Title: Adjust Pearson's $r$ to Measure Arbitrary Monotone Dependence
Venue: NeurIPS 2024
Accepted: yes
Keywords: Pearson's r

correlation coefficient

rearrangement inequality

nonlinear monotone dependence
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

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.

1 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 21st 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 statistics, 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, 2022). The use of Pearson's r 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 ρ (Spearman, 1904), Kendall's τ (Kendall, 1938), Hilbert-Schmidt Independence Criterion(HSIC) (Gretton et al., 2005), distance correlation(dCor) (Székely, Rizzo, and Bakirov, 2007), Maximal Information Coefficient(MIC) (Reshef et al., 2011), and Chatterjee's ξ (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 r for a breakthrough.

In their recent paper titled ' Myths About Linear and Monotonic Associations: Pearson's r , Spearman's ρ , and Kendall's τ ', van den Heuvel and Zhan (2022) challenged the widespread belief that Pearson's r is only a measure for linear dependence, proving this notion to be false. Their findings indicate that Pearson's r should not be ruled out a priori for measuring nonlinear monotone dependence. 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 a tighter inequality than Cauchy-Schwarz Inequality, leverage it to adjust Pearson's r to measure nonlinear monotone dependence. The adjusted version of Pearson's r 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 ) ; the variance of X as var ( X ) , and the variance of Y as var ( Y ) . We assume that 0 < var ( X ) < ∞ , 0 < var ( Y ) < ∞ . We define X ↑ = F -1 ( U ) , X ↓ = F -1 (1 -U ) . Here, U is a random variable with the uniform distribution on (0 , 1) , and F -1 is the inverse cdf or quantile function defined as F -1 ( u ) = inf { x ∈ R : F ( x ) ⩾ u } , u ∈ (0 , 1) . Similarly, Y ↑ = G -1 ( U ) , Y ↓ = G -1 (1 -U ) .

Let x = ( x 1 , · · · , x n ) and y = ( y 1 , · · · , y n ) be samples of X and Y , each with n elements. Neither x nor y is constant. We denote the sample covariance of x and y as s x,y ; the sample variance of x as s 2 x ; the sample variance of y as s 2 y . We define the increasing and decreasing rearrangement of x as x ↑ = ( x (1) , x (2) , · · · , x ( n ) ) and x ↓ = ( x ( n ) , x ( n -1) , · · · , x (1) ) respectively, with x (1) ⩽ x (2) ⩽ · · · ⩽ x ( n ) . Similarly, we define y ↑ = ( y (1) , y (2) , · · · , y ( n ) ) , y ↓ = ( y ( n ) , y ( n -1) , · · · , y (1) ) .

Definition 1. A subset S of R 2 is non-decreasing (resp. non-increasing) if and only if for all ( x 1 , y 1 ) , ( x 2 , y 2 ) in S , x 1 < x 2 implies y 1 ⩽ y 2 (resp. x 1 < x 2 implies y 1 ⩾ y 2 ). Random variables X and Y are called increasing (resp. decreasing) monotone dependent if ( X,Y ) lies almost surely in a non-decreasing (resp. non-increasing) subset of R 2 . Samples x and y are called increasing (resp. decreasing) monotone dependent if { ( x, y ) } is a non-decreasing (resp. non-increasing) subset of R 2 .

Definition 1 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 Y is almost surely a monotone function of the other '. This is primarily because it doesn't necessitate a one-to-one mapping. Also, linear dependence, i.e., P ( Y = αX + β ) = 1 at the population level or y = ax + b at the sample level, is special case of 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-Schwarz Inequality, the well-known covariance inequality can be directly derived as

thus the geometric mean of var ( X ) and var ( Y ) , i.e., √ var ( X ) var ( Y ) , mathematically provides a bound for covariance cov ( X,Y ) , which ensures that Pearson's Correlation Coefficient

always falls into the range [ -1 , +1] . Scaled by √ var ( X ) var ( Y ) , Pearson's r turns into a normalized covariance , which is dimensionless and bounded. It 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 .

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 (Lin, 1989; Zegers, 1986).

For example, with the Mean Inequality Series (Hardy, Littlewood, and Polya, 1952), it is immediate that

in the sense that geometric mean √ var ( X ) var ( Y ) is always less than or equal to arithmetic mean 1 2 ( var ( X ) + var ( Y )) for nonnegative values var ( X ) and var ( Y ) . Then we get a looser bound for covariance, i.e., 1 2 (var ( X ) + var ( Y )) , with which another measure can be defined as follows(Zegers, 1986):

r + ( X,Y ) is named early as Additivity Coefficient (Zegers, 1986) and later as Standardized Covariance (Andraszewicz and Rieskamp, 2014). It is proved that the capture range of r + ( X,Y ) is limited to additive relationships, i.e., Y = ± X + β , which are special cases of linear relationships, i.e., Y = αX + β , with α being fixed to ± 1 (Zegers, 1986).

