Simulation decomposition: Difference between revisions

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[[File:SimDec.gif |thumb|right | upright=2 | A typical SimDec output]]

'''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 [[Frequency (statistics)|distribution]] of the model output.<ref name="informs" /> This visual analytics approach exposes the underlying nature of the model behavior, including its nonlinear and multivariate [[Interaction (statistics) | interaction effects]].<ref name="kozlova_et_al_1" />

SimDec can be used in any range of science, engineering, and social domains. Existing applications include business<ref>Kozlova, M., Collan, M., & Luukka, P. (2017). Simulation decomposition: New approach for better simulation analysis of multi-variable investment projects.</ref> and environmental issues.<ref>Deviatkin, I., Kozlova, M., & Yeomans, J. S. (2021). Simulation decomposition for environmental sustainability: Enhanced decision-making in carbon footprint analysis. Socio-Economic Planning Sciences, 75, 100837.</ref><ref>Liu, Y. C., Leifsson, L., Pietrenko-Dabrowska, A., & Koziel, S. (2022). Analysis of Agricultural and Engineering Systems Using Simulation Decomposition. In International Conference on Computational Science (pp. 435-444). Springer, Cham.</ref>

== Method==

SimDec operates on [[Monte Carlo Method | 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,<ref name="Software">Simulation Decomposition GitHub https://github.com/Simulation-Decomposition </ref> 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.

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# '''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.<ref name="Software">Simulation Decomposition GitHub https://github.com/Simulation-Decomposition </ref> 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==

[[File:Distribution of Y.svg|thumb| upright=1.1 | A histogram built for an array of ''Y'' = {11, 12, 25, 28, 28, 29, 31, 35, 39, 41}]]

=== Histogram ===

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[[Histogram]] is an approximate representation of the [[Frequency (statistics)|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 distribution|probability]].<ref name="Kenney">{{cite book | last1 = Kenney | first1 = J. F. | last2 = Keeping | first2 = E. S. | title = Mathematics of Statistics, Part 1 | edition = 3rd | url = https://books.google.com/books?id=UdlLAAAAMAAJ | ___location = Princeton, NJ | publisher = [[John Wiley & Sons|Van Nostrand Reinhold]] | year = 1962}}</ref>

The distribution alone can supply only limited information about the data – its minimum, maximum, and shape (where the most of data occurs).

[[File:Simdec influence.svg|thumb|right|upright=1.5 |Different degrees of influence of ''X'' on ''Y'' on a scatter plot and SimDec histogram]]

=== 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-based sensitivity analysis|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''.<ref name="informs" />

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.

{| class="wikitable"

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| Strong effect || The border between sub-distributions is vertical, no overlap of the ''Y'' range. || If the high state of X can be achieved, it would guarantee high ''Y''.

|}

[[File:Simdec interactions.svg|thumb|upright=2.5 |Appearance of different interaction types on SimDec visualization]]

=== 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.

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|Various types of '''nonlinear interactions''' can occur in models. For example, one input variable has no effect on the output in one state of another variable (lying on top of each other red-shaded sub-distributions) but has a strong effect otherwise (shifted blue sub-distributions). Such effect, too, will show up in the non-zero second-order sensitivity index.<ref name="kozlova_et_al_1" />}}

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

== Limitations ==