Simulation Decomposition

A hybrid uncertainty-and-sensitivity visualization that partitions selected inputs into states, crosses those states into joint scenarios, and decomposes the sampled output distribution into scenario-labeled subdistributions.

Version

v3 · 2026-09-06 · History

Domain-specific #

2785

Origin domain

uncertainty analysis

Subdomain

global sensitivity visualization

Aliases

SimDec, Simulation-decomposition analysis

Core Idea

Simulation Decomposition, usually abbreviated SimDec, is a named method for showing how joint states of selected inputs contribute to the full distribution of a model output. Starting from row-linked input and output values—usually generated by Monte Carlo simulation—the analyst partitions a few inputs into discrete states, forms every joint combination of those states, assigns each observation to exactly one scenario, and preserves those labels when plotting the output distribution. A stacked, scenario-colored histogram then shows not only how uncertain the output is but which regions of input space produced different parts of that uncertainty.

The foundational investment-analysis formulation appeared as a method for interpreting multivariable simulations. Subsequent work applied it across geology, business, environmental science, system dynamics, agriculture, and engineering; extended it with quantitative sensitivity measures; and implemented it in open-source software.^{[1][2][3][4][5][6]}

Its defining move is a scenario-conditioned mixture decomposition, not an additive decomposition of the model equation. Let the retained dataset be

\[ D=\{(\mathbf x^{(r)},y^{(r)})\}_{r=1}^{N}. \]

For selected input indices \(J\), define mutually exclusive and exhaustive states \(B_{j1},\ldots,B_{jm_j}\) for each input \(X_j\). Their Cartesian combinations form scenarios

\[ \mathcal S=\prod_{j\in J}\{1,\ldots,m_j\}. \]

Every row receives a scenario label \(g(r)=s\) according to the states containing its selected input values. If \(N_s\) is the number of rows in scenario \(s\) and \(\widehat F_s\) is that scenario's empirical output distribution, then

\[ \widehat F(y)=\sum_{s\in\mathcal S}\frac{N_s}{N}\widehat F_s(y). \]

This identity is load-bearing. The overall output distribution is exactly reconstituted by weighted scenario subdistributions, subject to declared filtering and missing-data rules. Histogram colors are a visual encoding of that mixture, not decorative annotations.

Simulation Decomposition survives as an autonomous domain-specific abstraction at 0.99 confidence. Generic Decomposition supplies its whole-to-parts-to-recomposition skeleton; Partition supplies exclusive scenario membership; Monte Carlo supplies common input-output rows; sensitivity indices can choose influential inputs; and a histogram displays counts. None of those alone or as an unordered list entails the SimDec pipeline, conservation rule, joint-scenario encoding, or diagnostic interpretation. The method remains domain-specific because it is a concrete uncertainty/sensitivity-analysis technique with input-output tables, empirical distributions, state binning, and visual conventions—not a substrate-neutral prime.

Structural Signature

The recurring signature is:

row-linked uncertain inputs and output → selected explanatory inputs → mutually exclusive input states → Cartesian joint scenarios → exactly-one scenario assignment per retained row → scenario-conditioned output subdistributions → recomposed, scenario-colored output distribution → heterogeneity and decision interpretation

The mandatory roles are:

• Input-output ensemble. Each simulation run or empirical record retains the selected input values and the corresponding scalar output. An output sample detached from its generating inputs cannot be decomposed by SimDec.
• Selection rule. A small subset of inputs is selected manually for decision relevance or by a declared sensitivity-analysis rule. Modern implementations can compute or consume first- and higher-order sensitivity information, but no single ranking estimator defines the method.^[5][6]
• State partitions. Every selected numeric input is divided into intervals, often equal-frequency states; categorical inputs use declared category groupings. States must be operationally nonoverlapping and cover every retained row.
• Joint scenarios. The Cartesian product of selected input states supplies interpretable multivariable cases. With state counts \(m_j\), the maximum scenario count is \(\prod_{j\in J}m_j\), before empty cells are considered.
• Unique row assignment. Every retained observation maps to exactly one joint scenario. This is the invariant that prevents double counting and makes the component counts additive.
• Output distribution representation. Output values are binned or otherwise represented as a distribution while their scenario labels remain attached. For output bin \(H_b\), the scenario count is

