Dimensional analysis
is a well-established technique that some are familiar with. For those who are
familiar with it, dimensional analysis clearly presents an avenue for partially
(sometimes almost completely) solving for the relationship between inputs and
outputs of technical systems. The obvious application of dimensional analysis
to probabilistic methods would be to the development of a model that can then
be used for probabilistic analysis. However, it can also be used to augment DOE
for robustification and to determine how much information is required to
properly define random variability in input variables.
Because dimensional
analysis finds the collections of products and quotients that would be present
in an analytical model if it were possible to find, dimensional analysis
effectively solves for part of the model. All that is needed is the
relationship between the groups. This can be found through experimental methods
or thought experiments. Such a model allows us to now apply probabilistic methods
to optimize against negative random variability effects. You can read more
about dimensional analysis in this book and the combination with DOE in this
paper.
DOE for
robustification requires two things. First, you need to set an array of
variables (or parameters) that will be adjusted in turn so that the
contribution of each variable to the output can be determined. Second, an
‘outer array’ must be developed so that the affects of random variability can
be emulated, and the effects that parameter values can have on output random
variability. This typically requires a considerable experimental effort and the
effort increases geometrically with the number of parameters. Therefore, if we
can effectively reduce the number of parameters we can save considerably on
time and effort. How can we use dimensional analysis to do this? Dimensional
analysis identifies groupings of variables that will be present in the actual
relationship between the inputs and outputs. Because these groupings are
products and quotients, changing one variable is exactly the same as making an
ascertainable change to any other variable within that grouping. Therefore, we
no longer need to adjust each variable while performing the DOE, because we can
use the relationships within the grouping to determine the effect of changing
the other variables. What’s more, it does not matter which variable within the
group is changed. Therefore, we can choose to change the variables that are
easiest to adjust. This typically reduces the required experimental effort
considerably.
To properly predict
the random variability that the output will exhibit we need to have an adequate
description of the random variability of the inputs. This can lead to even more
uncertainty and effort. What can help here is a determination of what
information is actually required so that we don’t waste our effort finding
unnecessary information. This determination can be made by taking advantage of
the central limit theorem. Because the groups found when applying dimensional
analysis are products and quotients, the outcome is typically well approximated
by a Lognormal distribution. All that is needed to define this distribution is
the mean and standard deviation. Because multiplication is a linear process,
all we need is the mean and standard deviation of each variable. However, if
any of those variables are raised to a power greater than 1 then higher order
moments will be needed. The order required is related to the value of the
exponent. Therefore, we only need those identified moments to sufficiently
define the random variability. This is a bit different for random variables
that are in more than one group. These typically require a full distribution.
Thus, we can use dimensional analysis to determine what information is actually
required to properly define the random variability. I should point out that in
my experience the model has had a greater effect upon the output distribution
than the moments of a higher order than the variance. Therefore, if you can’t
get more than the mean and standard deviation you don’t need to worry about it
in most cases.
