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I am interested in whether or not a "correlation" of three variables is something, and if what, what would this be?

Pearson product moment correlation coefficient

$$\frac{\mathrm{E}\left[(X-\mu_X)(Y-\mu_Y)\right]}{\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)}}$$

Now the question for 3 variables: Is

$$\frac{\mathrm{E}\left[(X-\mu_X)(Y-\mu_Y)(Z-\mu_Z)\right]} {\sqrt{\mathrm{Var}(X)\mathrm{Var}(Y)\mathrm{Var}(Z)}}$$

anything?

In R it seems like something interpretable:

a <- rnorm(100); b <- rnorm(100); 
c <- rnorm(100)

mean((a-mean(a)) * (b-mean(b)) * (c-mean(c))) / 
     (sd(a) * sd(b) * sd(c))
[1] -0.3476942

We normally look at the correlation between 2 variables given a fixed third variable's value. Could someone clarify?

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    $\begingroup$ 1) In your bivariate Pearson formula, if "E" (mean in your code) implies division by n then st. deviations must also be based on n (not n-1). 2) Let all three variables be the same variable. In this case, we expect correlation to be 1 (as in bivariate case), but alas... $\endgroup$
    – ttnphns
    Commented Aug 13, 2013 at 10:03
  • $\begingroup$ For a trivariate normal distribution it's zero, regardless of what the correlations are. $\endgroup$ Commented Aug 13, 2013 at 18:51
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    $\begingroup$ @RayKoopman Agreed! According to Isserlis' theorem such a mixed-product moment of a multivariate Gaussian joint distribution will always have an expectation of zero when the number of variables is odd. $\endgroup$
    – Galen
    Commented Mar 19, 2022 at 20:03
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    $\begingroup$ In Cosmology, three point correlation is defined see the answer here $\endgroup$ Commented Sep 17, 2022 at 5:28
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    $\begingroup$ Wikipedia seems to call this coskewness. $\endgroup$
    – Rufflewind
    Commented Sep 17, 2022 at 8:41

3 Answers 3

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It is indeed something. To find out, we need to examine what we know about correlation itself.

  1. The correlation matrix of a vector-valued random variable $\mathbf{X}=(X_1,X_2,\ldots,X_p)$ is the variance-covariance matrix, or simply "variance," of the standardized version of $\mathbf{X}$. That is, each $X_i$ is replaced by its recentered, rescaled version.

  2. The covariance of $X_i$ and $X_j$ is the expectation of the product of their centered versions. That is, writing $X^\prime_i = X_i - E[X_i]$ and $X^\prime_j = X_j - E[X_j]$, we have

    $$\operatorname{Cov}(X_i,X_j) = E[X^\prime_i X^\prime_j].$$

  3. The variance of $\mathbf{X}$, which I will write $\operatorname{Var}(\mathbf{X})$, is not a single number. It is the array of values $$\operatorname{Var}(\mathbf{X})_{ij}=\operatorname{Cov}(X_i,X_j).$$

  4. The way to think of the covariance for the intended generalization is to consider it a tensor. That means it's an entire collection of quantities $v_{ij}$, indexed by $i$ and $j$ ranging from $1$ through $p$, whose values change in a particularly simple predictable way when $\mathbf{X}$ undergoes a linear transformation. Specifically, let $\mathbf{Y}=(Y_1,Y_2,\ldots,Y_q)$ be another vector-valued random variable defined by

    $$Y_i = \sum_{j=1}^p a_i^{\,j}X_j.$$

    The constants $a_i^{\,j}$ ($i$ and $j$ are indexes--$j$ is not a power) form a $q\times p$ array $\mathbb{A} = (a_i^{\,j})$, $j=1,\ldots, p$ and $i=1,\ldots, q$. The linearity of expectation implies

    $$\operatorname{Var}(\mathbf Y)_{ij} = \sum a_i^{\,k}a_j^{\,l}\operatorname{Var}(\mathbf X)_{kl} .$$

    In matrix notation,

    $$\operatorname{Var}(\mathbf Y) = \mathbb{A}\operatorname{Var}(\mathbf X) \mathbb{A}^\prime .$$

  5. All the components of $\operatorname{Var}(\mathbf{X})$ actually are univariate variances, due to the Polarization Identity

    $$4\operatorname{Cov}(X_i,X_j) = \operatorname{Var}(X_i+X_j) - \operatorname{Var}(X_i-X_j).$$

    This tells us that if you understand variances of univariate random variables, you already understand covariances of bivariate variables: they are "just" linear combinations of variances.


