Simulating Poisson processes using Google Sheets

-




How can we simulate Poisson processes on a computer or on Google Sheets?

Theory

Consider a Poisson process \(\{N(t)\}\). Given some parameter \(\lambda > 0\), we have the probability distribution given by

\[p_n(t) = \Pr(N(t) = n) = \frac{(\lambda t)^n e^{-\lambda t}}{n!}.\]

We assume that \(N(0) = 0\) almost surely. The value of \(N(t)\) increases by 1 over random periods of time. Let \(T_n\) denote the time at which \(N(t)\) becomes \(n\). Then, \(T_1, T_2, \cdots, T_n, \cdots\) are random variables. Next, let's define \(Q_n = T_n - T_{n-1}\). Of course, we assume \(T_0 = 0\); accordingly, \(Q_1 = T_1\). Since the Poisson process is a Markov process, \(Q_1, Q_2, \cdots, \) are i.i.d. (independent and identically distributed). Let's calculate the cumulative distribution function of \(Q_1 = T_1\).

\[\begin{eqnarray} \Pr(T_1 \leq t) &=& \Pr(N(t) \geq 1)\\ &=& p_1(t) + p_2(t) + \cdots\\ &=& 1 - p_0(t) \\ &=& 1 - e^{-\lambda t}. \end{eqnarray}\]

Thus, the density function \(\rho_{T_1}(t)\)of \(T_1\) is given by 

\[\rho_{T_1}(t) = \frac{d\Pr(T_1 \leq t)}{dt} = \lambda e^{-\lambda t}.\]

This is the density function of the exponential distribution with parameter \(\lambda\). Thus, \(Q_1, Q_2, \cdots\) all follow the same exponential distribution with parameter \(\lambda\). 

Simulation

Now, let's discretize the time. For some small \(\delta t > 0\), we consider the random values \(N(\delta t), N(2\delta 2), N(3\delta t), \cdots\). Since the time interval between one increment and the next follows the exponential distribution, we can determine whether to increment or not at each time step in the following manner.
  1. Generate a uniform random number \(r \in [0, 1)\).
  2. If \(r < 1 - e^{-\lambda \delta t}\), then increment by 1. Otherwise, do not increment.
The second step simply means that one increment occurs within the duration of \(\delta t\). Since the time till the increment follows the exponential distribution, its probability is \(\Pr(T_1 \leq \delta t) = 1 - e^{-\lambda \delta t}\). Thus, the algorithm for simulation is given as follows.
  1. Set \(N(0) = 0\), k = 0.
  2. Generate a uniform random number \(r \in [0, 1)\).
  3. If \(r < 1 - e^{-\lambda \delta t}\), then set \(N((k+1)\delta t) = N(k\delta t) + 1\); otherwise, set \(N((k+1)\delta t) = N(k\delta t)\).
  4. Set \(k = k + 1\). Go to step 2, and repeat.
Every time we run this simulation, a different sample path is generated. One example is shown at the top of this page. Other samples are found below.




Related resources

A Google Sheets implementation is available here.

An accompanying video is available on YouTube:



Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Newton's method

-
Image
Newton's method (the Newton-Raphson method) is a very powerful numerical method for solving nonlinear equations.  Suppose we'd like to solve a non-linear equation \(f(x) = 0\), where \(f(x)\) is a (twice) differentiable non-linear function. Newton's method generates a sequence of numbers \(c_1, c_2, c_3, \cdots\) that converges to a solution of the equation. That is, if \(\alpha\) is a solution (i.e., \(f(\alpha) = 0\)) then,  \[\lim_{n\to\infty}c_n = \alpha,\] and this sequence \(\{c_n\}\) is generated by a series of linear approximations of the function \(f(x)\). Theorem (Newton's method) Let \(f(x)\) be a function that is twice differentiable on an open interval \(I\) that contains the closed interval \([a, b]\) (i.e., \([a,b]\subset I\)) and satisfy the following conditions: \(f(a) < 0\) and \(f(b) > 0\); For all \(x\in [a, b]\), \(f'(x) > 0\) and \(f''(x) > 0\). Let us define the sequence \(\{c_n\}\) by \[ \begin{eqnarray} c_1 &
Read more