Why Gemfury?
Push, build, and install
RubyGems
npm packages
Python packages
Maven artifacts
PHP packages
Go Modules
Debian packages
RPM packages
NuGet packages
dany2018
Repository URL to install this package:
|
Version:
4.4.5.dfsg-3ubuntu2 ▾
|
<HTML>
<HEAD>
<TITLE>Poisson Distribution</TITLE>
</HEAD>
<BODY BGCOLOR="#FFFFFF" TEXT="#000000">
<P><font size="+2" color="green">Poisson distribution</font></P>
<P>
Assume that each data point has an error that is independently random and distributed as a Poisson
distribution. The log likelihood function, <code>L(p)</code>, as a function of the fit parameters,
<code>p</code>, is minimized using a Gauss-Newton method. Since logarithms are involved, a good
first approximation is required before starting the Poisson fit, so try a normal fit first, and
use the resultant parameter values to start off the Poisson fit.</P>
<P>
Weights do not have meaning, and so are not used, in a Poisson fit.</P>
<P>
Assume that each data point, <code>y<sub>k</sub></code>, has an error that is
independently random and distributed as a Poisson distribution, that is,</p>
<p>
<center><IMG SRC="FitS08I01.gif"></center></p>
<p>
We want to minimize:</P>
<P>
<IMG SRC="FitS08I02.gif"></P>
<P>
but <code>∑ln(y<sub>k</sub>!)</code> is a constant. So, the goal is to minimize</P>
<P>
<IMG SRC="FitS08I03.gif"></P>
<P>
Consider the Taylor expansion of <IMG SRC="FitS08I04.gif">:</P>
<P>
<IMG SRC="FitS08I05.gif"></P>
<P>
Define:</P>
<P>
<IMG SRC="FitS08I06.gif"></P>
<P>
Then:</p>
<p>
<IMG SRC="FitS08I07.gif"></p>
<p>
Linearize, and the problem reduces to solving the matrix equation</p>
<p>
<center><IMG SRC="FitS01I11.gif"></center></P>
<P>
<a href="FitS08S01.htm"><font size="+1" color="olive">Chi-square of the fit</font></a></P>
<P>
<a href="FitS07.htm"><img src="../shadow_left.gif">
<font size="+1" color="olive">Normal distribution</font></a>
</P>
</BODY>
</HTML>