\(n_{bs}=\sum_{r=1}^{N}\mathbf 1\{y^{(r)}\in H_b,\ g(r)=s\}.\)

Hence \(\sum_s n_{bs}=n_b\) and \(\sum_b\sum_s n_{bs}=N\) when the bins cover all retained outputs. - Scenario encoding. Colors and shades distinguish primary and nested input states while stacked heights preserve bin totals. Legend, stack order, and palette must be declared and accessible. - Conditional interpretation. Shifts, overlap, spread, tails, and scenario composition are read as empirical conditional behavior under the sampled input distribution. Causal conclusions require an additional identification or experimental warrant.

Equal-frequency states, a thousand runs, two or three selected inputs, and a particular color palette are common usability recommendations rather than identity invariants. A valid implementation may use substantive thresholds, more or fewer runs, or a non-histogram display if it preserves scenario assignment and the recomposable conditional-distribution logic. Removing joint scenarios or row-linked output components, however, collapses the method.

What It Is Not

Simulation Decomposition is not decomposition of a simulation model into submodels, spatial domains, processors, time steps, equations, or software components. Those methods restructure computation. SimDec leaves the simulator intact and decomposes its sampled output distribution by input-state scenarios.

It is not Monte Carlo simulation. Monte Carlo generates or approximates an output distribution through random sampling. SimDec is a postprocessing analysis that requires the input-output association and can also operate on suitable measured data.^[6]

It is not a sensitivity index. Sobol', regression, screening, or binning indices summarize influence numerically. SimDec can use such indices to select variables, but its distinctive result is the scenario-conditioned distribution and visual account of how an effect changes across other input states.

It is not variance decomposition, ANOVA, functional decomposition, or Shapley attribution. Those allocate a numerical target such as variance or prediction difference to terms or coalitions. SimDec partitions observations into cells and represents the output as a finite mixture. A modern SimDec workflow may add sensitivity indices, but the mixture identity and any variance allocation remain separate claims.

It is not an ordinary histogram. A histogram bins only output values. SimDec retains a second partition by joint input states and shows each scenario's count inside every output bin.

It is not scenario planning. SimDec scenarios are exhaustive combinations of discretized sampled inputs with empirical frequencies. Scenario planning constructs a small set of qualitative futures under deep uncertainty and need not assign probabilities.

It is not automatically a causal analysis. The visualization shows conditional associations in a simulation design or observed dataset. Causality follows only when the model semantics, sampling design, intervention, or external identification assumptions justify it.

Scope of Application

The home scope is uncertainty analysis, global sensitivity analysis, simulation analytics, and decision support for computational models. Original applications addressed multivariable investment projects. Cross-domain work subsequently used the method in geology, business, and environmental science; a carbon-footprint study decomposed uncertainty in a wooden-pallet life-cycle case; independent engineering researchers applied it to a nondestructive-evaluation system and an agricultural food-water-energy model.^[2][3][4]

System-dynamics work uses SimDec to connect uncertain drivers to urban-planning outcomes, while a quantitative extension tests heterogeneous effects in several computational decision models.^[7][5] The peer-reviewed Python implementation also describes use with empirical datasets and artificial-intelligence analyses, provided records retain predictors and a target variable.^[6]

The natural output is a scalar whose distribution has enough resolution for meaningful subdistributions. The method can be repeated for multiple outputs, but a single stacked histogram does not jointly characterize a multivariate output distribution. Binary or low-cardinality categorical outcomes allow scenario counts but offer little distributional geometry. Very expensive simulators can limit ensemble size; SimDec reuses an existing ensemble efficiently but does not eliminate the cost or design requirements of producing representative runs.