Further, we can find an even looser bound for covariance, in the sense that

and define a new measure as follows:

r = is named as Concordance Correlation Coefficient (Lin, 1989), and it is designed to measure identical relationship, i.e., Y = ± X . When X and Y are both positive, it 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, r , r + , and r = , 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 = αX + β ) towards additive ( Y = ± X + β ) and ultimately to identical ( Y = ± 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 a new measure with a broader capture range?

The bound in Pearson's r is intrinsically provided by Cauchy-Schwarz Inequality, which is one of the most widespread and useful inequalities in mathematics. Cauchy-Schwartz Inequality is so classic and reliable that one seldom tries to improve it. Both bounds in r + and r = are looser than that provided by Cauchy-Schwartz Inequality. To loosen Cauchy-Schwartz Inequality might be easy while to tighten such a classic inequality might be relatively difficult. However, we find that it is not impossible to improve the tightness of Cauchy-Schwarz Inequality. In other words, there exists sharper bound for covariance, which will be depicted in the next section.

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

where ⟨· , ·⟩ is the inner product. and ∥·∥ is the norm. The equality holds if and only if x and y are linearly dependent, i.e., y = ax for some constant a .

After defining an inner product on the set of random variables using the expectation of their product, i.e., ⟨ X,Y ⟩ = EXY , the Cauchy-Schwarz inequality becomes

Now, we will sharpen the Cauchy-Schwarz inequality. On the basis of the rearrangement theorems (Hardy, Littlewood, and Polya, 1952), we derive 6 theorems(corollaries/propositions) as follows.

Theorem 1. For random variables X and Y , we have

The equality on the left holds if and only if X and Y are monotone dependent, and the equality on the right holds if and only if Y d = αX , with sgn ( EXY ) = sgn ( α ) .

Here, d = denotes equality in distribution, and EX ↑ Y ↕ is defined as:

For the sake of conciseness, the proofs of Theorem 1 and theorems undermentioned are all included in Appendix A.1.

Theorem 2. For samples x and y we have

The equality on the left holds if and only if x and y 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 ↑ , y ↕ 〉 is defined as:

Corollary 1. For random variables X and Y , we have covariance inequality series as:

The first equality holds if and only if X and Y are monotone dependent, and the second equality holds if and only if Y d = αX + β , with sgn (cov ( X,Y )) = sgn ( α ) .

Here, cov ( X ↑ , Y ↕ ) is defined as:

Corollary 2. For samples x and y , we have covariance inequality series as

The first equality holds if and only if x and y are monotone dependent, and the second equality holds if and only if y is arbitrary permutation of ax + b , with sgn ( s x,y ) = sgn ( a ) .

Here, s x ↑ ,y ↕ is defined as:

2.4 The proposed Rearrangement Correlation

The inequality series in Corollary 1 and Corollary 2 provides sharper bounds for covariance at the population level and the sample level respectively. We will leverage them to define the so-called Rearrangement Correlation , which is the adjusted version of Pearson's r proposed here.

Definition 2. The Rearrangement Correlation of random variables X and Y is defined as:

Definition 3. The Rearrangement Correlation of samples x and y is defined as:

The new measure is named ' Rearrangement Correlation ' because its theoretical foundation is the rearrangement inequality, as shown in Theorem 1 and Theorem 2. We adopt the musical sharp symbol # to denote rearrangement correlation, signifying that this measure has sharp values because of its sharp bounds. Analogous to how C# is pronounced as C-sharp , r # is pronounced as r-sharp .

As for the relationship between r # ( x, y ) and r # ( X,Y ) , it is clear that r # ( x, y ) converges to r # ( X,Y ) when n →∞ 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 following hold:

∣ ∣ r # ( X,Y ) ∣ ∣ ⩽ 1 and the equality holds if and only if X and Y are monotone dependent.

∣ ∣ r # ( x, y ) ∣ ∣ ⩽ 1 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 ( X ↑ , Y ↕ )∣ ∣ , leads to the capture range expanding from linear dependence to (nonlinear) monotone dependence? The capture range is inherited from covariance itself. The capture range of covariance is limited neither to identical dependence as that of Concordance Correlation Coefficient , additive dependence as that of Additivity Coefficient , nor to linear dependence 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 r is also measuring nonlinear monotone dependence to some extent. The adjustment is nothing more than compensating for underestimation.

