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The propagated covariance is not the exact covariance of a nonlinear observation map

Statement refuted

Statement refuted: "if $U \subseteq R$ is open and convex, $h : U \to R$ is $C^{2}$ , $X : Ω \to U$ is square-integrable with $E [X] = μ \in U$ , and $Σ$ is its covariance matrix, then $Cov (h (X)) = D h (μ) Σ D h (μ)^{T}$ , that is, $Var (h (X)) = h^{'} (μ)^{2} Var (X)$ ."

This is an unsupported nonlinear extension of The covariance of the linearization is $A Σ A^{T}$ , which states the covariance formula for the affine linearization $T (X)$ and does not assert equality with $Cov (h (X))$ .

Facts & Assumptions

Given: The finite weighted set $Ω = {ω_{0}, ω_{1}, ω_{2}}$ with $w (ω_{0}) = 1 / 2$ , $w (ω_{1}) = w (ω_{2}) = 1 / 4$ ; the random variable $Y$ with $Y (ω_{0}) = 0$ , $Y (ω_{1}) = 1$ , $Y (ω_{2}) = 3$ ; and the function $h (y) = y^{3}$ on the open convex set $U = (- 1, 4)$ .

[L1]

On a finite probability space, expectations are the finite weighted sums of Expectation of a real random variable on a finite probability space, and the finite model determines the probability measure $P (A) = \sum_{ω \in A} w (ω)$ whose integral expectations agree with those sums. (Finite probability spaces, outcome weights, events, and event probabilities, Finite probability spaces are exactly finite full-power-set probability spaces)

[L2]

For a square-integrable random variable $X$ with mean $μ$ , the covariance of the linearization $x \mapsto h (μ) + h^{'} (μ) (x - μ)$ is $h^{'} (μ)^{2} Var (X)$ whenever $h$ is differentiable at $μ$ ; the derivative of $y^{3}$ is $3 y^{2}$ ; and $Var (Z) = E [Z^{2}] - E [Z]^{2}$ . (The covariance of the linearization is $A Σ A^{T}$ , Mean vector and covariance matrix of a square-integrable random vector, For a natural $n \geq 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $ι (n) x^{n - 1}$ ; for $n = 0$ it is the constant $1$ , with derivative $0$ ; for a natural $n \geq 1$ the function $x \mapsto x^{- n}$ is differentiable at every $x \neq 0$ with derivative $- ι (n) x^{- n - 1}$ ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term, Moments, variance, and covariance on a probability space)

[L3]

If $h : U \to R$ is $C^{2}$ on open convex $U$ , $∥ H_{h} (x) ∥ \leq B$ on $U$ , $X$ takes values in $U$ with $E [X] = μ \in U$ and finite second moment, then the linearization error satisfies $∣ E [R] ∣ \leq E ∣ R ∣ \leq \frac{B}{2} E ∣ X - μ ∣^{2}$ and $E [R^{2}] \leq \frac{B^{2}}{4} E ∣ X - μ ∣^{4}$ when the fourth moment is finite. (Mean, mean-square and covariance bounds for second-order linearization error)

Counterexample

1.1givenL1L2algebra

By [L1] the finite weighted set is a probability space and the expectations below are its finite weighted sums: $E [Y] = \frac{1}{2} \cdot 0 + \frac{1}{4} \cdot 1 + \frac{1}{4} \cdot 3 = 1$ , $E [Y^{2}] = \frac{1}{4} + \frac{9}{4} = \frac{5}{2}$ , and hence $Var (Y) = \frac{5}{2} - 1 = \frac{3}{2}$ by [L2].

2.1step 1.1givenL2algebra

The function $h (y) = y^{3}$ is $C^{2}$ on $U = (- 1, 4)$ , with $h^{'} (y) = 3 y^{2}$ , so $h^{'} (1) = 3$ and the propagated variance is $h^{'} (1)^{2} Var (Y) = 9 \cdot \frac{3}{2} = \frac{27}{2}$ .

2.2step 1.1givenL1L2algebra

The exact moments of $h (Y) = Y^{3}$ are $E [Y^{3}] = \frac{1}{4} \cdot 1 + \frac{1}{4} \cdot 27 = 7$ and $E [Y^{6}] = \frac{1}{4} \cdot 1 + \frac{1}{4} \cdot 729 = \frac{730}{4} = \frac{365}{2}$ , so $Var (h (Y)) = \frac{365}{2} - 7^{2} = \frac{365}{2} - \frac{98}{2} = \frac{267}{2}$ .

3.1step 2.1step 2.2algebra

Steps 2.1 and 2.2 give propagated value $\frac{27}{2}$ and exact variance $\frac{267}{2}$ ; these are unequal, so the refuted identity fails under the stated hypotheses.

4.1step 1.1step 3.1givenL3algebra

The comparison does not contradict [L3]: on $U = (- 1, 4)$ the second derivative of $h$ is $6 y$ , of absolute value at most $24$ , so $B = 24$ is admissible, and the linearization error $R = Y^{3} - 1 - 3 (Y - 1) = Y^{3} - 3 Y + 2$ satisfies $R = 2, 0, 20$ at $Y = 0, 1, 3$ , giving $E [R] = 6 \leq \frac{24}{2} \cdot \frac{3}{2} = 18$ and $E [R^{2}] = 102 \leq \frac{24^{2}}{4} \cdot \frac{9}{2} = 648$ , both bounds of [L3] holding strictly.

5.1step 3.1step 4.1∎

The hypotheses of the refuted statement hold, the propagated covariance $\frac{27}{2}$ differs from the exact variance $\frac{267}{2}$ , and the remainder bounds of [L3] remain satisfied; hence the propagated covariance controls the actual covariance only through those bounds.

Remarks

This witness is a genuine failure of equality, not of the domination bounds of Mean, mean-square and covariance bounds for second-order linearization error: the bounds hold strictly here. A consumer that reports the propagated covariance as an uncertainty of $h (Y)$ must carry the remainder qualification with it.

The propagated covariance is not the exact covariance of a nonlinear observation map

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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