Calibrating dependence between random elements

TL;DR

This paper explores and refines measures of dependence between random variables, introducing a calibrated scale based on the complexity of approximating functions, building on classical concepts from Gebelein and Rényi.

Contribution

It summarizes properties of Rényi's dependence measure and introduces a new calibrated scale based on function approximation complexity.

Findings

Provides new properties of Rényi dependence measure

Introduces a calibrated dependence scale

Links dependence to function approximation complexity

Abstract

Attempts to quantify dependence between random elements X and Y via maximal correlation go back to Gebelein (1941) and R\'{e}nyi (1959). After summarizing properties (including some new) of the R\'{e}nyi measure of dependence, a calibrated scale of dependence is introduced. It is based on the ``complexity`` of approximating functions of X by functions of Y.

Equations

$$({\varphi }_1,{\varphi }_2)=E\{{\varphi }_1(X,Y){\varphi }_2(X,Y)\}.$$

$$\mathrm{var}E\{\phi (X)|Y\}/\mathrm{var}\phi (X)$$

$$D(X:Y)=\underset{||\phi ||=1}{\sup }\mathrm{var}E\{\phi (X)|Y\}$$

$$\left(\begin{array}{cc}{V}_{11}& V_{12}\\ V_{21}& V_{22},\end{array}\right)$$

$$D(X:Y)=\sqrt{\lambda (\Sigma )},$$

$$D(X:Y)\ge \sqrt{\lambda (\Sigma )}.$$

$$D(X:Y+\lambda Z)\mathrm{increases}\mathrm{on}(-\infty .0)\mathrm{and}\mathrm{decreases}\mathrm{on}(0,\infty ).$$

$$D(X:Y)=\{R(X,Y){\}}^2$$

$$E\{\phi (X)|Y,Z\}=E\{\phi (X)|Y\}$$

$$D(X:Y+{\lambda }_{1}Z)=D(X:Y+{\lambda }_1(cZ+Z_c))=D(X:Y+{\lambda }_{2}Z+Z_c)\le D(X:(Y+{\lambda }_{2}Z,Z_c))$$

$$=D(X:Y+{\lambda }_{2}Z)$$

$$|\rho (\phi (X),\varphi (Y)){|}^2\le \mathrm{var}\varphi (Y)\mathrm{var}E\{\phi (X)|Y\}$$

$$E\{\phi (X)|Y\}=\mathrm{const}\Rightarrow P\{\phi (X)=\mathrm{const}\}=1.$$

$$D_m(X:Y)=\sup |V({\varphi }_0(Y),{\varphi }_1(Y),\dots ,{\varphi }_m(Y))|,$$

$$V({\varphi }_0(Y),{\varphi }_1(Y),\dots ,{\varphi }_m(Y))\le V({\phi }_0,{\phi }_1,\dots ,{\phi }_m),$$

$$c_1{\varphi }_0+\dots +c_{m+1}{\varphi }_m=0$$

$$c_1^{\prime }{\varphi }_1+\dots +c_m^{\prime }{\varphi }_m+c_{m+1}^{\prime }{\varphi }_m$$

$$\phi ={\phi }^{\prime }+{\phi }^{\prime \prime },{\phi }^{\prime }\in \mathrm{span}\{{\varphi }_1,\dots ,{\varphi }_{m+1}\},{\phi }^{\prime \prime }\in (\mathrm{span}\{{\varphi }_1,\dots ,{\varphi }_{m+1}\}{)}^{\perp },$$

$${\varphi }_1=E({\phi }_1|Y),\dots ,{\varphi }_{m+1}=E({\phi }_{m+1}|Y)$$

$${\mathcal{C}}_0\supset {\mathcal{C}}_1\supset {\mathcal{C}}_2\supset \dots ,$$

$$p(x|y)=p_0(x)+p_1(x)q_1(y)+\dots +p_m(x)q_m(y),$$

$$E(\phi (X)|Y)=\int \phi (x)p(x|Y)dx$$

Full text

*Keywords: Quantification, measures of dependence, projection

MSC 2010 classifications: Primary 60H99; secondary 62E10 *

1 Introduction

Let $X,\>Y$ be a pair of random elements taking values in a general probability space

$\{{\cal X}\times\>{\cal Y},\>{\cal A}\otimes{\cal B},\>P\}$ where $\cal A$ and $\cal B$ are $\sigma$-algebras of subsets of $\cal X$ and $\cal B$, respectively, and $P$ is a probability measure on ${\cal A}\otimes{\cal B}$.