The relationships among the above-mentioned Concordance Correlation Coefficient ( r = ), Additivity Coefficient ( r + ), Pearson's r , and the new proposed Rearrangement Correlation ( r # ) are depicted in Figure 1.

actualivalue

sharp bound

loose bound

loosestibound

[cov(x,r) ≤|cov(x,r)≤ √var(x)var(Y) ≤(var(x)+var(x)≤(var(x)+var(r)+|x-²)

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I

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linear andnonlinear

The following proposition reveals the relationship between Pearson's r and its adjusted version, i.e., Rearrangement Correlation:

Proposition 2. For random variables X , Y , and samples x , y , the following hold:

∣ ∣ r # ( X,Y ) ∣ ∣ ⩾ | r ( X,Y ) | and the equality holds if and only if Y d = αX + β , with sgn ( r ( X,Y )) = sgn ( α ) .

∣ ∣ r # ( x, y ) ∣ ∣ ⩾ | r ( x, y ) | and the equality holds if and only if y is arbitrary permutation of ax + b , with sgn ( r ( x, y )) = sgn ( a ) .

Proposition 2 shows that r # ( X,Y ) will revert to r ( X,Y ) if and only if Y d = αX + β , sgn ( r ( X,Y )) = sgn( α ) , and r # ( x, y ) to r ( x, y ) if and only if y is arbitrary permutation of ax + b , with sgn ( r ( x, y )) = sgn( a ) . It is clear that linear dependence is special case of these conditions. Thus, r # reverts to r in linear scenarios.

Another question to be asked is, do we need a new monotone measure given that rank-based measures such as Spearman's ρ can already measure monotone dependence? The answer is twofold:

On the one hand, r # has a 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 a certain loss of information. A small difference between two values may no longer be distinguished from a large difference. With sample size n , there are totally n 3 -n 6 possible ρ values between -1 and +1 , whatever raw values are and however correlated patterns differ. The resolution of Spearman's ρ might be inadequate. To take a simple example, let x = (4 , 3 , 2 , 1) and

y 1 = (5 , 4 , 3 , 2 . 00)

y 2 = (5 , 4 , 3 , 3 . 25)

y 3 = (5 , 4 , 3 , 3 . 50)

y 4 = (5 , 4 , 3 , 3 . 75)

y 5 = (5 , 4 , 3 , 4 . 50)

Obviously, y 1 and x behaves exactly in the same way, with their values getting small and small step by step. The behavior of y 2 , y 3 , y 4 , and y 5 are becoming more and more different from that of x . However, the ρ values are all the same for y 2 , y 3 and y 4 . In contrast, the r # values can reveal all these differences exactly.

r # ( x, y 1 ) = 1 . 00 , ρ ( x, y 1 ) = 1 . 00

r # ( x, y 2 ) = 0 . 93 , ρ ( x, y 2 ) = 0 . 80

r # ( x, y 3 ) = 0 . 85 , ρ ( x, y 3 ) = 0 . 80

r # ( x, y 4 ) = 0 . 76 , ρ ( x, y 4 ) = 0 . 80

r # ( x, y 5 ) = 0 . 38 , ρ ( x, y 5 ) = 0 . 40

On the other hand, r # is comparable with Pearson's r , while the latter is not. For nonlinear monotone dependence, the value of Spearman's ρ might be remarkably greater than the value of Pearson's r . One may attempt to search for nonlinear relationships in data by checking whether the value of ρ far exceeds that of r . However, it might be meaningless and even impossible to compare their values directly. In cases, ρ can be either greater or less than r , and their sign can even be different. Thus the difference | ρ | - | r | is confusing. On the contrary, the signs of r # and r are always the same, and ∣ ∣ r # ∣ ∣ is always greater than or equal to | r | . ∣ ∣ r # ∣ ∣ -| r | equals to 0 if and only if y is arbitrary permutation of ax + b . Its value increases with the degree of nonlinearity.

However, Spearman's ρ can also be superior to r # in the sense that the former is robust to outliers while the latter is not. Rearrangement correlation is a 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 to outliers.

To be more robust, we can also transform the raw data into their ranks before calculating r # . Interestingly, r # becomes equivalent to Spearman's ρ when calculated on ranks. Let P and Q be the ranks of x and y respectively. Then, in the sense that sd ( P, Q ) = sd ( P ↑ , Q ↕ ) = n ( n +1) 12 , r # ( P, Q ) = ρ ( P, Q ) . This explains why ρ can measure nonlinear monotone relationships while r only measures linear ones, despite them sharing a similar formula. The key is not just the ranking but achieving a sharp bound. Since ρ and r # are equivalent when applied to ranks, and r # can measure arbitrary monotone dependence (as proven in our manuscript), ρ can do the same.

