mathematics

Simulation decomposition

Simulation decomposition is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Simulation decomposition rather than just read about it. In short: SimDec, or Simulation decomposition, is a hybrid uncertainty and sensitivity analysis method, for visually examining the relationships between the output and input variables of a computational model. SimDec maps multivariable scenarios onto the distribution of the model output.

Key takeaways

Simulation decomposition belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Simulation decomposition to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Simulation decomposition from memory before moving on to harder problems.

Reference excerpt

SimDec, or Simulation decomposition, is a hybrid uncertainty and sensitivity analysis method, for visually examining the relationships between the output and input variables of a computational model. SimDec maps multivariable scenarios onto the distribution of the model output. This visual analytics approach exposes the underlying nature of the model behavior, including its nonlinear and multivariate interaction effects. SimDec can be used in any range of science, engineering, and social domains. Existing applications include business and environmental issues.

Method SimDec operates on Monte Carlo simulation (or measured) data where both output and input values are recorded. At least one thousand observations (or simulated iterations) are typically recommended to preserve the readability of the resulting histograms. An outline of the decomposition algorithm, which is readily available in multiple programming languages, proceeds as follows:

Select the input variables for decomposition. One can use sensitivity indices (see variance-based sensitivity analysis) to define the most influential variables for decomposition or choose them manually according to the decision-problem context (for example, only those input variables that the decision-maker can act upon). Two to three input variables, ordered by decreasing value of their sensitivity indices, usually provide the most meaningful decomposition results. Divide the inputs into states. The numeric ranges of the inputs are split into several intervals with an equal number of observations in each. For categorical variables, the categories represent states. Form scenarios. All combinations of states of the selected input variables produce unique scenarios or subsets of the data. For example, if the range of X2 is divided into low, medium and high, and X3 takes values of 1 or 2, six scenarios are formed: (i) X2 low & X3 = 1, (ii) X2 low & X3 = 2, (iii) X2 medium & X3 = 1, (iv) X2 medium & X3 = 2, (v) X2 high & X3 = 1, and (vi) X2 high & X3 = 2. Assign scenarios to each output value. The simulation data is used to define the scenario index for each simulation run. For example, if an X2 value falls into the low state and X3 is equal to 2, the corresponding scenario, defined in Step 3, is (ii). Color-code the output distribution. When all output values are assigned scenario indices, they are plotted as series in a stacked histogram, visually separated by color-coding. For ease of visual perception, the states of the most influential input variable are assigned distinct colors, and all the remaining partitions take shades of those colors (see Figure). All of these steps can be run automatically on the given data using the open-source SimDec packages currently available in Python, R, Julia, and Matlab. A SimDec template in Excel runs a Monte Carlo simulation of a spreadsheet model but possesses only a manual option for input selection.

How to read SimDec

Histogram Histogram is an approximate representation of the distribution of numerical data. Its horizontal axis shows the range of the variable of interest, and its vertical axis denotes count, also called frequency, or, if divided by the total number of data points, probability. The distribution alone can supply only limited information about the data – its minimum, maximum, and shape (where the most of data occurs).

Judging the importance of inputs If an input variable has no effect on the output, its states (e.g., low & high) would lie on top of each other on the SimDec histogram, occupying fully overlapping ranges of the output. If an input variable has a strong effect and explains most of the variance of the output, the border between its states on the SimDec histogram would be vertical. Such visualization has an important decision-making implication – e.g., if the high state of X can be achieved, it would guarantee a certain range of Y. All cases in-between with low-to-strong effects would show a diagonal border between the states. The less they overlap, the larger the effect of X on Y. While the horizontal displacement of sub-distributions on the SimDec histogram is the key to interpreting the results, the vertical disposition of sub-distributions is just a technical matter of the order of plotting the series of the stacked histogram.

Exploring the interaction of inputs When two or more input variables are used for decomposition, it becomes possible to examine their joint effects. A schematic visualization portrays how different types of joint effects of input variables on the output appear on SimDec visualization.

Understanding the nature of interaction effects in a computational model and its behavior in general is crucial for effective decision-making.

Limitations The SimDec method has several limitations:

It is based on Monte Carlo simulation and thus requires running a computational model a thousand of times or more. To models that take hours to evaluate once, it would be impossible to use SimDec (unless a supercomputer and/or large of time are available). SimDec is based on a histogram, thus, for binary or categorical output variables, the visualization would be very limited (e.g., only a few bins). The more input variables one selects for the decomposition, the less readable the histogram becomes. Only cases with two and three input variables are presented in.

References

See also Sensitivity analysis Data and information visualization Histogram Interaction (statistics) Uncertainty Decision making

Illustrations

Worked examples

Example 1 — a first encounter with Simulation decomposition

Start with the simplest possible case. Write down what Simulation decomposition claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Simulation decomposition before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Simulation decomposition ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Simulation decomposition

In research: Simulation decomposition appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Simulation decomposition in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Simulation decomposition is common in secondary-school and first-year university syllabi. It links to neighbouring topics Knowledge representation, Mathematical and quantitative methods (economics), Mathematical modeling, so understanding it makes those chapters shorter.
In everyday life: Look for Simulation decomposition outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Simulation decomposition in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Simulation decomposition means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Simulation decomposition out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Simulation decomposition in simple terms?

Why does Simulation decomposition matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Simulation decomposition?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Simulation decomposition.