Inhomogeneous Poisson Process

Inhomogeneous Poisson Process

The inhomogeneous Poisson Process is similar to an ordinary Poisson process. The main difference is that in an inhomogeneous Poisson Process the average rate of arrivals is allowed to vary with time. This is useful as many applications that generate random points in time are modelled more faithfully when these non-homogeneous processes. To understand the derivation of the inhomogeneous Poisson process we can start by introducing something akin to the jump rate matrix. However, because the rate of arrivals is now a function of time, the elements of this matrix are now functions rather than numbers as shown below. $$ \mathbf{Q}(t) = \left( \begin{matrix} -\lambda(t) & \lambda(t) & 0 & 0 & 0 & \dots \\ 0 & -\lambda(t) & \lambda(t) & 0 & 0 & \dots \\ 0 & 0 & -\lambda(t) & \lambda(t) & 0 & \dots \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots \end{matrix} \right) $$ This matrix can be inserted into the following differential equation: $$ \frac{\textrm{d}\mathbf{P}}{\textrm{d}t } = \mathbf{P}(t) \mathbf{Q}(t) $$ which is akin to the Kolomogorov equation. When the matrix equation above is multiplied above the resulting family of differential equations for the probabilities of having $n$ events during a time window of length $t$ can be solved giving: $$ p_n(t) = \frac{(\int_0^t \lambda(t) \textrm{d}t')^n}{n!} e^{-\int_0^t \lambda(t) \textrm{d}t'} $$

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