Domain recurrence does not make this a prime. Business cash flow, carbon footprint, structural reliability, groundwater management, and urban dynamics all enter through the same computational-model and input-output-table substrate. Outside that analytical substrate, “decomposing a simulation” normally means something else.

Clarity

A record qualifies as Simulation Decomposition when six questions have clear answers:

1. What is the row-linked ensemble? Identify the runs or records, selected inputs, output, sampling design, filters, and missing-data treatment.
2. Which inputs were selected, and why? State whether selection reflects sensitivity indices, decision control, scientific interest, or another rule.
3. How was each selected input partitioned? List interval boundaries or category mappings; “low” and “high” are insufficient without definitions.
4. Do joint states form a unique scenario partition? Every retained row must receive exactly one label, including boundary values.
5. Does the display conserve the output distribution? Scenario segments within each bin must sum to that bin's total, and all bins must sum to the retained sample.
6. Are conclusions bounded to the design? Conditional patterns must be tied to sampled ranges, probabilities, binning, and model validity.

The strongest diagnostic is the recomposition test. Remove the scenario colors and sum the stacked components. If the original output histogram is not recovered, the artifact is not a valid SimDec decomposition under its stated data scope. Conversely, arbitrary color-coding by observation order may recompose the histogram but fails the input-state scenario requirement.

The title needs contextual qualification. “Simulation decomposition” is also ordinary language for domain decomposition, co-simulation, model partitioning, and equation splitting. SimDec here always means the uncertainty/sensitivity visual-analytics method.

Manages Complexity

A probability distribution communicates output range, central tendency, spread, skewness, and tails, but it hides which input combinations populate those regions. A sensitivity index can rank inputs but compresses the shape of their effects into numbers. Scatter plots expose pairwise relationships but become difficult when several uncertain inputs interact. SimDec keeps the output distribution familiar while overlaying a finite joint-state account.

The method compresses a high-dimensional ensemble in two stages. First, variable selection limits attention to decision-relevant or influential inputs. Second, state binning maps continuous ranges into a small scenario lattice. The scenario-conditioned mixture then makes questions visually tractable: Which combinations dominate a desirable tail? Does one input shift the whole distribution, change only its spread, or matter only in a particular state of another input? Which adverse outcome region has several alternative generating scenarios?

The conservation invariant is especially valuable for communication. The audience never has to reconcile separate plots whose totals differ: the colored components remain parts of the observed output whole. The scenario table also supplies a stable bridge from visualization back to rows, so analysts can inspect the underlying model runs rather than treating the chart as a final explanation.

Compression also creates a combinatorial limit. Three inputs with three states each create 27 possible scenarios; four create 81. Empty or sparse cells destabilize conditional shapes and legends become unreadable. The common practice of choosing only a few inputs is therefore not cosmetic—it is the cost of maintaining scenario interpretability.

Abstract Reasoning

Conservation inference. If the scenario states are mutually exclusive and exhaustive, each output observation contributes exactly once. Counts and empirical probabilities add across scenarios. A mismatch indicates boundary, filtering, missingness, or implementation error.

No-effect diagnostic. If an input's states have identical population-conditional output distributions under the design, changing that state does not alter the represented output law. Finite samples can still show accidental differences, so overlap is evidence to quantify rather than a proof of no effect.

Heterogeneity diagnostic. Suppose the conditional effect of input \(A\) changes across states of input \(B\). Joint \((A,B)\) scenario subdistributions will change nonuniformly. That pattern identifies effect heterogeneity or interaction in the model response, even when a marginal plot of \(A\) averages it away.^[5]

Reachability diagnostic. If an output tail contains only certain scenario labels in a well-sampled design, those joint states are empirical routes to that tail. This supports targeted investigation or decision screening. It does not guarantee the outcome, prove necessity outside sampled ranges, or identify a causal intervention without further warrant.

Refinement inference. Splitting an input state refines the scenario partition. The refined components must recombine to the former component, but their apparent shapes can reveal heterogeneity hidden by coarser bins. Merging states coarsens the account and cannot create information, though it can improve sampling stability.

