Reading today about inhomogeneous Poisson Point Processes. How to even begin to describe what these things are? I'll start from what I know.
The poisson distribution can be built from the exponential distribution, which is a memoryless property:
\begin{eqnarray}
P(x) &=& \lambda e^{-\lambda x} \\
P(x >= t) &=& e^{-\lambda t}
\end{eqnarray}
One way of putting those two equations into words is, if the ``intensity'' or average number of subway trains to arrive per minute is $\lambda$, then the probability you have to wait more than $t$ minutes to grab a subway train goes down exponentially with $t$. But, it's important to note that if you've already been waiting 10 minutes, the Subway train ``doesn't care'' in the sense that your probability distribution stays the same. We can use Bayes' theorem to write this down.
\begin{eqnarray}
P(x > t+t_0 \vert x > t_0) &=& \frac{P(x > t_0 \vert x > t+t_0)P(x > t+t_0)}{P(x > t_0)} \\
&=& \frac{P(x > t+t_0)}{P(x > t_0)} \\
&=& P(x > t)
\end{eqnarray}
The first factor in our numerator is unity, since $t+t_0 > t_0$, and by the property of exponentials, the arguments cancel. So why do we care?
Well, let's take a step further and not that the sum of waiting times,
\begin{eqnarray}
T &=& \sum_{i=1}^n x_n
\end{eqnarray}
is drawn from the convolution of many exponential distributions, which is a Gamma distribution:
\begin{eqnarray}
T &\sim & \frac{T^{n-1}\lambda^n e^{-\lambda T}}{(n-1)!}
\end{eqnarray}
And now, if we want to construct the Poisson distribution, we ask ourselves, what is the probability that we had to wait time $T$ to see precisely $n$ arrivals -- be it of Subway cars, sandwiches, etc.
\begin{eqnarray}
P( T_{n+1} > t)-P( T_{n} > t) &=& \frac{(\lambda t)^n e^{-\lambda t}}{n!}
\end{eqnarray}
This is our Poisson distribution. One can get this quite easily by integrating the Gamma distribution by parts. But what if the ``intensity'' or underlying probability of a subway arrival time is not uniform throughout the day? What if $\lambda$ is in fact a unit of time?
Such a situation is related to an inhomogeneous Poisson process, which has certain properties. But before we move onto that, let's talk about a homogeneous Poisson process. I've learned from Statistical mechanics that the exponential distribution is the ``most random'' or ``most entropic'' or ``least informed'' of all distributions, corresponding to some expectation value $\lambda$ (which can be treated as a lagrange multiplier when maximizing the entropy of our probability distribution). This means that the Poisson distribution is the maximally entropic description of arrival counts up to some time $T$, given some arrival intensity $\lambda$.
Apparently, the Poisson process can be defined in three equivalent ways:
(1) The probability of an event in some infinitesimal window of time $dt$ is $\lambda dt$. This is reminiscent of the exponential distribution -- not a big deal
(2) The inter-arrival times are independent and obey the exponential distribution.
(3) The number of arrivals within some finite interval of time $T$ is drawn from Poisson.
These are all things we already know, that we have already built in, but good to review. One way to generate a Poisson process is to draw the total number of arrivals from a Poisson distribution $P(n,\lambda)$, then `sprinkle' $n$ points along some interval $T$ with a uniform distribution. It's important to note that two Poisson processes with intensities $\lambda_1,\lambda_2$ have an effective intensity $\lambda_1+\lambda_2$. Also, if one selects from this final Poisson process sample with a probability density $p(t)$, then the effective sample becomes representative of an underlying intensity $\lambda(t)$. In practice, this is how inhomogeneous Poisson processes are created.
For cosmological surveys, we use inhomogeneous Poisson point process to model Galaxy occupation numbers in redshift space. The underlying ``intensity'' or ``arrival rate'' of Galaxy occupation goes like the window function, $W(\mathbf{x})$.
The poisson distribution can be built from the exponential distribution, which is a memoryless property:
\begin{eqnarray}
P(x) &=& \lambda e^{-\lambda x} \\
P(x >= t) &=& e^{-\lambda t}
\end{eqnarray}
One way of putting those two equations into words is, if the ``intensity'' or average number of subway trains to arrive per minute is $\lambda$, then the probability you have to wait more than $t$ minutes to grab a subway train goes down exponentially with $t$. But, it's important to note that if you've already been waiting 10 minutes, the Subway train ``doesn't care'' in the sense that your probability distribution stays the same. We can use Bayes' theorem to write this down.
\begin{eqnarray}
P(x > t+t_0 \vert x > t_0) &=& \frac{P(x > t_0 \vert x > t+t_0)P(x > t+t_0)}{P(x > t_0)} \\
&=& \frac{P(x > t+t_0)}{P(x > t_0)} \\
&=& P(x > t)
\end{eqnarray}
The first factor in our numerator is unity, since $t+t_0 > t_0$, and by the property of exponentials, the arguments cancel. So why do we care?
Well, let's take a step further and not that the sum of waiting times,
\begin{eqnarray}
T &=& \sum_{i=1}^n x_n
\end{eqnarray}
is drawn from the convolution of many exponential distributions, which is a Gamma distribution:
\begin{eqnarray}
T &\sim & \frac{T^{n-1}\lambda^n e^{-\lambda T}}{(n-1)!}
\end{eqnarray}
And now, if we want to construct the Poisson distribution, we ask ourselves, what is the probability that we had to wait time $T$ to see precisely $n$ arrivals -- be it of Subway cars, sandwiches, etc.
\begin{eqnarray}
P( T_{n+1} > t)-P( T_{n} > t) &=& \frac{(\lambda t)^n e^{-\lambda t}}{n!}
\end{eqnarray}
This is our Poisson distribution. One can get this quite easily by integrating the Gamma distribution by parts. But what if the ``intensity'' or underlying probability of a subway arrival time is not uniform throughout the day? What if $\lambda$ is in fact a unit of time?
Such a situation is related to an inhomogeneous Poisson process, which has certain properties. But before we move onto that, let's talk about a homogeneous Poisson process. I've learned from Statistical mechanics that the exponential distribution is the ``most random'' or ``most entropic'' or ``least informed'' of all distributions, corresponding to some expectation value $\lambda$ (which can be treated as a lagrange multiplier when maximizing the entropy of our probability distribution). This means that the Poisson distribution is the maximally entropic description of arrival counts up to some time $T$, given some arrival intensity $\lambda$.
Apparently, the Poisson process can be defined in three equivalent ways:
(1) The probability of an event in some infinitesimal window of time $dt$ is $\lambda dt$. This is reminiscent of the exponential distribution -- not a big deal
(2) The inter-arrival times are independent and obey the exponential distribution.
(3) The number of arrivals within some finite interval of time $T$ is drawn from Poisson.
These are all things we already know, that we have already built in, but good to review. One way to generate a Poisson process is to draw the total number of arrivals from a Poisson distribution $P(n,\lambda)$, then `sprinkle' $n$ points along some interval $T$ with a uniform distribution. It's important to note that two Poisson processes with intensities $\lambda_1,\lambda_2$ have an effective intensity $\lambda_1+\lambda_2$. Also, if one selects from this final Poisson process sample with a probability density $p(t)$, then the effective sample becomes representative of an underlying intensity $\lambda(t)$. In practice, this is how inhomogeneous Poisson processes are created.
For cosmological surveys, we use inhomogeneous Poisson point process to model Galaxy occupation numbers in redshift space. The underlying ``intensity'' or ``arrival rate'' of Galaxy occupation goes like the window function, $W(\mathbf{x})$.
