A multiple changepoint model in continuous time is formulated as a continuous-time hidden Markov model, defined on a countable infinite state space. The new formulation of the multiple changepoint model allows the model complexities, i.e. the number of changepoints, to accrue unboundedly upon the arrivals of new data. Inference on the number of changepoints and their locations is based on a collapsed Gibbs sampler. We suggest a new version of forwardfiltering backward-sampling (FFBS) algorithm in continuous time for simulating the full trajectory of the latent Markov chain, i.e. the changepoints. The FFBS algorithm is based on a randomized time-discretization for the latent Markov chain through uniformization schemes, combined with a discrete-time version of FFBS algorithm. It is shown that, desirably, both the computational cost and the memory cost of the FFBS algorithm are only quadratic to the number of changepoints. The new formulation of the multiple changepoint models allows varying scale of run lengths of changepoints to be characterized. We demonstrate the methods through simulations and a real data example for earthquakes.

Due to the importance and complexity of photo assimilate transport in raffinose family oligosaccharide (RFO)-transporting plants such as melon, it is important to study the features of the transport structure (phloem) particularly of the lateral branches connecting the source leaves and the sink fruits, and its responses to environmental challenges. Currently, it is unclear to what extents the cold environmental temperature stress would alter the phloem ultrastructure and RFO accumulation in RFO-transporting plants. In this study, we firstly utilized electron microscopy to investigate the changes in the phloem ultrastructure of lateral branches and RFO accumulation in melons after being subjected to low night temperatures (12°C and 9°C). The results demonstrated that exposure to 9°C and 12°C altered the ultrastructure of the phloem, with the effect of 9°C being more obvious. The most obvious change was the appearance of plasma membrane invaginations in 99% companion cells and intermediary cells. In addition, phloem parenchyma cells contained chloroplasts with increased amounts of starch grains, sparse cytoplasm and reduced numbers of mitochondria. In the intermediary cells, the volume of cytoplasm was reduced by 50%, and the central vacuole was present. Moreover, the treatment at 9°C during the night led to RFO accumulation in the vascular bundles of the lateral branches and fruit carpopodiums. These ultrastructural changes of the transport structure (phloem) following the treatment at 9°C represented adaptive responses of melons to low temperature stresses. Future studies are required to examine whether these responses may affect phloem transport.

In this paper, we introduce one type of Markov-Modulated Poisson Process (MMPP) whose arrival times are associated with state-dependent marks. Statistical inference problems including the derivation of the likelihood, parameter estimation through EM algorithm and statistical inference on the state process and the observed point process are addressed. A goodness-of-fit test is proposed for MMPP with statedependent marks by utilizing the theories of rescaling marked point process. We also perform some numerical simulations to indicate the effects of different marks on the efficiencies and accuracies of MLE. The effects of the attached marks on the estimation tend to be weakened for increasing data sizes. Then we apply these methods to characterize the occurrence patterns of New Zealand deep earthquakes through a second-order MMPP with state-dependent marks. In this model, the occurrence times and magnitudes of the deep earthquakes are associated with two levels of seismicity which evolves in terms of an unobservable two-state Markov chain.

Summary In this paper, we perform a sparse filtering recursion for efficient changepoint detection for discrete‐time observations. We attach auxiliary event times to the chronologically ordered observations and formulate multiple changepoint problems of discrete‐time observations into continuous‐time observations. Ideally, both the computational and memory costs of the proposed auxiliary uniformisation forward‐filtering backward‐sampling algorithm can be quadratically scaled down to the number of changepoints instead of the number of observations, which would otherwise be prohibitive for a long sequence of observations. To avoid model bias, a time‐varying changepoint recurrence rate across different segments is assumed to characterise diverse scales of run lengths of the changepoints. We demonstrate the methods through simulation studies and real data analysis.