Inference for cluster point processes with over- or under-dispersed cluster sizes

Abstract

Cluster point processes comprise a class of models that have been used for a wide range of applications. While several models have been studied for the probability density function of the offspring displacements and the parent point process, there are few examples of non-Poisson distributed cluster sizes. In this paper, we introduce a generalization of the Thomas process, which allows for the cluster sizes to have a variance that is greater or less than the expected value. We refer to this as the cluster sizes being over- and under-dispersed, respectively. To fit the model, we introduce minimum contrast methods and a Bayesian MCMC algorithm. These are evaluated in a simulation study. It is found that using the Bayesian MCMC method, we are in most cases able to detect over- and under-dispersion in the cluster sizes. We use the MCMC method to fit the model to nerve fiber data, and contrast the results to those of a fitted Thomas process.

Appendices

Parameter combinations giving identical intensity and K-function

Recall that for the TGCS

$$\begin{aligned} \rho&= \kappa \frac{\theta }{1-\lambda } \quad \quad \text { and } \\ K(r)&= \pi r^2 + \frac{1 + \frac{\lambda }{\theta }\left( 1-\frac{1}{1-\lambda }\right) }{\kappa }\left[ 1-\exp \left( -\frac{r^2}{4\omega ^2}\right) \right] . \end{aligned}$$

Clearly, for two TGCSs to have the same K-function, their kernel standard deviations must be equal. Now assume that $Y_0 \sim \text{ TGCS }( \kappa _0, \omega , \lambda _0, \theta _0)$ and $Y_1 \sim \text{ TGCS }( \kappa _1, \omega , \lambda _1, \theta _1)$, with

$$\begin{aligned} \theta _1&= \left( \theta _0 + \lambda _0\left( 1+\frac{1}{1-\lambda _0}\right) \right) \frac{1-\lambda _1}{1-\lambda _0}-\lambda _1\left( 1+\frac{1}{1-\lambda _1}\right) \end{aligned}$$

(11)

$$\begin{aligned} \kappa _1&= \frac{\theta _0}{\theta _1}\frac{1-\lambda _1}{1-\lambda _0}\kappa _0. \end{aligned}$$

(12)

To show that $Y_0$ and $Y_1$ have the same intensity and K-function, we need to show that $\rho _0 = \rho _1$, where $\rho _i = \kappa _i \frac{\theta _i}{1-\lambda _i}$, $i = 0,1$, and $C_0 = C_1$, where

$$\begin{aligned} C_i = \frac{1 + \frac{\lambda _i}{\theta _i}\left( 1-\frac{1}{1-\lambda _i}\right) }{\kappa _i}, \quad i = 0,1. \end{aligned}$$

Starting with the intensity we have that

$$\begin{aligned} \rho _1&= \frac{\theta _1}{1-\lambda _1}\kappa _1 = \frac{\theta _1}{1-\lambda _1} \frac{\theta _0}{\theta _1} \frac{1-\lambda _1}{1-\lambda _0}\kappa _0 = \frac{\theta _0}{1-\lambda _0}\kappa _0 = \rho _0 \end{aligned}$$

To show that $C_0 = C_1$ we let $\rho = \rho _0 = \rho _1$ and start from (11):

$$\begin{aligned}&\theta _1 = \left( \theta _0 + \lambda _0\left( 1+\frac{1}{1-\lambda _0}\right) \right) \frac{1-\lambda _1}{1-\lambda _0}-\lambda _1\left( 1+\frac{1}{1-\lambda _1}\right) \Leftrightarrow \\&\frac{\theta _1 + \lambda _1\left( 1+ \frac{1}{1-\lambda _1}\right) }{1-\lambda _1}= \frac{\theta _0 + \lambda _0\left( 1+ \frac{1}{1-\lambda _0}\right) }{1-\lambda _0} \Leftrightarrow \\&\frac{1 + \frac{\lambda _1}{\theta _1}\left( 1+ \frac{1}{1-\lambda _1}\right) }{\frac{(1-\lambda _1)\rho }{\theta _1}} = \frac{1 + \frac{\lambda _0}{\theta _0}\left( 1+ \frac{1}{1-\lambda _0}\right) }{\frac{(1-\lambda _0)\rho }{\theta _0}} \Leftrightarrow \\&\frac{1 + \frac{\lambda _1}{\theta _1}\left( 1+ \frac{1}{1-\lambda _1}\right) }{\kappa _1} = \frac{1 + \frac{\lambda _0}{\theta _0}\left( 1+ \frac{1}{1-\lambda _0}\right) }{\kappa _0} \Leftrightarrow \\&C_1 = C_0, \end{aligned}$$

where all the lines are equivalent to each other.

