the renowned ent scaling bounds yield coefficients with different capture ranges, and interestingly, tighter bounds actually expand these ranges. We derive a tighter inequality than Bauchy-Schward Inequality, leverage refine Pearson's r, and propose a new correlation coefficient, i.e., rearrangement correlation. This coefficient is able to capture arbitrary monotone relationships, both linear and nonlinear ones. reverts to Pearson's 's in linear scenarios. Simulation experiments and real-life investi- gations show that the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.
Introduction
Proposed in the late 19th century, Pearson's r (Pearson, 1896) has been one of the main tools for scientists and engineers to study bivariate dependence during the 20th century. It is remarkably unaffected by the passage of time (Lee Rodgers and Alan Nice Wander, 1988) and still goes strong in the st century (Puccetti, 2022). It has been, and probably still is, the most used measure for statistical associations, and generally accepted as the measure of dependence, not only in statisties, but also in most applications of natural and social sciences (Tjøstheim, Otneim, and Støve, 2022).
Despite its popularity, Pearson's r has a number of shortcomings, and the most serious issue might be that it can only capture linear dependence, as stated in classical textbooks (Wasserman, 2004) and contemporary literatures (Armstrong, 2019 Tjøstheim, Otneim, and Støve, (002).. The use of Pearson's 's has been strongly discouraged for forms of associations other than linear ones (Speed, 2011).
Numerous nonlinear alternative coefficients have been proposed to address this deficiency, such as Spearman's p (Spearman, 1904), Kendall's 7 (Kendall, 1938). Hilbert-Schmidt Independence Criterion(HSIC) (Gretton et al., 2005), distance correlation( (CCor) (Székely, Rizzo, and Bakirov, 2007 Maximal Information (M))) (Reshef al., 2011), and Chatterjee's & (Chatterjee, 2021 Their capture ranges are extending from linear dependence to monotone dependence, and then to non-monotone dependence. Without exception, all these coefficients adopt radically different approaches for nonlinear dependence, rather than following the original way of Pearson's 's for breakthrough.
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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In their recent paper titled "Myths About Linear and Monotonic Associations: Pearson 's Spear- man's p. and Kendall's r", van den Heuvel and Zhan (2022) challenged the widespread belief that Pearson's, 8s is only measure for linear dependence, proving this notion to be false. Their findings indicate that Pearson's 's r should not be ruled out a priori for measuring nonlinear monotone depen- dence. Although this potential has been recognized, the specific approach to using Pearson's r for accurate measurement of nonlinear monotone dependence remains unresolved.
Pearson's r is essentially a scaled covariance, with the renowned Cauchy-Schwarz Inequality as its mathematical foundation. We find that different scaling bounds yield coefficients with different capture ranges, and interestingly, tighter bounds actually expand these ranges. We derive tighter inequality than Cauchy-Setie Schwarz Inequality, leverage adjust Pearson's 's r to measure nonlinear monotone dependence. The adjusted version of Pearson's 's is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.
2 Methods
2.1 Definitions and notations
Consider real-valued random variables X and Y with cdf's (cumulative distribution functions) F and G respectively. We denote the covariance of X and Y as cov (X.Y): Y); the variance of X as var (X), and the variance of Y as var (Y). We assume that 0 < var (X) oc, 0 < var (Y) < 00. We define X F=1 (U), X = F-1 (1 (1. Here, U is random variable with the uniform distribution on (0, 1), and F-1 is the inverse edf or quantile function defined as F-1 (u) inf (a :: (x) > u)
(0, 1). Similarly, Y G (U).
Y (U) G-1 (1-U). U).
Let (x1, :.n) and y = (y)... .......... : yo be samples of X and Y, each with n elements. Neither nor y is constant. We denote the sample covariance of x and y as Sx.y' the sample variance of x as 2 the sample variance of y as syy We define the increasing and decreasing rearrangement of x as x (I(1), ((2), Immm) and x (I()). 1)--1),- x(1) respectively, with 2(1) F(2) & < I(()). Similarly, we define y (y(1) -12) ()m)) y² (y(n)) 1)=-1): .V()))
Definition 1. subset S of non-decreasing (resp. non-increasing) if and only for all (x1, 31). (#2, in S, xı < implies y1 < 122 (resp. Iı < x2 implies y1 2 42). Random variables X and Y are called increasing (resp. decreasing) monotone dependent if (X, lies almost surely non-decreasing (resp. non-increasing) subset of RR? Samples z and y are called increasing (resp. decreasing) monotone dependent if {(x,y)} is a non-decreasing (resp. con-increasing) subset of R2.
Definition is sourced from (Mikusinski, Sherwood, and Taylor, 1991 Clearly Definition 1 is symmetrical with respect to X and Y. The monotone dependence outlined here encompasses a broader scope than definitions like the one in (Kimeldorf and Sampson, 1978), where "each of X and is almost surely a monotone function of the other". This is primarily because it doesn' 't necessitate one-to-one mapping. Also, linear dependence, i.e., P PY ax + B) = 1 at the population level or y = az + b at the sample level, is special case monotone dependence, and we will refer to dependence which is monotone but not linear as "nonlinear monotone dependence".
2.2 Different bounds lead to different capture ranges
With Cauchy-Schward Inequality, the well-known covariance inequality can be directly derived as
ccov (X, Y) & / var (X) var (Y),
thus the geometric mean of var (X) and var (Y), i.e., v var (X) var (Y), mathematically provides bound for covariance cov (X, Y), which ensures that Pearson's 's Correlation Coefficient
/var
cov (X,
(X,Y) var (X) var (Y)
always falls into the range [-1, +1]. Scaled by /va (X) var (Y), Pearson's turns into a normalized covariance, which dimensionless and bounded. possesses significant advantage over the original covariance in the sense that its value will not be affected by the change in the units of X and Y.

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A crucial issue that has been neglected so far is that boundedness doesn't ensure optimum. Scaling cov (X, Y) to the range [-1, +1] is not the only thing that matters. In fact, different bounds can be utilized to scale covariance to be bounded coefficients, as reported in previous works (L.in, 1989 Zegers. 1986).
For example, with the Mean Inequality Series (Hardy, Littlewood, and Polya. 1952). it is immediate that
1
Icov (X, & V var (X) var (Y) & (var (X) + var (Y))
in the sense that geometric mean / var (X) var (Y) is always less than or equal to arithmetic mean (var (X) + var (Y)) for nonnegative values var (X) and var (Y). Then we get looser bound for covariance, i.e., (var (X) + var (Y)), with which another measure can be defined as follows(Zegers, 1986):
I Cov (X, Y)
+ (X,Y)=
2 (var (X) + var (Y))
r+ (X.Y) is named early as Additivity Coefficient (Zegers, 1986) and later as Standardized Covari- ance (Andraszewicz and Rieskamp, 2014). It is proved that the capture range of r+ (X.Y) Y) is limited to additive relationships, i.e., Y =X + 3. which are special cases of linear relationships, i.e., Y= = aX + B, with a being fixed to 1 (Zegers, 1986
Further, we can find an even looser bound for covariance, in the sense that
(var (X) var (Y)) < var (X) var (Y) + X - 817).
and define new measure as follows:
cov (X,
(X,Y)=
var (X) + var (Y) + 1 11²²)
r= is named Concordance Correlation Coefficient (Lin, 1989), and it is designed to measure identical relationship, i.e., Y +XX. When X and are both positive, can be utilized to evaluate their agreement by measuring the variation from the 45" line through the origin (Lin, 1989).
As for the abovementioned three measures, T, r+, and they share the same numerator, cov (X, Y), the differences lie in their denominators. These denominators serve as bounds for cov (X, Y). Different bounds lead to different capture ranges. With the bounds being looser, their capture ranges are shrinking from linear (Y = a. + B) towards additive (Y = = + B) and ultimately to identical +X) relationships. The looser the bound, the narrower the capture range.
Up until now, all the efforts have only led to looser bounds and measures with narrower capture ranges. Could we possibly explore breakthroughs by approaching the problem from the opposite direction, aiming to achieve a tighter bound and consequently, devise new measure with broader capture range?
The bound in Pearson's is intrinsically provided by Cauchy-Schwaar Inequality, which is one the most widespread and useful inequalities in mathematics. Sauuch--Schwarte Inequality is classic and reliable that one seldom tries to improve it. Both bounds in r+ and r are looser than that provided by Cauchy-Schwarr while to tighten such a classic inequality might be relatively difficult. However, we find that is not impossible improve the tightness of -Scwway- Inequality. In other words, there exists sharper bound for covariance, which will be depicted in the next section.
Inequality. To loosen Cauchy-Schwartz Inequality might be easy
2.3 New inequality tighter than Cauchy-Schwarz Inequality
Before deriving the new inequality, we will briefly review the classic Cauchy-Schwarz inequality, which is common in textbooks. The Cauchy-Schwarz inequality states that for x and y, we have
(z, y <
||>||| Ilvll
where ( is the inner product. and |I-I| is the norm. The equality holds if and only if and y are linearly dependent, i.c., y az for some constant
3

