Calculating Sum of Three Correlated Gaussian Distributions

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Discussion Overview

The discussion revolves around calculating the sum of three correlated Gaussian distributions, focusing on how to derive the new mean and standard deviation when combining these distributions. Participants explore the implications of correlation between the data sets and seek clarification on the appropriate formulas to use.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant outlines the method for summing two Gaussian distributions, providing formulas for variance and standard deviation based on correlation.
  • Another participant suggests that summing three distributions can be approached by first adding two of them, though this lacks specificity regarding correlation.
  • A participant highlights the complexity introduced by different correlations between each data set, questioning how to determine the correlation for the combined distributions.
  • One participant later provides a formula for the variance of three distributions, incorporating all relevant correlations, indicating a potential resolution to their initial query.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the correct approach to summing three correlated distributions, with some disagreement on how to handle the correlations between the sets. The discussion does not reach a consensus on the best method.

Contextual Notes

Participants mention that the correlations between the data sets are not uniform, which complicates the calculations. There is also an acknowledgment that the correlation values provided are specific to pairs of sets, raising questions about how to apply them in the context of summing three distributions.

Who May Find This Useful

This discussion may be useful for individuals working with correlated data in statistical analysis, particularly those interested in Gaussian distributions and their properties in combined datasets.

mpoirier
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I have three sets of data that I’ve used to create three Gaussian distributions which have different means and standard deviations. The data sets are also correlated as the data is dependent on time. I want to compare the sum of two distributions with the sum of three distributions to find which sets of distribution produce the best stdev as a percentage of the mean.

Now, I think I know how to sum two of the distributions together, but how do I sum the three?

To find the new mean of two distributions, simply add the two initial means together.
To find the new stdev of the two distributions, use the following formulas.
First, find the variance.
(1) Var(X1, X2) = stdev1^2 + stdev2^2 + 2cov(X1, X2)
(2) Correl(X1, X2) = cov(X1, X2) / (stdev1 * stdev2)
And you obtain… (2) into (1)
Var(X1, X2) = stdev1^2 + stdev2^2 + 2 * Correl(X1,X2) * (stdev1 * stdev2)

Then the stdev is simple the square root of the variance.

None of the three samples are completely independent or dependent. In other words, the correlation is not equal to 0 or 1.

Any help would be greatly appreciated.
 
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Now, I think I know how to sum two of the distributions together, but how do I sum the three?
The same way you would add three numbers together: you start by adding two of them.
 
Sorry, I don't think I explained the question properly.
The problem is that there is a different correlation between each data set.
For example:
Set 1 and Set 2 have a correl of .71
Set 1 and Set 3 have a correl of .80
Set 2 and Set 3 have a correl of .70

So, if I simply solve it out twice, what do I use for the correl between the solution of Set 1/2 and Set 3? That's if I solve it the way you just suggested.
 
My bad, I didn't catch the entire problem. :frown:
 
Ok, I figured out how to answer my own question. You just plug in all the numbers into the following equation.
Var(X1, X2, X3) = stdev1^2 + stdev2^2 + stdev3^2 + 2 * Correl(X1,X2) * (stdev1 * stdev2) + 2 * Correl(X1,X3) * (stdev1 * stdev3) + 2 * Correl (X2,X3) * (stdev2 * stdev3)

Anyways, thanks Hurkyl for trying to help.
 

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