updated description of how we symmetrize covariance matrices when we generate synthetic data

Author: jeremymanning

paper/diff.pdf

@@ -1,7 +1,7 @@
 \documentclass[english]{article}
 %DIF LATEXDIFF DIFFERENCE FILE
 %DIF DEL old.tex    Wed Jan 13 12:48:03 2021
-%DIF ADD main.tex   Thu Jun 10 12:09:39 2021
+%DIF ADD main.tex   Thu Jun 10 12:23:36 2021
 \usepackage{graphicx}
 \usepackage{amsmath}
 \usepackage{hyperref}
@@ -1345,18 +1345,20 @@ \subsubsection*{Synthetic data: simulating dynamic high-order
   }\end{align*}%DIFAUXCMD
 %DIFDELCMD <   %%%
 \DIFdel{We can then use repeated applications of
-  Equations~\ref{eqn:highorder-gen1} and~\ref{eqn:highorder-gen2} in
-  order to obtain a synthetic dataset.
-}%DIFDELCMD < 
+  Equations~\ref{eqn:highorder-gen1} and ~\ref{eqn:highorder-gen2} in
+  order to
+obtain a synthetic dataset.  }%DIFDELCMD < 
 
 %DIFDELCMD <  %%%
-\DIFdel{When the template first-order correlations are constructed to
-exhibit different
+\DIFdel{When the template first-order correlations are constructed to exhibit different
  temporal profiles (e.  g., using the constant, random, ramping, and
  event procedures described above)  , }\DIFdelend \DIFaddbegin \DIFadd{(for $n > 1$) by taking a draw from
 $\mathcal{N}\left(0, m_n\right)$ and reshaping the resulting vector to
-have square dimensions.  Intuitively, }\DIFaddend the \DIFdelbegin \DIFdel{resulting }\DIFdelend \DIFaddbegin \DIFadd{re-shaped matrix will look
-like a noisy version of the template matrix, $m_{n-1}$.  (When
+have square dimensions.  To force the resulting matrix to be
+symmetric, we remove its lower triangle, and replace the lower
+triangle with (a reflected version
+of) its upper triangle.  Intuitively, }\DIFaddend the resulting \DIFaddbegin \DIFadd{re-shaped matrix will look
+like a noisy (but symmetric) version of the template matrix, $m_{n-1}$.  (When
 $n = 1$, no re-shaping is needed; the resulting $K$-dimensional vector
 may be used as the observation at the given timepoint.)  After
 independently drawing each timepoint's order $n-1$ correlation matrix

paper/diff.tex

@@ -1218,8 +1218,11 @@ \subsubsection*{Synthetic data: simulating dynamic high-order
 at a given timepoint is $m_n$, then we can generate an order $n-1$
 correlation matrix (for $n > 1$) by taking a draw from
 $\mathcal{N}\left(0, m_n\right)$ and reshaping the resulting vector to
-have square dimensions.  Intuitively, the re-shaped matrix will look
-like a noisy version of the template matrix, $m_{n-1}$.  (When
+have square dimensions.  To force the resulting matrix to be
+symmetric, we remove its lower triangle, and replace the lower
+triangle with (a reflected version
+of) its upper triangle.  Intuitively, the resulting re-shaped matrix will look
+like a noisy (but symmetric) version of the template matrix, $m_{n-1}$.  (When
 $n = 1$, no re-shaping is needed; the resulting $K$-dimensional vector
 may be used as the observation at the given timepoint.)  After
 independently drawing each timepoint's order $n-1$ correlation matrix