Up to Var[estimate] completed

This commit is contained in:
Aaron Huber 2019-06-10 13:36:43 -04:00
parent 06c5001235
commit 170a5ff54b
2 changed files with 43 additions and 10 deletions

View file

@ -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}

View file

@ -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}}