Computed software examples

Spatial fields, stochastic simulations, replicated comparisons and incomplete-data reconstruction using released packages.

These examples use released versions of OP System, OP Engine, trade-study and VBPCApy. All data are synthetic. The plots show computations rather than findings from an application study; their captions state the assumptions and what is checked.

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Spatial diffusion

OP System compiles a named-axis expression for a periodic diffusion stencil; OP Engine integrates the resulting field with fourth-order Runge–Kutta (RK4). The stencil is supplied explicitly on 64 equally spaced cells. Faster spatial variation fades sooner than slower variation.

A 64-cell periodic diffusion field and profiles at four times, computed from an OP System specification with RK4 integration in OP Engine. Spatial discretization is supplied explicitly; all inputs are synthetic.
A periodic spatial field at 121 output times and four selected profiles. The spatial mean is conserved.

The code checks the full trajectory against the exact Fourier-mode solution of the discrete spatial stencil, with a maximum absolute difference below 2e-10. This measures temporal integration error for the chosen grid. It does not measure error against a continuum PDE or establish accuracy for other spatial discretizations.

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Stochastic transfers

OP Engine runs 512 independent realizations of a two-state transfer process using the exact stochastic simulation algorithm (SSA), with one event clock per run. Each starts with 40 units; each remaining unit transfers at rate 0.6. The example retains the accepted firing counts and checks conservation and the correspondence between counts and transferred units.

512 independent exact stochastic transfer simulations, their empirical mean and pointwise 5th–95th percentile range, and the final distribution alongside analytic mean and binomial references. Synthetic demonstration.
Individual paths vary. The reference mean and final binomial distribution are known for this transfer process. The shaded range is pointwise variability across runs, not uncertainty in the estimated mean.

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Replicated designs

trade-study evaluates ten sample-mean designs using 512 paired replicates. The measurement standard deviation is either 1 or 1.5. The noisier measurement is assigned a lower cost; the standard measurement has a fixed setup cost. Those costs are illustrative assumptions. The circles mark designs for which no alternative improves both mean error and assumed cost.

Ten synthetic sample-mean designs compared across 512 paired replicates. Nondominated mean error and assumed cost are marked, and empirical errors are compared with analytic variances. Costs are declared assumptions, not measured field costs.
Increasing the sample count reduces estimation error in this toy problem. The right panel compares the Monte Carlo estimates with the known variance of a sample mean.

The table reads the same generated data as the figure. Error estimates depend on these 512 simulated replicates. A design is nondominated when no other design is at least as good on both stated objectives and better on at least one. Choosing a design for an application also requires preferences or constraints relevant to that application.

On smaller screens, scroll the table to see cost and nondominance.

Ten synthetic designs, 512 paired replicates each. Lower error and cost are preferred. Bars use a separate linear scale in each column.
Samples Measurement SD Mean squared error Assumed cost Nondominated
8 1.0 0.137 16.0 Yes
8 1.5 0.308 4.4 Yes
16 1.0 0.065 24.0 No
16 1.5 0.146 8.8 Yes
32 1.0 0.029 40.0 Yes
32 1.5 0.064 17.6 Yes
64 1.0 0.015 72.0 Yes
64 1.5 0.034 35.2 Yes
128 1.0 0.008 136.0 Yes
128 1.5 0.019 70.4 Yes

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Incomplete data

VBPCApy fits a two-component model to observed entries of a noisy synthetic matrix. Roughly 35% of entries are withheld, shown in gray. The generating signal uses two patterns; that choice is supplied to the fitted model.

A synthetic two-pattern signal, noisy observed entries with unobserved cells in gray, and a VBPCA reconstruction. Held-out observations are excluded from fitting. This illustration does not establish rank selection or uncertainty calibration.
Left: generating signal. Middle: available noisy observations. Right: fitted reconstruction. All panels share a color scale.

The root mean squared reconstruction error on withheld noisy observations is about 0.22 for this seeded example; the generating noise standard deviation is 0.2. This is one reconstruction check with a known component count. Choosing the rank and assessing uncertainty calibration require separate studies.

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