Sampling-distribution inference. Scenario weights \(N_s/N\) depend on how inputs were sampled. Changing marginal distributions, dependencies, or experimental design changes the mixture even if the deterministic model is unchanged. SimDec therefore describes the model under a declared input distribution, not an invariant ranking independent of uncertainty assumptions.

Knowledge Transfer

The method transfers literally wherever four data conditions remain intact: paired input-output records, defensible input states, enough coverage of joint scenarios, and an output representation capable of preserving scenario membership. A financial analyst can label simulated net-present-value outcomes by cost and demand states; an environmental analyst can label carbon-footprint outcomes by material and process states; an engineer can label reliability outcomes by load and material-property states. The application vocabulary changes, while row assignment, mixture recomposition, and interpretation remain identical.

Transfer requires recalibration rather than mechanical reuse. Quantile states make scenario counts comparable but can split a meaningful regulatory or physical threshold. Substantive states improve decision meaning but may leave sparse scenarios. Input selection by variance importance can miss variables important only for tails or particular policies. Output bins appropriate for financial loss may obscure a safety threshold.

The transferable residue above the method belongs to existing primes. Decomposition explains whole-to-parts-to-recomposition. Partition explains mutually exclusive and exhaustive scenario membership. Classification explains row labeling. Uncertainty explains the sampled input state. Those abstractions help reason about SimDec, but they do not replace its domain-specific analytical protocol.

Examples

Canonical pure-interaction example. Let independent inputs \(A\) and \(B\) each take \(-1\) or \(+1\) with equal probability, and let \(Y=AB\). The four joint scenarios are equally weighted. Scenarios with equal signs produce \(Y=+1\); scenarios with opposite signs produce \(Y=-1\). Marginally,

\[ P(Y=1\mid A=1)=P(Y=1\mid A=-1)=\tfrac12, \]

and likewise for \(B\), so either input considered alone appears uninformative. A SimDec output with joint scenario colors shows that the entire output is determined by their combination. The example demonstrates interaction visibility; it does not imply that every visual separation has this exact algebraic form.

Carbon-footprint uncertainty. Deviatkin, Kozlova, and Yeomans apply Simulation Decomposition to a wooden-pallet carbon-footprint model. The method preclassifies uncertain inputs into states, forms multivariable scenarios, and projects their contributions onto the simulated footprint distribution.^[3] The output distribution remains the uncertainty account, while the colored components supply a decision-facing explanation of which combinations populate different regions.

Agricultural and engineering systems. Liu and colleagues—an application team distinct from the method's original authors—use Simulation Decomposition on a tutorial model, a nondestructive-evaluation testing system, and an agricultural food-water-energy system.^[4] This establishes recurrence beyond investment and beyond the originating research group. The same scenario partition operates on different model inputs and outputs.

Heterogeneous sequential decision effects. The quantitative extension examines cases where an input's influence changes with another input or with model structure. In one sequential decision example, reward sensitivity differs by solver state; a single overall index would not express that conditional shape.^[5] SimDec's value is not merely “another ranking” but a map of the contexts in which influence appears.