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After defning an inner product on the set of random variables using the expectation of their product, ie., (X, Y EXY, the Cauchy-Schwer Schhwarz inequality becomes
EXY| & VEX-EY2
Now, we will sharpen the Cauchy-Schwarr inequality. On the basis of the rearrangement theorems (Hardy, Littlewood, and Polya, 1952), we derive 6 thereess(crollaiis/ppooiiii as follows.
Theorem 1. For random variables X and Y, we have
EXY & EX & VEXEEY'.
The equality on the left holds if and only if X and Y monotone dependent, and the equality the right holds if and only if = =xX, with sgn (EXY) =sgn sg).
Here, denotes equality in distribution, and Exty is defined as:
Ex Extyt, if EXY > 0 EXTYL, EXY <0
For the sake of conciseness, the proofs of Theorem Tand theorems undermentioned all included in Appendix [.
Theorem 2. For samples æ and y we have
(er, < (xt (
< ||>||| Ilvll
The equality the left holds if and only if x and are monotone dependent, and the equality on the right holds if and only if y is arbitrary permutation of ax, with sgn ((x, y)) = sgn (a).
Here, (x .²) is defined as:
(5).0))=
{{
(xt, ,if if (a, > (0) ,8 .if (I, <
Corollary 1. |cov (X,Y) & cov (xt,rt)
For random variables X and Y, we have covariance inequality series as:
< var (X) var (Y)
(var (X) + var (Y))
var (X) + var (Y) + XX 1 8P)
The first equality holds if and only if X and are monotone dependent, and the second equality holds if and only if aX 3. with sgn (cov (X,Y)) sgn (a).
Here, cov (X1, Y²) is defined as:
cov (xt, (xt,rt)= cov (xt,yt), is cav (X,Y) > 0 con (xt,y)) if cov (X,Y) < 0
Corollary 2. For samples x and y, have covariance inequality series as
|sa.vl < ||xtt,
V $2
$ + y
2
8g
The first equality holds if and only if x and y are monotone dependent, and the second equality holds if and only if is arbitrary permutation of ax + b, with sgn (sz.y) sgn (a).
Here, 82 is defined as:
8xt sat if 82.y 2 Sut s if Sx. <0
Szt .yt

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2.4 The proposed Rearrangement Correlation
The inequality series in Corollary [ and Corollary provides sharper bounds for covariance the population level and the sample level respectively. We will leverage them to define the so-called Rearrangement Correlation, which the adjusted version of Pearson's 's r proposed here.
Definition 2. The Rearrangement Correlation of random variables X and Y is defined as:
cov (X,Y)
(X,Y) coov (Xt, Y+)
Definition 3.
The Rearrangement Correlation of samples and is defined as:
85,8
(x.
1
The new measure is named "Rearrangement Correlation" because its theoretical foundation the rearrangement inequality, shown in Theorem I # to denote rearrangement correlation, signifying that this measure has sharp values because of its sharp bounds. Analogous how C# is pronounced C-sharp, y pronounced as r-sharp.
and Theorem 2 We adopt the musical sharp symbol
As for the relationship between # (x, y) and r (X, is clear that # (x. y) converges to # (X, Y) when + oc according to their definitions.
The capture range of rearrangement correlation is no longer limited to linear dependence but monotone dependence, which is revealed by the next proposition.
Proposition 1. For random variables X, Y, and samples x, y, the the following hold:
YX < 1 and the equality holds if and only if X and Y are monotone dependent.
, (x,y) < and the equality holds if and only if x and y are monotone dependent.
An interesting question might arise in one's mind: how can the simple adjustment, replacing /var (X) var (Y) with |cov (Xt, | leads to the capture range expanding from linear depen- dence to (nonlinear) monotone dependence? The capture range is inherited from covariance itself. The capture range of covariance is limited neither identical dependence as that of Concordance Correlation Coefficient, additive dependence that of Additiviry Coefficient, as that of Pearson's ' r. In fact, it can potentially detect and measure arbitrary monotone dependence, if scaled properly. In other words, Pearson's is also measuring nonlinear monotone dependence to some extent. The adjustment is nothing more than compensating for underestimation.
to linear dependence
The relationships among the above-mentioned Concordance Correlation Coefficient (r²), Additivity Coefficient (r+), Pearson's r, and the new proposed Rearrangement Correlation (x are depicted in Figure
actuall value
sharp
bound
loose bound
looser hound
loosest bound
()) |cov(x",, (x),
/vrr (ar(( (x) var(r) (r) + var (r)
cov(X.))
(x.(X.))
cov(XX.)
cov(xX,)
coov| X xx]
(var( (or(x)vvr(?
{(var[X)+ var(Y))
(x.Y)
r(xx))
(xxx))
(x.r)
monotone Y (x) linear and nonlinear
linear
additive y ==*X+
identical Y 1X
y=aX+>
Figure 1: Covariance inequality series, correlation coefficients and their capture ranges
5

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The following proposition reveals the relationship between Pearson's and its adjusted version, i.c. Rearrangement Correlation: Proposition 2. For random variables X, Y, and samples x, y, the following hold:
r (X, > Ir (X,Y) and the equality holds if and only if Y ax + B with (x, sgn (ax)
(x, > Ir (Ir w) and the equality holds if and only if is arbitrary permutation of ar + b, with sgn (r (z.u)) = sgn (a).
Proposition shows that r# (X, Y) will revert to r (X if and only if Y ax + B, sgn (r (X, )) = sgn (a), and r# (x,, y) to r (x, y) if and only if is arbitrary permutation of aæ + b, with sgm (r (a, y)) sgn (a). It is clear that linear dependence is special case of these conditions. Thus, + reverts to in linear scenarios.
Another question to asked is, do we need a new monotone measure given that rank-based measures such Spearman's 's can already measure monotone dependence? The answer is twofold:
On the one hand, y* has higher resolution and is more accurate. Without exception, all measures designed for monotone dependence are utilizing the order information. However, what we utilize here is the original information, rather than the ranks. Mapping numerical values to their ranks does of course produce certain loss of information. A small difference between two values may no longer be distinguished from large difference. With sample size n, there totally n²-n between 1 and + 1, whatever raw values are and however correlated patterns differ. The resolution of Spearman's 's p might be inadequate. To take simple example, let (4,3, 2, 1) and
6 possible p values
8/1 = (,,4,3,2 2.000
1/2 3,,4,3,3.25)
ys 35,4,3,3.50
* 54 35,4,3,3.75
= (5, 3.500
Obviously, Y1 and behaves exactly in the same way. with their values getting small and small step by step. The behavior of 1/. ys. ya, and ys are becoming more and more different from that of r. However, the p values all the same for yz, Us and y4. In contrast, the r# values can reveal all these differences exactly.
# (a, v1) 1.00, p (I. vn) 1.00
# (2, y2) 0.93, p (I. U2) 0.80
# (x, y3) 0.85, p (I, y3) 0.80
# (a, y4) 0.76, P (x. y4) = 0.80
(z, ys) = 0.38, p (Ix y6) = 0.40
On the other hand, # is comparable with Pearson's r, while the latter is not. For nonlinear monotone dependence, the value of Spearman's p might be remarkably greater than the value of Pearson's 's One may attempt search for nonlinear relationships in data by checking whether the value of p far exceeds that of r. However, it might be meaningless and even impossible to compare their values directly. In cases, p can be either greater or less than r, and their sign can even be different. Thus the difference |p| - |r| is confusing. On the contrary, the signs of rit and r are always the same, and (r is always greater than equal to |r]. # - |r| equals to 0 if and only if is arbitrary permutation of az b. Its value increases with the degree of nonlinearity.
However, Spearman's 's can also be superior to rf in the sense that the former is robust to outliers while the latter not. Rearrangement correlation is scaled covariance, and the limitation of being non-robust to outliers is inherited from covariance itself. In fact, concordance correlation coefficient, additivity coefficient, and Pearson's r are also scaled covariance measures, and none of them are robust outliers.

