Recently I've encountered an article in our local astronomical popularization magazine. It is about a well known task of Sgr A*-black hole's mass estimation. The article is written as a step by step guide for young researchers. For example, the orbital parameters of the S2 star should be found using a drawing on a sheet of paper.

That is why I believed there is no need for very precise calculations for this task.

However, in addition to the $ x $, $ y $ coordinates of the S2 star, I was also given the corresponding uncertainty values $ \Delta x $, $ \Delta y $ (19 coordinate measurements for the time period from 1992 to 2003):

\begin{array}{|c|c|c|c|} \hline time & x & \Delta x & y & \Delta y \\ \hline 1992.226 & 0.104 & 0.003 & -0.166 & 0.004 \\ 1994.321 & 0.097 & 0.003 & -0.189 & 0.004 \\ ... & .. & .. & .. & \\ 2003.353 & 0.077 & 0.002 & -0.030 & 0.002 \\ 2003.454 & 0.081 & 0.002 & -0.036 & 0.002\\ \hline \end{array}

Using direct least squares method for S2 orbital fitting by an ellipse I've found parameters $a,\ b,\ c,\ d,\ e,\ f$ of the conic equation: $$ax^2+bxy+cy^2+ dx+ey+f=0$$ I've also decided to simulate the error propagation in a Monte Carlo fashion to find the uncertainty in the estimates of $a,\ b,\ c,\ d,\ e,\ f$. And I succeeded in that.

But, here is the question: is there any other appropriate, "analytical" way (not a Monte Carlo method) to find the uncertainty in the estimates of $a,\ b,\ c,\ d,\ e,\ f$ which may be used in usual astronomers' practice?

  • $\begingroup$ For a good statistical analysis answer, you might consider posting this in crossvalidated stackexchange instead. It's certainly possible you'll get a good answer here, but you might get it a lot faster there. They even have a tag specifically for error-propagation! $\endgroup$
    – uhoh
    Dec 31 '17 at 17:29
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    $\begingroup$ Analytical error propagation is usually possible when you know the underlying distribution of the measurments. As in all generality for all astrophysical sources this distribution is unknown and will also never be the same it is common practice to 'smash' this problem with the 'MC' Hammer. $\endgroup$ Dec 31 '17 at 17:52
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    $\begingroup$ Unless you can linearise the model, there is no analytic error estimate. $\endgroup$
    – ProfRob
    Dec 31 '17 at 20:34

ProfRob's comment is actually already the answer:

Unless you can linearise the model, there is no analytic error estimate.

I consider error estimation very important, so I would like to show a bit more details here: A way of estimating the error of a model is by Gaussian error propagation, see in particular the part of the Wikipedia entry on non-linear combinations. Let me briefly summarize the idea behind as I find the Wikipedia article not so easy to follow. I will also use different variable names so that they do not overlap with those in your question.

We start with a function $\varphi$ which depends on different variables $x_1, x_2, \ldots$, meaning $\varphi = \varphi(x_1, x_2, \ldots)$. We assume that we can linearize around a certain point $\tilde{\bf x} = (\tilde{x}_1, \tilde{x}_2, \ldots)$, which mainly means writing the function at that point as a Taylor series.

Then, we can determine the Gaussian (maximal) error in the vicinity of ${\bf \tilde{x}}$.

$$\left. \Delta \varphi \right|_{\bf \tilde{x}} = \left. \frac{\partial \varphi}{\partial x_1} \right|_{(\tilde{x}_1,\, \tilde{x}_2,\, \ldots )} \!\!\!\!\cdot \Delta x_1 + \left. \frac{\partial \varphi}{\partial x_2} \right|_{(\tilde{x}_1,\, \tilde{x}_2,\, \ldots )} \!\!\!\!\cdot \Delta x_2+\cdots$$

In this formula you calculate partial derivatives $\frac{\partial \varphi}{\partial x_i}$ with respect to each variable $x_i$ and plug in the values of the variables at the central point ${\bf \tilde{x}}$. The $\Delta x_i$ is the error estimate for each individual variable $x_i$.

How to we apply that to your problem where you have fitted an implicit equation? Since you excluded the Monte-Carlo approach (which is the usual method for this case), I suggest the following recipe:

  1. Formulate explicit functions $a(x,y), b(x,y), \ldots f(x,y)$, e.g. $$ a = - \frac{bxy + cy^2+ dx+ ey + f}{x^2}$$
  2. Linearize the functions around $x_0,y_0$. The $x_0$ is called $x$ in your table, and respectively $y_0$ is in the $y$-column.
  3. Determine $\Delta a, \ldots, \Delta f$ by partial differentiation.
  4. Plugin in $\Delta x$ and $\Delta y$ as also given in your table.

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