Some questions on correlations

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In summary, correlations between uncertainties can be found when two random variables are not independent. This information is useful for understanding the slope of a linear function between the two variables.
  • #1
ChrisVer
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I am always confused when reading about correlation between uncertainties.
I have some basic questions:
1. Can statistical and systematic uncertainties be correlated? [Well they can but I don't understand how this can be interpreted]
2. What are correlations telling us ? obviously a correlation is not giving us any relation [or causation].
3. One example which has confused me:
the transverse momentum of charged particles in ATLAS for example can be measured by looking at the charged particle trajectories within the magnetic field.
The magnetic field's uncertainty [itex]\sigma_B[/itex] is then common for all particles, and this makes their transverse momenta measurement correlated. How does this affect the analyses?
 
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  • #2
ChrisVer said:
I am always confused when reading about correlation between uncertainties.

It's difficult to answer questions about "uncertainties" from the point of view of a forum on "Set Theory, Logic, Probablity, Statistics" because "uncertainties" in laboratory experiments seems to be a topic that results from an interaction between mathematics and bureaucracy. For example, http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3387884/ says:

Systematic error (bias) can, at least theoretically, be eliminated from the result by an appropriate correction.
...
Random errors may be analysed statistically while systematic errors are resistant to statistical analysis. Systematic errors are generally evaluated by non-statistical procedures.
...
but also says:
The uncertainty in the reported value of the measurand comprises the uncertainty due to random errors and the uncertainty of any corrections for systematic errors.

I suppose a bureaucratic document needs feel no shame in applying the term "uncertainty" to "corrections of systematic errors" while also saying that systematic errors are "are generally evaluated by non-statistical procedures". However that kind of language renders the relation of "systematic uncertainties" to anything defined in statistics completely ambiguous.

People doing a specific experiments probably figure out how to apply bureaucratic standards in reporting their work by copying what predecessors have done or by negotiating with the people that write the standards.


I have some basic questions:
1. Can statistical and systematic uncertainties be correlated? [Well they can but I don't understand how this can be interpreted]

I know what "correlated" means from the viewpoint of "Set Theory, Logic, Probability, Statistics". It is a term that is applicable to random variables. But what do bureaucratic documents mean by "correlated"?

2. What are correlations telling us ? obviously a correlation is not giving us any relation [or causation].

The existence of a non-zero correlation between two random variables tells us they are not independent. If the random variables represent "noise" added to deterministic variables that are approximately related by a linear function then the correlation tells us something about the slope of linear function. If the deterministic variables are related by a very non-linear function, what correlation tells us is not clear.

3. One example which has confused me:
the transverse momentum of charged particles in ATLAS for example can be measured by looking at the charged particle trajectories within the magnetic field.
The magnetic field's uncertainty [itex]\sigma_B[/itex] is then common for all particles, and this makes their transverse momenta measurement correlated. How does this affect the analyses?

Some other forum member may be familiar with what "the analysis" is. Is the physics complicated ?
 
  • #3
Hey ChrisVer.

Correlation acts on a random variable (or random vector to be a bit more precise) and that vector can contain any distribution.

You can correlate things that are not just raw value data like uncertainties but the data itself will have that structure.

If you have vectors of uncertainties then you can correlate them.

Usually though - the way to assess relations between variances and uncertainties in general is via the correlation and/or covariance "matrix" in a general capacity.
 

1. What is a correlation?

A correlation is a statistical measure that shows the relationship between two variables. It indicates the extent to which two variables are related and how they change together. A correlation can range from -1 to +1, with 0 indicating no relationship and -1 or +1 indicating a strong negative or positive relationship respectively.

2. How do you calculate correlation?

Correlation is calculated using a mathematical formula that takes into account the values of two variables and their deviations from their respective means. The most commonly used measure of correlation is the Pearson correlation coefficient, which is calculated by dividing the covariance of the two variables by the product of their standard deviations.

3. What is the difference between correlation and causation?

Correlation does not imply causation. Just because two variables are correlated, it does not necessarily mean that one causes the other. There could be other factors at play that are influencing the relationship between the two variables. To establish causation, further research and experimentation are needed.

4. Can correlation be used to make predictions?

Yes, correlation can be used to make predictions, but it does not necessarily mean that the predictions will be accurate. Correlation only shows the relationship between two variables, but it does not consider other factors that may affect the relationship. Therefore, it is important to use caution when using correlation to make predictions.

5. What are some limitations of correlation?

Correlation can be a useful tool, but it also has its limitations. For example, correlation does not indicate the direction of the relationship between two variables, and it does not account for causation. Additionally, correlation only shows the relationship between two variables and does not take into account other factors that may influence this relationship. Therefore, it is important to interpret correlation results carefully and consider other factors before drawing conclusions.

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