From 17c7de727061923940321e7fb90e1182e1b8938e Mon Sep 17 00:00:00 2001 From: Mark Wells Date: Fri, 9 Oct 2020 13:54:40 -0400 Subject: [PATCH 1/3] uncertainties and variances typos --- straightline/straightline.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/straightline/straightline.tex b/straightline/straightline.tex index aa9f1f1..4a73fe1 100644 --- a/straightline/straightline.tex +++ b/straightline/straightline.tex @@ -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 @@ -1766,11 +1766,11 @@ \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 +$\sigma_{yi}=S$, that is, ignoring the uncertainties and replacing +them all with the same value $S$. What uncertainty $S$ 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 $\allsigmay$? +median of the uncertainty $\allsigmay$? \end{problem} \begin{figure}[htbp] From c88ca428559d621a9e1f6e020c09388e71161e20 Mon Sep 17 00:00:00 2001 From: Mark Wells Date: Fri, 9 Oct 2020 14:12:07 -0400 Subject: [PATCH 2/3] fixed typos --- straightline/straightline.tex | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/straightline/straightline.tex b/straightline/straightline.tex index 4a73fe1..0b28d62 100644 --- a/straightline/straightline.tex +++ b/straightline/straightline.tex @@ -1766,11 +1766,11 @@ \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 -$\sigma_{yi}=S$, that is, ignoring the uncertainties and replacing -them all with the same value $S$. What uncertainty $S$ would +$\sigma_{yi}^2=S$, that is, ignoring the uncertainties and replacing +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 $\allsigmay$? +median of the uncertainty variances $\allsigmay$? \end{problem} \begin{figure}[htbp] From 2785293664e8aff076bd10e146c00a4ac92806c2 Mon Sep 17 00:00:00 2001 From: Mark Wells Date: Sat, 10 Oct 2020 12:29:10 -0400 Subject: [PATCH 3/3] Update straightline.tex Using \sigma_{xyi} for covariance is rather confusing as \sigma is used for uncertainties while \sigma^2 represents variance. Covariance should be of the same "order" as variance. --- straightline/straightline.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/straightline/straightline.tex b/straightline/straightline.tex index 0b28d62..bdc1d2d 100644 --- a/straightline/straightline.tex +++ b/straightline/straightline.tex @@ -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 @@ -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