The study of dependence of $X$ from $Y$ attracted much attention; see, e. g., Székely, Rizzo and Bakirov (2007), Székely and Rizzo (2009), Reshef et al.(2011), Reimherr and Nicolae (2013). In Rényi (1959) natural (“necessary”) requirements to measures of dependence were stated. Essential features of dependence of $X$ from $Y$ are contained in properties of the conditional expectation $E\{\varphi(X)|Y\}$ defined for all $\varphi(X)$ with $E\{|\varphi(X)|\}<\infty.$ If one reduces the domain to $\varphi(X)$ with $E\{|\varphi(X)|^{2}\}<\infty$, the conditional expectation becomes the projection of the subspace $L^{2}(X)$ into the subspace of $L^{2}(Y)$ of $L^{2}(X,\>Y)$, the Hilbert space of functions $\phi(X,\>Y)$ with $E\{|\phi(X,\>Y)|^{2}\}=\int_{\cal X}\int_{\cal Y}|\phi(x,y)|^{2}dP(x,y)<\infty$ and the standard inner product

[TABLE]

According to Rényi (1959),in the classical monograph Kolmogorov (1933) introduced the following ratios of variances to measure dependence between $X$ and $Y$:

[TABLE]

Its supremum

[TABLE]

possesses many properties required from a measure of dependence. These properties are summarized in the following.

Theorem 1**.**

• (i)

$0\leq D(X:Y)\leq 1$.

• (ii)

$D(X:Y)=0$* iff $X$ and $Y$ are independent.*

• (iii)

$\>D(X:Y)=1$* if $X=h(Y)$ (but not iff).*

• (iv)

$D(X:Y)$* is monotone in the following sense: for any random element*

$Z,\>D(X:(Y,\>Z))\geq D(X:Y).$**

• (v)

If $Z$ is independent of the pair $(X,\>Y)$, then $D(X:(Y,\>Z))=D(X:Y)$.

• (vi)

For a bivariate Gaussian vector $(X,\>Y)$ with corr$(X,\>Y)=\rho$, $D(X:Y)=|\rho|$.

• (vii)

For any bivariate vector $(X,\>Y)$ with corr$(X,\>Y)=\rho$, $D(X:Y)\geq|\rho|$.

• (viii)

For an $(m+n)$-dimensional Gaussian random vector $(X,\>Y)$ with variance-covariance matrix

[TABLE]

assuming the submatrices $V_{11}={\rm var}(X),\>V_{22}={\rm var}(Y)$ non-singular,

[TABLE]

where $\lambda(\Sigma)$ is the maximal eigenvalue of the matrix $\Sigma=V_{11}^{-1/2}V_{12}V_{22}^{-1}V_{21}V_{11}^{-1/2}.$

• (ix)

For an arbitrary $(m+n)$-dimensional vector $(X,\>Y)$ with the above variance-covariance matrix

[TABLE]

• (x)

If a self-decomposable $Z$ is independent of $(X,\>Y)$, then as a function of $\lambda$

[TABLE]

• (xi)

[TABLE]

where $R(X,Y)=\sup\rho(\varphi(X),\>\phi(Y)$, $\rho$ is the classical (Pearson) correlation coefficient and the supremum is taken over all $\varphi(X),\>\phi(Y)$ with $0<{\rm var}\varphi(X)<\infty,\\ 0<{\rm var}\phi(Y)<\infty$.

• (xii)

$D(X:Y)=D(Y:X)$.

Proofs. Properties (i)-(iv) and (vi)-(ix) are well known, (v) follows from that if $Z$ is independent of $(X,\>Y)$ then for any $\varphi(X)$ with finite expectation

[TABLE]

(see, e. g., Meyer (1966), Theorem 61). Actually, (v) holds if $X$ and $Z$ are conditionally independent given $Y$ which is a weaker condition than independence of $Z$ and $(X,\>Y)$.

To prove (x), recall that a random variable $Z$ is called self-decomposable if for any $c,\>0<c<1$ there is an independent of $Z$ random variable $Z_{c}$ such that $Z$ is equidistributed with $cZ+Z_{c},\>Z\cong cZ+Z_{c}$. All random variables having stable distributions are self-decomposable. See Lukacs (1970), Ch. 5 for properties of self-decomposable random variables.

