Estimating the covariance

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In summary, the Poisson ratio is a measurement of the correlation between two variables and the covariance is a measure of the variability of two variables. The covariance is important when the value of one variable affects the value of another.
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JulienB
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Hi everybody! I have to write a protocole for our last experiment about elasticity and torsion (in physics), and as an extra question I am asked to calculate the Poisson ratio and to calculate the correlated error by estimating the covariance. Unfortunately I have never done that before, and I don't really understand what the covariance is. Here is the formula for the Poisson ratio:

##\mu = \frac{E}{2G} - 1##

I imagine that ##Cov(X,Y)## refers to ##Cov(E,G)## in our case. The problem is that we have only one value for E and for G (determined experimentally), which is probably why we were asked to estimate the covariance. I've seen that ##0 < Cov(X,Y) < 1## when the two values grow together, and that's the result we got:

##G = \frac{E}{2(1 + \mu)} = kE = 0.3708 E## (that's the value we got for the Poisson ratio)

Since ##\mu## is supposed to be a constant, a graph of ##G## in function of ##E## should be linear. Can I then estimate ##Cov(E,G) \approx 1##? Or did I completely misunderstand it?Thanks a lot in advance for your answers.Julien.
 
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I'm not sure I understand the use of covariance in the context of your experiment which is not of a statistical nature since you have only one measurement.. Since G =kE you only have one independent variable E which determines G and for your case G changes by an amount proportional to E, There exists a statistical quantity called the linear correlation coefficient which is one for you case. But again you have only one measurement so it is indeterminate.

The covariance is not constrained to have a value of less than one.

Let me explain a bit.

The uncertainty in the value of a function F is related to the statistical uncertainties in the values of the variables xi by the equation

σF2 = ∑i σI2(∂F/∂xi)2 +2⋅ Σij σij2(∂F/∂xi)(∂F/∂xj)

where σi2 is the variance of xi i.e. the square of the statistical uncertainty in xi

where σij2 is the covariance of xj and xj where σij2 = < (xi- < xi>)⋅( xj- <xj>) >

The brackets < quantity > refer to the average value of quantity,

The covariance is important when the value of one variable affects the value of another. If they are totally independent then in the limit of large number of measurements the covariance approaches zero.

Obviously the the variances and the covariance can only be determined if you have two or more values of the variables.

For you case as a guess since the covariance is the average of the product of the uncertainties of the variables E and G the covariance might be taken as ΔE⋅ΔG your estimated experimental uncertainties in E and G. and the error associated with this is 2∂μ/∂E⋅∂μ/∂GΔEΔG.

Perhaps someone can critique this approach.

If you are continuing in the experimental physical sciences I highly recommend that you find a copy of "Data Reduction and Error Analysis for the Physical Sciences" by Bevington ( original version) or the revised edition by Bevington and Robinson.
 
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Hey JulienB.

You can pick up a good multi-variate statistics book for the actual estimator but if you want to derive it you are going to have use something like MLE.

You can also just use the sample estimates for the different components (equating sample mean to actual mean) and doing a degree of freedom correction.
 
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Thanks a lot to both of you for your answers. At the end I used the inequality ## \frac{1}{3} E < G \leq \frac{1}{2} E ## to state that ##0 \leq Cov(E,G) \leq 1##. I calculated the propagation of error for both extremes and found that the correlation had a negligible impact on the uncertainty. I think that's what was asked, as our advisor specifically said that we didn't have to calculate the covariance.

thank you for your reading recommendations too.Julien.
 

What is estimating the covariance?

Estimating the covariance is a statistical technique used to measure the relationship between two or more variables. It calculates the degree to which two variables tend to vary together, and is commonly used in data analysis and modeling.

Why is estimating the covariance important?

Estimating the covariance allows us to understand the degree of correlation between variables. This information can be used to make predictions, identify patterns, and make decisions in fields such as finance, economics, and social sciences.

How is the covariance calculated?

The covariance is calculated using a specific formula that takes into account the values of each variable and how they vary together. The formula is: cov(X, Y) = Σ (xi - x̄)(yi - ȳ) / (n - 1), where xi and yi are the individual values of variables X and Y, x̄ and ȳ are the mean values of X and Y, and n is the number of data points.

What is the difference between covariance and correlation?

Covariance and correlation are both measures of the relationship between variables, but they differ in their scales. Covariance can take on any value, indicating the strength and direction of the relationship, while correlation is a standardized measure that ranges from -1 to 1, with 0 indicating no relationship.

What are some limitations of estimating the covariance?

One limitation of estimating the covariance is that it can be affected by the scale of the variables. It is also sensitive to outliers in the data, which can skew the results. Additionally, the covariance does not indicate causation, only a relationship between variables.

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