MCMC algorithm

We let ${\mathbf {y}}$ denote the observed pattern and ${\mathbf {x}}$ the current parent pattern and their point counts $N({\mathbf {y}})$ and $N({\mathbf {x}})$, respectively. Recall that $\kappa$ is the intensity of the parent process, $\omega$ the standard deviation of the Gaussian kernel for the offspring displacements and $\lambda$ and $\theta$ are the parameters to the GPD for the cluster size distribution. The priors for $\kappa$, $\omega$ and $\theta$ are gamma-distributions, and we let $\alpha _\cdot$ and $\beta _\cdot$ denote the shape and rate parameters to the priors, respectively, where the subindex indicates which parameter the prior is for. Furthermore, let ${\mathbf {p}}(y) = \{p_y\}_{y\in Y}$ be the connections, i.e. $p_y = x$ means that offspring y belongs to parent $x \in {\mathbf {x}}$.

1.1 Full conditional density

The full conditional density of the unobserved entities is then given by

$$\begin{aligned} \begin{aligned} p( {\mathbf {x}}, \kappa , \omega , \lambda , \theta , {\mathbf {p}}(y)| {\mathbf {y}} ) \propto&p( {\mathbf {y}}| {\mathbf {x}}, {\mathbf {p}}(y), \omega ) p( {\mathbf {p}}(y)| \omega , \lambda , \theta , {\mathbf {x}})\\ {}&p( {\mathbf {x}}| \kappa )p( \kappa )p( \omega )p(\lambda ) p( \theta ), \end{aligned} \end{aligned}$$

(13)

up to a normalizing constant.

The first factor is given by the product of the bivariate Gaussian density at the distances between offspring and their respective parents, i.e

$$\begin{aligned} p( {\mathbf {y}}| {\mathbf {x}}, {\mathbf {p}}(y), \omega ) {=} (2\pi \omega ^2)^{-N({\mathbf {y}})} \exp \left( -\frac{1}{2\omega ^2}\sum _{y\in {\mathbf {y}}}{||y - p_y||^2} \right) \end{aligned}$$

(14)

Moreover, since $X|\kappa$ is a Poisson process with intensity $\kappa$, the density with respect to the unit rate Poisson process is given by

$$\begin{aligned} p( {\mathbf {x}}|\kappa ) = \kappa ^{N({\mathbf {x}})}\exp \left( |W|(1-\kappa ) \right) , \end{aligned}$$

(15)

and the last four densities are the priors for the respective parameters. The conditional density $p({\mathbf {p}}(y)|\omega , \lambda , \theta , {\mathbf {x}})$ depends on the number of offspring of each parent generated by the connections ${\mathbf {p}}(y)$, and takes a little more work to derive.

First, let ${\mathbf {o}}(y) = \{o_x(y)\}_{ x\in {\mathbf {x}}}$ be the observed cluster sizes. Note that we can obtain ${\mathbf {o}}(y)$ from ${\mathbf {p}}(y)$ by computing, for each x,

$$\begin{aligned} o_x(y) = \sum _{y \in {\mathbf {y}}} {\mathbf {1}}\{ p_y = x \}, \end{aligned}$$

where ${\mathbf {1}}$ is the indicator function. Having $o_x(y) = k$ means that parent x has k offspring in the observation window, W, but that it could have more offspring that are not observed. We denote the total number of offspring per parent by ${\mathbf {o}}(y)^* = \{O^*_x\}_{ x\in {\mathbf {x}}}$. The number of offspring per parent follows a generalized Poisson distribution, i.e. $o_x(y)^* \sim \text{ GPD }( \lambda , \theta )$, and the probability that an offspring of a parent x is inside the observation window is $\nu _W(x; \omega )$, the integral of the kernel centered at x over the observation window is denoted by $\nu _W( x; \omega )$, i.e

$$\begin{aligned} \nu _W(x;\omega ) = \int _W \frac{1}{2\pi \omega ^2} \exp \left( -\frac{||x-z||^2}{2\omega ^2}\right) dz. \end{aligned}$$

