<p><a href="https://astronomy.stackexchange.com/users/2531/profrob">ProfRob</a>'s comment is actually already the answer:</p>
<blockquote>
<p>Unless you can linearise the model, there is no analytic error estimate.</p>
</blockquote>
<p>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 <a href="https://en.wikipedia.org/wiki/Propagation_of_uncertainty" rel="nofollow noreferrer">Gaussian error propagation</a>, see in particular the <a href="https://en.wikipedia.org/wiki/Propagation_of_uncertainty#Non-linear_combinations" rel="nofollow noreferrer">part of the Wikipedia entry on non-linear combinations</a>. 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.</p>
<p>We start with a function <span class="math-container">$\varphi$</span> which depends on different variables <span class="math-container">$x_1, x_2, \ldots$</span>, meaning <span class="math-container">$\varphi = \varphi(x_1, x_2, \ldots)$</span>. We assume that we can linearize around a certain point <span class="math-container">$\tilde{\bf x} = (\tilde{x}_1, \tilde{x}_2, \ldots)$</span>, which mainly means writing the function at that point as a <a href="https://en.wikipedia.org/wiki/Taylor_series" rel="nofollow noreferrer">Taylor series</a>.</p>
<p>Then, we can determine the <strong>Gaussian (maximal) error</strong> in the vicinity of <span class="math-container">${\bf \tilde{x}}$</span>.</p>
<p><span class="math-container">$$\left. \Delta \varphi \right|_{\bf \tilde{x}} = \left. \frac{\partial \varphi}{\partial x_1} \right|_{(\tilde{x}_1,\, \tilde{x}_2,\, \ldots )} \!\!\!\!\cdot \Delta x_1 + \left. \frac{\partial \varphi}{\partial x_2} \right|_{(\tilde{x}_1,\, \tilde{x}_2,\, \ldots )} \!\!\!\!\cdot \Delta x_2+\cdots$$</span></p>
<p>In this formula you calculate partial derivatives <span class="math-container">$\frac{\partial \varphi}{\partial x_i}$</span> with respect to each variable <span class="math-container">$x_i$</span> and plug in the values of the variables at the central point <span class="math-container">${\bf \tilde{x}}$</span>. The <span class="math-container">$\Delta x_i$</span> is the error estimate for each individual variable <span class="math-container">$x_i$</span>.</p>
<p>How to we apply that to your problem where you have fitted an <em>implicit</em> equation? Since you excluded the Monte-Carlo approach (which is the usual method for this case), I suggest the following recipe:</p>
<ol>
<li>Formulate explicit functions <span class="math-container">$a(x,y), b(x,y), \ldots f(x,y)$</span>, e.g.
<span class="math-container">$$ a = - \frac{bxy + cy^2+ dx+ ey + f}{x^2}$$</span></li>
<li>Linearize the functions around <span class="math-container">$x_0,y_0$</span>. The <span class="math-container">$x_0$</span> is called <span class="math-container">$x$</span> in your table, and respectively <span class="math-container">$y_0$</span> is in the <span class="math-container">$y$</span>-column.</li>
<li>Determine <span class="math-container">$\Delta a, \ldots, \Delta f$</span> by <a href="https://www.wolframalpha.com/input/?i=diff+%28bxy+%2B+cy%5E2%2B+d+x%2B+ey+%2B+f%29%2Fx%5E2" rel="nofollow noreferrer">partial differentiation</a>.</li>
<li>Plugin in <span class="math-container">$\Delta x$</span> and <span class="math-container">$\Delta y$</span> as also given in your table.</li>
</ol>

