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15.3.1 The Poisson distribution

• $B==J,$KD9$$;~4VFb(B(=$BO"B3$7$?;~4VFb(B)$B$K$"$k;v>]$,(Bk$B2sH/@8$9$k3NN(J,I[(B
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  $$\begin{eqnarray*}p(k;\lambda) = e^{-\lambda} \frac{\lambda^k}{k!} \end{eqnarray*}$$
  
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  • $B$"$k:YJ,$5$l$?;~4VFb$K;v>]$,$*$-$k3NN($O;~4V$ND9$5$KHfNc(B
  • $B;~4V$rHs>o$K:Y$+$/:YJ,$9$l$P!":YJ,$5$l$?;~4VFb$K$=$N;v>](B $B$,(B2$B2s0J>e5/$-$k3NN($OL5;k$G$-$k(B
  • $B8D!9$N:YJ,$5$l$?;~4VFb$K5/$-$k;v>]$O!"B>$N;~4V$K5/$-$k;v>](B $B$HFHN)(B
• binomial distribution $\rightarrow$ poisson distribution
  
  $$\begin{eqnarray*}\lim_{n \to \infty, p \to 0} b(k;n,p) = p(k;\lambda)\ \ \ (\lambda = np) \end{eqnarray*}$$
  
  ($B>ZL@(B)
  
  $$\begin{eqnarray*}b(k;n,\frac{\lambda}{n}) &=& \frac{n(n-1)\cdots(n-(k-1))}{k!} ... ...bda}}\Bigr\}^{-\lambda} \Bigl(1-\frac{\lambda}{n}\Bigr)^{k} \\ \end{eqnarray*}$$
  
  $n\rightarrow \infty$ $B$N;~(B
  
  $$\begin{eqnarray*}\Bigl(1-\frac{\lambda}{n}\Bigr)^{\frac{-n}{\lambda}} \rightarro... ...ac{2}{n},\cdot \frac{k-1}{n},\frac{\lambda}{n} \rightarrow 0\\ \end{eqnarray*}$$
  
  $B$h$C$F(B
  
  $$\begin{eqnarray*}\lim_{n \to \infty} b(k;n,\frac{\lambda}{n}) = e^{-\lambda} \frac{\lambda^k}{k!} \end{eqnarray*}$$
  
  
• $B4|BTCM(B,$BJ,;6(B $E(p) = Var(p) = \lambda$
• Term distribution model $B$XE,MQ(B
  
  $$\begin{eqnarray*}P_i(k) = p(k;\lambda_i) = e^{-\lambda_i} \frac{\lambda_i^k}{k!} \end{eqnarray*}$$
  
  $\lambda_i$ $B$O(B term w_i $B$N(B1 document $B$"$?$j$NJ?6Q=P8=2s?t(B, ( $\lambda_i = \frac{cf_i}{N}$)
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  1.
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  1.

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  4.

  $B$3$N(B text(document) $B$K(B w_i $B$,(B k $B2s=P8=$9$k3NN($O(B, binominal distribution ($BA0Ds(B3)
  

  $$\begin{eqnarray*}b(k;n,\frac{\lambda}{n}) = { }_nC_k \Bigl(\frac{\lambda}{n}\Bigr)^k \Bigl(1-\frac{\lambda}{n}\Bigr)^{n-k} \end{eqnarray*}$$

  
  

  5.

  $n\rightarrow \infty$ $B$N6K8B$r$H$k$H(B poisson distribution
  

  $$\begin{eqnarray*}p(k;\lambda) = e^{-\lambda_i} \frac{\lambda_i^k}{k!} \end{eqnarray*}$$

  
  

  $BNc(B

  • Table 15.6, 6$B$D$NC18l$r(B possion distribution $B$G?dDj(B
  • $\hat{df_i} = N(1-p(0;\lambda))$ $B$O(B df_i $B$N?dDjNL(B
  • follows, transformed $BEy$N(B no-content word($BHsFbMF8l(B,$BHs=EMW8l(B,keyword $B$K$J$j$K$/$$8l(B) $B$O(B, $B$[$\?dDj$I$&$j(B
  • soviet, students $BEy$N(B content word ($BFbMF8l(B,$B=EMW8l(B,keyword $B$K$J$j$d$9$$8l(B)$B$O(B $B?dDjCM$H3+$-$,$"$k(B
  • $B?ML>$O?7J9$G:G=i$K(B1$B2s$@$1;H$o$l$d$9$$(B $\rightarrow$ james
  • freshly $B$O(B no-content word $B$K$b4X$o$i$:(B, $B$"$kFCDj$N(B document $B$K=P8=(B
  • possion distribution $B$O$9$Y$F$NC18l$rJ?Ey$K07$C$F$7$^$&$N(B $B$,LdBj(B, content-word $B$O(B poisson distribution $B$K=>$$$K$/$$(B
  • $B=EMW8l$O(B, $B$"$kFCDj$N(B document $B$K=8Cf$7$d$9$$(B $\rightarrow$ burstiness , term clustering
  • Document size $B$O0lDjNL$G$O$J$$$3$H$,LdBj(B