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| \documentclass{article}[11pt] \usepackage[onehalfspacing]{setspace} \usepackage{times} \usepackage{amsmath} \usepackage{psfrag} \usepackage{graphicx} \usepackage{epsfig} \usepackage{geometry} \newtheorem{corollary1}{Corollary} \newtheorem{definition1}{Definition} \newtheorem{example1}{Example} \newtheorem{lemma1}{Lemma} \newtheorem{remark1}{Remark} \newtheorem{theorem1}{Theorem} \newtheorem{algorithm1}{Algorithm} \newenvironment{corollary}{\begin{corollary1} \rm}{\end{corollary1}} \newenvironment{definition}{\begin{definition1} \rm}{\end{definition1}} \newenvironment{example}{\begin{example1} \rm}{\end{example1}} \newenvironment{lemma}{\begin{lemma1} \rm}{\end{lemma1}} \newenvironment{remark}{\begin{remark1} \rm}{\end{remark1}} \newenvironment{theorem}{\begin{theorem1} \rm}{\end{theorem1}} \newenvironment{algorithm}{\begin{algorithm1} \rm}{\end{algorithm1}} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \geometry{left=1in,right=1in,top=1in,bottom=1in} \begin{document} \section{Dynamic Max Count} |
= Dynamic Max Count = |
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| \subsection{Concept} | == Concept == |
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| Instead of using Hyper-buckets that have discrete boundaries and densities which can not be updated reasonably using the MaxCountProgramNotes ideas, we propose a probabilistic method where by we put probability densities in space. Each probability density will need the following properties: |
Instead of using Hyper-buckets that have constant densities which can not be updated reasonably using the MaxCountProgramNotes ideas, we propose a probabilistic method where by we define a probability density function in each hyper-bucket. In a sense we are not trying to minimize skew in the bucket creation process, but recognizing and modeling skew in each bucket. The added advantage to this concept is that a region equivalent to a hyper-bucket containing no points may be excluded from the index. Consequently the algorithm searches a smaller space. For example if it becomes necessary to shrink the size of a hyper-bucket to [[latex2($10~unit^{6}$)]] size in a [[latex2($10000~unit^6$)]] space we will have [[latex2($1 \times 10^{18}$)]] buckets. With points concentrated in specific regions we will only have to track a fraction of these in the index. |
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| \begin{enumerate} \item Parameters that define the distribution e.g. \begin{enumerate} \item Center location \item Spatial size \item Standard deviation \item A measure of symmetry or skew \end{enumerate} \item A multi-dimensional probability function preferably a function that uses types functions as parameters e.g. $p(x_u(t),x_l(t),y_u(t),y_l(t)[,z_u(t),z_l(t)])$ \item A theory to update ({\em delete} or {\em insert} points) the distributions based on changes to points. \end{enumerate} |
Here is a description of the index: |
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| Based on this last item, we must maintain a database of 4-dimensional points that we index using 4-dimensional, probability buckets. |
1. Index defines 1. Spatial dimensions 1. Bucket Dimensions 1. Histogram divisions that determine the level of approximation in each hyper-bucket (see below) 1. A multi-dimensional probability function preferably a function that uses functions as parameters e.g. [[latex2($$p(x_u(t),x_l(t),y_u(t),y_l(t)[,z_u(t),z_l(t)])$$)]] 1. A theory to ''update'', ''delete'' or ''insert'' points and the distributions based on changes to points. |
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| '''The above has been implemented in C# 8/15/2005.''' | |
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\end{document} |
Thus we maintain a database of 4-dimensional points that we index using 6-dimensional, probability buckets. |
Dynamic Max Count
This contains the ideas and notes for a Dynamic Max Count (Dynamic Max-in-time) aggregate operator
Concept
Instead of using Hyper-buckets that have constant densities which can not be updated reasonably using the MaxCountProgramNotes ideas, we propose a probabilistic method where by we define a probability density function in each hyper-bucket. In a sense we are not trying to minimize skew in the bucket creation process, but recognizing and modeling skew in each bucket. The added advantage to this concept is that a region equivalent to a hyper-bucket containing no points may be excluded from the index. Consequently the algorithm searches a smaller space. For example if it becomes necessary to shrink the size of a hyper-bucket to latex2($10~unit^{6}$) size in a latex2($10000~unit^6$) space we will have latex2($1 \times 10^{18}$) buckets. With points concentrated in specific regions we will only have to track a fraction of these in the index.
Here is a description of the index:
- Index defines
- Spatial dimensions
- Bucket Dimensions
- Histogram divisions that determine the level of approximation in each hyper-bucket (see below)
A multi-dimensional probability function preferably a function that uses functions as parameters e.g. latex2($$p(x_u(t),x_l(t),y_u(t),y_l(t)[,z_u(t),z_l(t)])$$)
A theory to update, delete or insert points and the distributions based on changes to points.
The above has been implemented in C# 8/15/2005.
Thus we maintain a database of 4-dimensional points that we index using 6-dimensional, probability buckets.