Uncertainty Intervals May Grow or Shrink over Time:
An Illustrative Example with the Flower Growth Model
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Appendix J of Modeling the Environment describes a formalized method of estimating the tolerance intervals associated with model projections. Figure J.2 shows an example for the first four years of behavior of the flower growth model from chapter 6. During these years, the flowered area is growing rapidly as there is plenty of empty area to accomodate the spread of flowers. The diagram shows the 75% and 95% confidence bounds estimated from a collection of 100 simulations using the advanced version of the Vensim software. Latin Hypercube Sampling was used to generate 100 simulations with uncertainty in both the intrinsic growth rate and the decay rate of the flowers. The figure shows a narrow band of uncertainty at the start, but the band grows larger and larger over time. Appenedix J explains that this pattern is often seen in systems that are dominated by positive feedback. Uncertainty in the parameters translates into uncertainty in the grain around the positive feedback loop, and uncertainty in the gain leads to a growing band of uncertainty over time.

 Growth in Uncertainty Over Time:
This is Figure J.2 from the book.
It shows confidence bounds for the first four years of simulated growth in flowered area.

The next chart shows how the band of uncertainty changes if the simulations are continued beyond the first four years. As the flowers spread over the suitable area, the negative feedback in the system becomes increasingly important. The confidence bounds are no longer growing over time. After the 8th year of the simulations, they are actually shrinking over time. By the end of the time period, we arrive at a situation with a 95% confidence bound ranging from around 700 to 900 acres, and the width of the confidence interval is constant over time.

Appendix J explains that the narrowing of uncertainty during the middle years of this example is typical of systems that are dominated by negative feedback. Differences in the input parameters are eventually overridden by the actions of the negative feedback loop causing the uncertainty to shrink over time.

 
Uncertainty Shrinks Over Time.
This is Figure J.3 from the book.
It shows the confidence bounds for the entire simulation.