Thus, by the independence of the offspring displacements, we get that the probability that a parent x has o offspring inside W is

$$\begin{aligned}&P(o_x(y) = o| x, \lambda , \theta ,\omega ) \\&\quad =\sum _{k = o}^\infty {k \atopwithdelims ()o}\nu _W(x;\omega )^{o} (1-\nu _W(x;\omega ))^{k-o} p_{\text{ GPD }}( k| \lambda , \theta ), \end{aligned}$$

where $p_{\text{ GPD }}(\cdot ; \lambda , \theta )$ is the PMF of the GPD. When computed, this sum is evaluated numerically, by truncating it at a value of k such that $p_{\text{ GPD }}(k; \lambda , \theta ) < 10^{-6}$. The probability of observing a certain connection configuration, ${\mathbf {p}}(y)$, given a parent pattern ${\mathbf {x}}$, but not considering the displacements of the offspring, is given by the probability of the cluster sizes, ${\mathbf {o}}(y)$, generated by ${\mathbf {p}}(y)$, divided by the number of possible configurations with the same ${\mathbf {o}}(y)$, i.e.

$$\begin{aligned} \begin{aligned} P( {\mathbf {p}}(y)&= {\mathbf {p}}(y) | \omega , \lambda , \theta , {\mathbf {x}} ) =\\&= \frac{\prod _{x \in {\mathbf {x}}} o_x(y)!}{\left( \sum _{x\in {\mathbf {x}}} o_x(y)\right) !} \prod _{x\in {\mathbf {x}}} P( o_x(y) = o_x(y) | \omega , \lambda , \theta ). \end{aligned} \end{aligned}$$

(16)

1.2 Updating scheme

For many proposals we use the log-normal distribution. We say that $Z \sim \log {\mathcal {N}} (\mu , \sigma )$ if the density is given by

$$\begin{aligned} f_Z(z) = \frac{1}{\sqrt{2\pi }\sigma z}\exp \left( -\frac{(\log z - \mu )^2}{2\sigma ^2} \right) . \end{aligned}$$

All updates are performed using a Metropolis-Hastings step.

1.3 Updating $\kappa$

The full conditional density of $\kappa$ is given by

$$\begin{aligned} p( \kappa |\ldots ) \propto p( {\mathbf {x}}| \kappa ) p( \kappa ), \end{aligned}$$

and given the current value, $\kappa ^{(i)}$, the proposal, $\kappa ^*$, is log-normal with mean $\log \kappa ^{(i)}$ and standard deviation $\sigma _\kappa$. This, together with the prior, gives the Hastings ratio as

$$\begin{aligned} r_\kappa (\kappa ^*|\kappa ^{(i)})&= \left( \frac{\kappa ^*}{\kappa ^{(i)}} \right) ^{N({\mathbf {x}})+1}\exp \left( -|W|(\kappa ^*-\kappa ^{(i)})\right) \times \\&\left( \frac{\kappa ^*}{\kappa ^{(i)}}\right) ^{\alpha _\kappa -1}\exp \left( -\beta _\kappa (\kappa ^*-\kappa ^{(i)})\right) . \end{aligned}$$

1.4 Updating $\omega$

Given the current value, $\omega ^{(i)}$, the proposal, $\omega ^*$, is log-normal, with mean $\log \omega ^{(i)}$ and standard deviation $\sigma _\omega$. The full conditional density of $\omega$ is given by

$$\begin{aligned} p( \omega | \ldots ) \propto p( {\mathbf {y}}| {\mathbf {x}}, {\mathbf {p}}(y), \omega ) p( {\mathbf {p}}(y)|{\mathbf {x}}, \omega , \lambda , \theta )p(\omega ). \end{aligned}$$

