# PCE - Why is coefficient of variance and standard deviation so high?

I am using the PCE methodology. There are three input variables. Two inputs are defined as having a gaussian distribution and the third variable is defined as having a uniform distribution. The coefficient of variance and standard deviation are unreasonably high, as in 700%… why?

Dear @Pilgrim

Can you please provide more information about the problem you are solving and in particular the distribution parameters you are using?

Best regards
Styfen

Dear Styfen,
Thank you for the response I appreciate it. I am using the PCE method to approximate the ignition delay time (tau). The equation is as follows:
tau = Aphiexp(E/(R*T))
A is defined as having a Gaussian distribution [0.00112 0.0017]
phi = constant
E is defined as having a Gaussian distribution [28.0 3.96]
R = constant
T is defined as having a uniform distribution [1243 1276]
Note: the temperature was measured as 1243 K with an uncertainty of 2.66%, hence the defined bounds for the uniform distribution.

Background: ignition delay time was measured at 17 different temperatures, the ignition delay time measurements were plotted against temperature, a curve fit was applied to the data in order to calculate A and E.

Thank you for the advising.
Pilgrim

Correction:
bounds for uniform distribution of T are [1210 1276]

Dear Pilgrim,
just a quick comment on my side: according to your post, your model is

with \phi constant, and \exp(...) being a positive quantity. However, according to your post you are using a Gaussian A with mean 1.12\cdot 10^{-3} and standard deviation 1.7\cdot 10^{-3}, hence a large coefficient of variation there of approx 1.5.
Now, the Gaussian distribution is symmetric around the mean, which means that more than 1/4 of the values of A will be negative. This means that regardless on the distribution of the exponential part of your equation, \tau will have a lot of negative values, and therefore a relatively low mean. At the same time, it will also have a very large variance due to the change of sign. This may easily explain your observed large coefficient of variation.
From a quick numerical investigation (I don’t have either \phi nor R), it looks like R affects strongly the variance of the model. E.g. with \phi = 1, R = 30, I get a CoV of 700%, while with \phi=1, R=50 the CoV goes down to 1.5.

Finally, please note that about 25% of your times \tau will be negative, regardless on the distribution of your remaining parameters.

All of this discussion has nothing to do with PCE, but with the problem formulation itself.

I wish you a good day
Stefano