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To be more robust, we can also transform the raw data into their ranks before calculating r Interestingly, r becomes equivalent to Spearman's when calculated on ranks. Let P and Q? be the ranks of x and y respectively. Then, the sense that sd (P.Q) sd (F* r#hh (P.Q) - p (IP..) Q). This explains why p can measure nonlinear monotone relationships while only measures linear ones, despite them sharing similar formula. The key is not just the ranking but achieving sharp bound. Since and y are equivalent when applied to ranks, and r# can measure arbitrary monotone dependence (as proven our manuscript), P can do the same.
m(n+1)
12
Experiments
3.1
Performance metrics
The main purpose of proposing Rearrangement Correlation to provide a measure of dependence strength for nonlinear monotone relationships, rather than to simply serve test statistic for testing independence. Thus, our performance metrics focus on strength measurement.
The basic question to be asked when measuring any attribute is how accurate is this measurement, and there should be no exception for dependence measurement. ISO 5725 uses two terms "trueness" " and "precision" to describe the accuracy of measurement method. Trueness refers to the closeness of agreement between the mean median of measured results and the true accepted reference value. Precision refers to the closeness of agreement between measured values (ISO, (194). The comprehensive performance of trueness and precision can be represented as the mean absolute error (MAE for short), and calculated as the mean of the absolute values of the difference between the measured value and the conventional true value. On the whole, a measure with lower MAE value better.
We evaluate the performance of different measures in a supervised way. We employ the coefficient of determination, It², which is defined as the proportion of variance for one variable explained by the other variable, as the ground truth of strength of dependence, which common in practices (Reshef et al., 201I). Further, we take its square root, R, as the conventional true value of the relationship strength. simple evidence that Pearson's 's r, as the golden standard for measuring linear dependence, is equivalent to R, as long as the relationship is linear. Thus, it is reasonable to adopt R as the reference value.
3.2 Simulation procedure
We investigate the accuracy performance of # along with r+, Pearson's r. Spearman's Kendall's and four other leading dependence measures, i.e., HSIC, dCor, MIC and Chatterjee's E in the following way: for each scenario = (x), we generated 512 pairs of (z, u) from the regression model y = f (x) + E, and computed the values of different measures between and y at different R levels. In the regression model, the r sample is uniformly distributed on the unit interval (0, 1), and the noise is normally distributed as N (0,o), with a controlling R a certain level.
R
var (Y)
For the sake of robustness, the computation process is repeated 10 times for each measure at each R level, and the mean value is adopted.
Simulation procedure is implemented in the recor R package, which is available in supplemental materials. The workflow is to call accuracy db() firstly, accuracy results fom__lfmaabr secondly and accuracy_ pllttppoo_____ee plot_lite() finally. To reproduce the results, just keep all the parameters as default. More details available in the package help files.
3.3 Performance in simulated scenarios
The simulation is conducted up to 50 types different monotone scenarios, including all basic elementary functions, lots of composite functions and several special functions. To our knowledge, our research explores the most extensive and representative range of scenarios. Detailed descriptions of these scenarios can be found in Appendix A.3 and the results are shown in Figure 2

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Figure 2: Performance of different measures in 50 simulated scenarios
Figure 2 shows the scatter plots of conventional true value versus measured values in different scenarios. The nine investigated measures are located in nine panels respectively. In each panel, there are totally 50 transparent green lines, representing the measured value (Y-axis) with respect to conventional true value (X-axis for 50 scenarios. The prefixes I- and H- refer to Inverse and Hyperbolic respectively. For example. I-H-Tangent stands for Inverse Hyperbolic Tangent function. In all these panels, the dashed diagonal lines represent an ideal measure, the score of which is exactly the same as the conventional true value. Apparently, the closeness to this reference line reflects the performance of a measure for a certain scenario. The median values of each measure among scenarios at different R levels are also calculated and denoted by the non-transparent red line.
We first look at the extreme values on both sides. It is expected that a measure will score nearly zero when X and Y are randomly generated, i.e., R O, and score one when there is perfect monotone relationship. i.e., R = 1. Only Spearman's 's p. Kendall's and the adjusted 7 meet this requirement. MIC also scores +1 when R 1, however, tends to overestimate the strength when is near zero, as also reported in other literature before (Chatterjee, )00I)) The remaining five measures, i.e., r+, HSIC, dCor and &, underestimate the strength of nonlinear relationships. and never converge +1 even when R approaches to I.
Now let's take close look the intermediate values. It can be seen in Figure 2 that the non- transparent red line of r# is the closest one to the dashed line, which means the measured values by , possess the minimum error. To further quantify the accuracy, we add four boxplots at four representative R levels (approximately, 0.25, 0.50, 0.75, and 1.00) for each measure. y# has the highest trueness in all these representative levels. As for precision, the y# also outperforms all other measures except HSIC and MIC. Although HSIC and MIC possesses the best precision, they suffer from lower trueness. HSIC tends to underestimate the strength severely, and MIC is also biased measure, tending to overestimate the strength when the signal is weak, and underestimate it when the signal is strong, as shown in Figure
The overall performance in terms of MAE is ordered as ,# (0.060) > / (0.102) > dCor (0.127) > r (0.150) > (0.157) > IC (0.196) > E (0.206) + (0.263) > HHSC (OS1C)))))) ## possesses significant accuracy advantage over all other measures.
3.4 Performance in real-life scenarios
In addition to simulated scenarios, we also investigate the performance of these measures on real life scenarios provided by NIST (National Institute of Standards and Technology. (2003). There are five available monotone scenarios: Chwirutl, Hahn1, Rat43, Roszmanl, and Thurber. Details for these scenarios are available in Appendix A.4
The performance of nine measures in these five scenarios are depicted in Figure 3 Bar plots illustrate the measured values, the conventional true value verified by NIST is annotated the top of each
8

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0.90
Spuurmart
Kendalft
duscic
acter
MIC
Chatojee'
Figure 3: Performance of Different Measures in 5 Real-life Scenarios
scenario group, And the differences between the measured value and the true value are mapped as error bars.
It can be seen from Figure that r possesses minimum error, or best accuracy performance among all these five scenarios, with its MAE value as 0.00141, followed by p(0.0159), MIC(0.0249)., dCor((.0575), (0.07779) r(0.0916), 1(0.166), HSIC(0.891) and (0.956)..
3.5 Performance in non-monotone scenarios
It's worth noting that the aforementioned scenarios only monotone To evaluate perfor- mance in typical non-monotone contexts, we conducted experiments in 16 scenarios, encompassing those outlined (Reshef al., (011]) and (Simon and Tibshirami, 2014). Details for these scenarios are available in Appendix A.5
As anticipated, the MAE value of # reaches significant 0.418, notably inferior those of 5(0.141) MIC (0.157) and dCor (0.364). In essence, y* struggles to accurately measure non-monotone dependence. This limitation stems from its reliance covariance, which inherently fails detect non-monotone relationships. To illustrate, consider a standard introductory text book example, i.e., сov (X, Y) 0 despite Y being totally dependent X via Y X2, Attempts tighten its bound proves futile.
However, the performance of ff is also superior to those of Spearman's 's p(0.431), Pearson's r(0.437) and Kendall's 's 70.4611) As for accuracy performance, r outperforms the three classical correlation coefficients in not only monotone, but also non-monotone scenarios.
4 Conclusion and discussion
We proposed here an adjusted version of Pearson's r, i.e., rearrangement correlation, which can be treated counterpart of Pearson's 's for nonlinear monotone dependence.
The basic idea of rearrangement correlation is simple and straightforward. Its mathematical foundation sharpened version of the famous Ktuchy-Schwald Inequality. Tighter bound leads to wider capture range. With the adjustment, the capture range of Pearson's 's r extended from linear dependence to (nonlinear) monotone dependence. Simulated and real-life investigations demonstrate that the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients and other more recently proposed dependence measures.
We may draw the conclusion that: Pearson's 's is undoubtedly the gold measure for linear dependence. Now, it might be the gold measure also for nonlinear monotone dependence, if adjusted.