Given the prior and the update proposal density we get the Hastings-ratio for $\omega$ as

$$\begin{aligned} r_\omega ( \omega ^*|\omega ^{(i)})&= \exp \left[ {-}\left( \frac{1}{2(\omega ^*)^2}{-}\frac{1}{2(\omega ^{(i)})^2}\right) \sum _{y\in {\mathbf {y}}}||y{-}p_y||^2\right] \times \\&\left( \frac{\omega ^{(i)}}{\omega ^*}\right) ^{2N({\mathbf {y}}) {-} \alpha _\omega }\prod _{x\in {\mathbf {x}}} \frac{P(o_x(y)|x, \lambda , \theta \omega ^*)}{P(o_x(y)| x, \lambda , \theta , \omega ^{(i)})} \times \\&\exp [ -\beta _\omega (\omega ^* - \omega ^{(i)})]. \end{aligned}$$

1.5 Updating $\lambda$ and $\theta$

In this step we accept or reject $\lambda$ and $\theta$ jointly. The full conditional density of the pair $(\lambda , \theta )$ is given as

$$\begin{aligned} p( \lambda , \theta | \ldots )\propto p( {\mathbf {o}}(y)| \lambda , \theta , X, \omega )p(\lambda )p(\theta ). \end{aligned}$$

Given the current values $\theta ^{(i)}$ and $\lambda ^{(i)}$, the proposal $\theta ^*$ is log-normal with mean $\log \theta ^{(i)}$ and standard deviation $\sigma _\theta$. The proposal $\lambda ^*$, given $\lambda ^{(i)}$ and $\theta ^*$, is uniform on $[ l^*, u^*]$, where the limits are given by

$$\begin{aligned} l^*&= \max \left\{ -1, \lambda ^{(i)} - \varDelta \lambda , - \frac{\theta ^*}{\max _{x\in {\mathbf {x}}} o_x(y)}\right\} \\ u^*&= \min \left\{ 0.99, \lambda ^{(i)} + \varDelta \lambda \right\} , \end{aligned}$$

where $\varDelta \lambda$ is a chosen constant. We denote the proposal density by $q_\lambda ( \lambda ^*| \lambda ^{(i)}, \theta ^*)$. These limits are chosen to assure that we do not propose a value that is outside the support of the prior (recall that the prior for $\lambda$ is uniform on $[-1, 0.99]$) and that the current cluster sizes are possible with the proposed parameter combination $(\lambda ^*, \theta ^*)$. In the same way, we let

$$\begin{aligned} l&= \max \left\{ -1, \lambda ^{*} - \varDelta \lambda , - \frac{\theta ^{(i)}}{\max _{x\in X} o_x(y)}\right\} \\ u&= \min \left\{ 0.99, \lambda ^{*} + \varDelta \lambda \right\} , \end{aligned}$$

i.e. the limits for the proposal interval for $q_{\lambda }(\lambda ^{(i)}|\lambda ^*, \theta ^{(i)})$. Of course, it could happen that for the proposed $\theta ^*$ we get $u^*\le l^*$, in which case we immediately reject. From this we get that the Hastings-ratio for the pair $(\lambda , \theta )$ is given by

$$\begin{aligned} r_{(\lambda , \theta )}((\lambda ^*, \theta ^*)|(\lambda ^{(i)}, \theta ^{(i)}))&= \frac{\prod _{x\in {\mathbf {x}}} p(o_x(y)|\lambda , \theta ^*, \omega ^*)}{\prod _{x\in {\mathbf {x}}} p(o_x(y)|\lambda , \theta ^{(i)}, \omega ^{(i)})} \\&\quad \times \exp \left( -\beta _\theta (\theta ^* -\theta ^{(i)})\right) \\&\quad \times \left( \frac{\theta ^*}{\theta ^{(i)}}\right) ^{\alpha _\theta } \frac{u^*-l^*}{u-l}. \end{aligned}$$

1.6 Updating ${\mathbf {p}}(y)$

The full conditional density of ${\mathbf {p}}(y)$ is given by

$$\begin{aligned} p( {\mathbf {p}}(y)| \ldots ) \propto&p( {\mathbf {y}}| {\mathbf {x}}, \omega , {\mathbf {p}}(y) )p( {\mathbf {p}}(y)| {\mathbf {x}}, \omega , \lambda , \theta ), \end{aligned}$$

which is according to the (13) and (16) equal

$$\begin{aligned}&=(2\pi \omega ^2)^{-N({\mathbf {y}})} \exp \left( -\frac{1}{2\omega ^2}\sum _{y \in {\mathbf {y}}}||y - {p_y}||^2 \right) \\&\quad \frac{o_1! \cdot \ldots \cdot o_{N({\mathbf {x}})}!}{N({\mathbf {y}})!} \prod _{x \in {\mathbf {x}}} p( o_x(y)|x, \omega , \lambda , \theta ). \end{aligned}$$