Structural Tensions

• Balanced states vs. meaningful states. Equal-frequency bins stabilize counts, while domain thresholds carry decision meaning. Diagnostic: compare quantile and substantive partitions and report conclusions that change.
• Joint visibility vs. combinatorial explosion. Adding inputs can expose higher-order heterogeneity but multiplies scenarios and colors. Diagnostic: inspect cell counts and stop refining when sparse cells or legend complexity defeat interpretation.
• Visual compression vs. quantitative precision. A stacked distribution communicates shape quickly but does not supply uncertainty intervals or a unique effect magnitude. Diagnostic: pair visible claims with conditional statistics or sensitivity estimates when decisions require numerical precision.
• Reuse of an ensemble vs. quality of the ensemble. SimDec adds little model-evaluation cost to existing runs, but cannot repair biased sampling or inadequate coverage. Diagnostic: audit design, convergence, dependence, and effective scenario sample sizes before reading patterns.
• Variable selection vs. omitted interaction. Restricting the display improves readability but can conceal an input that matters only conditionally or in a tail. Diagnostic: compare selection rules and inspect higher-order or tail-sensitive evidence.
• Association vs. causation. Scenario membership identifies conditional provenance of outputs, not necessarily the effect of intervening on an input. Diagnostic: ask whether sampling and model structure support an interventional contrast.
• Stable method vs. analytic variants. Manual selection and visual-only SimDec coexist with automated sensitivity-index extensions. Diagnostic: state the version and keep the scenario-mixture invariant separate from optional quantitative modules.
• Autonomy vs. component closure. The method uses Decomposition, Partition, Monte Carlo, and visualization, yet its ordered protocol and exact mixture account add a reusable reasoning object. Diagnostic: if joint input states are not mapped back onto the full output distribution, the result is only one component, not SimDec.

Structural–Framed Character

Simulation Decomposition is mixed structural–framed. Its structural core is unusually crisp for a visual method: selected input states induce a partition; Cartesian products create joint cells; every retained record receives one label; scenario subdistributions recombine exactly to the empirical output distribution. These invariants are independent of palette and application domain.

Its interpretation is strongly framed by analytical choices. The input distribution controls scenario weights. Variable selection controls which heterogeneity can appear. State boundaries control conditional groups. Output-bin edges control visible shape. Stack order and color hierarchy control perceptual salience. A chart can be arithmetically correct yet substantively misleading if these choices are hidden or unstable.

Accordingly, the method is not a free-form infographic. It is a structurally disciplined partition displayed through a framed visual encoding. Reference-grade use reports both halves.

Structural Core vs. Domain Accent

The structural core is: partition a whole by a declared classification, analyze the parts, and recombine them without loss. That core is a strict specialization of Decomposition and relies on Partition for exactly-one membership. It transfers to any count-preserving labeled mixture.

The domain accent binds that core to uncertainty and sensitivity analysis: inputs and outputs remain row-linked; selected input ranges become states; state products become simulation scenarios; empirical conditional distributions are stacked inside the output distribution; and patterns are interpreted as model-response heterogeneity under a sampling design. These roles are not optional metaphors.

Because the method is named, algorithmic, and software-supported, its broad disciplinary use should not be mistaken for prime-level substrate independence. Its applications all pass through computational or empirical input-output data. The prime-worthy pattern is already represented by Decomposition and Partition; SimDec is the domain-specific instrument built from them.

Instantiates / Related Primes

Decomposition is the minimal live parent. The whole is the empirical output distribution; parts are scenario-conditioned subdistributions; part-level inspection reveals structure; and weighted recomposition exactly restores the whole. Simulation Decomposition therefore strictly specializes the live prime rather than merely resembling it.

Partition supplies the nonoverlap, exhaustiveness, and single-valued membership invariants for input states and joint scenarios. It is a load-bearing related prime but need not be a second parent because the Decomposition edge already captures the method-level whole/parts/recomposition identity.

Monte Carlo Simulation commonly supplies the ensemble, but SimDec can analyze measured input-output rows and does not itself generate random samples. Sensitivity Analysis (in Operations Research) is a nearby catalog method and modern SimDec can incorporate global sensitivity indices, but the live prime is centered on post-optimality analysis of solved optimization models. SimDec neither requires an optimum nor produces shadow prices or stability ranges.

Classification appears when observations receive scenario labels. Uncertainty appears in the input distribution. Statistical Inference can be added when conditional differences are generalized beyond the finite ensemble, but descriptive SimDec does not automatically perform population inference.

Relationships to Other Abstractions

Current abstraction Simulation Decomposition Domain-specific

Parents (1) — more general patterns this builds on

• Simulation Decomposition is a kind of Decomposition Prime

  Decomposition is the minimal live parent.

