Measurements and Fisher information

Parameters, directions, and design variables

Write the physical model as

\[\dot x=f(x,\theta,v,t),\qquad x(t_0)=x_0,\]

where $x\in\mathbb R^{n_x}$, $\theta\in\mathbb R^{n_\theta}$ are the selected physical parameters, and $v$ denotes physical controls or other design choices. In Julia, both physical parameters and controls occupy entries of the original vector problem.p of length $n_p$; params selects the indices of $\theta$ in the requested order. Tunable/Constant annotations independently determine which values can vary during design optimization. Unannotated physical parameters are fixed at nominal values under the OED constructor's default fixed_parameters policy.

The matrix $B\in\mathbb R^{n_\theta\times n_s}$ defines parameter directions. With no reduction, $n_s=n_\theta$ and $B=I$ on the selected coordinates. The package integrates $G\in\mathbb R^{n_x\times n_s}$:

\[\dot G=f_xG+f_\theta B,\qquad G(t_0)=0.\]

For $B=I$, $G$ contains state sensitivities with respect to the selected physical parameters, holding the design schedule fixed. The column count is set by these selected coordinates and the basis, not by the number of scheduled trainable values. The reduced case is explained in SVD reduction. Initial conditions are assumed independent of these estimated parameters. Free initial values can be design variables, but params does not select estimated initial conditions or initialize their sensitivities.

These sensitivity columns are distinct from the AD gradient of a design objective with respect to optimizer variables. AD differentiates through the augmented numerical solve, including its sensitivities and information, as the design changes.

Measurement grids and outputs

A measurement is ContinuousMeasurement(id, tpoints, observed) or DiscreteMeasurement(id, tpoints, observed), with observed(x, p, t) returning a scalar or vector. The observation map must accept symbolic inputs and keep its output dimension fixed. A vector observation has one shared weight for all its output rows; use separate measurements for independently weighted outputs.

Identifiers must be unique and distinct from model symbols. Grids are validated when constructing the OED layer: times must be finite, sorted, unique, and inside problem.tspan. A discrete grid must be nonempty; an empty continuous grid is valid.

Weights are tunable in $[0,1]$ and initially one. Continuous densities are piecewise constant, taking the outgoing interval's value at a switch. Discrete weights are also represented by a right-continuous parameter schedule, but only the values at the declared events contribute information or discrete budgets. The first event's weight is stored on the initial interval even if the first event is later than $t_0$. There is no extra discrete sample before that event.

Discrete event times are added to the saved grid. At a shooting reset, both incoming and outgoing states are retained, and discrete information selects the last saved sample at that time, the outgoing/post-reset state. General observed_equations and sensitivities outputs include both saved sides. Parameter histories are right-continuous on either saved side; see the Corleone trajectory guide.

Observation sensitivities and information

For measurement $m$ with $r_m$ output rows, define $h_m:\mathbb R^{n_x}\times\mathbb R^{n_p}\times\mathbb R\to\mathbb R^{r_m}$. The implemented observation-sensitivity matrix has size $r_m\times n_s$:

\[J_m=h_{m,x}G+h_{m,\theta}B.\]

Direct dependence on estimated parameters matters. An observation such as (x, p, t) -> p[1] can provide information even if its state derivative is zero. The supplied map's rows are all included; no normalization by output count is applied.

Let $\mathcal C$ and $\mathcal D$ be continuous and discrete measurements, and $T_m$ the event grid of a discrete measurement. Terminal information is

\[F=F_{\mathrm{prior}} +\sum_{m\in\mathcal C}\int_{t_0}^{t_f}w_m(t)J_m(t)^\top J_m(t)\,dt +\sum_{m\in\mathcal D}\sum_{t\in T_m}w_m(t)J_m(t)^\top J_m(t).\]

Here $F\in\mathbb R^{n_s\times n_s}$ and st.F_init supplies the prior in these coordinates. The ODE integrates the upper triangle of the continuous contribution, initially zero. Discrete contributions are accumulated from saved states on demand. fisher_information(design, trajectory) combines them with the current prior without another differential-equation solve.

Information is linear in the measurement weight, $wJ^\top J$, rather than quadratic in that weight. This is a sampling-design weighting convention, not multiplication of the observation by $w$ before forming its Jacobian.

The formula assumes independent, whitened observation errors with unit covariance. To represent a known, fixed positive-definite covariance $R_m=L_mL_m^\top$, supply the whitened map $L_m^{-1}h_m$. Scalar standard deviations are handled by dividing the observation by the standard deviation. Correlations across distinct measurements/events and parameter-dependent noise information are not assembled by this formula.

Budgets and information gains

The corresponding budgets are

\[b_m=\int_{t_0}^{t_f}w_m(t)\,dt\quad(m\in\mathcal C),\qquad b_m=\sum_{t\in T_m}w_m(t)\quad(m\in\mathcal D).\]

Continuous budgets integrate the current piecewise-constant schedule exactly by interval length; discrete budgets sum event weights. They need no additional ODE solve. sampling_sums and optimizer bounds M order continuous measurements first, then discrete measurements, retaining declaration order within each kind. Multiple experiments concatenate that order experiment by experiment.

local_information_gain returns unweighted outer products of individual rows of $J$ at every saved time. global_information_gain returns $(J_iF^{-1})^\top(J_iF^{-1})$ for each row $i$. For symmetric nonsingular information, its trace gives the magnitude of the row's marginal improvement in A-optimality before interval length/event weighting. For a vector measurement, sum its rows to obtain its shared weight's contribution. These routines do not select only the discrete events.

Criteria

All criteria are minimized. Their exact definitions are:

julia> using CorleoneOED, LinearAlgebra

julia> F = diagm([2.0, 4.0]);

julia> (ACriterion()(F), DCriterion()(F), ECriterion()(F))
(0.75, 0.125, 0.5)

julia> (FisherACriterion()(F), FisherDCriterion()(F), FisherECriterion()(F))
(-6.0, -8.0, -2.0)

These are not log-determinant or dimension-normalized criteria. Matrix inputs are wrapped in Symmetric; supply symmetric information. No diagonal regularization or positive-definiteness check is added by the criteria themselves.