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Spike Density Estimation

- Histogram Method

- Kernel Method

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$$P\left(\textrm{one spike in}\left[t,t+\Delta\right)\right)=\lambda\left(t\right)\Delta+o\left(\Delta\right)$$

$$P\left(\textrm{more than one spike in}\left[t,t+\Delta\right)\right)=o\left(\Delta\right)$$

$$P\left(\textrm{no spike in}\left[t,t+\Delta\right)\right)=1-\lambda\left(t\right)\Delta+o\left(\Delta\right)$$ (2.1)

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$$\Lambda\left( t\right) =\int_{0}^{t}\lambda\left( u\right) du %$$ (2.2)

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$$g\left( z\right) =\exp\left[ -z\right]$$

$B$K=>$&!% $\Lambda_{i}=\Lambda\left( t_{i}\right)$ $B$@$+$i(B

$$z\left( t_{i}\vert t_{i-1},\lambda_{t_{i-1}:t_{i}}\right) \equiv\... ...mbda\left( t_{i-1}\right) =\int_{t_{i-1}}^{t_{i}}% \lambda\left( u\right) du$$

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4. $t_{i}\geq T$ $B$J$i$P=*N;!%(B

   $t_{i}<T$ $B$J$i$P(B $i\leftarrow i+1.\ 2$ $B$X!%(B

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$$p\left( t_{i}\vert t_{i-1},\lambda_{t_{i-1}:t_{i}}\right) =\left\vert \frac{dz}{dt_{i}}\right\vert g\left( z\vert\lambda_{t_{i-1}:t_{i}}\right)$$

$$=\lambda\left( t_{i}\right) \exp\left[ -\int_{t_{i-1}}^{t_{i}} \lambda\left( u\right) du\right]%$$ (2.3)

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4. $t_{i}\geq T$ $B$J$i$P=*N;!%(B

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$$r\left( t\right) =\lambda\left( t\right)%$$ (2.4)

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$$p\left( t_{i}\vert t_{i-1};\lambda_{t_{i-1}:t_{i}}\right) =\lambd... ...}\right) \exp\left[ -\int_{t_{i-1}}^{t_{i}}\lambda\left( u\right) du\right].$$

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$$p\left(t_{1},\ldots,t_{n}\cap N_{T}=n\right)\Delta^{n} = {\displaystyle\prod\limits_{i=1}^{n}} p\left( t_{i}\vert t_{i-1}\right) P\left( t_{n}<T\vert t_{n-1}\right)$$

$$=% {\displaystyle\prod\limits_{i=1}^{n}} \lambda\left( t_{i}\ri... ...\right) du\right] \exp\left[ -\int_{t_{n}}^{T}\lambda\left( u\right) du\right]$$

$$=% {\displaystyle\prod\limits_{i=1}^{n}} \lambda\left( t_{i}\right) \exp\left[ -\int_{t_{0}}^{T}\lambda\left( u\right) du\right].$$

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$$p\left( \left\{ t_{i}\right\} _{i=1}^{n}\right) \Delta^{n}$$

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$$\rightarrow {\displaystyle\prod\limits_{i=1}^{n}} \lambda\left( t_{i}\right) \Delta % \Delta\rightarrow0 %$$

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$${\displaystyle\prod\limits_{i=1}^{K}} \left( 1-\lambda_{i\Delta}\Delta\right) =\exp\left[ \sum_{i=1}^{K}% \log\left( 1-\lambda\left( j\Delta\right) \Delta\right) \right]$$

$$=\exp\left[ \sum_{i=1}^{K}\left[ -\lambda\left( j\Delta\right) \... ...\left\{ \lambda\left( j\Delta\right) \Delta\right\} ^{2}+\cdots\right] \right]$$

$$\rightarrow\exp\left[ -\int_{0}^{T}\lambda\left( t\right) dt\right] \Delta\rightarrow0 %$$

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$$p\left( \left\{ t_{i}\right\} _{i=1}^{n}\right) =% {\displaystyl... ...left( t_{i}\right) \exp\left[ -\int_{0}^{T}\lambda\left( u\right) du\right].$$