3 Experiments

3.1 Performance metrics

The main purpose of proposing Rearrangement Correlation is to provide a measure of dependence strength for nonlinear monotone relationships, rather than to simply serve as a 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 a measurement method. Trueness refers to the closeness of agreement between the mean or median of measured results and the true or accepted reference value. Precision refers to the closeness of agreement between measured values (ISO, 1994). 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 is better.

We evaluate the performance of different measures in a supervised way. We employ the coefficient of determination, R 2 , 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 is common in practices (Reshef et al., 2011). Further, we take its square root, R , as the conventional true value of the relationship strength. A simple evidence is that Pearson'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 r # , along with r + , Pearson's r , Spearman's ρ , Kendall's τ and four other leading dependence measures, i.e., HSIC, dCor, MIC and Chatterjee's ξ in the following way: for each scenario y = f ( x ) , we generated 512 pairs of ( x, y ) from the regression model y = f ( x ) + ε , and computed the values of different measures between x and y at different R levels. In the regression model, the x sample is uniformly distributed on the unit interval (0 , 1) , and the noise is normally distributed as ε ∼ N (0 , σ ) , with σ controlling R to a certain level.

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_frm_db () secondly and accuracy_plot_lite () finally. To reproduce the results, just keep all the parameters as default. More details are available in the package help files.

3.3 Performance in simulated scenarios

The simulation is conducted in up to 50 types of 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.

d seeds

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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 Iand Hrefer 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 ≈ 0 , and score one when there is perfect monotone relationship, i.e., R = 1 . Only Spearman's ρ , Kendall's τ and the adjusted r # meet this requirement. MIC also scores +1 when R = 1 , however, it tends to overestimate the strength when R is near zero, as also reported in other literature before (Chatterjee, 2021). The remaining five measures, i.e., r + , r , HSIC, dCor and ξ , underestimate the strength of nonlinear relationships, and never converge to +1 even when R approaches to 1.

Now let's take a close look at the intermediate values. It can be seen in Figure 2 that the nontransparent red line of r # is the closest one to the dashed line, which means the measured values by r # 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. r # has the highest trueness in all these representative levels. As for precision, the r # 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 a biased measure, tending to overestimate the strength when the signal is weak, and underestimate it when the signal is strong, as shown in Figure 2.

The overall performance in terms of MAE is ordered as r # (0 . 060) ≻ ρ (0 . 102) ≻ dCor (0 . 127) ≻ r (0 . 150) ≻ τ (0 . 157) ≻ MIC (0 . 196) ≻ ξ (0 . 206) ≻ r + (0 . 263) ≻ HSIC (0 . 518) . r # 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: Chwirut1, Hahn1, Rat43, Roszman1, 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 on the top of each

R = 0.98997

R = 0.9999

R = 0.98719

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scenario group. And the differences between the measured value and the true value are mapped as error bars.

It can be seen from Figure 3 that r # possesses minimum error, or best accuracy performance among all these five scenarios, with its MAE value as 0.00141, followed by ρ (0.0159), MIC(0.0249), dCor(0.0575), τ (0.0779), r (0.0916), ξ (0.166), HSIC(0.891) and r + (0.956).

3.5 Performance in non-monotone scenarios

It's worth noting that the aforementioned scenarios only cover monotone cases. To evaluate performance in typical non-monotone contexts, we conducted experiments in 16 scenarios, encompassing those outlined in (Reshef et al., 2011) and (Simon and Tibshirani, 2014). Details for these scenarios are available in Appendix A.5.

As anticipated, the MAE value of r # reaches a significant 0.418, notably inferior to those of ξ (0.141), MIC (0.157) and dCor (0.364). In essence, r # struggles to accurately measure non-monotone dependence. This limitation stems from its reliance on covariance , which inherently fails to detect non-monotone relationships. To illustrate, consider a standard introductory text book example, i.e., cov ( X,Y ) = 0 despite Y being totally dependent on X via Y = X 2 . Attempts to tighten its bound proves futile.

However, the performance of r # is also superior to those of Spearman's ρ (0.431), Pearson's r (0.437) and Kendall's τ (0.461). 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 as counterpart of Pearson's r for nonlinear monotone dependence.