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A Appendix
A.1 Proofs of theorems, corollaries and propositions
Theorem 1. For random variables X and Y, we have
EXY & Exty & VEX.EY?.
The equality on the left holds if and only if X and Y are monotone dependent, and the equality the right holds if and only Y _h aX, with sgn (EXY)= = =sgn (ag)
Here, denotes equality in distribution, and EX Y v is defined as:
Ex EX if EXY 00 EX EXY <0
Proof. The proof will be completed in two parts.
The proof of |XXY| < |ExTY*| is mainly based on the rearrangement theorem for functions, i.e., Theorem 378 on page 278 of (Hardy, Littlewood, and Polya, 1952): Let ft, g" denote increasing rearrangements and f², g decreasing rearrangements of and on (0, 1] as defined on page 276 of (Hardy, Littlewood, and Polya. 1952). Then we have
So 5 (t) (u)du & S.' g < L.d (1) (u)du.
Let II (F, G) be the set of all joint edf's on R22 having F and as marginal edf's. For arbitrary edf H E II there exists (X,Y): Y): [0, 1] + I2 such that [X (U), (U) has cdf H. We can let f (u)
X (u) and g (u) Y (u) so that EXY The increasing and decreasing rearrangements of f and g are just f [w) = F (u). ft (u) F- u). 9d (u) = G-1 (u), and g (u) G (1 u). as stated in (Whitt, 1976). Thus, we have
So (w) (u)du.
Ex < EXY ExYY!
The hinht-hand (resp. left-hand) equality holds if and only if (X,Y) ( (resp. (X, Y) d (F-1 (U), " -
U))). The equality conditions can be equivalently expressed as X and Y are increasing (resp. decreasing) monotone dependent (Mikusinski, Sherwood, and Taylor, 1991).
EXX > 0, have Exty+ Extyt > EXY >> O, which implies that EX Extyt > EXY |EXY| J, and the equality holds if and only if X and Y increasing monotone dependent. If EXY O, we have Extyt EXty+ & EXY O, which implies that 1xx
Exty+ > -XXY = FXXY) and the equality holds if and only if X and are decreasing monotone dependent. Either way, we have |EtYy|] > EXY|, , and the equality holds if and only X and are monotone dependent.
The proof of |EXTY1| & VEX is mainly based on Cauchy-Schwarz Inequality: If EXY > 0, (ExtY1 = |ExtY*| < E(())
VEX=EY" in the sense that xt = and Y _ Y. And the equality holds if and only if if = axt equivalently,
Y d aX, for some constant a 2 Similarly, If E (XY) < 0, E 2 = E <
((Xt))E(YU) = VEX** in the sense that X _ X, and Y _ Y. And the equality holds if and only yt ax, equivalently, Y _ ax for some constant a 0. Either way, have |Extyt & VEX=EY2 and the equality holds if and only if Y d aX, with
sgn (EXY) sgn (a).

---

Theorem 2. For samples a and y have
Ke, 3)1 < (xt .x))))
||=|||
llvll
The equality the left holds if and only if x and are monotone dependent, and the equality on the right holds if and only if y is arbitrary permutation of az, with sgn ((x, y)) = sgn (a).
Here, (x) u is defined as:
xt (0')²²)
-
(o), (1).0)) if (x,0) > 0 (xt (ut) is (1,1) < 0
Proof. The proof will also be completed in two parts.
The proof of (a, v)1 < (xx, r is mainly based on another rearrangement theorem for nite sets, i.e., Theorem 368 on page 261 of (Hardy, Littlewood, and Polya, 1952): With x =
((2))"" 8" (3(1)) B(2)-" -B(m)) and y (y(m): 1):-1):" - . .(1) we have
(i) (in-it1) i+1)
Tilk
T()(
1=1
That is,
(x < (z, 2)
y²)
and the right-hand (resp. left-hand) equality holds if and only if x and y are similarly(resp. oppositely) ordered. According to the definitions of "similarly( oppositely) ordered" on page 43 in (Hardy, Littlewood, and Polya, 1952). the equality conditions can be equivalently expressed as and y are increasing resp.. decreasing) monotone dependent.
If u) > O, we have (xx y²) = (x1, (xt,y") > (x. y) > O, which implies Kxt. .t))l (x > (x. v) and the equality holds if and only if x and are increasing
monotone dependent.If (2r, v) < O, we have (xt, y²) (xt,, vt) & (Ir,y) 0, which implies [<=,, v = (x xt) > - (z. v) Kr,, u) and the equality holds if and only x and y are decreasing monotone dependent. Either way, we have Kx* (x,,>²> > (2, v) and the equality holds if and only x and y are monotone dependent.
Inequality: in the ||21|| = ||ж||| and ||>0|| = =|1x^|| = |lvll- If (xr, v) >> O, we have Kx*,, y²) = K v²> | | || [|v||. and the equality holds if and only y ax², or equivalently, y is arbitrary permutation of ax for some constant 2 0. If (ar, v) O, we have (xt - (x , (,",,,""))
The proof of (x,,, x²> & ||>||| I |lvll is mainly based on sense that norm ||-|| is permutation invariant, we have
< [[=*]]] || = ||>||| ||vlll and the equality holds if and only if yt ax", or equivalently, y is arbitrary permutation of ar for some constant
O. Either way. we have (xt. < | [|yl], and the equality holds if and only if y is arbitrary permutation of az,, with sgn ((x, y)) sgn (a).
Corollary 1. For random variables X and Y, we have covariance inequality series as: lcov (X, Y,,Y & cov (xt, var (X) var (Y)
(var (X) + var (Y))
var (X) + var (Y) + 1 1 (())
The first equality holds if and only if X and Y are monotone dependent, and the second equality d holds if and only if = a. 3, with sgn (cov (X,Y)) sgn (a).
12

---

Here, cov (x1, (2) is defined as:
cov (xt,r)) { con (xt, rt). if cav (X, Y) > 0
cov (xt, y.) if cav (x,
Proof. lcov (x; Y) diate from Theorem I
cov (Xt, / v var (X) var (Y), and the equality conditions are imme- The remaining parts of the inequality series are obvious.
Corollary 2. For samples a and we have covariance inequality series as
sxyl & (
$2
(5 + 8g)
1
2 s 5x
5²)
The first equality holds if and only if x and are monotone dependent, and the second equality holds and only if y is arbitrary permutation of ax + b, with sgn (8x,g) sgn (a).
Here, af az is defined as:
Sut if 82. Syt st, 82.9 < 0
821, yt =
Proof. [$2.y] & 1= remaining parts of the inequality series are obvious.
sy. and the equality conditions are immediate from Theorem 2 The
Proposition For random variables X, Y, and samples x, y, the the following hold:
(x,Y) & 1 and the equality holds if and only X and Y are monotone dependent.
t (r,y) < 1 and the equality holds if and only if x and y are monotone dependent.
Proof. The proposition is immediate from Corollary 1 and Corollary
Proposition 2. For random variables X, Y, and samples y. the following hold:
# (x, > (xxY) and the equality holds if and only if Y ax + B, with (xX, sgn (ax)
(I,w) rr (Ir, w)1 and the equality holds if and only if is arbitrary permutation of ar + b, with sgm (r (z, v)) = sgn (a).
Proof. The proposition is immediate from Corollary and Corollary
A.2 Experiments settings
All the experiments are implemented with the language (R Core Team, 2024), along with several add-on packages. The following are lists of packages and functions for the implementation of the nine measures involved our study:
recor:loose (cco::looseppearsonn)
T, stats:ccor(
recor:sharrp_pent p
P. stats::cor)), with the argument method set as "spearman"
7, stats::cor), with the argument method set as "kendall"
HSIC, dHSiC::dhsic)
dCor, energy::cor()
13