CC BY-SA 4.0
2021-03-18T13:00:05.780Z
2021-03-18T14:44:02.277Z
2021-03-18T14:44:02.277Z
B--rian
<p>Recently I've encountered an article in our local astronomical popularization magazine. It is about a well known task of <a href="https://en.wikipedia.org/wiki/Sagittarius_A*" rel="nofollow noreferrer">Sgr A*</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.</p>
<p>That is why I believed there is no need for <em>very precise</em> calculations
for this task.</p>
<p>However, in addition to the <span class="math-container">$ x $</span>, <span class="math-container">$ y $</span> coordinates of the S2 star, I was also given the corresponding uncertainty values <span class="math-container">$ \Delta x $</span>, <span class="math-container">$ \Delta y $</span> (19  coordinate measurements for the time period from 1992 to 2003):</p>
<p><span class="math-container">\begin{array}{|c|c|c|c|}
\hline
time    &amp; x      &amp;   \Delta x    &amp; y      &amp; \Delta y \\ \hline
1992.226 &amp; 0.104 &amp; 0.003 &amp; -0.166 &amp; 0.004 \\
1994.321 &amp; 0.097  &amp; 0.003 &amp; -0.189 &amp; 0.004 \\
...     &amp; ..     &amp; ..    &amp; ..     &amp;       \\
2003.353 &amp; 0.077  &amp; 0.002 &amp; -0.030 &amp; 0.002 \\
2003.454 &amp; 0.081  &amp; 0.002 &amp; -0.036 &amp; 0.002\\
\hline
\end{array}</span></p>
<p>Using <a href="http://cseweb.ucsd.edu/%7Emdailey/Face-Coord/ellipse-specific-fitting.pdf" rel="nofollow noreferrer">direct least squares method</a> for S2 orbital fitting by an ellipse I've found parameters
<span class="math-container">$a,\ b,\ c,\ d,\ e,\ f$</span> of the conic equation: <span class="math-container">$$ax^2+bxy+cy^2+ dx+ey+f=0$$</span>  I've
also decided to simulate the <strong>error propagation</strong> in a Monte Carlo fashion to find
the uncertainty in the estimates of <span class="math-container">$a,\ b,\ c,\ d,\ e,\ f$</span>. And I succeeded in that.</p>
<p>But, here is <strong>the question</strong>: is there any <strong>other</strong> appropriate,
&quot;analytical&quot; way (not a Monte Carlo method) to find the uncertainty in
the estimates of <span class="math-container">$a,\ b,\ c,\ d,\ e,\ f$</span> which may be used in usual astronomers' practice?</p>

CC BY-SA 3.0
2017-12-31T17:29:49.883Z
For a good statistical analysis answer, you might consider posting this in [crossvalidated stackexchange](https://stats.stackexchange.com/help/on-topic) 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`](https://stats.stackexchange.com/questions/tagged/error-propagation)!
uhoh
CC BY-SA 3.0
2017-12-31T17:52:31.160Z
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.
AtmosphericPrisonEscape
CC BY-SA 3.0
2017-12-31T20:34:47.090Z
Unless you can linearise the model, there is no analytic error estimate.
ProfRob
CC BY-SA 4.0
2021-12-13T23:23:30.447Z
I should also add, that the orbit of S2 (famously) is not an ellipse.
ProfRob
<orbit><mathematics><sagittarius-a>
Error propagation methods for orbit parameters
Vladimir Belyaev
# Error propagation methods for orbit parameters

(Asked by: username_0 on 2017-12-31)

Recently I've encountered an article in our local astronomical popularization magazine. It is about a well known task of [Sgr A\*](https://en.wikipedia.org/wiki/Sagittarius_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](http://cseweb.ucsd.edu/%7Emdailey/Face-Coord/ellipse-specific-fitting.pdf) 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?

**Tags:** orbit, mathematics, sagittarius-a

> **Comments:**
> **username_1**: For a good statistical analysis answer, you might consider posting this in [crossvalidated stackexchange](https://stats.stackexchange.com/help/on-topic) 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`](https://stats.stackexchange.com/questions/tagged/error-propagation)!  
> **username_2**: 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.  
> **username_3**: Unless you can linearise the model, there is no analytic error estimate.  
> **username_3**: I should also add, that the orbit of S2 (famously) is not an ellipse.  

---

## 1 answers

### Answer by username_4 on 2021-03-18. Score: 1

[username_3](https://astronomy.stackexchange.com/users/2531/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](https://en.wikipedia.org/wiki/Propagation_of_uncertainty), see in particular the [part of the Wikipedia entry on non-linear combinations](https://en.wikipedia.org/wiki/Propagation_of_uncertainty#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](https://en.wikipedia.org/wiki/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](https://www.wolframalpha.com/input/?i=diff+%28bxy+%2B+cy%5E2%2B+d+x%2B+ey+%2B+f%29%2Fx%5E2).
4. Plugin in $\Delta x$ and $\Delta y$ as also given in your table.


---