To update ${\mathbf {p}}(y)$ we chose a point, $y'$, of ${\mathbf {y}}$ uniformly and propose a new parent for it, uniformly over the current parent pattern. We denote by $p_{y'}^*$ the proposed parent, and by $o_{p_{y'}}$ and $o_{p_{y'}^*}$ the current offspring counts of the current parent and the proposed parent, respectively. Moreover, we let $o_{p_{y'}}^*$ and $o_{p_{y'}^*}^*$ be the same offspring counts with the proposed parent for $y'$, i.e. $o_{p_{y'}}^* = o_{p_{y'}}-1$ and $o_{p_{y'}^*}^* =o_{p_{y'}^*}+1$. With this in place, we get the Hastings ratio for updating ${\mathbf {p}}(y)$ as

$$\begin{aligned} r_{{\mathbf {p}}(y)}&( {\mathbf {p}}(y)^*| {\mathbf {p}}(y)) = \exp \left( -\frac{||y'-p_{y'}^*||^2-||y'-p_{y'}||^2}{2\omega ^2}\right) \times \\&\frac{p\left( o_{p_{y'}^*}^*|{\mathbf {x}}, \omega , \lambda , \theta \right) p\left( o_{p_{y'}}^*|{\mathbf {x}}, \omega , \lambda , \theta \right) }{p\left( o_{p_{y'}^*}|{\mathbf {x}}, \omega , \lambda , \theta \right) p\left( o_{p_{y'}}|{\mathbf {x}}, \omega , \lambda , \theta \right) } \times \frac{o_{p_{y'}^*}^*!o_{p_{y'}}^*!}{o_{p_{y'}^*}!o_{p_{y'}}!}. \end{aligned}$$

1.7 Updating ${\mathbf {x}}$

When updating ${\mathbf {x}}$ by a birth (adding a point) or death (removing a point), we must also update ${\mathbf {p}}(y)$. Both birth and death proposals are uniform, i.e. the location of a new parent is uniform on W and the parent to remove at a death is uniform on ${\mathbf {x}}$.

At a birth, we propose a new set of connections, ${\mathbf {p}}(y)^*$, by proposing offspring to “steal” from other parents. Given that we propose to add a parent, $x^*$, the probability that $y \in {\mathbf {y}}$ is stolen depends on a Gaussian kernel at the distances between parent and offspring with the current and updated connection, as

$$\begin{aligned} P(P_y^* = x^* ) = \frac{\exp \left( -\frac{||y-x^*||^2}{2\sigma _{bd}^2}\right) }{\exp \left( -\frac{||y-x^*||^2}{2\sigma _{bd}^2}\right) +\exp \left( -\frac{||y-p_y||^2}{2\sigma _{bd}^2}\right) } := b_y(x^*), \end{aligned}$$

where $\sigma _{bd}$ is a tuning parameter.

When a point, $x_0 \in {\mathbf {x}}$, is proposed for a death, the probability that $y\in {\mathbf {y}}$ is instead assigned to $x'$ is given by

$$\begin{aligned} P( P_y^* = x' ) = \frac{\exp \left( -\frac{||y-x'||^2}{2\sigma _{bd}^2}\right) }{\sum _{x\in {\mathbf {x}}\setminus x_0} \exp \left( -\frac{||y-x||^2}{2\sigma _{bd}^2}\right) } := d_y( x'), \end{aligned}$$

where we use the same standard deviation for the Gaussian kernel as in the birth updates.

