Point processes provide a powerful framework
for modeling the distribution and interactions
of events in time or space. Their flexibility
has given rise to a variety of sophisticated
models in statistics and machine learning, yet
model diagnostic and criticism techniques re-
main underdeveloped. In this work, we pro-
pose a general Stein operator for point pro-
cesses based on the Papangelou conditional
intensity function. We then establish a kernel
goodness-of-fit test by defining a Stein dis-
crepancy measure for general point processes.
Notably, our test also applies to non-Poisson
point processes whose intensity functions con-
tain intractable normalization constants due
to the presence of complex interactions among
points. We apply our proposed test to sev-
eral point process models, and show that it
outperforms a two-sample test based on the
maximum mean discrepancy.
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Scattering Statistics of Generalized Spatial Poisson Point Processes
We present a machine learning model for the analysis of randomly generated discrete signals, modeled as the points of an inhomogeneous, compound Poisson point process. Like the wavelet scattering transform introduced by Mallat, our construction is naturally invariant to translations and reflections, but it decouples the roles of scale and frequency, replacing wavelets with Gabor-type measurements. We show that, with suitable nonlinearities, our measurements distinguish Poisson point processes from common self-similar processes, and separate different types of Poisson point processes.
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- Award ID(s):
- 1845856
- NSF-PAR ID:
- 10336640
- Date Published:
- Journal Name:
- ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)
- Page Range / eLocation ID:
- 5528 - 5532
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We say that one point process on the line R mimics another at a bandwidth B if for each n ≥ 1 the two point processes have n-level correlation functions that agree when integrated against all band-limited test functions on bandwidth [−B, B]. This paper asks the question of for what values a and B can a given point process on the real line be mimicked at bandwidth B by a point process supported on the lattice aZ. For Poisson point processes we give a complete answer for allowed parameter ranges (a,B), and for the sine process we give existence and nonexistence regions for parameter ranges. The results for the sine process have an application to the Alternative Hypothesis regarding the scaled spacing of zeros of the Riemann zeta function, given in a companion paper.more » « less
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Abstract We develop a prior probability model for temporal Poisson process intensities through structured mixtures of Erlang densities with common scale parameter, mixing on the integer shape parameters. The mixture weights are constructed through increments of a cumulative intensity function which is modeled nonparametrically with a gamma process prior. Such model specification provides a novel extension of Erlang mixtures for density estimation to the intensity estimation setting. The prior model structure supports general shapes for the point process intensity function, and it also enables effective handling of the Poisson process likelihood normalizing term resulting in efficient posterior simulation. The Erlang mixture modeling approach is further elaborated to develop an inference method for spatial Poisson processes. The methodology is examined relative to existing Bayesian nonparametric modeling approaches, including empirical comparison with Gaussian process prior based models, and is illustrated with synthetic and real data examples.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.1109/ICASSP43922.2022.9746382