The basic idea of rearrangement correlation is simple and straightforward. Its mathematical foundation is a sharpened version of the famous Cauchy-Schwarz Inequality. Tighter bound leads to wider capture range. With the adjustment, the capture range of Pearson's r is 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 r 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

The equality on the left holds if and only if X and Y are monotone dependent, and the equality on the right holds if and only if Y d = αX , with sgn ( EXY ) = sgn ( α ) .

Here, d = denotes equality in distribution, and EX ↑ Y ↕ is defined as:

Proof. The proof will be completed in two parts.

The proof of | EXY | ⩽ ∣ ∣ EX ↑ Y ↕ ∣ ∣ is mainly based on the rearrangement theorem for functions, i.e., Theorem 378 on page 278 of (Hardy, Littlewood, and Polya, 1952): Let f ↑ , g ↑ denote increasing rearrangements and f ↓ , g ↓ decreasing rearrangements of f and g on [0 , 1] as defined on page 276 of (Hardy, Littlewood, and Polya, 1952). Then we have

Let ∏ ( F, G ) be the set of all joint cdf's on R 2 having F and G as marginal cdf's. For arbitrary cdf H ∈ ∏ there exists ( X,Y ) : [0 , 1] → R 2 such that [ X ( U ) , Y ( U )] has cdf H . We can let f ( u ) = X ( u ) and g ( u ) = Y ( u ) so that EXY = ∫ 1 0 f ( u ) g ( u ) du . The increasing and decreasing rearrangements of f and g are just f ↑ ( u ) = F -1 ( u ) , f ↓ ( u ) = F -1 (1 -u ) , g ↑ ( u ) = G -1 ( u ) , and g ↓ ( u ) = G -1 (1 -u ) , as stated in (Whitt, 1976). Thus, we have

The right-hand ( resp. left-hand) equality holds if and only if ( X,Y ) = ( F -1 ( U ) , G -1 ( U ) ) d

d

( resp. ( X,Y ) = ( F -1 ( U ) , G -1 (1 -U ) ) ). The equality conditions can be equivalently expressed as X and Y are increasing ( resp. decreasing) monotone dependent (Mikusinski, Sherwood, and Taylor, 1991).

If EXY ⩾ 0 , we have EX ↑ Y ↕ = EX ↑ Y ↑ ⩾ EXY ⩾ 0 , which implies that ∣ ∣ EX ↑ Y ↕ ∣ ∣ = EX ↑ Y ↑ ⩾ EXY = | EXY | , and the equality holds if and only if X and Y are increasing monotone dependent. If EXY < 0 , we have EX ↑ Y ↕ = EX ↑ Y ↓ ⩽ EXY < 0 , which implies that ∣ ∣ EX ↑ Y ↕ ∣ ∣ = -EX ↑ Y ↓ ⩾ -EXY = | EXY | , and the equality holds if and only if X and Y are decreasing monotone dependent. Either way, we have ∣ ∣ EX ↑ Y ↕ ∣ ∣ ⩾ | EXY | , and the equality holds if and only if X and Y are monotone dependent.

The proof of ∣ ∣ EX ↑ Y ↕ ∣ ∣ ≤ √ EX 2 EY 2 is mainly based on Cauchy-Schwarz Inequality: If EXY ⩾ 0 , ∣ ∣ EX ↑ Y ↕ ∣ ∣ = ∣ ∣ EX ↑ Y ↑ ∣ ∣ ⩽ √ E ( X ↑ ) 2 E ( Y ↑ ) 2 = √ EX 2 EY 2 in the sense that X ↑ d = X , and Y ↑ d = Y . And the equality holds if and only if Y ↑ = αX ↑ , equivalently, Y d = αX , for some constant α ≥ 0 . Similarly, If E ( XY ) < 0 , ∣ ∣ EX ↑ Y ↕ ∣ ∣ = ∣ ∣ EX ↑ Y ↓ ∣ ∣ ⩽ √ E ( X ↑ ) 2 E ( Y ↓ ) 2 = √ EX 2 EY 2 in the sense that X ↑ d = X , and Y ↓ d = Y . And the equality holds if and only if Y ↓ = αX ↑ , equivalently, Y d = αX , for some constant α < 0 . Either way, we have ∣ ∣ EX ↑ Y ↕ ∣ ∣ ⩽ √ EX 2 EY 2 and the equality holds if and only if Y d = αX , with sgn ( EXY ) = sgn ( α ) .

Theorem 2. For samples x and y we have

The equality on the left holds if and only if x and y 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 ↑ , y ↕ 〉 is defined as:

Proof. The proof will also be completed in two parts.