With this notation we get the following proposal density for a birth

$$\begin{aligned} q_b( {\mathbf {x}}^*, {\mathbf {p}}(y)^*| {\mathbf {x}}, {\mathbf {p}}(y) ) = |W|^{-1} \prod _{y:p_y^* = x^*} b_y( x^* ) \prod _{y: p_y^* \ne x^*} (1-b_y(x^*)), \end{aligned}$$

where ${\mathbf {x}}^* = {\mathbf {x}}\cup x^*$ and ${\mathbf {p}}(y)^*$ is obtained by proposing offspring to steal. The proposal density for a death is given by

$$\begin{aligned} q_d( {\mathbf {x}}^*, {\mathbf {p}}(y)^*| {\mathbf {x}}, {\mathbf {p}}(y) ) =N({\mathbf {x}})^{-1} \prod _{y:p_y = x_0} d_y( p_y^* ), \end{aligned}$$

where ${\mathbf {x}}^* = {\mathbf {x}}\setminus x_o$ and ${\mathbf {p}}(y)^*$ is obtained by reassigning the current offspring of $x_0$ according to the above. This yields the Hastings-ratios for a proposed birth update, where the dimensions in products change, as

$$\begin{aligned}&r_{birth}( {\mathbf {x}}^*, {\mathbf {p}}(y)^*| {\mathbf {x}}, {\mathbf {p}}(y)) \\&\quad =\exp \left( -\frac{\sum _{y: p^*_y = x^*}(||y - x^*||^2 - ||y-p_y||^2)}{2\omega ^*}\right) \\&\qquad \times \frac{\prod _{x\in {\mathbf {x}}^*}p(o_x(y)^*|x, \lambda , \theta , \omega )o_x(y)^*!}{\prod _{x\in {\mathbf {x}}}p(o_x(y)|x, \lambda , \theta , \omega )o_x(y)!}\\&\qquad \times \kappa \frac{q_d({\mathbf {x}}, {\mathbf {p}}(y)|{\mathbf {x}}^*, {\mathbf {p}}(y)^* ) }{q_b({\mathbf {x}}^*, {\mathbf {p}}(y)^*|{\mathbf {x}}, {\mathbf {p}}(y) )}, \end{aligned}$$

and for a proposed death update as

$$\begin{aligned}&r_{death}( {\mathbf {x}}^*, {\mathbf {p}}(y)^*| {\mathbf {x}}, {\mathbf {p}}(y) ) \\&\quad =\exp \left( -\frac{1}{2\omega ^2}\sum _{y: p_y = x_0}(||y - p_y^*||^2 - ||y-x_0||^2)\right) \\&\qquad \times \frac{\prod _{x\in {\mathbf {x}}^*}p(o_x(y)^*|x, \lambda , \theta , \omega )o_x(y)^*!}{\prod _{x\in {\mathbf {x}}}p(o_x(y)|x, \lambda , \theta , \omega )o_x(y)!} \\&\qquad \times \kappa ^{-1} \frac{q_b({\mathbf {x}}, {\mathbf {p}}(y)|{\mathbf {x}}^*, {\mathbf {p}}(y)^* ) }{q_d({\mathbf {x}}^*, {\mathbf {p}}(y)^*|{\mathbf {x}}, {\mathbf {p}}(y) )}. \end{aligned}$$

When proposing a move of a point, x, of the current parent pattern, we propose a new location, $x^*$, according to a Gaussian kernel with standard deviation $\sigma _m$ around x restricted to the observation window, i.e.

$$\begin{aligned} q_{move}( x^*|x) \propto \frac{\exp \left( -\frac{1}{2\sigma _m^2}||x^*-x||^2 \right) }{\nu _W(x; \sigma _m)}{\mathbf {1}}\{ x^* \in W \}, \end{aligned}$$

and since only one point of the pattern is proposed to be moved, the Hastings ratio is given by

$$\begin{aligned}&r_{move}( x^*|x ) = \exp \left( -\frac{1}{2\omega ^2}\sum _{y:p_y = x}(||y-x^*||^2-||y-x||^2)\right) \\&\quad \frac{\nu _W(x;\sigma _m)}{\nu _W(x^*;\sigma _m)}. \end{aligned}$$

1.8 Irreducibility of Markov chain

A Markov chain is said to be irreducible if it is possible to get to any state from any state. Irreducibility of x, $\kappa$ and $\omega$ are straightforward. The problem is in that for negative $\lambda$ not all configurations of ${\mathbf {p}}(y)$ are accessible. But there is a positive probability that the Markov chain gets to the state of $\lambda$ and $\theta$ from which the desired configuration of ${\mathbf {p}}(y)$ is already accessible. The same holds for state $\lambda$ and $\theta$ which are inaccessible from certain configuration of ${\mathbf {p}}(y)$. Thus the whole Markov chain is irreducible.