# Standard deviation in principal component space

I am running a large number of simulations in which I have a 3D parameter space, and for each set of parameters (point in the 3D space) I run 100 simulations. I then use 14 measures to quantify the results of the simulations, so for each point in parameter space I have the mean and standard deviation of each of these 14 measures.

I want to test which of the 14 measures are most useful for discriminating between my 3 initial parameters, so I'm using Principal Component Analysis. I can discriminate pretty well between all the mean values of my measures using 2 or 3 principal components. My question is, how do I convert the standard deviations on each of the measures at each point into principal component space so I can tell if they are well spaced enough to differentiate between the different parameters?

Also, I'm an astrophysicist, so please keep your maths clear and explanations free of jargon/unexplained symbols.

• In mainstream statistics, parameters are unknown constants you are estimating, which does not sound like your situation, so this might be as hard for a non-astrophysicist to understand as what statistical people might say in return. – Nick Cox Apr 14 '16 at 14:20
• Well actually later in my program they are unknows that I am estimating by comparing a real event to the simulations I have and seeing which simulations it resembles (using the PCA) so I can see what parameters best describe the real event. But if my terminology is confusing, I'm happy to edit it. What term would you suggest? – FJC Apr 14 '16 at 14:45
• I can't be very constructive here because I couldn't follow your story. I see already a better response. – Nick Cox Apr 14 '16 at 14:59

Consider a matrix $X$ of dimensions $r\times c$, you can create a space out of the eigenvectors of $X$ and have your $r$ points in it. What you can do to estimate "variance" for those $r$ points is use a resampling method like bootstrap. With that method you can generate $n$ matrices $X^{(i)}$ with $i \in [1, n]$. If you project those matrices onto your space (as supplementary individuals/variables) you can have for each $r$ points a cloud of $n$ points.
Something we do in sensometrics to decide if two points (from the initial $X$ matrix) are well-discriminated is this : plot for each of the $r$ points an ellispoid in which there is at least 95% (or any other value) of the $n$ points associated. If two ellispoids are overlaping then it may be a sign that they are not that different.
Either way, you generated a $n + 1$ sample for each of the $r$ points so you can estimate mean and variance and conduct comparison test such as Student or Hotelling.