Let now $\lambda_{1}>\lambda_{2}>0,\>\lambda_{2}=c\lambda_{1}$ for some $0<c<1$ and let $Z_{c}$ be an independent of $X,\>Y,\>Z$ random variable such that $Z\cong cZ+Z_{c}$. Then

[TABLE]

due to monotonicity (iv) and (v).

Property (xi) follows from $\rho(\varphi(X),\>\phi(Y))=\rho(E\{\varphi(X)|Y\},\>\phi(Y))$ and the Cauchy-Schwarz inequality

[TABLE]

with the equality sign holding for $\phi(Y)=cE\{\varphi(X)|Y\}$.

Some special forms of dependence are worth studying. One example may be $X,\>Y$ such that

[TABLE]

If $p(x|y)$ is the conditional probability density function of $X$ given $Y=y$, (2) is what in statistics is known as completeness of the family $\{p(x|y),\>y\in{\cal Y}\}$ with $y$ as a family parameter.

Plainly, if $X$ is fully dependent of $Y$, i. e.,$X=h(Y)$, the relation (2) holds.

For independent $X,\>Y,\>E\{\varphi(X)|Y\}={\rm const}$ for any $\varphi(X)$ with finite expectation, so that (2) moves $X,\>Y$ far away from independence.

2 The main result

Here a family $D_{m}(X:Y),\>m=0,\>1,\>2,\ldots$ of measures of dependence is defined that generalizes $D(X:Y)$ in the following direction.

A random element $X$ is called $m$-dependent of $Y$ if for any $\varphi(X)\in L^{2}(X)$, the conditional expectation $E\{\varphi(X)|Y\}=\phi(Y)$ belongs to an $(m+1)$-dimensional subspace of $L^{2}(Y)$. It is proved that $X$ is $m$-dependent of $Y$ if and only if $D_{m}(X:Y)=0$. For $m=0,\\ D_{0}(X:Y)=D(X:Y)$ so that [math]-dependence means independence when for any $\varphi(X),\>E\{\varphi(X)|Y\}$ belongs to the one-dimensional subspace of constants.

Denote by $V(\varphi_{0},\ldots,\varphi_{m})$ the covariance matrix of the $(m+1)$-dimensional vector $(\varphi_{0}(X),\ldots,\varphi_{m}(X))$ of random variables in $L^{2}(X)$ and by $|V|$ the determinant of $V$ (sometimes referred to as the generalized variance of the vector with covariance matrix $V$.

For $\varphi(X)\in L^{2}(X)$ set $\phi(Y)=E\{\varphi(X)|Y\}$.

Definition 1**.**

The index of $m$-dependence of $X$ from $Y$ is defined as

[TABLE]

*where the supremum is taken over all $\varphi_{0}(X),\varphi_{1}(X),\ldots,\varphi_{m}(X)$ with

$V(\varphi_{0},\varphi_{1},\ldots,\varphi_{m})={\rm Id}_{m+1}$, the identity matrix of order $m+1$.*

The matrix $V(\phi_{0}(Y),\phi_{1}(Y),\ldots,\phi_{m}(Y))$ is non-negative definite and, as well known,

[TABLE]

so that $0\leq D_{m}(X:Y)\leq 1.$

Plainly, $D_{m}(X:Y)$ is monotone in the same sense as $D(X:Y)$ (see (iv)).

Since $D_{0}(X:Y)=\sup_{\varphi:{\rm var}\varphi(X)=1}{\rm var}E\{\varphi(X)|Y\}$, $D_{0}(X:Y)=0$ if and only if for any $\varphi\in L^{2}(X)$, $E(\varphi|Y)$ belongs to the one-dimensional subspace of constants in $L^{2}(Y)$. A similar property of $D_{m}(X:Y)$ is proved in the next theorem.

Theorem 2**.**

$D_{m}(X:Y)=0$* if and only if for any $\varphi\in L^{2}(X)$, $E(\varphi|Y)$ belongs to a subspace of $L^{2}(Y)$ of dimension $\leq m+1$.