The expression in the question is perfectly analogous: the variables $X_i$ have been standardized as in $(1)$. We can understand what it represents by considering what it means for any variable, standardized or not. We would replace each $X_i$ by its centered version, as in $(2)$, and form quantities having three indexes,

$$\mu_3(\mathbf{X})_{ijk} = E[X_i^\prime X_j^\prime X_k^\prime].$$

These are the central (multivariate) moments of degree $3$. As in $(4)$, they form a tensor: when $\mathbf{Y} = \mathbb{A}\mathbf{X}$, then

$$\mu_3(\mathbf{Y})_{ijk} = \sum_{l,m,n} a_i^{\,l}a_j^{\,m}a_k^{\,n} \mu_3(\mathbf{X})_{lmn}.$$

The indexes in this triple sum range over all combinations of integers from $1$ through $p$.

The analog of the Polarization Identity is

$$\eqalign{&24\mu_3(\mathbf{X})_{ijk} = \\ &\mu_3(X_i+X_j+X_k) - \mu_3(X_i-X_j+X_k) - \mu_3(X_i+X_j-X_k) + \mu_3(X_i-X_j-X_k).}$$

On the right hand side, $\mu_3$ refers to the (univariate) central third moment: the expected value of the cube of the centered variable. When the variables are standardized, this moment is usually called the skewness. Accordingly, we may think of $\mu_3(\mathbf{X})$ as being the multivariate skewness of $\mathbf{X}$. It is a tensor of rank three (that is, with three indices) whose values are linear combinations of the skewnesses of various sums and differences of the $X_i$. If we were to seek interpretations, then, we would think of these components as measuring in $p$ dimensions whatever the skewness is measuring in one dimension. In many cases,

  • The first moments measure the location of a distribution;

  • The second moments (the variance-covariance matrix) measure its spread;

  • The standardized second moments (the correlations) indicate how the spread varies in $p$-dimensional space; and

  • The standardized third and fourth moments are taken to measure the shape of a distribution relative to its spread.

To elaborate on what a multidimensional "shape" might mean, observe that we can understand principal component analysis (PCA) as a mechanism to reduce any multivariate distribution to a standard version located at the origin and equal spreads in all directions. After PCA is performed, then, $\mu_3$ would provide the simplest indicators of the multidimensional shape of the distribution. These ideas apply equally well to data as to random variables, because data can always be analyzed in terms of their empirical distribution.


Reference

Alan Stuart & J. Keith Ord, Kendall's Advanced Theory of Statistics Fifth Edition, Volume 1: Distribution Theory; Chapter 3, Moments and Cumulants. Oxford University Press (1987).


Appendix: Proof of the Generalized Polarization Identity

Let $x_1,\ldots, x_n$ be algebraic variables. There are $2^n$ ways to add and subtract all $n$ of them. When we raise each of these sums-and-differences to the $n^\text{th}$ power, pick a suitable sign for each of those results, and add them up, we will get a multiple of $x_1x_2\cdots x_n$.

More formally, let $S=\{1,-1\}^n$ be the set of all $n$-tuples of $\pm 1$, so that any element $s\in S$ is a vector $s=(s_1,s_2,\ldots,s_n)$ whose coefficients are all $\pm 1$. The claim is

$$2^n n!\, x_1x_2\cdots x_n = \sum_{s\in S} \color{red}{s_1s_2\cdots s_n}(s_1x_1+s_2x_2+\cdots+s_nx_n)^n.\tag{1}$$

A prettier way of writing this equality helps explain the factor of $2^n n!$ that appears: upon dividing by $2^n$ we obtain the average of the terms on the right side (since $S$ has $|S|=2^n$ elements) and the $n!$ counts the distinct ways to form the monomial $x_1\cdots x_n$ from products of its components--namely, it counts the elements of the symmetric group $\mathfrak{S}^n.$ Thus, upon abbreviating $s_1s_2\cdots s_n=\chi(\mathbf s)$ and letting $\mathbf{s}\cdot \mathbf{x} = s_1x_1+ \cdots + s_nx_n$ be the (usual) dot product of vectors,

$$\sum_{\sigma\in\mathfrak{S}^n} x_{\sigma(1)}x_{\sigma(2)}\cdots x_{\sigma(n)} = \frac{1}{|S|}\sum_{\mathbf s\in S} \color{red}{\chi(\mathbf s)}(\mathbf{s}\cdot \mathbf{x} )^n.\tag{1}$$

Indeed, the Multinomial Theorem states that the coefficient of the monomial $x_1^{i_1}x_2^{i_2}\cdots x_n^{i_n}$ (where the $i_j$ are nonnegative integers summing to $n$) in the expansion of any term on the right hand side is

$$\binom{n}{i_1,i_2,\ldots,i_n}s_1^{i_1}s_2^{i_2}\cdots s_n^{i_n}.$$

In the sum $(1)$, the coefficients involving $x_1^{i_1}$ appear in pairs where one of each pair involves the case $s_1=1$, with coefficient proportional to $ \color{red}{s_1}$ times $s_1^{i_1}$, equal to $1$, and the other of each pair involves the case $s_1=-1$, with coefficient proportional to $\color{red}{-1}$ times $(-1)^{i_1}$, equal to $(-1)^{i_1+1}$. They cancel in the sum whenever $i_1+1$ is odd. The same argument applies to $i_2, \ldots, i_n$. Consequently, the only monomials that occur with nonzero coefficients must have odd powers of all the $x_i$. The only such monomial is $x_1x_2\cdots x_n$. It appears with coefficient $\binom{n}{1,1,\ldots,1}=n!$ in all $2^n$ terms of the sum. Consequently its coefficient is $2^nn!$, QED.

