From 24cc338697084f6ecf294330d1748bc7b6961099 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Wed, 19 Jun 2019 14:08:03 -0400 Subject: [PATCH] Suggested Changes 061919 --- analysis.tex | 30 +++++++++++++++--------------- macros.tex | 8 +++++--- 2 files changed, 20 insertions(+), 18 deletions(-) diff --git a/analysis.tex b/analysis.tex index 392ee2b..cc7e71a 100644 --- a/analysis.tex +++ b/analysis.tex @@ -48,31 +48,31 @@ Note that four-wise independence is assumed across all four random variables of \end{equation} it can be seen that for $\wOne, \wOneP \in \pw$ and $\wTwo, \wTwoP \in \pw'$, all four random variables in \eqref{eq:polar-product} take their values from $\pw$, although we have iteration over two separate sets $\pw$. Thus, there are four possible sets of $\wVec$ variable combinations, namely: \begin{align*} -&\distPattern{1}:&\cOne\\ -&\distPattern{2}:&\cTwo \textit{*} \\ -&\distPattern{3}:&\cThree \textit{*} \\ -&\distPattern{4}:&\cFour \textit{*}\\ -&\distPattern{5}:&\cFive +&\distPattern{1}:&\forElems{\cOne}&\\ +&\distPattern{2}:&\forElems{\cTwo}& \textit{*} \\ +&\distPattern{3}:&\forElems{\cThree}& \textit{*} \\ +&\distPattern{4}:&\forElems{\cFour}& \textit{*}\\ +&\distPattern{5}:&\forElems{\cFive}& \end{align*} $$\text{ }^*\textit{(and all variants of the respective pattern)}$$ We are interested in those particular cases whose expecation does not equal zero, since an expectation of zero will not add to the summation of \eqref{eq:var-sum-w}. In expectation we have that \begin{align} -&\expect{%\sum_{\substack{\elems \\ +\forAllW{\distPattern{1}}&\rightarrow\expect{%\sum_{\substack{\elems \\ %\st \cOne}} - \polarProdEq \st \cOne} = 1 \label{eq:polar-prod-all}\\ -&\expect{%\sum_{\substack{\elems \\ + \polarProdEq} = 1 \label{eq:polar-prod-all}\\ +\forAllW{\distPattern{2}}&\rightarrow\expect{%\sum_{\substack{\elems \\ %\st \cTwo}} - \polarProdEq \st \cTwo} = 1 \label{eq:polar-prod-two-and-two}\\ -&\expect{%\sum_{\substack{\elems \\ + \polarProdEq} = 1 \label{eq:polar-prod-two-and-two}\\ +\forAllW{\distPattern{3}}&\rightarrow\expect{%\sum_{\substack{\elems \\ %\st \cThree}} - \polarProdEq \st \cThree} = 0 \nonumber \\ -&\expect{%\sum_{\substack{\elems \\ + \polarProdEq} = 0 \nonumber \\ +\forAllW{\distPattern{4}}&\rightarrow\expect{%\sum_{\substack{\elems \\ %\st \cFour}} - \polarProdEq \st \cFour} = 0 \nonumber \\ -&\expect{%\sum_{\substack{\elems \\ + \polarProdEq} = 0 \nonumber \\ +\forAllW{\distPattern{5}}&\rightarrow\expect{%\sum_{\substack{\elems \\ %\st \cFive}} - \polarProdEq \st \cFive} = 0 \nonumber + \polarProdEq} = 0 \nonumber \end{align} Only equations \eqref{eq:polar-prod-all} and \eqref{eq:polar-prod-two-and-two} influence the $\var$ computation. diff --git a/macros.tex b/macros.tex index 121014f..c5bc4c3 100644 --- a/macros.tex +++ b/macros.tex @@ -43,10 +43,12 @@ %4-way cases %%%%%%%%%%%%%%%% \newcommand{\polarProdEq}{\sketchPolarParam{\wVec_1}\cdot\sketchPolarParam{\wVec_2}\cdot\sketchPolarParam{\wVecPrime_1}\cdot\sketchPolarParam{\wVecPrime_2}} -\newcommand{\elems}{\wOne, \wOneP, \wTwo, \wTwoP \in \pw} -\newcommand{\lab}[1]{\textit{#1}} -\newcommand{\distPattern}[1]{\lab{Set}{\textit{ {#1}}}} +\newcommand{\elems}{\wOne, \wOneP, \wTwo, \wTwoP} +\newcommand{\forAllW}[1]{\forall \elems \in {#1}} +\newcommand{\lab}[1]{#1} +\newcommand{\distPattern}[1]{\lab{S_{#1}}} \newcommand{\vCase}[1]{\lab{Variant }{#1}} +\newcommand{\forElems}[1]{\{\elems \st {#1}\}} \newcommand{\cOne}{\wOne = \wOneP = \wTwo = \wTwoP} \newcommand{\cTwo}{\wOne = \wOneP \neq \wTwo = \wTwoP} \newcommand{\cThree}{\wOne = \wOneP = \wTwo \neq \wTwoP}