Proof. First notice that the image of $\varphi\equiv 1\in L^{2}(X)$ is $\phi\equiv 1\in L^{2}(Y)$. For any $\varphi\in L^{2}(X)$ with $E(\varphi(X))=0$ its image $E(\varphi|Y)=\phi(Y)$ also has $E(\phi(Y))=0$. We shall show that $D_{m}(X:Y)=0$ if and only if for any $\varphi\in L^{2}(X)$, with $E(\varphi)=0$ its image $E(\varphi|Y)$ belongs to a subspace of $L^{2}(Y)$ of dimension $\leq m.$

Now take elements $\varphi_{1},\ldots,\varphi_{m+1}$ in $L^{2}(X)$ with zero expectations and the identity covariance matrix (that is why $\varphi\equiv 1$ is treated separately).

Due to $D_{m}(X:Y)=0$, the covariance matrix $V(\phi_{1},\ldots,\phi_{m+1})$ of their images $\phi_{1}(Y)=E(\varphi_{1}|Y),\ldots,\phi_{m+1}(Y)=E(\varphi_{m+1}|Y)$ is degenerate and thus

[TABLE]

with probability 1 for some constants $c_{1},\ldots,c_{m+1}$.

Assuming without loss in generality $\phi_{1},\ldots,\phi_{m}$ linearly independent and thus $c_{m+1}\neq 0$, $D_{m}(X:Y)=0$ implies that for the chosen $\varphi_{1},\ldots,\varphi_{m+1}$ their images belong to the $m$-dimensional subspace span$\{\phi_{1},\ldots,\phi_{m}\}$. If $\varphi\in{\rm span}\{\varphi_{1},\ldots,\varphi_{m+1}\}$, then plainly $E(\varphi|Y)=\phi\in{\rm span}\{\phi_{1},\ldots,\phi_{m}\}$.

Let now $\varphi\in L^{2}(X)$ belong to the orthocomplement of span$\{\varphi_{1},\ldots,\varphi_{m+1}\},\varphi\in({\rm span}\{\varphi_{1},\ldots,\varphi_{m+1}\})^{\perp}$. Assuming var$\varphi(X)=1$, consider the vector $(\varphi_{1},\ldots,\varphi_{m},\varphi)$. The covariance matrix $V(\phi_{1},\ldots,\phi_{m},\phi)$ where $\phi=E(\varphi|Y)$, is degenerate and for some constants $c^{\prime}_{1},\ldots,c^{\prime}_{m+1}$

[TABLE]

so that $\phi\in{\rm span}\{\phi_{1},\ldots,\phi_{m}\}$.

Since any $\varphi\in L^{2}(X)$ can be represented as

[TABLE]

the above proves the sufficiency part of Theorem 2.

To prove necessity, take any set $\varphi_{1},\ldots,\varphi_{m+1}$ of elements of $L^{2}(Y)$ with zero expectations and identity covariance matrix. If their images

[TABLE]

belong to an $m$-dimensional subspace of $L^{2}(Y)$ and thus are linearly dependent, their covariance matrix is degenerate and $|V(\phi_{1},\ldots,\phi_{m+1})|=0.$

The images of constants form an one-dimensional subspace of constants and for any $\varphi(X),\>\varphi=E(\varphi)+[\varphi-E(\varphi)]$ proving the necessity part of Theorem 2.

A trivial corollary is $D_{m}(X:Y)=0\Rightarrow D_{m+1}(X:Y)=0$.

Starting with the class ${\cal C}_{0}$ of independent $X,\>Y$, the indices $D_{m}$ lead to a scale of classes ${\cal C}_{m}$,

[TABLE]

where $(X,\>Y)\in{\cal C}_{m}\Leftrightarrow D_{m}(X:Y)=0$.

The classes ${\cal C}_{m}$ are non-empty. If the conditional probability density function $p(x|y)$ of $X$ given $Y$ has a form of

[TABLE]

with $q_{1}\in L^{2}(Y),\ldots,q_{m}\in L^{2}(Y)$, then

[TABLE]

belongs to the $(m+1)$-dimensional subspace span $\{1,q_{1}(Y),\ldots,q_{m}(Y)\}$ of $L^{2}(Y)$, so that $(X,\>Y)\in{\cal C}_{m}$.

Of some interest may be an approach that starts not at independent $X,\>Y$ but at the other end when $X$ and $Y$ are arbitrarily dependent. Let us call $X$ $m$-codependent of $Y$ if for any $\varphi\in L^{2}(X),\>E(\varphi|Y)$ belongs to a subspace of $L^{2}(Y)$ of codimension $m$. As $m$ increases, $X$ and $Y$ become less and less dependent, in a sense. How to quantify the codependence?