# What are correlated errors and why are they important?

I am looking for help on correlated systematic errors, and their meaning. I have some quantities $$x,y,z$$ which determine a function I need to calculate. These 3 quantities are determined by a measurement, and the uncertainties on the measurement translate into an uncertainty on the function they are used to calculate.

I think that somehow, these 3 observables can be varied individually to determine some type of error correlation, or covariance matrix. But I'm not sure I understand the use or meaning of error correlation. Let me be more concrete...

I am calculating a certain function, say $$N = \int f(x,y,z) ~dx ~dy~ dz$$ using a Monte Carlo, for a certain experiment; so I can sample randomly over $$x , y , z$$ and calculate $$N$$.

Now the measured $$x,y,z$$ have some experimental uncertainties, and I wish to estimate the impact of these uncertainties on $$N$$; i.e. $$N \rightarrow N \pm \delta N$$. We can take the systematic errors on $$x,y,z$$ to be gaussian with widths $$\delta x, \delta y, \delta z$$.

So I can take my Monte Carlo data set $$\{x_i,y_i,z_i\}$$ and smear each entry randomly within their widths to produce a new Monte Carlo Data set $$\{x_j,y_j,z_j\}$$, and then calculate $$N_j$$. I can repeat this process to produce a set of $$\{N\} = \{N_1,N_2...\}$$. I can then take $$N \pm \delta N = {\rm mean}(\{N\}) \pm \rm{Std. Dev}(\{N\})$$.

Now I think this would be a result with fully decorrelated errors. My question: what does error correlation mean, how does it apply in general (or for my problem), and what is the purpose? Am I over/under-estimating the error? If so, how do I properly check if there is correlation and deal with it properly?

If anyone has any good sources that would be greatly appreciated. I can find some useful info (https://arxiv.org/pdf/1507.08210.pdf), but I'm still quite confused. This question (How do I propagate correlated errors numerically?) seems to be related. Thank you in advance

Edit: There may be some confusion on terminology as pointed out in the comments by whuber. The types of measurement errors which I called "systematic errors" may be better called "random errors".

• Are you perhaps using the term "systematic errors" to mean random errors? Systematic errors, by their very definition, would not be modeled with a probability distribution as you propose.
– whuber
Mar 10 at 20:42
• Hi whuber, thanks for your comment. No I don't think I mean random (aka statistical?) errors. In my field I popularly see these errors to be Sqrt{N}, which I think comes from a poissonian distribution (N is counts of a pixel, basically). The systematic errors I am talking about are something like the following: I measure something to be a foot long, but my eyes aren't too good, so I say it's a foot $\pm$ 1 inch. Mar 10 at 21:12
• That's a measurement error. It is not considered "systematic." It is one component of the random errors.
– whuber
Mar 10 at 22:17
• Okay fair enough, I see your point. In particle physics we call these systematic errors to distinguish them from statistical errors which occur even for a perfect experiment. Nevertheless, there can be correlations between these measurement errors, that I am trying to understand Mar 11 at 1:35
• Okay. Thank you for acknowledging a confusion on terminology. I'll add an edit / disclaimer to my question to avoid confusion Mar 13 at 18:50

You can have correlation in two ways in settings like this:

• the error in $$x_i$$ is correlated with the error in $$y_i$$
• the error in $$x_i$$ is correlated with the error in $$x_{i+1}$$

In both situations, the effect of correlation is on how the positive and negative errors cancel (or reinforce) each other. If the errors in $$x_i$$ and $$y_i$$ are positively correlated, the error in $$x_i+y_i$$ is larger and the error in $$x_i-y_i$$ is smaller than if they were independent. Similarly, if the errors in $$x_i$$ and $$x_{i+1}$$ are positively correlated, the error in the sum (or average) will be higher than if they are independent.

In the real world there are reasons why correlated measurement errors are plausible

• $$x$$, $$y$$, and $$z$$ are all measured on the same physical sample, which might not be perfectly representative (air pollution, soil sampling)
• $$x_i$$ and $$x_{i+1}$$ are measured in the same location at different times, and that location is high/low compared to the average
• measured in the same lab (lab drift/batch effects)
• negative correlation because $$x$$, $$y$$, and $$z$$ add up to a fixed total (% calories from different sources)
• measurements derived from the same imperfectly accurate theoretical model
• etc, etc

In your case, then, you have the questions:

• are your measurements positively or negatively correlated?
• is your function $$N$$ more like an average or more like a difference? (low-pass or high-pass in engineering terms)

These questions aren't how you calculate -- you do that by simulating appropriately correlated errors -- but they are useful for thinking about what you should expect.

• Thanks for your answer Thomas. Let me ask a follow up. For me the situation is the case that the error on $x$ might be correlated with $y$. But how can I understand if they are correlated? Currently I model this as independent fluctuations around the minima for each variable $x,y,z$ - in other words, the smeared value of $x$ does not depend on what $y$ is. But they could both affect $N$ non-trivially. Mar 10 at 20:25