---

MIC, minerva::iine stat()
5. :ICoo::calculaeee((
For convenience, developed an package recor, which encapsulates these measures as cor cor_XXX() functions. The recor package is available as recor 1.0.2.tar.zz in supplemental materials. For latest version, please visit http://thhbb.coo//bbbbbbberr
Hardware environment configuration for this study was: DELL OptiPlex 7070 Tower, equipped with 8-core CPU Core 17-9700 @ 3.00GHz, 24G DDR4 2666MHz RAM. Under this configuration, it took about 5 days to complete all the experiments.
A.3 Simulated scenarios
We carry out experiments 50 simulated scenarios, including all basic elementary functions, lots of composite functions and several typical special functions.
1. Linear: y 2x 1, [0,
2. Quadratic [asymmetry]: = 22, [0.1]
3. Square Root: = Vz,2 [0.
4. Cubic: x², x E [0, 1]
5. Reciprocal: y = x x E [0,
1
6. Exponential: y = e", with x [0
7. Logarithm: y In., [0, 1]
8. Sine [quarter period]: y sin (z), [0.
9. Cosine [quarter period]: y cos (z), E [0,
Tangent [half period]: y = tan (z), [0,
11. Cotangent [half period]: y = cot (z), E [0,
12. Inverse Sine: y arcsin (z)
[0
13. Inverse Cosine: y arccos (x) x E [0, 1]
14. Inverse Tangent: y = arctan (r) x E [0,1]
15. Inverse Cotangent: y = arccot (x) / [0,1]
16. Secant [quarter period]: see (x) x E [0,
17. Cosecant [quarter period]: y ese (x) x [0.
18. Hyperbolic Sine: y = sinh x 8² --
2 x [0,
Hyperbolic Cosine: y = cosh r 5*
x E [0, 1]
2
20. Hyperbolic Tangent: y = tanh x . x E [0, 1]
21. Hyperbolic Cotangent: = coth x: = -== x E [0,
+1
22. Hyperbolic Secant: y = sech (x) 6t - x E [0, 100)
23. Hyperbolic Cosecant: y esch (x) ---- 20 100
24. Inverse Hyperbolic Sine: y aresinh (x) In (x x 2
, [0,
25. Inverse Hyperbolic Cosine: y ar arccosh (x:) In (x + Vx -
[1,2]
26. Inverse Hyperbolic Tangent: y = arctanh (x) In
x [0.1]
1-2
27. Inverse Hyperbolic Cotangent: y = arccoth (x) = _ In 2+1
x E [1,2]
28. Inverse Hyperbolic Secant: = aresech (x) In + x E [0,
14

---

29. Inverse Hyperbolic Cosccant: = arcesch (x) = In ( +++
V /++++1
+1) x E [0.1
30. Hook: y = a. x: a = 1,b 1, E [0,1]
31. Rational: y/
x+1 x- [0,1]
32. Hoerl: y = x=ee","" , a = -1, E [0.1]
33. Sigmoid: y = Ite 2, [-0.5, 0.5]
34. Logit: y In -- x E [0,1]
0,if < x 1, <<<
35. Step: y/-
0,
i < <
49
100
36. Piecewise [Sigmoid]:
50
2
1
+
2. if
100
19
< 2 <
100
51
if
<
<
100
37. Linear + Periodic, High Freq: 1/ To sin (10.6 (2x 1)) + H/ (2x - 1), x [0,1]
38. Sine Function: Sk, (x)) (mlmT(=kh)//l hb a(x-kN)//
k 0, 1, x [0.1]
39. Einstein Function: Einsteing (I)
(c--1)2 E [0
x² x²c"
40. Exponential Integral: Ei (x) 1/ dit, [0, 1]
41. Hyperbolic Sine Integral: Shi (x) = S0" sinht dt, x E [0, 1]
42. Hyperbolic Cosine Integral: Chi (x) = 7 + In + Lo oosh dt, x E [0, 1]. Here y is Euler's Constant
43. Error Function: er (x) =
v J S0 - = dt, [0.1]
44. Inverse Error Function: inver / = 1 + 30 1 t= 2 x E [0.1]
45. Gamma Function: (xx) 50 1 dt [0.1]
gh+1 46. Psi Function: va (z) क In r (x)
r'xx)
[(a] x E [0, 1], =
47. Riemann Zeta Function: (xx
x [0,1]
48. Bessel Function: Y, (x) = co=)))))))) Jo (x) = =(xx) Σ (-)* (+*2)*
sin(
(T+++++1
k=0
U == 0, x E [0
49. Beta Function: B (I, w)
[w)) T(ato)
W = 1, E [0.1]
50. Dirichlet Eta Function: (x) Σ
-1n-1
n= x E [0.1]
A.4
Real-life scenarios
All the five real life scenarios are provided by NIST (National Institute of Standards and Technology,
2003 as follows:
Chwirutl: ultrasonic calibration, with Y as ultrasonic response, and X as metal distance.
Hahml:
thermal expansion of copper, with Y as the coefficient of thermal expansion, and X as temperature in degrees kelvin.
* Rat43:
sigmoid growth, with Y as dry weight of onion bulbs and tops, and X as growing time.
quantum defects in iodine atoms, with as the number of quantum defects, and X as the excited energy state.
Roszmanl:
15

---

Thurber: semiconductor electron mobility, with Y as measure of electron mobility, and X as the natural log of the density.
Data and details about these scenarios are available publicly at: wtt..iiiii...ooooiivv/////nnnnni
A.5 Non-monotone scenarios
We conducted our experiments on 16 non-monotone scenarios, comprehensively covering all the scenarios from (Reshef al., 2011) and (Simon and Tibshirani, (2014).
1. Quadratic [symmetry]: y 4x x [-2. 2.
2. Cubic 2: y = 128(x ))²² 48 })
12 12 ()) [0.1]
3. Sine, High Freq: y sin (167x) x E [0,
4. Cosine [High Freq]: y cos (14Tx), E [0,
200x, if
200
5. Lopsided L-shaped:
1998r + 100 if
199
& x 100
200
y5 if
& x x
100
6. Circle: V 1-(2x 1)², E [0, 1]
7. Linear + Periodic, Medium Freq: = =in (10Tx) +0, E [0
8. Cubie 3: y y= 4x + x² 4x, x E 1.33
9. Cubic, Y-stretched: y = (4x* + x² 4r) x E [-1.1,
10. Sine [Two periods]: sin (4Tx) E [0, 1]
11. Sine [Low Freq]: y = sin (8T4), x [0. 1]
12. Sine, Non-Fourier Freq [Low]: y sin (9Tz)) x E [0, 1]
13. Cosine, Non-Fourier Freq [Low]: y cos (7m.) x [0.
14. Sine, Varying Freq [Medium]: y = sin (6Zæ (1 .))
[0.
15. Cosine, Varying Freq [Medium]: y = cos (5 (1 + x)) E [0.1]
16. Linear + Periodic, High Freq 2: sin (10.6 (2x - 1)) + # (2x 1) x E [0.1]
16