The proof of |⟨ x, y ⟩| ⩽ ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ is mainly based on another rearrangement theorem for finite sets, i.e., Theorem 368 on page 261 of (Hardy, Littlewood, and Polya, 1952): With x ↑ = ( x (1) , x (2) , · · · , x ( n ) ) , y ↑ = ( y (1) , y (2) , · · · , y ( n ) ) , and y ↓ = ( y ( n ) , y ( n -1) , · · · , y (1) ) , we have

That is,

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(resp. oppositely) ordered ' on page 43 in (Hardy, Littlewood, and Polya, 1952), the equality conditions can be equivalently expressed as x and y are increasing(resp. decreasing) monotone dependent .

If ⟨ x, y ⟩ ⩾ 0 , we have 〈 x ↑ , y ↕ 〉 = 〈 x ↑ , y ↑ 〉 ⩾ ⟨ x, y ⟩ ⩾ 0 , which implies ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ = 〈 x ↑ , y ↑ 〉 ⩾ ⟨ x, y ⟩ = |⟨ x, y ⟩| , and the equality holds if and only if x and y are increasing monotone dependent.If ⟨ x, y ⟩ < 0 , we have 〈 x ↑ , y ↕ 〉 = 〈 x ↑ , y ↓ 〉 ⩽ ⟨ x, y ⟩ < 0 , which implies ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ = -〈 x ↑ , y ↓ 〉 ⩾ -⟨ x, y ⟩ = |⟨ x, y ⟩| , and the equality holds if and only if x and y are decreasing monotone dependent. Either way, we have ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ ⩾ |⟨ x, y ⟩| and the equality holds if and only if x and y are monotone dependent.

The proof of ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ ⩽ ∥ x ∥ ∥ y ∥ is mainly based on Cauchy-Schwarz Inequality: in the sense that norm ∥·∥ is permutation invariant, we have ∥ ∥ x ↑ ∥ ∥ = ∥ x ∥ and ∥ ∥ y ↑ ∥ ∥ = ∥ ∥ y ↓ ∥ ∥ = ∥ y ∥ . If ⟨ x, y ⟩ ⩾ 0 , we have ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ = ∣ ∣ 〈 x ↑ , y ↑ 〉∣ ∣ ⩽ ∥ ∥ x ↑ ∥ ∥ ∥ ∥ y ↑ ∥ ∥ = ∥ x ∥ ∥ y ∥ , and the equality holds if and only if y ↑ = ax ↑ , or equivalently, y is arbitrary permutation of ax for some constant a ≥ 0 . If ⟨ x, y ⟩ < 0 , we have ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ = ∣ ∣ 〈 x ↑ , y ↓ 〉∣ ∣ ⩽ ∥ ∥ x ↑ ∥ ∥ ∥ ∥ y ↓ ∥ ∥ = ∥ x ∥ ∥ y ∥ , and the equality holds if and only if y ↓ = ax ↑ , or equivalently, y is arbitrary permutation of ax for some constant a < 0 . Either way, we have ∣ ∣ 〈 x ↑ , y ↕ 〉∣ ∣ ⩽ ∥ x ∥ ∥ y ∥ , and the equality holds if and only if y is arbitrary permutation of ax , , with sgn ( ⟨ x, y ⟩ ) = sgn ( a ) .

Corollary 1. For random variables X and Y , we have covariance inequality series as:

The first equality holds if and only if X and Y are monotone dependent, and the second equality holds if and only if Y d = αX + β , with sgn (cov ( X,Y )) = sgn ( α ) .

Here, cov ( X ↑ , Y ↕ ) is defined as:

Proof. | cov ( X,Y ) | ⩽ ∣ ∣ cov ( X ↑ , Y ↕ )∣ ∣ ⩽ √ var ( X ) var ( Y ) , and the equality conditions are immediate from Theorem 1. The remaining parts of the inequality series are obvious.

Corollary 2. For samples x and y , we have covariance inequality series as

The first equality holds if and only if x and y are monotone dependent, and the second equality holds if and only if y is arbitrary permutation of ax + b , with sgn ( s x,y ) = sgn ( a ) .

Here, s x ↑ ,y ↕ is defined as:

Proof.

Proof. | s x,y | ⩽ ∣ ∣ s x ↑ ,y ↕ ∣ ∣ ⩽ √ s 2 x s 2 y , and the equality conditions are immediate from Theorem 2. The remaining parts of the inequality series are obvious.

Proposition 1. For random variables X , Y , and samples x , y , the following hold:

∣ ∣ r # ( X,Y ) ∣ ∣ ⩽ 1 and the equality holds if and only if X and Y are monotone dependent.

