wiki:MatrixElement

Version 4 (modified by md987, 8 years ago) (diff)

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Matrix Element Method

The Matrix Element Method consist in minimizing a likelihood.

The likelihood for N events is defined as $ L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)$

The best estimate of the parameter $\alpha$ is obtained through a maximisation of the likelihood. It is common practice to minimize $-ln(L(\alpha))$ with respect to $\alpha$, $-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$

In general, the probability that an event is accepted depends on the characteristics of the measured event, and not on the process that produced it. The measured probability density $\bar{P}(x,\alpha)$ can be related to the produced probability density $P(x,\alpha)$:

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The command:

'/usr/bin/pdflatex -interaction=nonstopmode f04536e1f28716cfeaabaf7f87c44cc561dca9e3.tex'
failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./f04536e1f28716cfeaabaf7f87c44cc561dca9e3.tex\nLaTeX2e <2011/06/27>\nBabel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo\naded.\n(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo))\n(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/usr/share/texlive/texmf-dist/tex/latex/cmap/cmap.sty) (./type1ec.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1cmr.fd))\n(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap>>)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty)\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/preview.sty\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/prtightpage.def)\n) (./f04536e1f28716cfeaabaf7f87c44cc561dca9e3.aux)\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n<<ot1.cmap>> (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Missing } inserted.\n<inserted text> \n                }\nl.14 $\\bar{P}(x,\\alpha){{{Acc(x) P(x,\\alpha)$\n                                             \n! Missing } inserted.\n<inserted text> \n                }\nl.14 $\\bar{P}(x,\\alpha){{{Acc(x) P(x,\\alpha)$\n                                             \n! Missing } inserted.\n<inserted text> \n                }\nl.14 $\\bar{P}(x,\\alpha){{{Acc(x) P(x,\\alpha)$\n                                             \nPreview: Tightpage -32891 -32891 32891 32891\n[1{/usr/share/texlive/texmf/fonts/map/pdftex/updmap/pdftex.map}]\n(./f04536e1f28716cfeaabaf7f87c44cc561dca9e3.aux) )\n(see the transcript file for additional information)</usr/share/texlive/texmf-d\nist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fo\nnts/type1/public/amsfonts/cm/cmr10.pfb>\nOutput written on f04536e1f28716cfeaabaf7f87c44cc561dca9e3.pdf (1 page, 18546 b\nytes).\nTranscript written on f04536e1f28716cfeaabaf7f87c44cc561dca9e3.log.\n"

where $ Acc(x)$ is the detector acceptance, which depends only on $ x $. So the quantity that we have to minimize is
TracMath macro processor has detected an error. Please fix the problem before continuing.


The command:

'/usr/bin/pdflatex -interaction=nonstopmode 56c05a207a38a4f6eb352518e3d09bd39caca1f2.tex'
failed with the following output:
"This is pdfTeX, Version 3.1415926-2.5-1.40.14 (TeX Live 2013)\n restricted \\write18 enabled.\nentering extended mode\n(./56c05a207a38a4f6eb352518e3d09bd39caca1f2.tex\nLaTeX2e <2011/06/27>\nBabel <v3.8m> and hyphenation patterns for english, dumylang, nohyphenation, lo\naded.\n(/usr/share/texlive/texmf-dist/tex/latex/base/article.cls\nDocument Class: article 2007/10/19 v1.4h Standard LaTeX document class\n(/usr/share/texlive/texmf-dist/tex/latex/base/size10.clo))\n(/usr/share/texlive/texmf-dist/tex/latex/base/inputenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/utf8.def\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/ot1enc.dfu)\n(/usr/share/texlive/texmf-dist/tex/latex/base/omsenc.dfu)))\n(/usr/share/texlive/texmf-dist/tex/latex/cmap/cmap.sty) (./type1ec.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1cmr.fd))\n(/usr/share/texlive/texmf-dist/tex/latex/base/fontenc.sty\n(/usr/share/texlive/texmf-dist/tex/latex/base/t1enc.def)<<t1.cmap>>)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsmath.sty\nFor additional information on amsmath, use the `?' option.\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amstext.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsgen.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsbsy.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsmath/amsopn.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/amscls/amsthm.sty)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amssymb.sty\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/amsfonts.sty))\n(/usr/share/texlive/texmf-dist/tex/latex/tools/bm.sty)\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/preview.sty\n(/usr/share/texlive/texmf-local/texmf-compat/tex/latex/preview/prtightpage.def)\n) (./56c05a207a38a4f6eb352518e3d09bd39caca1f2.aux)\nPreview: Fontsize 10pt\nPreview: PDFoutput 1\n<<ot1.cmap>> (/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsa.fd)\n(/usr/share/texlive/texmf-dist/tex/latex/amsfonts/umsb.fd)\n! Extra }, or forgotten $.\nl.14 $-ln (\\tilde{L})}\n                      }}-\\sum_{i=1}^{N} ln(P(x_i;\\alpha)) + N \\int Acc(x) P(...\n\n! Extra }, or forgotten $.\nl.14 $-ln (\\tilde{L})}}\n                       }-\\sum_{i=1}^{N} ln(P(x_i;\\alpha)) + N \\int Acc(x) P(...\n\n! Extra }, or forgotten $.\nl.14 $-ln (\\tilde{L})}}}\n                        -\\sum_{i=1}^{N} ln(P(x_i;\\alpha)) + N \\int Acc(x) P(...\n\nPreview: Tightpage -32891 -32891 32891 32891\n[1{/usr/share/texlive/texmf/fonts/map/pdftex/updmap/pdftex.map}]\n(./56c05a207a38a4f6eb352518e3d09bd39caca1f2.aux) )\n(see the transcript file for additional information)</usr/share/texlive/texmf-d\nist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fo\nnts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/ty\npe1/public/amsfonts/cm/cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/pub\nlic/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/ams\nfonts/cm/cmr7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm\n/cmsy10.pfb>\nOutput written on 56c05a207a38a4f6eb352518e3d09bd39caca1f2.pdf (1 page, 51545 b\nytes).\nTranscript written on 56c05a207a38a4f6eb352518e3d09bd39caca1f2.log.\n"

where the term $-\sum_{i=1}^N ln(Acc(x_i))$ has been omitted since it does not depend on $\alpha$.

Definition of the Weight

The Matrix Element Method associates a weight to each experimental event $ P( x || \alpha)=\frac{1}{\sigma_{ \alpha}} \int d \phi( y) ||M_{ \alpha}||^2 ( y) dw_1 dw_2 f_1(w_1) f_2(w_2) W(x, y) $ where

  1. $ x $ is the set of information describing the events in the detector (momenta,tag,...)
  2. $ \alpha $ describe a theoretical hyppothesis
  3. $\sigma_{ \alpha}$ is the cross section of this theoretical hyppothesis
  4. $M_{ \alpha}$ is the aplitude linked to this theoretical framework
  5. $f_i(w_i)$ is the parton distribution function associate to the initial parton
  6. $W(x, y)$ is the TransferFunction

Computation of those elements

The MadWeight has created a series of tool to compute the transfer function, the weight, the cross-section, the likelihood,... Some of these tools have their own specific page/

  1. TransferFunction
  2. Computation of the Weight
  3. AcceptanceTerm

-- Main.OlivierMattelaer - 22 May 2009

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