Skip to content
New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

uncertainties and variances typos #43

Open
wants to merge 3 commits into
base: master
Choose a base branch
from
Open

uncertainties and variances typos #43

Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
17c7de7
uncertainties and variances typos
m-wells Oct 9, 2020
c88ca42
fixed typos
m-wells Oct 9, 2020
2785293
Update straightline.tex
m-wells Oct 10, 2020
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
12 changes: 6 additions & 6 deletions straightline/straightline.tex
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -367,7 +367,7 @@ \section{Standard practice}\label{sec:standard}
optimization.} This, of course, is only one possible meaning of the
phrase ``best fit''; that issue is addressed more below.

When the uncertainties are Gaussian and their variances $\sigma_{yi}$
When the uncertainties are Gaussian and their variances $\sigma_{yi}^2$
are correctly estimated, the matrix
$\inverse{\left[\mAT\,\mCinv\,\mA\right]}$ that appears in
\equationname~(\ref{eq:lsf}) is just the covariance matrix (Gaussian
Expand Down Expand Up @@ -1767,7 +1767,7 @@ \section{Goodness of fit and unknown uncertainties}\label{sec:goodness}
\begin{problem}\label{prob:chi2}
Re-do the fit of \problemname~\ref{prob:easy} but setting all
$\sigma_{yi}^2=S$, that is, ignoring the uncertainties and replacing
them all with the same value $S$. What uncertainty variance $S$ would
them all with the same value $\sqrt(S)$. What uncertainty variance $S$ would
make $\chi^2 = N-2$? Relevant plots are shown in
\figurename~\ref{fig:chi2}. How does it compare to the mean and
median of the uncertainty variances $\allsigmay$?
Expand Down Expand Up @@ -1814,7 +1814,7 @@ \section{Arbitrary two-dimensional uncertainties}\label{sec:twod}
uncertainties in \emph{both} directions (in both $x$ and $y$). You
might not know the amplitudes of these uncertainties, but it is
unlikely that the $x$ values are known to sufficiently high accuracy
that any of the straight-line fitting methods given so far is valid.
that any of the straight-line fitting methods given so far are valid.
Recall that everything so far has assumed that the $x$-direction
uncertainties were negligible. This might be true, for example, when
the $x_i$ are the times at which a set of stellar measurements are
Expand All @@ -1823,12 +1823,12 @@ \section{Arbitrary two-dimensional uncertainties}\label{sec:twod}
ascensions of the star.

In general, when one makes a two-dimensional measurement $(x_i,y_i)$,
that measurement comes with uncertainties $(\sigma_{xi}^2,\sigma_{yi}^2)$
in both directions, and some covariance $\sigma_{xyi}$ between them.
that measurement comes with uncertainties $(\sigma_{xi},\sigma_{yi})$
in both directions, and some covariance $S_{x_iy_i}$ between them.
These can be put together into a covariance tensor $\mS_i$
\begin{equation}
\mS_i \equiv \left[\begin{array}{cc}
\sigma_{xi}^2 & \sigma_{xyi} \\ \sigma_{xyi} & \sigma_{yi}^2
\sigma_{xi}^2 & S_{x_iy_i} \\ S_{x_iy_i} & \sigma_{yi}^2
\end{array}\right] \quad .
\end{equation}
If the uncertainties are Gaussian, or if all that is known about the
Expand Down