We need take only half of each pair associated with $x_1$: that is, we can restrict the right hand side of $(1)$ to the terms with $s_1=1$ and halve the coefficient on the left hand side to $2^{n-1}n!$ . That gives precisely the two versions of the Polarization Identity quoted in this answer for the cases $n=2$ and $n=3$: $2^{2-1}2! = 4$ and $2^{3-1}3!=24$.

Of course the Polarization Identity for algebraic variables immediately implies it for random variables: let each $x_i$ be a random variable $X_i$. Take expectations of both sides. The result follows by linearity of expectation.

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    $\begingroup$ Well done on explaining so far! Multivariate skewness kind of makes sense. Could you perhaps add an example that would show the importance of this multivariate skewness? Either as an issue in a statistical models, or perhaps more interesting, what real life case would be subject to multivariate skewness :)? $\endgroup$ Commented Mar 12, 2017 at 18:55
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Hmmm. If we run...

a <- rnorm(100);
b <- rnorm(100);
c <- rnorm(100)
mean((a-mean(a))*(b-mean(b))*(c-mean(c)))/
  (sd(a) * sd(b) * sd(c))

it does seem to center on 0 (I haven't done a real simulation), but as @ttnphns alludes, running this (all variables the same)

a <- rnorm(100)
mean((a-mean(a))*(a-mean(a))*(a-mean(a)))/
  (sd(a) * sd(a) * sd(a))

also seems to center on 0, which certainly makes me wonder what use this could be.

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    $\begingroup$ The nonsense apparently comes from the fact that sd or variance is a function of squaring, as is covariance. But with 3 variables, cubing occurs in the numerator while denominator remains based on originally squared terms $\endgroup$
    – ttnphns
    Commented Aug 13, 2013 at 11:26
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    $\begingroup$ Is that the root of it (pun intended)? Numerator and denominator have the same dimensions and units, which cancel, so that alone doesn't make the measure poorly formed. $\endgroup$
    – Nick Cox
    Commented Aug 13, 2013 at 13:12
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    $\begingroup$ @Nick That's right. This is simply one of the multivariate central third moments. It is one component of a rank-three tensor giving the full set of third moments (which is closely related to the order-3 component of the multivariate cumulant generating function). In conjunction with the other components it could be of some use in describing asymmetries (higher-dimensional "skewness") in the distribution. It's not what anyone would call a "correlation," though: almost by definition, a correlation is a second-order property of the standardized variable. $\endgroup$
    – whuber
    Commented Mar 10, 2017 at 20:56
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    $\begingroup$ You are sampling (hypothetically) from an IID standard normal vector. Normal distributions are symmetric, so $\mathbb{E}[XYZ] = -\mathbb{E}[XYZ] = 0$ is the result. The Isserlis' theorem is related as well. $\endgroup$
    – Galen
    Commented Sep 20, 2022 at 4:36
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If You need to calculate "correlation" between three or more variables, you could not use Pearson, as in this case it will be different for different order of variables have a look here.

If you are interesting in linear dependency, or how well they are fitted by 3D line, you may use PCA, obtain explained variance for first PC, permute your data and find probability, that this value may be to to random reasons. I've discuss something similar here (see Technical details below).

Matlab code

% Simulate our experimental data
x=normrnd(0,1,100,1);
y=2*x.*normrnd(1,0.1,100,1);
z=(-3*x+1.5*y).*normrnd(1,2,100,1);
% perform pca
[loadings, scores, variance]=pca([x,y,z]);
% Observed Explained Variance for first 
% principal component
OEV1 = variance(1)/sum(variance)
% perform permutations
permOEV1=[];
for iPermutation=1:1000
    permX=datasample(x,numel(x), 'replace', 
                            false);
    permY=datasample(y,numel(y), 'replace', 
                            false);
        permZ  = datasample(z, numel(z), 
                    'replace', false);
[loadings,  scores, variance] = pca([permX, 
                          permY, permZ]);
        permOEV1(end+1) = variance(1) / 
                             sum(variance);
end
    
% Calculate p-value
    p_value=sum(permOEV1>=OEV1)/(numel(permOEV1)+1)  
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