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Conditional moments of one Poisson tau-leap increment

Statement

Fix a state $x \in E$ and step $h \geq 0$ in Explicit Poisson tau-leaping with an invalid-state status. Let $Δ Y = \sum_{j = 1}^{R} ν_{j} K_{j}$ , where conditionally independent $K_{j} \sim Pois (a_{j} (x) h)$ . Then $E [Δ Y ∣ x] = h \sum_{j = 1}^{R} a_{j} (x) ν_{j},$ $Cov (Δ Y ∣ x) = h \sum_{j = 1}^{R} a_{j} (x) ν_{j} ν_{j}^{T},$ and $E [∥ Δ Y ∥^{2} ∣ x] = h \sum_{j = 1}^{R} a_{j} (x) ∥ ν_{j} ∥^{2} + h^{2} {∥ \sum_{j = 1}^{R} a_{j} (x) ν_{j} ∥}^{2} .$

Facts & Assumptions

Given: A state $x \in E$ , a step $h \geq 0$ , the product count law in Explicit Poisson tau-leaping with an invalid-state status, and the Poisson count laws below.

[F1]

The scalar Poisson count law: for $λ \geq 0$ , a $Pois (λ)$ count has masses $e^{- λ} λ^{k} / k!$ (with a point mass at zero when $λ = 0$ ).

[F2]

Normalization, moments, and generating function of the Poisson count law: if $K \sim Pois (λ)$ , then $E K = λ$ and $Var (K) = λ$ .

[F3]

Explicit Poisson tau-leaping with an invalid-state status: the channel counts are conditionally independent and the state increment is $Δ Y = \sum_{j} ν_{j} K_{j}$ .

Proof

Given: The inputs above.

1.1givenF1F2F3algebra

Each parameter $a_{j} (x) h$ is finite, so [F1] defines the corresponding Poisson law and [F2] gives $E [K_{j} ∣ x] = a_{j} (x) h$ . By [F3], conditional expectation is linear, hence $E [Δ Y ∣ x] = \sum_{j} ν_{j} a_{j} (x) h$ .

1.2F2F3algebra

Conditional independence in [F3] makes cross-channel covariances zero. The covariance of $ν_{j} K_{j}$ is $ν_{j} ν_{j}^{T} Var (K_{j} ∣ x) = h a_{j} (x) ν_{j} ν_{j}^{T}$ by [F2]. Summing proves the covariance formula.

2.1step 1.1step 1.2algebra∎

For a square-integrable vector, $E ∥ Δ Y ∥^{2} = tr Cov (Δ Y) + ∥ E Δ Y ∥^{2}$ . Substitution of steps 1.1 and 1.2 gives the asserted identity.

Depends on

Used by

Dependency tree · two levels

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Sources

Daniel T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems (standard reference, not scraped)
Tiejun Li, Lecture 21: Application in chemical reaction kinetics, Applied Stochastic Analysis course notes (standard reference, not scraped)