Hierarchy path (1) — routes to 1 parentless root

• Simulation Decomposition → Decomposition

Neighborhood in Abstraction Space

Simulation Decomposition sits in a sparse region of the domain-specific corpus (89^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (1565 abstractions)

Nearest neighbors

• Nonrecursive (FIR) Filter — 0.81
• Generalized Semi-Infinite Programming — 0.80
• Language of Temporal Ordering Specification — 0.79
• Particle Filter — 0.78
• Probability Bounds Analysis — 0.78

Computed from structural-signature embeddings · 2026-09-08

Not to Be Confused With

• Generic simulation/model decomposition: splitting a computational model, spatial domain, equation, or workload into components.
• Decomposition: the substrate-neutral whole-to-parts pattern; SimDec is one analytical specialization.
• Partition: the exactly-one-block structure used inside SimDec, not the full method.
• Monte Carlo simulation: generation of random model runs, not their scenario-conditioned visual decomposition.
• Sensitivity analysis: the broader family of methods for studying input influence; indices may select SimDec inputs but do not reproduce its output mixture.
• Variance decomposition/Sobol' analysis: allocation of output variance to effects, not partition of observations into output subdistributions.
• Histogram: output bin counts without joint input-state provenance.
• Scenario planning: qualitative alternative futures rather than exhaustive sampled state combinations.
• Stratified sampling: a design for drawing observations from strata; SimDec usually stratifies an existing input-output ensemble after generation.
• Regression, decision trees, clustering, or classification models: alternative predictive or grouping procedures with different objectives and loss or split rules.
• Causal decomposition: attribution under an identified causal model; SimDec alone supplies conditional association.

References

[1] Kozlova, Mariia, Mikael Collan, and Pasi Luukka. “Simulation Decomposition: New Approach for Better Simulation Analysis of Multi-Variable Investment Projects.” Fuzzy Economic Review 21, no. 2 (2017): 3–18. Foundational named-method formulation. registry ↩

[2] Kozlova, Mariia, and Julian Scott Yeomans. “Monte Carlo Enhancement via Simulation Decomposition: A ‘Must-Have’ Inclusion for Many Disciplines.” INFORMS Transactions on Education 22, no. 3 (2022): 147–159; published online 2020. Establishes the method's cross-domain Monte Carlo role and pedagogical applications. registry ↩a ↩b

[3] Deviatkin, Ivan, Mariia Kozlova, and Julian Scott Yeomans. “Simulation Decomposition for Environmental Sustainability: Enhanced Decision-Making in Carbon Footprint Analysis.” Socio-Economic Planning Sciences 75 (2021): 100837. Primary environmental application and explicit state–scenario–subdistribution description. registry ↩a ↩b ↩c

[4] Liu, Yen-Chen, Leifur Leifsson, Anna Pietrenko-Dabrowska, and Slawomir Koziel. “Analysis of Agricultural and Engineering Systems Using Simulation Decomposition.” In Computational Science—ICCS 2022, 435–444. Independent applications to nondestructive evaluation and a food-water-energy system. registry ↩a ↩b ↩c

[5] Kozlova, Mariia, Robert J. Moss, Julian Scott Yeomans, and Jef Caers. “Uncovering Heterogeneous Effects in Computational Models for Sustainable Decision-Making.” Environmental Modelling & Software 171 (2024): 105898. Primary quantitative extension, algorithm description, comparison with neighboring sensitivity methods, and heterogeneous-effect cases. registry ↩a ↩b ↩c ↩d ↩e

[6] Roy, Pamphile T., and Mariia Kozlova. “Simulation Decomposition in Python.” Journal of Open Source Software 9, no. 98 (2024): 6713. Peer-reviewed software paper documenting automatic scenario formation, sensitivity integration, visualization, empirical-data scope, and open implementation. registry ↩a ↩b ↩c ↩d

[7] Yeomans, Julian Scott, and Mariia Kozlova. “Extending System Dynamics Modeling Using Simulation Decomposition to Improve the Urban Planning Process.” Frontiers in Sustainable Cities 5 (2023): 1129316. Open primary application to system dynamics and urban planning under uncertainty. registry ↩