From 170a5ff54b745cf918450c2d90253e6f8c0e0117 Mon Sep 17 00:00:00 2001 From: Aaron Huber Date: Mon, 10 Jun 2019 13:36:43 -0400 Subject: [PATCH] Up to Var[estimate] completed --- analysis.tex | 37 +++++++++++++++++++++++++++++-------- macros.tex | 16 ++++++++++++++-- 2 files changed, 43 insertions(+), 10 deletions(-) diff --git a/analysis.tex b/analysis.tex index a4ef2a0..86f3ca2 100644 --- a/analysis.tex +++ b/analysis.tex @@ -23,8 +23,8 @@ The first step is to show that the expectation of the estimate of a tuple t's me \wVecPrime~|~\sketchHashParam{\wVecPrime} = j,\\ \wVec = \wVecPrime}} \kMapParam{\wVec} \cdot \sketchPolarParam{\wVec} \cdot \sketchPolarParam{\wVecPrime}} \nonumber \\ &\phantom{{}\big[}\textit{(by uniform distribution in the second summation)}\\ -=& \sum_{\substack{j \in [B],\\ - \wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec} +=& \estExp \sum_{\substack{j \in [B],\\ + \wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec} \label{eq:estExpect} \end{align} For the next step, we show that the variance of an estimate is small.$$\varParam{\estimate}$$ @@ -72,7 +72,7 @@ Only equation \eqref{eq:polar-prod-all} (which maps to $\cOne$) and \eqref{eq:po Thus, when considering $\distPattern{1}$ the variance results in \begin{equation} -\sum_{\wVec \in \pw} \kMapParam{\wVec}^2 +\distPatOne\label{eq:distPatOne} \end{equation} For the distribution pattern $\cTwo$, we have three variants to consider. @@ -83,12 +83,33 @@ For the distribution pattern $\cTwo$, we have three variants to consider. \end{align*} When considered separately, the variants have the following $\var$. \begin{align} -\cTwo&=\sum_{\wOne \neq \wTwo}\kMapParam{\wOne} \cdot \kMapParam{\wTwo}\\ -\cTwoV{\wOne}{\wTwo}{\wOneP}{\wTwoP}&=\sum_{\substack{\wOne \neq \wOneP,\\ - \wOne = \wTwo,\\ - \sketchHashParam{\wOne} = \sketchHashParam{\wOneP}}} \big| \sketchHashParam{\wOne}\neq \sketchHashParam{\wOneP} \big|\cdot \kMapParam{\wOne}\cdot \kMapParam{\wTwo}\\ -\cTwoV{\wOne}{\wTwoP}{\wOneP}{\wTwo}&=\sum_{\wOne \neq \wTwo} \kMapParam{\wOne} \cdot \kMapParam{\wTwo} +\cTwo&= \variantOne \label{eq:variantOne}\\ +\cTwoV{\wOne}{\wTwo}{\wOneP}{\wTwoP}&=\variantTwo \label{eq:variantTwo}\\ +\cTwoV{\wOne}{\wTwoP}{\wOneP}{\wTwo}&=\variantThree\label{eq:variantThree} \end{align} +Note that at the start of the analysis of $\var$, the second term (expectation \eqref{eq:estExpect} squared) of the $\var$ calculation was not considered. This is because it is cancelled out by \eqref{eq:distPatOne} and \eqref{eq:variantOne}. +\begin{equation*} +\big(\estExp\big)^2 = \distPatOne + \variantOne +\end{equation*} +With only \eqref{eq:variantTwo} and \eqref{eq:variantThree} remaining, we have + +\begin{multline*} +\varParam{\estimate} = \\ +\variantTwo ~+ \\ +\variantThree +\end{multline*} + +Converting terms into their space requirements yields +\begin{align} +&\variantTwo \Rightarrow\numWorldsP \cdot \frac{\numWorlds - 1}{\sketchCols}\label{eq:spaceOne}\\ +&\variantThree \Rightarrow \numWorldsP \cdot \frac{\numWorldsP - 1}{\sketchCols}\label{eq:spaceTwo} +\end{align} +\eqref{eq:spaceOne} and \eqref{eq:spaceTwo} further reduce to +\begin{equation} +\frac{2^{2N}(\prob + \prob^2)}{\sketchCols} - \numWorlds(\frac{\prob}{\sketchCols} + \prob) +\end{equation} + + diff --git a/macros.tex b/macros.tex index 5aab8f7..c88c270 100644 --- a/macros.tex +++ b/macros.tex @@ -21,7 +21,8 @@ \newcommand{\st}{~|~} \newcommand{\pw}{W} \newcommand{\numWorlds}{2^N} -\newcommand{\numWorldsP}{\numWorlds \cdot p} +\newcommand{\prob}{p} +\newcommand{\numWorldsP}{\numWorlds\prob} \newcommand{\numWorldsSum}{\sum_{\wVec \in \pw}\kMap{t}[\wVec]} \newcommand{\numTup}{N} %\newcommand{\kMap}{v_t} @@ -58,6 +59,9 @@ \newcommand{\multLineExpect}{\mathop{\mathbb{E}}} \newcommand{\var}{Var} \newcommand{\varParam}[1]{Var\bigParamBox{#1}} +%%%%%%%%%%%%%%%%% +% Equations +%%%%%%%%%%%%%%%%% \newcommand{\polarFuncSum}[1][]{\sum_{\substack{\wVecPrime ~|~ \\ \sketchHash\left[\wVecPrime\right] = j\\ {#1}}}\sketchPolarParam{\wVecPrime}} @@ -67,6 +71,14 @@ \newcommand{\estTwo}{\sum_{\substack{j \in [B],\\ \wVec \in \pw~|~ \sketchHash{[\wVec]} = j,\\ \wVec[w']\in \pw~|~ \sketchHash{[\wVec[w']]} = j} } v_t[\wVec] \cdot s_i[\wVec] \cdot s_i[\wVec[w']]} +\newcommand{\estExp}{ \sum_{\substack{j \in [B],\\ + \wVec~|~\sketchHashParam{\wVec}= j,\\}} \kMapParam{\wVec}} +\newcommand{\distPatOne}{\sum_{\wVec \in \pw} \kMapParam{\wVec}^2} +\newcommand{\variantOne}{\sum_{\wOne \neq \wTwo}\kMapParam{\wOne} \cdot \kMapParam{\wTwo}} +\newcommand{\variantTwo}{\sum_{\substack{\wVec \neq \wVecPrime,\\ + \sketchHashParam{\wVec} = \sketchHashParam{\wVecPrime}}} \big| \sketchHashParam{\wVec} = \sketchHashParam{\wVecPrime} \big|\cdot \kMapParam{\wVec}^2} +\newcommand{\variantThree}{\sum_{\substack{\wOne \neq \wTwo,\\ + \sketchHashParam{\wOne} = \sketchHashParam{\wTwo}}} \kMapParam{\wOne} \cdot \kMapParam{\wTwo}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % COMMENTS %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -209,7 +221,7 @@ \newcommand{\bestG}[1]{BestGuess(#1)} \newcommand{\uadb}{\db_{UA}} \newcommand{\bgdb}{D_{bg}} -\newcommand{\prob}{P} +%\newcommand{\prob}{P} \newcommand{\probOf}[1]{\prob(#1)} \newcommand{\pTupMult}[2]{\prob({#1}, \geq {#2})} \newcommand{\rowsmr}{\mathcal{K}_{uncert}}