PhyslibAlpha.AlgebraicFramework.Algebra.Statistics

Statistics of a linear functional on a bilinear algebra

i. Overview

Second moments and covariance use only a bilinear multiplication and a linear functional. They do not intrinsically require an order, positivity, associativity, commutativity, or the Jordan identity. This file defines them at that level so ordered Jordan algebras and self-adjoint parts of star algebras can share one canonical construction.

For `omega : E ->_l[R] R`, the covariance is packaged as the bilinear form

`(a, b) |-> omega (a * b) - omega a * omega b`.

Commutativity of the product is used only to prove symmetry. A unit and the normalization `omega 1 = 1` are used only for centering identities. Positivity is deliberately left to the ordered specialization.

ii. Key definitions and results

`LinearMap.secondMomentForm`
`LinearMap.covarianceForm`
`LinearMap.variance`
`LinearMap.centered`
`LinearMap.apply_centered`
`LinearMap.apply_centered_mul_centered`

iii. Table of contents

A. Second moments and covariance
B. Centering a normalized functional

A. Second moments and covariance

B. Centering a normalized functional

10 declarations

definition

Second-moment bilinear form $\omega(ab)$

#secondMomentForm

Given a linear functional $\omega : E \to \mathbb{R}$ on an algebra $E$ over the real numbers, the second-moment bilinear form is the mapping $E \times E \to \mathbb{R}$ defined by $(a, b) \mapsto \omega(a \cdot b)$ for all $a, b \in E$ .

theorem

The second-moment bilinear form evaluated at $(a, b)$ equals $\omega(a \cdot b)$

#secondMomentForm_apply

Given a linear functional $\omega : E \to \mathbb{R}$ on an algebra $E$ over the real numbers, the second-moment bilinear form evaluated at elements $a, b \in E$ is equal to $\omega(a \cdot b)$ .

definition

Covariance bilinear form $(a, b) \mapsto \omega(ab) - \omega(a)\omega(b)$

#covarianceForm

For a linear functional $\omega : E \to \mathbb{R}$ on a bilinear algebra $E$ over the real numbers, the covariance bilinear form is the mapping from $E \times E$ to $\mathbb{R}$ defined by $(a, b) \mapsto \omega(a \cdot b) - \omega(a)\omega(b)$ . This construction generalizes the statistical notion of covariance to any algebra equipped with a linear functional, without requiring properties such as associativity or commutativity.

theorem

$\text{covarianceForm}(\omega)(a, b) = \omega(ab) - \omega(a)\omega(b)$

#covarianceForm_apply

Let $E$ be an algebra over the real numbers and $\omega: E \to \mathbb{R}$ be a linear functional. For any elements $a, b \in E$ , the value of the covariance bilinear form of $\omega$ at $(a, b)$ is given by $\omega(a \cdot b) - \omega(a)\omega(b)$ .

theorem

Commutativity of multiplication implies symmetry of the covariance form

#covarianceForm_isSymm

Let $E$ be a bilinear algebra over $\mathbb{R}$ and $\omega : E \to \mathbb{R}$ be a linear functional. If the multiplication in $E$ is commutative (i.e., $a \cdot b = b \cdot a$ for all $a, b \in E$ ), then the covariance bilinear form defined by $(a, b) \mapsto \omega(a \cdot b) - \omega(a)\omega(b)$ is symmetric.

definition

Variance of $a$ with respect to $\omega$ ( $\omega(a^2) - \omega(a)^2$ )

#variance

For a linear functional $\omega : E \to \mathbb{R}$ on a bilinear algebra $E$ over the real numbers and an element $a \in E$ , the variance of $a$ with respect to $\omega$ is defined as $\omega(a^2) - (\omega(a))^2$ . This is equivalent to the diagonal of the covariance bilinear form, evaluated at $(a, a)$ .

theorem

$Cov_{\omega}(a, a) = Var_{\omega}(a)$

#covarianceForm_self

Let $E$ be a bilinear algebra over $\mathbb{R}$ and $\omega: E \to \mathbb{R}$ be a linear functional. For any element $a \in E$ , the covariance bilinear form evaluated at $(a, a)$ is equal to the variance of $a$ with respect to $\omega$ . That is, $\text{covarianceForm}(\omega)(a, a) = \text{variance}(\omega, a)$ .