---

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What we proposed here is an adjusted version of Pearson's ' This study simply provides theoretical results for measuring dependence and does not involve societal impacts.
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Reviews:
[{"review_id": "BeucC80xva", "reviewer": "Reviewer_4BVE", "summary": "This paper proposes a new variant of Pearson's r coefficient to capture arbitrary monotone relationships between random variables. \n\nTraditionally, Pearson's r is used to capture linear dependence between variables. The relation of \"being linearly dependent\" is stronger than the relation of \"being monotone dependent (i.e. (X, Y) almost surely lies in a non-decreasing or non-increasing subset of R^2). The capturing power of linear relations by Pearson's r coefficient is explained by equality condition in Cauchy-Schwarz inequality. The authors of this paper demonstrate that strengthening of Cauchy-Schwarz inequality using rearrangement theorems by Hardy, Littlewood and Polya provides a new adjusted Pearson's coefficient, which authors call rearragement correlation coefficient that is able to capture monotonic dependences.", "questions": "Can this method be generalized beyond one-dimensional data? I.e., is there an analog of \"rearrangement correlation coefficient\" for a proper definition of monotone dependency between X and Y in R^d?", "limitations": "-", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "The problem studied in this paper is quite natural and has various practical applications. \n\nPrior work proposed modifications to Pearson's r coefficient but \"in opposite direction\" to capture more strict relations: concordance correlation coefficient (Lin 1989) to measure identical relationship, additivity coefficient (Zegers 1986) to capture additive relations Y = +-X+b. This work instead moves in the opposite direction and proposes a coefficient to capture more \"loose\" monotone relation. Therefore, this paper can be seen as a continuation of that line of work.\n\nThe paper is well-writtena and clearly presented. Proofs are simple, elegant and easy to follow.\nTheoretical results are supported by experiment on simulated data and real-world datasets.", "weaknesses": "-", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Q:Can this method be generalized beyond one-dimensional data? I.e., is there an analog of \"rearrangement correlation coefficient\" for a proper definition of monotone dependency between $X$ and $Y$ in $R^d$?\n\n\nA:Thank you for your comments. Your suggestion means a lot to us. We believe that there should be a multivariable version of the rearrangement correlation. We can draw inspiration from several successful precedents. For example, distance correlation can test and measure the dependence of two random vectors in arbitrary dimension (Székely et al. 2007). Pearson's $r$ has already been generalized to measure correlation between random vectors (Puccetti, 2022). Unfortunately, we do not yet have an analog of rearrangement correlation for random vectors. We have incorporated this into our research agenda. Thank you again for your valuable advice.\n\n---\n\nSzékely, Gábor J., Maria L. Rizzo, and Nail K. Bakirov (2007). Measuring and Testing Dependence by Correlation of Distances. In: Annals of Statistics 35.6, pp. 2769–2794. \n\nPuccetti, G. (2022). Measuring linear correlation between random vectors. Information Sciences 607, 1328-1347", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "sqzOG4WPMz", "reviewer": "Reviewer_Uy7T", "summary": "The paper proposes an adjustment to Pearson’s r to measure nonlinear monotone relationships, resulting in a new coefficient called the rearrangement correlation. The rearrangement correlation can capture both linear and nonlinear monotone relationships more accurately than traditional measures.", "questions": "1. It would be better to analyze some special cases under which how much the proposed correlation can improve the traditional Pearson one.\n2. The title is confusing. Linear monotone dependence also fits in the paper's scope.", "limitations": "1. It would be better to analyze some special cases under which how much the proposed correlation can improve the traditional Pearson one.\n2. The title is confusing. Linear monotone dependence also fits in the paper's scope.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "This paper rresents a new inequality tighter than the Cauchy-Schwarz Inequality. The proposed Rearrangement Correlation is well-defined but could use more intuitive explanations.", "weaknesses": "1. It would be better to analyze some special cases under which how much the proposed correlation can improve the traditional Pearson one.\n2. The title is confusing. Linear monotone dependence also fits in the paper's scope.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Comment: This paper presents a new inequality tighter than the Cauchy-Schwarz Inequality. The proposed Rearrangement Correlation is well-defined but could use more intuitive explanations.\n\nResponse: I am deeply grateful for your kind acknowledgement, and we will resort to permutahedron to explain the rearrangement correlation intuitively. Please kindly find the intuitive explanation in the global official comments.\n\n---\nQ: It would be better to analyze some special cases under which how much the proposed correlation can improve the traditional Pearson one.\n\nA: We have investigated 50 monotonic and 16 non-monotonic simulated cases, and 5 real-life cases to compare the performance of r♯ and other coefficients, including Pearson's r. The results are shown in Figure 2 and Figure 3.\n\nTo the best of our knowledge, our research explores the most extensive and representative range of scenarios. In fact, the number of scenarios in our research is much higher than those in other similar studies, such as (Reshef et al., 2011) [9 simulated functional relationships], (Simon and Tibshirani, 2014) [8 simulated scenarios], and (Chatterjee, 2021) [6 simulated cases]. \n\n\nReshef, David N. et al. (2011). “Detecting Novel Associations in Large Data Sets”. In: Science 334.6062, pp. 1518–1524.\n\nSimon, Noah and Robert Tibshirani (2014). Comment on \"Detecting Novel Associations In Large Data Sets\" by Reshef Et Al, Science Dec 16, 2011. arXiv: 1401.7645.\n\nChatterjee, Sourav (2021). “A New Coefficient of Correlation”. In: Journal of the American Statistical Association 116.536, pp. 2009–2022.\n\n---\nQ:The title is confusing. Linear monotone dependence also fits in the paper's scope.\n\nA: Thank you very much for your comments. According to your suggestion, we will change our title as *Adjust Pearson's r to Measure Arbitrary Monotone Dependence*", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QfM8C9MzXW", "reviewer": "Reviewer_Z7Nv", "summary": "The authors refined Pearson’s r, and proposed a new correlation coefficient, i.e., rearrangement correlation. They showed that this coefficient is able to capture arbitrary monotone relationships, both linear and nonlinear ones. With simulation, they showed the rearrangement correlation is more accurate in measuring nonlinear monotone dependence than the three classical correlation coefficients, and other recently proposed dependence measures.", "questions": "As mentioned above, Theorem 1, which was already proposed in the existing literature, seems enough to motivate this rearrangement correlation. I am not clear about the authors' significant contributions.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "The introduction of this correlation coefficient, i.e., rearrangement correlation, can measure some nonlinear dependence unlike the classical  Pearson's. The simulation also showed it performs better than some other correlation coefficient in the literature.", "weaknesses": "The paper's weakness lies in some aspects. First, the contribution of this rearrangement correlation doesn't seem significant enough or at least the authors can clarify more on this. For example, Theorem 1, which was already proposed in the existing literature, seems enough to motivate this rearrangement correlation thus lacking some significant theoretical contribution. In other way, in terms of application, the paper failed to clarify the importance of this rearrangement correlation in some meaningful statistical problems/machine learning algorithms other than just introduce this new concept.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Comments:\nThe contribution of this rearrangement correlation doesn't seem significant enough or at least the authors can clarify more on this. For example, Theorem 1, which was already proposed in the existing literature, seems enough to motivate this rearrangement correlation thus lacking some significant theoretical contribution. \n\n---\n\nResponse:\n\nThank you for your valuable feedback. Indeed, there seem to be some misconceptions that require clarification. \n\nIt is crucial to note that Theorem 1 was meticulously deduced and originally proposed within our manuscript. We have thoroughly researched the existing literature, and to the best of our knowledge, Theorem 1 has not been previously reported, making it highly unlikely that it appears elsewhere.\n\nThe paramount contribution of Theorem 1 lies in forging a novel connection between two famous inequalities: the Cauchy-Schwarz Inequality and the Rearrangement Inequality. Our proposition herein asserts that the Rearrangement Inequality offers a tighter bound compared to the Cauchy-Schwarz Inequality. If you happen to be aware of any similar theorem in the existing literature that also explores the intricate relationship between these two inequalities, we would be immensely grateful if you could furnish us with the precise source reference.\n\nIt is imperative to clarify that our contribution does not lie in the Rearrangement Inequality itself nor its probabilistic variant. Both the original Rearrangement Inequality and its probabilistic counterpart have been well-established in the literature, as documented in (Hardy, Littlewood, and Polya, 1952) and (Whitt, 1976), respectively, and have been duly cited within our manuscript. Rather, our novel contribution stems from uncovering the fact that the Rearrangement Inequality offers a tighter bound for covariance compared to the Cauchy-Schwarz Inequality. This tighter bound, in turn, paves the way for the introduction of the novel correlation.\n\nWe express our heartfelt gratitude for the valuable time and diligent effort you have dedicated to reviewing our manuscript. We sincerely hope that our endeavors will be duly recognized and appreciated.", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "b2NtLNtQdm", "reviewer": "Reviewer_61Js", "summary": "The authors here introduce a new correlation statistic here they call the\n\"rearrangement correlation:\" $r^{\\sharp}$. The work argues this statistic\ncaptures non-linear, monotonic relationships between two samples. The\nauthors prove a couple theoretical results for this new statistic and then\nconduct a large empirical analysis comparing this statistic to many\nalternatives that measure the strength of relationship between two variables.", "questions": "Q1: Given this measure is better suited to handle non-linear relationships\nbetween data, how does it compare to computing the Pearson correlation\nbetween the quantiles of two samples? Or, e.g, does computing some summary\nstatistic of a QQ-plot encode some of the same information being picked up\nby $r^{\\sharp}$?\n\nQ2: How does the new correlation measure $r^{\\sharp}$ encode different\nrelationships in data? E.g., if we have three datasets $x = (x_1, ...,\nx_n)$, $y^1 = (y^1_1, ..., y^1_n)$ and $y^2 = (y^2_1, ..., y^2_n)$, what is\nimplied if $r^{\\sharp}(x, y^1) > r^{\\sharp}(x, y^2)$? Does this signify a stronger\nrelationship between $x$ and $y^1$ than $x$ and $y^2$? How should one really\ninterpret this for correlations not close to 1?