# Interpretation of the elements of the error matrix as inverse of hessian matrix [duplicate]

In a report I am reading at work, the error matrix is calculated as the inverse of the hessian matrix and used to calculate the error ellipse angle and axes with a not theoretically correct formula.

They start from a function of 2 variables: $$f=f(x, y)$$ (I won't use the real function, for the purpose of the question a generic one should be enough) and calculate the hessian matrix:

$$H = \begin{pmatrix} \partial^2_x f & \partial_x \partial_y f \\\\ \partial_y \partial_x f & \partial^2_y f \end{pmatrix}$$

and the error matrix as its inverse:

$$E = H^{-1} = \begin{pmatrix} E_{xx} & E_{xy} \\\\ E_{yx} & E_{yy} \\ \end{pmatrix}$$

Written in this way it seems to me that the error associated to the variable $$x$$ is the one located in position $$E[0, 0]$$, but if I calculate explicitly the inverse of the hessian:

$$H^{-1} = \frac{1}{\mid H\mid} \begin{pmatrix} \partial^2_y f & -\partial_x\partial_y f \\\\ -\partial_y\partial_x f & \partial^2_x f \\ \end{pmatrix}$$

the term in position $$[0, 0]$$ is something related to the $$y$$ variable and not $$x$$.

And finally the error matrix is used to calculate the orientation of the error ellipse, $$\vartheta$$, and the size of the major ($$e_1$$) and minor ($$e_2$$) axes with the not theoretically correct formula:

$$\tan{\vartheta} = \frac{E_{xy}}{E_{xx}}$$

$$e^2_1 = \frac{E^3_{yy} + E^2_{xy} (2E_{yy} + E_{xx}) }{E^2_{xy} + E^2_{yy} }$$

$$e^2_2 = \frac{E_{yy}( E_{xx} E_{yy} - E^2_{xy}) }{E^2_{xy} + E^2_{yy} }$$

Now I am confused on what the elements of the error matrix are.

Of course call something $$x$$ or $$y$$ is just a convention, but when the error matrix is calculated as inverse of the hessian matrix, its first line is associate to which variable? $$x$$ or $$y$$? I.e. is $$E_{xx}$$ the variance related to the $$x$$ even if contain term related to the $$y$$ variable, or it is the $$y$$ variance?

Did anyone see these equations before and know where I can find any info about them? I couldn't find them anywhere, I always used the eigenvalues and eigenvectors of the matrix never this approximated formula.

I know these are questions to the people who wrote this report, but they are gone leaving just this really confusing document and I don't have anyone else to ask.

• Re "the term in position [0,0] is something related to the y variable and not x." Not so. That term usually does depend on $x;$ moreover, you have neglected the division by $|H|,$ which explicitly depends on $x.$ Re "ever see these equations before": this is called the Fisher Information matrix.
– whuber
Sep 1, 2022 at 13:02
• @whuber thanks for you comment, I will check the Fisher Information matrix better. Yes you are right, of course the $|H|$ depends on $x$. When I asked if anyone saw these equations I meant the equations for the ellipse angle, the major and the minor axes, I always calculated them using the eigenvalues and eigenvectors of the error matrix. The ones in the report seems to me some sort of strange approximation that I don't recognise. Sep 1, 2022 at 16:06
• Thank you for the clarification. You can obtain those angles in several ways. Two geometric solutions are indicated in my discussion of ellipses at stats.stackexchange.com/a/71303/919, although i do not stop to write them down (that wasn't the focus of the thread). The formulas you quote look like long-winded renditions of the matrix formulas.
– whuber
Sep 1, 2022 at 17:41
• @whuber thanks a lot for your help, that's a really interesting. Sep 2, 2022 at 7:37