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Endpoint sensor times are D-optimal for an intercept-slope model

Example

For the linear time-response model $F_{t} (θ) = θ_{0} + θ_{1} t$ with candidate sensor times $t \in {0, 1, 2}$ and unit precision, the equal-weight endpoint design is D-optimal. Starting from uniform weights and using a one-update budget, the multiplicative procedure returns a positive equivalence violation instead of claiming exact optimality.

Facts & Assumptions

Given: Candidate features $f_{t} = (1, t)^{T}$ at $t = 0, 1, 2$ , parameter dimension $p = 2$ , unit precision, and initial approximate design $ξ^{(0)} = (1 / 3, 1 / 3, 1 / 3)$ in that order.

[F1]

Approximate designs and linear-model optimality criteria: $M (ξ) = \sum_{t} ξ_{t} f_{t} f_{t}^{T}$ and D-optimality maximizes $\log \det M (ξ)$ among nonsingular approximate designs.

[F2]

Finite-budget multiplicative update for approximate D-optimal design: with $s_{t} = f_{t}^{T} M^{- 1} f_{t}$ , the update is $ξ_{t}^{+} = ξ_{t} s_{t} / p$ and the returned violation is $\max_{t} s_{t} - p$ .

[F3]

Finite-budget multiplicative design updates preserve the simplex and nonsingularity: each exact update preserves the approximate-design simplex and nonsingularity.

[F4]

The D-equivalence violation bounds the finite-budget design's log-determinant gap: zero violation certifies D-optimality, and a positive violation bounds every nonsingular competitor's log-determinant gap.

[F5]

The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t gives $\log^{'} (x) = 1 / x > 0$ for $x > 0$ . For $0 < a < b$ , the logarithm is continuous on $[a, b]$ and differentiable inside, so The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ gives $\log b - \log a = (b - a) / c > 0$ for some $c \in (a, b)$ . Thus $\log$ is strictly increasing on $(0, \infty)$ .

Verification

1.1givenF1F5algebra

For an arbitrary approximate design $ξ$ , write $μ = \sum_{t} ξ_{t} t$ and $ν = \sum_{t} ξ_{t} t^{2}$ . Direct multiplication gives $M (ξ) = [\begin{array}{cc} 1 & μ \\ μ & ν \end{array}]$ and $\det M (ξ) = ν - μ^{2}$ . Since $0 \leq t \leq 2$ , $(t - 1)^{2} \leq 1$ , so $ν - μ^{2} = \sum_{t} ξ_{t} (t - 1)^{2} - (μ - 1)^{2} \leq 1$ . The equal endpoint design $ξ^{*} = (1 / 2, 0, 1 / 2)$ has $M (ξ^{*}) = [\begin{array}{cc} 1 & 1 \\ 1 & 2 \end{array}]$ and determinant $1$ , so it maximizes the determinant and hence $\log \det M$ among nonsingular designs because $\log$ is strictly increasing by [F5].

1.2givenF2F3algebra

For $ξ^{(0)}$ , $M_{0} = [\begin{array}{cc} 1 & 1 \\ 1 & 5 / 3 \end{array}]$ and $M_{0}^{- 1} = [\begin{array}{cc} 5 / 2 & - 3 / 2 \\ - 3 / 2 & 3 / 2 \end{array}]$ . Thus $(s_{0}, s_{1}, s_{2}) = (5 / 2, 1, 5 / 2)$ and $v_{0} = 1 / 2$ . One update gives $ξ^{(1)} = (5 / 12, 1 / 6, 5 / 12)$ ; the update is defined by [F2] and remains nonsingular by [F3].

2.1step 1.2F2F3algebra

The updated matrix is $M_{1} = [\begin{array}{cc} 1 & 1 \\ 1 & 11 / 6 \end{array}]$ , with inverse $M_{1}^{- 1} = [\begin{array}{cc} 11 / 5 & - 6 / 5 \\ - 6 / 5 & 6 / 5 \end{array}]$ . Therefore $(s_{0}, s_{1}, s_{2}) = (11 / 5, 1, 11 / 5)$ and $v_{1} = 11 / 5 - 2 = 1 / 5$ . With budget $N = 1$ , the procedure returns the updated design with status budget-exhausted and violation $1 / 5$ .

3.1step 1.1step 2.1F1F4algebra∎

For the endpoint design $ξ^{*}$ , $M (ξ^{*})^{- 1} = [\begin{array}{cc} 2 & - 1 \\ - 1 & 1 \end{array}]$ , so its sensitivities at $0, 1, 2$ are $2, 1, 2$ and its violation is zero. By [F4] this is an exact D-optimality certificate; [F4] also bounds the log-determinant gap of the one-update output by $1 / 5$ . The weights are approximate-design weights; integer run counts are a separate problem by [F1].

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Sources

Stephen Boyd and Lieven Vandenberghe, Convex Optimization, §7.5 (standard reference, not scraped)
Holger Dette, Andrey Pepelyshev, and Anatoly Zhigljavsky, Improving Updating Rules in Multiplicative Algorithms for Computing D-optimal Designs, §2 (standard reference, not scraped)