\n\nQ3: In what situations should one prefer this statistic over another? E.g.,\nfor approximately linear relationships, the Pearson correlation should be\nbetter than this measure. In what settings and under what assumptions should\nthe reader opt to use this statistic?", "limitations": "NA", "rating": 3, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "The paper provides a simple summary statistic (called the rearrangement\ncorrelation $r^{\\sharp}$) that aims to capture correlation between two\nvariables that may just be monotonic, rather than linear. This statistic is\nsimple to compute (requires $O(n \\log n)$ to sort each sample) and is 1 if\nand only if one sample is increasing monotone dependent in the other. They\nalso show that it is bounded below by the Pearson correlation statistic.\n\nThe authors provide some empirical analysis to suggest that this statistic\nis well-suited to capture the coefficient of determination between two\nsamples even when the relationship is non-linear. They perform this analysis\nwith both simulated and real-world data and compare their new statistic to\nmany other alternatives.\n\nOriginality:\n\n[+]\n\n+ The idea of introducing a different correlation statistic is not completely\n  new, but to the best of the reviewers knowledge, this statistic is not\n  known. This appears to be a new statistic.\n\n+ The authors are able to lower bound this new statistic with the Pearson\n  correlation statistic, which is a more well known statistic.\n\nQuality:\n\n[+]\n\n+ The paper is mostly well-written and easy to follow.\n\n+ Figure 1 does a nice job illustrates a hierarchy of statistics in the case\n  that the correlation coefficient is 1.\n\nClarity:\n\n[+]\n\n+ Most of the paper is easy to follow, barring some parts of the\n  experimental section.\n\nSignificance:\n\n[+]\n\n+ The authors have introduced a new correlation statistic that appears to\n  encode some mutual information between two paired samples.\n\n+ When the relationship between two samples is roughly monotonic, the\n  statistic appears better then many alternatives to estimates the $R^2$\n  between those two samples.", "weaknesses": "The main weakness in the current form of the paper is when to utilize this\nstatistic. The paper mentions that it is best as a proxy for \"strength\nmeasurement,\" but its main utility seems to be using it to measure the\nstrength of a monotonic relationship between two artibrary samples. This is\nuseful, but if this is the core utility, it is not clear how much better it\nis compared to using the RSS of a LOWESS regression or some summary\nstatistic of a QQ-plot.\n\nIf there were more theory on the topology of this statistic, one may be able\nto use it as a measure of how well a model fits a dataset, i.e., instead of\nthe Pearson coefficient of $y$ and $\\hat{y}$ (which is basically the $R^2$\nfor the model) one could compute $r^{\\sharp}$ instead. However, there is no\nreal analysis in the paper of whether a larger $r^{\\sharp}$ actually means\nit is a better fit in some sense.\n\nOverall, I think there is something interesting about this statistic, but it\nfeels a bit unfinished. E.g., understanding robustness to outliers,\nhow sensitive it is if one sample changed monotonically, or if it could be\nused in hypothesis testing would be huge reasons to consider this statistic\nmore broadly.\n\nOriginality:\n\n[-]\n\n- Outside the introduction of this statistic, much is still relatively\n  unknown. E.g., what is it limiting distribution as $n$ goes to infinity?\n  Is it robust to outliers? Is it sensitive to monotonic transformations?\n\nQuality:\n\n[-]\n\n- It would be nice to move the proof of Theorem 1 to the appendix, as it\n  isn't really necessary to understand the rest of the paper and there\n  already are proofs in the appendix.\n\n- What exactly does the population value $|r^{\\sharp}(X, Y)|$ measure? Is\n  there any intuitive way to understand its value?\n\nClarity:\n\n[-]\n\n- The experiment in Section 3 is not described in full detail. Is the\n  experiment computing correlations between $f(x)$ and $y$? Is it computing\n  correlations between $x$ and $y$? It would be helpful to spell this out a\n  bit more clearly. I think its the latter but it would be better to be\n  explicit.\n\n- Section 3.1 spends a lot of time discussing \"trueness\" and \"precision,\"\n  when it suffices to just suggest MAE as a statistic. There is no real need\n  to explain bias and variance to this audience :)\n\nSignificance:\n\n[-]\n\n- The authors mention is it meant to be used as a \"strength measurement,\"\n  but the context for when to use this measurement is not entirely\n  clear. E.g., the authors demonstrate that it can be used to measure the\n  amount of noise present between two samples $x$ and $y$ where $y$ is\n  roughly a (monotone) function of $x$, but if this is the use case, why\n  couldn't one just use a locally-weighted regression (e.g. LOWESS) to\n  estimate $f$ and thus the $R^2$? The true benefits of using this statistic\n  are still a bit unclear.\n\n- There are still some questions that remain regarding this statistic: e.g.,\n  is there some proof that explains roughly under what $f$, $\\sigma$, error\n  distribution, etc. will yield reasonable estimates for the $R^2$?\n\nDetailed Comments:\nL36: No need to mention the title of the paper, you can just mention the authors and provide the citation.\nL51-56: IMHO, most of these definitions are common enough to omit. One could just state \"X and Y are assumed to be random variables with bounded second moment.\"", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "---\nQ:The main weakness in the current form of the paper is when to utilize this statistic... it is not clear how much better it is compared to using the RSS of a LOWESS regression.\n\nA:Yes, the main utility of the proposed coefficient is to measure the strength of monotonic relationships. The roles of correlation and regression are different. Pearson's correlation coefficient should not be omitted even if we can fit linear regression between variables. And the same is for rearrangement correlation. \n1. Correlation analysis precedes regression modeling in most cases. One can utilize correlation analysis to determine whether there is a relationship between the independent variable and the dependent variable. And those most correlated independent variables are usually selected for further modelling. \n2. Rearrangement correlation are more general than certain monotonic regression models. For example, if the rearrangement correlation between X and Y is 0.8, it can take different monotonic forms, such as exponential function Y = e^kX or polynomial function Y = aX^3 + bX^2 + cX + d. And one has to try and fit those specific models.\n3. Rearrangement correlation can determine the strength between X and Y. However, it can not predict Y with X. If we select the right model and extract the signal from the observations, then the RSS might be similar to rearrangement correlation coefficient.\n\n---\nQ: Given this measure is better suited to handle non-linear relationships between data, how does it compare to computing the Pearson correlation between the quantiles of two samples? Or, e.g, does computing some summary statistic of a QQ-plot encode some of the same information being picked up by r♯?\n\nA:This is a great question! It is true that we can get r♯ by computing the Pearson correlation between the quantiles of two samples in the following way:\n\n\nr♯(x, y)=r(x, y) / r(quantile(x), quantile(y)) if r(x, y)>=0; and r♯(x, y)=r(x, y) / r(quantile(x), rev(quantile(y))) if r(x, y)<0. We can verify that it is equivalent to our definition of r♯.\n\nAlso, we can verify it with lines of R code:\n```\n> library(recor)\n> set.seed(2024)\n> x <- 10*rnorm(100)\n> noises <- 30*rnorm(100)\n#The positive case\n> y <- x^3 + noises\n> cor(x, y)\n[1] 0.7427739\n> recor(x, y)\n[1] 0.9997153\n> cor(x, y) /abs(cor(quantile(x, seq(0, 1, len = 100)), quantile(y, seq(0, 1, len = 100))))\n[1] 0.9997153\n#The negative case\n> y <- -x^3 + noises\n> cor(x, y)\n[1] -0.7429709\n> recor(x, y)\n[1] -0.9996488\n> cor(x, y) /abs(cor(quantile(x, seq(0, 1, len = 100)), quantile(y, seq(1, 0, len = 100))))\n[1] -0.9996488\n```\n\n---\nQ:There is no real analysis in the paper of whether a larger r♯ actually means it is a better fit in some sense.\n\nA: Yes, a larger r♯ actually means it is a better fit. We have done simulations and shown the results in Figure 2.\n\n1. In Figure 2, the x-axis is R, which stands for how well the model can be fitted, and the y-axis is the correlation coefficient. The non-transparent red line in the r♯ panel shows that r♯ increases with R.\n2. It is to be noted that R is the square root of coefficient of determination. R is NOT the same as Pearson's r, except for unary linear regression.\n\n---\nQ: How does the new correlation measure r♯ encode different relationships in data? ...\n\nA:Suppose that x=(x1,...,xn), y1=(y11,...,yn1)=f(x)+noise1 and y2=(y12,...,yn2)=f(x)+noise2. Then r♯(x,y1)>r♯(x,y2) implies that the noise level in y2 is higher than that in y1. Namely, there is a stronger relationship between x and y1 than x and y2. As the noise level rises, the correlation coefficient will stray further and further away from 1.\n\n---\nQ: In what situations should one prefer this statistic over another?\n\nA: \n1. For non-monotonic scenarios, the proposed r♯ is inferior to other coefficients such as MIC and dCor; For monotonic scenarios, r♯ outperforms all other coefficients.\n2. For non-linear scenarios, rearrangement correlation is superior to Pearson's r. For linear scenarios, rearrangement revert to Pearson's r.\n3. In most cases, we will try different coefficients and compare their values for further conclusion. For example, if we obtain r♯(X,Y)=1, then we can judge that the relationship is at least monotonic. To determine whether this monotonic relationship is also linear, we will further calculate the value of r(X,Y). \n\n---\nQ:Is there any intuitive way to understand its value?\n\nA:YES. We resort to permutahedron for intuitive explanation. Please kindly find it in the global rebuttal.\n\n---\nQ:Is the experiment computing correlations between f(x) and y? Is it computing correlations between x and y? \n\nA:Yes. We compute correlations between x and y. We will describe it explicitly in the main text, and provide more details in the appendix. \n\n---\nQ:Overall, I think there is something interesting about this statistic, but it feels a bit unfinished. E.g., understanding robustness to outliers,...\n\nA:It is indeed challenging to encapsulate the entirety of its theoretical underpinnings within a single conference paper. Hence, we have meticulously outlined the rationale behind its development, elucidated the methodology employed, and presented the experimental outcomes. This work is grounded on six pivotal theorems (inclusive of corollaries and propositions), which form the cornerstone of its theoretical framework. Furthermore, we have validated its practical efficacy through rigorous experiments conducted in both simulated and real-world settings, thereby ensuring a comprehensive narrative that encapsulates both the theoretical foundations and empirical validations.\n\n\n---\nOnce again, thank you for your comments. In addition to the aforementioned clarifications and revisions, we will also take into account all of your other suggestions.\n* to move the proof of Theorem 1 to the appendix\n* to simplify the statements evaluation indices\n* to follow your Detailed Comments about L36 and L51-56\n* ...", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:
# Main theoretical contributions

