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Abstract:
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We propose a general framework of using multi-level log-Gaussian Cox processes to model repeatedly observed point processes with complex structures. A novel nonparametric approach is developed to consistently estimate the covari- ance kernels of the latent Gaussian processes at all levels. Consequently, multi-level functional principal component analysis can be conducted to investigate the vari- ous sources of variations in the observed point patterns. In particular, to predict the functional principal component scores, we propose a consistent estimation proce- dure by maximizing the conditional likelihoods of super-positions of point processes. We further extend our procedure to the bivariate point process case where poten- tial correlations between the processes can be assessed. Asymptotic properties of the proposed estimators are investigated, and the effectiveness of our procedures is illustrated by a simulation study and an application to a stock trading dataset.
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