A Bayesian multiple changepoint model for marked poisson processes with applications to deep earthquakes

Abstract

A multiple changepoint model for marked Poisson process is formulated as a continuous time hidden Markov model, which is an extension of Chib’s multiple changepoint models (J Econ 86:221–241, 1998). The inference on the locations of changepoints and other model parameters is based on a two-block Gibbs sampling scheme. We suggest a continuous time version of forward-filtering backward-sampling algorithm for simulating the full trajectories of the latent Markov chain without utilizing the uniformization method. To retrieve the optimal posterior path of the latent Markov chain, i.e. the maximum a posteriori estimation of changepoint locations, a continuous-time version of Viterbi algorithm (CT-Viterbi) is proposed. The set of changepoint locations is obtainable either from the CT-Viterbi algorithm or the posterior samples of the latent Markov chain. The number of changepoints is determined according to a modified BIC criterion tailored particularly for the multiple changepoint problems of a marked Poisson process. We then perform a simulation study to demonstrate the methods. The methods are applied to investigate the temporal variabilities of seismicity rates and the magnitude-frequency distributions of medium size deep earthquakes in New Zealand.

Appendix

Poisson process attached by exponential marks belongs to a two-parameter exponential family with canonical form:

$$\begin{aligned} \frac{d^2F_j}{dudv}=e^{\theta _ju-\Psi (\theta _j)+\eta _jv-\Phi (\eta )}. \end{aligned}$$

After fixing changepoints $\tau$, the log-likelihood can be written in a second order Taylor series around the maximum likelihood estimate $\hat{\varvec{\Theta }}(\tau )=arg\max \limits _{\varvec{\Theta }}l(\varvec{\Theta }, \tau )$, such that:

$$\begin{aligned} l(\varvec{\Theta }, \tau )\approx l(\hat{\varvec{\Theta }}(\tau ), \tau ) + (\varvec{\Theta }-\hat{\varvec{\Theta }}(\tau ))'H(\hat{\varvec{\Theta }}(\tau ), \tau )(\varvec{\Theta }-\hat{\varvec{\Theta }}(\tau ))/2, \end{aligned}$$

(16)

where H(.) is the Hessian matrix of the log-likelihood and $\varvec{\Theta }=(\theta , \eta )\in {\mathcal {R}}^2.$ Therefore, after exponentiating $l(\varvec{\Theta }, \tau )$ and treating it as a Gaussian kernel, under the Uniform priors for the changepoint locations, the marginal likelihood of the model ${\mathcal {M}}_m$ is given by

$$\begin{aligned} P({\mathcal {M}}_m|Y[0,T])=\int _{{\mathcal {D}}_m}\int _{{\mathcal {R}}^{2m}}e^{l(\varvec{\Theta }, \tau )}\frac{m!}{T^m}\,d\varvec{\Theta }d\tau =C\int _{{\mathcal {D}}_m}e^{l( {\hat{\varvec{\Theta }}}(\tau ), \tau )}|H({\hat{\varvec{\Theta }}}(\tau ), \tau )|^{-\frac{1}{2}}\frac{m!}{T^m}\,d\tau , \end{aligned}$$

(17)

where ${\mathcal {D}}_m=\{(t_0,t_1,\ldots ,t_{m+1}):0=t_0<t_1<\cdots <t_{m+1}=T\}$ and C is a normalizing constant. The data points in the sample are independent,

$$\begin{aligned} l(\varvec{\Theta }, \tau )=\sum _{i=1}^{m+1}\left\{ \theta _i(S_{\tau _i}^u-S_{\tau _{i-1}}^u)-(\tau _i-\tau _{i-1})\Psi (\theta _i) + \eta _i(S_{\tau _i}^v-S_{\tau _{i-1}}^v)-(\tau _i-\tau _{i-1})\Phi (\eta _i)\right\} , \end{aligned}$$

where $S_{\tau _i}^u$ denotes the sum of u from time 0 to $\tau _i$.

Obviously, the Hessian matrix is a diagonal matrix $H(\hat{\Theta }(\tau ), \tau )=diag(\tau _1\ddot{\Psi }(\hat{\theta }_1(\tau )), (\tau _2-\tau _1)\ddot{\Psi }(\hat{\theta }_2(\tau )), \cdots , (T-\tau _m)\ddot{\Psi }(\hat{\theta }_m(\tau )), \tau _1\ddot{\Phi }(\hat{\eta }_1(\tau )), (\tau _2-\tau _1)\ddot{\Phi }(\hat{\eta }_2(\tau )), \cdots , (T-\tau _m)\ddot{\Phi }(\hat{\eta }_m(\tau )))$ and

$$\begin{aligned} |H(\hat{\Theta }(\tau ), \tau )|=\prod \limits _{i=1}^{m+1}(\tau _i-\tau _{i-1})^2\prod \limits _{i=1}^{m+1} \ddot{\Psi }(\hat{\theta }_i(\tau ))\ddot{\Phi }(\hat{\eta }_i(\tau )). \end{aligned}$$

(18)

In the denominator of Bayes factor, the marginal likelihood of the model ${\mathcal {M}}_0$ is simply given by

$$\begin{aligned} P({\mathcal {M}}_0|Y[0,T])=\int e^{l_0(\varvec{\Theta })}\,d\varvec{\Theta }=\frac{1}{2\pi } e^{l_0\left( \hat{\varvec{\Theta }}_0\right) }|\ddot{l}_0\left( \hat{\varvec{\Theta }}_0\right) |^{-\frac{1}{2}}, \end{aligned}$$

(19)

where $\hat{\varvec{\Theta }}_0=arg\max \limits _{\varvec{\Theta }}l_0(\varvec{\Theta })$ and $|\ddot{l}_0(\hat{\varvec{\Theta }}_0)|=T^2\ddot{\Psi }(\hat{\theta }_0)\ddot{\Phi }(\hat{\eta }_0)$.

Ignoring constant factors, the Bayes factor has approximation:

$$\begin{aligned} \int _{{\mathcal {D}}_m}e^{(l(\hat{\Theta }(\tau ), \tau )-l_0(\hat{\varvec{\Theta }}_0))}\frac{1}{T^{m-1}}\prod \limits _{i=1}^{m}(\tau _i-\tau _{i-1})^{-1}\,d\tau . \end{aligned}$$

(20)

In above equation, the log of the integrant at the maximum likelihood value $\hat{\tau }$ is the Modified BIC. To prove the Modified BIC given in (11), it is necessary to show that the remainder term

$$\begin{aligned} \int _{{\mathcal {D}}_m}e^{l\left( \hat{\Theta }(\tau ), \tau \right) -l\left( \hat{\Theta }(\hat{\tau }), \hat{\tau }\right) }\prod \limits _{i=1}^{m}\left( \hat{\tau }_i-\hat{\tau }_{i-1}\right) /\prod \limits _{i=1}^{m}\left( \tau _i-\tau _{i-1}\right) \,d\tau . \end{aligned}$$

(21)

is uniformly bounded in T, see the proof in Zhang and Siegmund (2007).