The main theoretical contribution is Theorem 1 and Theorem 2, along with their corollaries and propositions.

Theorem 1 establishes the connection between two famous inequalities, i.e., the Cauchy-Schwarz Inequality and the Rearrangement Inequality. It is revealed here that the Rearrangement Inequality provides a tighter bound for covariance than the Cauchy-Schwarz Inequality. This sharper bound leads to a new correlation coefficient, which outperforms others in measuring monotonic dependence.


# Intuitive explanation of rearrangement correlation
We have added a new figure (Fig. N) to provide an intuitive explanation of rearrangement correlation. Please refer to the PDF file for the figure.

The crux of the proposed rearrangement correlation lies in the rearrangement process, and the permutahedron offers a systematic representation of this rearrangement along with the corresponding correlation coefficients.

In Fig. N, we consider x = (1,2,3,4) and let y be an arbitrary permutation of {1,2,4,5}. With the reference x = (1,2,3,4) fixed in ascending order, there are two perfectly covarying y sequences, namely, (1,2,4,5) and (5,4,2,1), which are in similar and opposite order respectively compared to x. We place all permutations of y and the corresponding  r♯(x,y) values on a permutahedron P4 as depicted in Fig. N. The vertices of the permutahedron are in a one-to-one correspondence with the permutations. There are a total of n!=4!=24 covarying states, which coincides with the number of permutations of y relative to the fixed x. 

With the permutahedron, it is evident that covariance quantifies the extent of all the covarying states exactly. Through any path forward, the covariance will decrease to negative ones, and ultimately reach the lower limit, ${s_{{x^ \uparrow },{y^ \downarrow }}}$; conversely, through any path backward, the covariance will increase to positive ones, and ultimately reach the upper limit ${s_{{x^ \uparrow },{y^ \uparrow }}}$. Thus, ${s_{{x^ \uparrow },{y^ \downarrow }}}$ and ${s_{{x^ \uparrow },{y^ \uparrow }}}$ prove to be the infimum and supremum and for ${s_{x,y}}$. As standardized version of covariance, a correlation statistic should increase to +1 if the covarying status approaches the supremum node, and decrease to −1 as the covarying status nears the infimum node.

If we designate these two endpoints as ±1, the magnitude of the covarying status can intuitively be represented as the percentage of the maximum attainable covariance that is indeed realized, or alternatively, as the ratio of the actual covariance to the maximum possible covariance in the same direction. Given that the absolute values of the supremum and infimum may not necessarily coincide, we will scale the actual covariance with either the supremum or the infimum, contingent upon the sign of the covariance. This rationale underpins our definitions of rearrangement correlation as outlined in Definitions 2 and 3.

As shown in Fig.N, following the path indicated by the arrows, the covarying status is from similar order, through complete disorder, to opposite order, and the r♯(x,y) values will correspondingly vary from positive one (+1), through near zero, to negative one (−1), expressing directly and thoroughly the covarying status between x and y in a standardized way.

Meta-review: The paper proposes a very simple parameter-free measure of monotone dependence, which appears to be new, and possibly sharper than any existing ones,  and provides an elegant justification via relaxations of Cauchy-Schwarz inequality, and Hardy-Littlewood-Polya rearrangement inequality.  The scores for this paper are borderline, with a wide range of opinion, ranging from accept to reject.  While all of the reviewers thought the proposal is new and interesting, there were concerns that (i) it is unclear how and where it applies to ML -- why is it important,  (ii) theoretical results are interesting but narrow, not providing a comprehensive overview of the properties of the rearrangement coefficient.  (iii) some lack of comparison with other measures of dependence like the strength of association with Lowess regression.  

While I also think that the paper could be strengthened in many ways -- I think it's rare to have simple elegant proposals for such basic elemental notions as dependence measures, so I give it the benefit of the doubt and recommend for acceptance.  I also think that comparison to Lowess may not be strictly apples-to-apples, as Lowess is a complex procedure involving many parameter choices (kernel width, etc), while the method here has none.  In terms of theory -- the authors did add new discussion on comparing and contrasting it with other monotone dependence measures (Spearman, Kendall, e.t.c),  robustness to outliers, and relation to QQ plots.  There's
much more to be developed -- but the hope that the initial spark will ignite some interest, and more thorough analysis will follow.
Decision: Accept (poster)