∣ ∣ r # ( x, y ) ∣ ∣ ⩽ 1 and the equality holds if and only if x and y are monotone dependent.

The proposition is immediate from Corollary 1 and Corollary 2.

Proposition 2. For random variables X , Y , and samples x , y , the following hold:

∣ ∣ r # ( X,Y ) ∣ ∣ ⩾ | r ( X,Y ) | and the equality holds if and only if Y d = αX + β , with sgn ( r ( X,Y )) = sgn ( α ) .

∣ ∣ r # ( x, y ) ∣ ∣ ⩾ | r ( x, y ) | and the equality holds if and only if y is arbitrary permutation of ax + b , with sgn ( r ( x, y )) = sgn ( a ) .

Proof. The proposition is immediate from Corollary 1 and Corollary 2.

A.2 Experiments settings

All the experiments are implemented with the R 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 in our study:

r + , recor::loose_pearson()

r , stats::cor()

r # , recor::sharp_pearson()

ρ , stats::cor(), with the argument method set as 'spearman'

τ , stats::cor(), with the argument method set as 'kendall'

HSIC, dHSIC::dhsic()

dCor, energy::dcor()

MIC, minerva::mine_stat()

ξ , XICOR::calculateXI()

For convenience, we developed an R package recor , which encapsulates all these measures as cor_XXX () functions. The recor package is available as recor_1.0.2.tar.gz in supplemental materials. For a latest version, please visit https://github.com/byaxb/recor.

Hardware environment configuration for this study was: DELL OptiPlex 7070 Tower, equipped with 8-core CPU Core i7-9700 @ 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 our experiments on 50 simulated scenarios, including all basic elementary functions, lots of composite functions and several typical special functions.

Linear: y = 2 x +1 , x ∈ [0 , 1]

Quadratic [asymmetry]: y = x 2 , x ∈ [0 , 1]

Square Root: y = √ x, x ∈ [0 , 1]

Cubic: y = x 3 , x ∈ [0 , 1]

Reciprocal: y = 1 x , x ∈ [0 , 1]

Exponential: y = e x , with x ∈ [0 , 1]

Logarithm: y = ln x, x ∈ [0 , 1]

Sine [quarter period]: y = sin ( x ) , x ∈ [ 0 , π 2 ]

Cosine [quarter period]: y = cos ( x ) , x ∈ [ 0 , π 2 ]

Tangent [half period]: y = tan ( x ) , x ∈ [ 0 , π 2 ]

Cotangent [half period]: y = cot ( x ) , x ∈ [ 0 , π 2 ]

Inverse Sine: y = arcsin ( x ) , x ∈ [0 , 1]

Inverse Cosine: y = arccos ( x ) , x ∈ [0 , 1]

Inverse Tangent: y = arctan ( x ) , x ∈ [0 , 1]

Inverse Cotangent: y = arccot ( x ) , x ∈ [0 , 1]

Secant [quarter period]: y = sec ( x ) , x ∈ [ 0 , π 2 ]

Cosecant [quarter period]: y = csc ( x ) , x ∈ [ 0 , π 2 ]

Hyperbolic Sine: y = sinh x = e x -e -x 2 , x ∈ [0 , 1]

Hyperbolic Cosine: y = cosh x = e x + e -x 2 , x ∈ [0 , 1]

Hyperbolic Tangent: y = tanh x = e 2 x -1 e 2 x +1 , x ∈ [0 , 1]

Hyperbolic Cotangent: y = coth x = e 2 x +1 e 2 x -1 , x ∈ [0 , 1]

Hyperbolic Secant: y = sech ( x ) = 2 e x + e -x , x ∈ [0 , 100]

Hyperbolic Cosecant: y = csch ( x ) = 2 e x -e -x , x ∈ [0 , 100]

Inverse Hyperbolic Sine: y = arcsinh ( x ) = ln ( x + √ x 2 +1 ) , x ∈ [0 , 1]

Inverse Hyperbolic Cosine: y = arccosh ( x ) = ln ( x + √ x 2 -1 ) , x ∈ [1 , 2]

Inverse Hyperbolic Tangent: y = arctanh ( x ) = 1 2 ln ( 1+ x 1 -x ) , x ∈ [0 , 1]

Inverse Hyperbolic Cotangent: y = arccoth ( x ) = 1 2 ln ( x +1 x -1 ) , x ∈ [1 , 2]

Inverse Hyperbolic Secant: y = arcsech ( x ) = ln ( 1 x + √ 1 x 2 -1 ) , x ∈ [0 , 1]

