How you go about finding correlations?

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Discussion Overview

The discussion revolves around the methodology for determining correlation matrices between various systematic uncertainties (Nuisance Parameters, NPs) in experimental analyses. Participants explore how to quantify correlations among systematics like the tau/jet energy scale (TES), jet energy scale (JES), and jet energy resolution (JER), focusing on the implications of these correlations for data analysis and interpretation.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant inquires about constructing a correlation matrix for systematics, suggesting a table format for displaying correlation coefficients.
  • Another participant suggests consulting experts who created the NPs for insights on correlations, noting that ideally, there should be no correlation between their effects on final results.
  • Concerns are raised about the potential for statistics to create misleading correlations, particularly when varying multiple NPs simultaneously.
  • A participant expresses confusion about applying correlation concepts to a set of events, questioning how correlations can be defined when multiple variables are involved.
  • Another participant explains that correlation can be studied by examining how variations in one NP might affect another, using examples related to detector material and its impact on measurements.
  • Discussion includes the idea that if the same components are used in different parts of a detector, correlations may arise between NPs related to those components.

Areas of Agreement / Disagreement

Participants express differing views on the nature of correlations among NPs, with some suggesting that ideally, no correlations should exist, while others explore the complexities and potential for correlations in practice. The discussion remains unresolved regarding the best approach to quantify these correlations.

Contextual Notes

Participants acknowledge limitations in their understanding of how correlations apply to multiple NPs and the statistical implications of varying them simultaneously. There is also mention of the need for clarity on the definitions and assumptions underlying correlation in this context.

ChrisVer
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Suppose you have several systematics [as happens in any analysis].
How can you go and determine the correlation matrix between each of their NPs?
so for example if you have (let's say roughly) 3 systematics: TES, JES , JER (tau/jet energy scale and jet energy resolution), how could you make a table that hase:

\begin{tabular}{c|c|c|c}
TES JES JER
TES
JES
JER
\end{tabular}

and entries the correlation coefficients [itex]\rho \in [-1, 1][/itex]?
Of course the diagonal will have +1 ...
 
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A correlation matrix between the actual NP? Ask the experts who made them.
A correlation between their effect on the final numbers? Ideally there is no such correlation (otherwise the NP are not well chosen), but you can vary more than one at the same time to study correlations. Keep in mind that statistics will can fake nonexistent correlations.
 
mfb said:
A correlation matrix between the actual NP? Ask the experts who made them.
I think that's the case... huh...

mfb said:
A correlation between their effect on the final numbers? Ideally there is no such correlation (otherwise the NP are not well chosen), but you can vary more than one at the same time to study correlations. Keep in mind that statistics will can fake nonexistent correlations.
Well that would be a way to go if I actually had the ability to produce events with more than just 1 NP altered per time.

In fact what I don't understand is that [as far as I know] correlation is between two variables which might be random distributed... I don't understand how it can be applied to the case of a whole bunch of "events". In fact they change the experiment [eg 1 TES has the ability to change the amount of the detector matterial I think].
 
The random distribution is in the value of the NP. You hope that the nominal value is correct, but it could also be a bit higher or lower.

As an example, let's say two NP correspond to the amount of material in your detector, relevant for scattering or showering. Maybe one NP for the barrel, one for the endcaps, or something like that. You can now study two things:

- if we overestimate the amount of material in the barrel, do we also overestimate it in the endcap? This could happen if the same components are used in both detector parts. It would correlate the NP itself.
- how does our measured value change if we (a) vary the barrel NP up and down within its uncertainties, (b) vary the endcap NP up/down, (c) vary both at the same time? Ideally this is linear, if it is not things can get interesting.
 

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