Probabailistic techniques in practice

Reducing the number of inputs

Some models that are set up to predict the effects of uncertainty and assist with risk management have a phenomenal number of inputs. This is often a result of dealing with large projects or business models where the basic model structure is the same as for smaller cases, but the number of inputs is much larger.

If a distribution is allocated to each input, then there will be a significant load placed on the simulation software. This in turn slows down the speed of simulation. It also increases the chance of an error during the building of the model that will cause errors in the results. For these reasons it would be ideal if it were possible to create a smaller model (with fewer inputs), but that still gave correct results.

One way of doing this is to take advantage of analytical techniques to partially solve the model first. As mentioned in previous posts, one should always consider the central limit theorem and how it can be used. By finding parts of a model that are collections of additions (or subtractions) we identify a section of the model that can be replaced with a simple analytical surrogate. For example a large number of loss lines in a P&L can be converted into a total loss represented by a single Normal distribution. If this proves to be a significant contributor to variability then the analytical surrogate can be easily analyzed to find the largest contributor(s) within.

Therefore, when you are creating your model be vigilant for opportunities to simplify the model by utilizing the central limit theorem. It will increase speed and reduce the chance of error. There are other cases where it is possible to reduce the number of inputs by reconsidering the nature of the model that is needed. But I will talk about that later.

Is it the model or the distributions?

In a previous post I made a brief (but probably controversial to some) statement. I stated that often the type of input distributions used is not that important because the nature of the model has a much more significant affect upon the shape of the output distribution. I therefore stated that when dealing with the input distribution you probably should not worry too much about the moments other than the mean or the standard deviation for input variables.

 This is a fairly bold statement, and it probably requires more support. So why do I believe that you often don’t need to worry too much about the distribution type? This conviction is a result of what I experienced during my PhD. During my PhD I had to model a number of manufacturing operations. This was to ascertain heuristics that could be used to allocate a distribution to variable once the associated method of manufacture was known. I frequently found that it was the phenomena underlying the manufacturing method (and thus the model of that operation) that had the greatest affect upon the calculated distribution. This was because the central limit theorem would often be present because multiplicative and additive operations are so common. In fact, it is almost impossible to develop a model of any real system that does not include a number of additive or multiplicative operations.

This is certainly not always the case. I found that in cases where there a small number of random variables there was a reduced opportunity for the central limit theorem to express itself, and the type of distribution became very important. However, the majority of cases considered in the real world have a large number of random variables. Therefore, the central limit theorem is more common, and the distribution type is less important in the majority of real life cases. Still, be aware of cases where this might not be the case, especially when there are only a small number of random variables.

If you wish to verify this contention of mine, consider the models that you often deal with. You will most likely find that there are a number of additive and multiplicative functions within any of these models. You can also try changing the distribution types to check that they do not matter that much. Of course, keep the mean and the standard deviation the same.