definition

Centering of $a$ with respect to $\omega$ ( $a - \omega(a) \cdot 1$ )

#centered

Given a linear functional $\omega : E \to \mathbb{R}$ on a unital algebra $E$ and an element $a \in E$ , the centering of $a$ with respect to $\omega$ is defined as $a - \omega(a) \cdot 1$ , where $1$ is the identity element of $E$ .

theorem

A normalized functional maps centered elements to zero

#apply_centered

Let $E$ be a unital algebra over $\mathbb{R}$ and $\omega : E \to \mathbb{R}$ be a linear functional. If $\omega$ is normalized such that $\omega(1) = 1$ , then for any element $a \in E$ , the value of the functional on the centered element is zero. That is, $\omega(a - \omega(a) \cdot 1) = 0$ .

theorem

$\omega(\text{centered } a \cdot \text{centered } b) = \omega(ab) - \omega(a)\omega(b)$

#apply_centered_mul_centered

Let $E$ be a unital algebra over $\mathbb{R}$ and let $\omega : E \to \mathbb{R}$ be a linear functional such that $\omega(1) = 1$ . Suppose that the identity element $1$ satisfies $1 \cdot x = x$ and $x \cdot 1 = x$ for all $x \in E$ . For any $a, b \in E$ , let the centering of an element $x$ with respect to $\omega$ be defined as $x - \omega(x) \cdot 1$ . Then the covariance of $a$ and $b$ , defined by the bilinear form $\text{Cov}_\omega(a, b) = \omega(ab) - \omega(a)\omega(b)$ , is equal to the value of the functional on the product of the centered elements: $\omega\big((a - \omega(a) \cdot 1)(b - \omega(b) \cdot 1)\big) = \omega(ab) - \omega(a)\omega(b)$

PhyslibAlpha.AlgebraicFramework.Algebra.Statistics

Statistics of a linear functional on a bilinear algebra

i. Overview

ii. Key definitions and results

iii. Table of contents

A. Second moments and covariance

B. Centering a normalized functional

Second-moment bilinear form ω(ab)\omega(ab)ω(ab)

The second-moment bilinear form evaluated at (a,b)(a, b)(a,b) equals ω(a⋅b)\omega(a \cdot b)ω(a⋅b)

Covariance bilinear form (a,b)↦ω(ab)−ω(a)ω(b)(a, b) \mapsto \omega(ab) - \omega(a)\omega(b)(a,b)↦ω(ab)−ω(a)ω(b)

covarianceForm(ω)(a,b)=ω(ab)−ω(a)ω(b)\text{covarianceForm}(\omega)(a, b) = \omega(ab) - \omega(a)\omega(b)covarianceForm(ω)(a,b)=ω(ab)−ω(a)ω(b)

Commutativity of multiplication implies symmetry of the covariance form

Variance of aaa with respect to ω\omegaω (ω(a2)−ω(a)2\omega(a^2) - \omega(a)^2ω(a2)−ω(a)2)

Covω(a,a)=Varω(a)Cov_{\omega}(a, a) = Var_{\omega}(a)Covω​(a,a)=Varω​(a)

Centering of aaa with respect to ω\omegaω (a−ω(a)⋅1a - \omega(a) \cdot 1a−ω(a)⋅1)

A normalized functional maps centered elements to zero

ω(centered a⋅centered b)=ω(ab)−ω(a)ω(b)\omega(\text{centered } a \cdot \text{centered } b) = \omega(ab) - \omega(a)\omega(b)ω(centered a⋅centered b)=ω(ab)−ω(a)ω(b)

Second-moment bilinear form $\omega(ab)$

The second-moment bilinear form evaluated at $(a, b)$ equals $\omega(a \cdot b)$

Covariance bilinear form $(a, b) \mapsto \omega(ab) - \omega(a)\omega(b)$

$\text{covarianceForm}(\omega)(a, b) = \omega(ab) - \omega(a)\omega(b)$

Variance of $a$ with respect to $\omega$ ( $\omega(a^2) - \omega(a)^2$ )

$Cov_{\omega}(a, a) = Var_{\omega}(a)$

Centering of $a$ with respect to $\omega$ ( $a - \omega(a) \cdot 1$ )

$\omega(\text{centered } a \cdot \text{centered } b) = \omega(ab) - \omega(a)\omega(b)$