Undergrad Why Does Covariance Matrix Change with Different Functions?

[Silviu]

Hello! I have to calculate the covariance between 2 parameters from a fit function. I found this package in Python called iminuit that did a good fit and also calculate the covariance matrix of the parameters. I tested the package on a simple function and I am not sure I understand the result. When the function I put is x^2+y^2, which has the minimum for x=y=0, I obtain ((1.0, 0.0), (0.0, 1.0)), as a covariance matrix. When I use (x-y)^2 I obtain ((250.24975024975475, 249.75024975025426), (249.75024975025426, 250.24975024975475)), as a covariance matrix. I don't understand why do I get value of covariance greater than 1 and why in the first case I get 0 on the sides and 1 on the main diagonal? It is the first time I encounter covariance so I am not sure I got it right. Thank you!

 

[gleem]

The definition of covariance is

If two variable are independent we would expect in the limit of large n that the covariance would be 0. If they are dependent the covariance is not limited to 1.

Perhaps you are thinking of the correlation matrix call it Σ where the diagonal elements ∑_II are always 1 and the off diagonal elements are -1 ≤ Σ_ij ≤ 1

Your second function does not have an unique minimum but an infinitely long trough for values x=y.

 

[mathman]

covariance is not limited to 1. Correlation, the ratio of the covariance to the square root of the product of the variances, is limited (in magnitude) to 1.