MMP.9:4 - Solution

Choose the retained result → derive its change → expose the unclosed contribution → construct a replacement or bound → use it over the required horizon → revise the part that changes the answer.

The work may end at a useful bound. It may also show that the smaller description would cost more than continuing with the source model.

MMP.9:4.1 - Express the retained result and its change

Name the source state z, its initial possibilities and the admitted inputs u. Specify the answer and horizon: a quantity at time T, a threshold crossing, a response to an input, or a distributional feature. Construct the retained description x=r(z) from those needs. A.3.3.PI:4.1-4.2 tests whether merged source states can still answer that question; use an already sufficient result from that test.

For a discrete source update z_next=F(z,u), substitute it into the retained description:

x_next = r(F(z,u)).

For a differentiable r and a differential law z_dot=F(z,u), the chain rule gives:

x_dot = Dr(z) F(z,u).

Dr is the derivative of r. If r also depends explicitly on time, include its time derivative. A discontinuous readout or event needs its own change relation rather than this differentiable formula. A.3.3.TR supplies composition of the relevant changes.

Rewrite the result using x, the admitted inputs and whatever other information the smaller model proposes to carry. The part still depending on discarded quantities is the unclosed contribution. One convenient decomposition is x_dot=f0(x,u)+c(z,u), where f0 is the part you will compute directly and c is the remaining contribution. Another decomposition can be useful; what matters is the complete retained law and what information obtains each term.

If every term is obtainable from the retained information, construct the closed update through A.3.3.PI and compare it through C.29.1. Continue here when the replacement itself still needs construction.

MMP.9:4.2 - Derive the discarded contribution before approximating it

Try to express the discarded variables through their own evolution, initial conditions and the retained history. In a discrete law, iterate the discarded update and substitute each earlier term until the dependence has a usable form. In a differential law, solve or integrate the discarded equation under the supplied retained history, then substitute the result into the retained equation.

For constant compatible matrices and supplied initial values, consider:

x_dot=A*x+B*y, y_dot=C*x+D*y.

Solving the second equation while treating x(s) as its input gives:

y(t)=exp(D*t)*y0 + integral_0^t exp(D*(t-s))*C*x(s) ds.

Substitution gives the retained law:

x_dot(t)=A*x(t)+B*exp(D*t)*y0 + integral_0^t B*exp(D*(t-s))*C*x(s) ds.

The first added term carries the discarded initial condition. The integral carries the past influence of x through y. With a forcing term in the discarded equation, its propagated contribution also appears in the integral. These terms identify what a proposed memory approximation would replace. Unknown y0 remains an initial uncertainty; it cannot be set to zero solely because y is being removed.

Look for a less costly way to compute the derived expression. Equal decay modes can be combined; a sum of exponentials can be updated through a few auxiliary variables. For example, v(t)=integral_0^t exp(-lambda*(t-s))*x(s) ds satisfies v_dot=x-lambda*v, v(0)=0. This replaces storage of the whole history with an evolving value. Carry a nonzero initial contribution separately or incorporate its matching initial condition. Count the required auxiliary values and update work before claiming a computational saving.

For nonlinear discarded dynamics the same elimination question remains, but the response to retained history may require solving a nonlinear problem. Use the derived dependence to select an approximation, retained variable or bound; an implicit formula that still requires the original computation has not yet supplied a cheaper model.

MMP.9:4.3 - Choose and construct a useful replacement

Use the expression just derived to decide which information or calculation earns its cost.

Retain a quantity that obtains the missing contribution. If c depends on a small additional observable v, derive v’s update from the source law and repeat the closure test for the pair (x,v). This operation can reveal a useful finite system or a growing hierarchy. For members obeying Z_dot=-Z^2, the mean m has m_dot=-mean(Z^2). Retaining q=mean(Z^2) gives q_dot=-2*mean(Z^3). The new equation exposes the next assumption needed; adding q alone does not finish the closure.

Compute the effect through memory. Use the history expression from :4.2. To truncate old history, bound its omitted contribution for the admitted histories. To replace the memory kernel by a simpler one, bound or estimate the resulting difference on those histories, then propagate that difference through the retained evolution. A slowly decaying kernel can make distant history consequential. An auxiliary-state representation can be cheaper than truncation when a few modes express that kernel.

Bound the requested output. If the use asks for a threshold or interval, derive bounds directly from the source law and available initial information. For a population, solve or bound the member response as a function of its initial value, then average using known ranges or moments. Monotonicity, convexity or conservation can give a bound without choosing a complete distribution. The nonlinear example in :5.2 constructs such bounds. Use C.29.1:4.5 to turn them into the corresponding decision, or to expose the still-unresolved range.

Approximate a contribution whose accumulated effect is small. Identify the parameter and the class over which smallness is claimed: initial values, inputs, horizon and required error. Derive or bound the contribution integrated over that horizon. A small coefficient can multiply a large hidden value, and a fast transient can shift the later state. If a transient only matters near the start, retain its effect in an adjusted initial value and state when the later approximation begins. :5.3 shows both constructions.

A learned closure is another possible approximation to the missing contribution. Specify what its inputs contain and how its output enters the retained update. Pairs of retained inputs and source contributions can support fitting, but different hidden states can give different contributions at the same retained input. A fitted conditional mean then answers a distribution-dependent question; it is not an all-cases replacement of those contributions. When probability is needed, MMP.7 helps construct the law of the sampled or recorded training cases. Use the resulting closure at its intended inputs and horizon before drawing the corresponding predictive conclusion.

Compare the candidates through the existing characterization and choice methods: the required output or bound, obtaining cost, initial information and conditions. There is no need to develop every alternative. C.11.DUA helps choose whether more derivation, data or computation can change the decision.

MMP.9:4.4 - Evaluate the replacement inside the retained evolution

Write the model that will actually be used, including its initial values, any auxiliary variables, input rule and observation interpretation. Construct the requested output from that model. Substituting the closure into known source trajectories tests a different computation from letting it generate successive retained states.

Compare the retained source result and the reduced result for the same admitted initial case and input. For an identity, use the derivation to establish the covered equality. For an approximation, propagate the discrepancy to the requested output and horizon through C.29.1:4.5. A numerical scheme adds its own approximation; include it when it can change the answer. A.3.3.PI:4.5 supplies the repeated-use and changed-condition questions.

For a probabilistic model, choose the distributional feature the work needs. Agreement of a mean can coexist with different variance, correlations or event probabilities. A further observable is worth checking when it can change the planned use. State which result the comparison supports rather than extending one fitted statistic to the whole model.

When inputs are chosen from observations, retain the information the choice rule uses. If reduction removes that information, formulate the revised choice under MMP.8 before claiming that the same intervention or control method remains available. A changed intervention also requires the corresponding source law or mechanism under C.28.MR.

MMP.9:4.5 - Return the result and revise the construction when needed

Return the reduced law, sufficient bound or located obstacle with the assumptions that affect its use. An existing derivation and calculation can carry this information. Explain which discarded contribution was replaced, how to initialize and run the replacement, and which question it answers.

If a bound settles the working question, use it. If the result is too weak, locate why: uncertain initial influence, unresolved higher moment, long memory, error amplification, or information unavailable to an action. Improve that contribution or retain more of the source model. A mathematical construction can also show that the proposed simplification offers no saving.

Reopen the affected construction when the source law, initial class, inputs, observation, horizon or required output changes. Keep conclusions whose conditions still hold. In a model of a working method, ME can use the comparison to design a different method or representation; the actual work must still provide the quantities and relations assumed in the model.