From 3137e4478621446dc9592688f4ec342459bc4749 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 25 Jun 2020 15:06:56 -0400
Subject: Shorted the fitting response to ref 1.

---
 ref_response.tex | 53 +++++++++++------------------------------------------
 1 file changed, 11 insertions(+), 42 deletions(-)

(limited to 'ref_response.tex')

diff --git a/ref_response.tex b/ref_response.tex
index 69764dd..78524bc 100644
--- a/ref_response.tex
+++ b/ref_response.tex
@@ -86,49 +86,18 @@ As suggested, an inset with a zoom on the critical region has been added to the
 }\\
 
 In order to check the agreement in our fit, we performed the fit with a moving
-temperature window that cuts off at $T_\text{max}$. Our fits' parameters are
-$x_i$ for $i = 1, \ldots, 5$ for $C_0 = x_1 - x_2 (T / \mathrm K)$, $x_3 =
-b^2/a$, $x_4 = b^2/Dq_*^4$, and $x_5 = b \sqrt{-g/u}$.  The variation of these
-parameters as a function of $T_\text{max}$ are shown on the top of
-Figure \ref{fig:parameter_cutoff}. The parameter $x_1$ is fairly stable at all
-temperature cutoffs, while the rest vary by 1.5--$2.5\times$ their $275\,\mathrm K$ value
-down to cutoffs of $\sim90\,\mathrm K$. The fit functions that result from varying the cutoff are shown in Figure \ref{fig:parameter_curve}.
+temperature window that cuts off at $T_\text{max}$. The fit functions that
+result from varying the cutoff are shown in Figure \ref{fig:parameter_curve}.
+The result is fairly robust until $T_\text{max}$ is reduced to around
+$90\,\mathrm K$.  As with any nonlinear least-squares fit, some linear
+combinations of parameters are stiff and well-constrained while others are
+sloppy and very poorly constrained. The parameters that vary most with the
+temperature window are those that belong to the sloppiest linear combinations.
+Notably, the parameters that make up the bare modulus $C_0$---and therefore
+those responsible for transforming the experimental data in Figure 2(d)---are
+robust as a function of the temperature cutoff and therefore so is the modified
+Curie--Weiss behavior observed in that figure.\\
 
-\begin{figure}
-  \centering
-  \includegraphics[width=0.7\textwidth]{referee_response_cutoff_parameters.pdf}
-  \includegraphics[width=0.7\textwidth]{referee_response_cutoff_eigenvectors.pdf}
-  \caption{
-    Fit parameters as a function of the cutoff temperature $T_\text{max}$.
-    (Top) Bare fit parameters corresponding to ratios of Landau coefficients.
-    (Bottom) Linear combinations of bare fit parameters corresponding to
-    eigendirections of the covariance matrix at $T_\text{max}=275\,\mathrm K$.
-  }
-  \label{fig:parameter_cutoff}
-\end{figure}
-
-More insight into the consistency of the fit comes from examining the linear
-combinations of parameters that form eigenvectors of the fit covariance matrix,
-since---unlike the natural parameters of the mean field theory---these have
-uncorrelated uncertainties. For the fit including all temperatures (up to
-$275\,\mathrm K$), these are (in order of fit uncertainty):
-\begin{align*}
-  y_1 &= -0.0020 x_1 + 2.2 \times 10^{-6} x_2 - 1.0 x_3 - 0.0023 x_4 - 0.0056 x_5 \\
-  y_2 &= -0.015 x_1 + 0.000042 x_2 - 0.0055 x_3 - 0.021 x_4 + 1.0 x_5 \\
-  y_3 &= -0.64 x_1 + 0.0020 x_2 + 0.0032 x_3 - 0.77 x_4 - 0.025 x_5 \\
-  y_4 &= -0.77 x_1 + 0.0066 x_2 + 0.000075 x_3 + 0.64 x_4 + 0.0014 x_5 \\
-  y_5 &= 0.0064 x_1 + 1.0 x_2 - 4.3 \times 10^{-6} x_3 - 0.0027 x_4 - 4.9 \times 10^{-7} x_5
-\end{align*}
-The variation of these parameter combinations as a function of $T_\text{max}$
-are shown on the bottom of Figure \ref{fig:parameter_cutoff}. The parameter
-$y_1$, which is principally $x_3 = a/b^2$, varies the most with the cutoff, at
-most around $2\times$ its value until $\sim90\,\mathrm K$. The parameter $y_2$,
-which is principally $x_5 = b \sqrt{-g/u}$, varies at most around $1.25\times$
-its value until $\sim90\,\mathrm K$. The other three parameters are stable at
-any cutoff, and are mixed combinations of $x_1$, $x_2$, and $x_4$.
-Notably, $x_1$ and $x_2$ are the only parameters involved in transforming the
-experimental data in Figure 2(d), and their stability as a function of the data
-window means that transformation is likewise stable.\\
 
 \begin{figure}
   \centering
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