Inverse Hyperbolic Cosecant: y = arccsch ( x ) = ln ( 1 x + √ 1 x 2 +1 ) , x ∈ [0 , 1]

Hook: y = ax + b x , a = 1 , b = 1 , x ∈ [0 , 1]

Hoerl: y = x a e x , a = -1 , x ∈ [0 , 1]

Logit: y = ln x 1 -x , x ∈ [0 , 1]

Linear + Periodic, High Freq: y = 1 10 sin (10 . 6 (2 x -1)) + 11 10 (2 x -1) , x ∈ [0 , 1]

Einstein Function: Einstein 1 ( x ) = x 2 e x ( e x -1) 2 , x ∈ [0 , 1]

Exponential Integral: E 1 ( x ) = ∫ ∞ x e -t t dt, x ∈ [0 , 1]

Hyperbolic Sine Integral: Shi ( x ) = ∫ x 0 sinh t t dt, x ∈ [0 , 1]

Hyperbolic Cosine Integral: Chi ( x ) = γ + ln x + ∫ x 0 cosh t -1 t dt, x ∈ [0 , 1] . Here γ is Euler's Constant

Inverse Error Function: inverf ( x ) = t + 1 3 t 3 + 7 30 t 5 + · · · , t = 1 2 √ πx, x ∈ [0 , 1]

Gamma Function: Γ( x ) = ∫ ∞ 0 t x -1 e -t dt, x ∈ [0 , 1]

Psi Function: ψ ( x ) = d k +1 dx k +1 ln Γ ( x ) = Γ ′ ( x ) Γ( x ) , x ∈ [0 , 1] , k = 0

Riemann Zeta Function: ζ ( x ) = ∞ ∑ n =1 1 n x , x ∈ [0 , 1]

Beta Function: B ( x, w ) = Γ( x )Γ( w ) Γ( x + w ) , w = 1 , x ∈ [0 , 1]

Dirichlet Eta Function: η ( x ) = ∞ ∑ n =1 -1 n -1 n x , x ∈ [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:

Chwirut1: ultrasonic calibration, with Y as ultrasonic response, and X as metal distance.

Hahn1: 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.

Roszman1: quantum defects in iodine atoms, with Y as the number of quantum defects, and X as the excited energy state.

Thurber: semiconductor electron mobility, with Y as a measure of electron mobility, and X as the natural log of the density.

Data and details about these scenarios are available publicly at: https://www.itl.nist.gov/div898/strd/nls/nls_main.shtml

A.5 Non-monotone scenarios

We conducted our experiments on 16 non-monotone scenarios, comprehensively covering all the scenarios from (Reshef et al., 2011) and (Simon and Tibshirani, 2014).

Quadratic [symmetry]: y = 4 x 2 , x ∈ [ -1 2 , 1 2 ]

Sine, High Freq: y = sin (16 πx ) , x ∈ [0 , 1]

Cosine [High Freq]: y = cos (14 πx ) , x ∈ [0 , 1]

Linear + Periodic, Medium Freq: y = sin (10 πx ) + x, x ∈ [0 , 1]

Sine [Two periods]: y = sin (4 πx ) , x ∈ [0 , 1]

Sine [Low Freq]: y = sin (8 πx ) , x ∈ [0 , 1]

Sine, Non-Fourier Freq [Low]: y = sin (9 πx ) , x ∈ [0 , 1]

Cosine, Non-Fourier Freq [Low]: y = cos (7 πx ) , x ∈ [0 , 1]

Sine, Varying Freq [Medium]: y = sin (6 πx (1 + x )) , x ∈ [0 , 1]

Cosine, Varying Freq [Medium]: y = cos (5 πx (1 + x )) , x ∈ [0 , 1]

Linear + Periodic, High Freq 2: y = 1 5 sin (10 . 6 (2 x -1)) + 11 10 (2 x -1) , x ∈ [0 , 1]

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The answer NA means that the paper does not include experiments.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\nThe 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.\"\n\nQUESTIONS:\nQ1: 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?\n\nLIMITATIONS:\nNA", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-01"}

comments: 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.

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.

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
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 2
DATE: 2024-07-09

comments: 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.

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.
2. The title is confusing. Linear monotone dependence also fits in the paper's scope.

QUESTIONS:
1. It would be better to analyze some special cases under which how much the proposed correlation can improve the traditional Pearson one.
2. 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.
2. The title is confusing. Linear monotone dependence also fits in the paper's scope.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-12

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS:\nThe 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.\n\nWEAKNESSES:\n-\n\nQUESTIONS:\nCan 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\nLIMITATIONS:\n-", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}