# Spherical Poisson Point Process Intensity Function Modeling and Estimation with Measure Transport

arXiv: https://arxiv.org/abs/2201.09485 · HTML: https://mlsanity.com/paper/2201.09485/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 45 references that are in the map.
- First posted on arXiv 24 Jan 2022.

## Builds on (most cited first, 23 of 23)

- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [On the Expressive Power of Deep Neural Networks](https://mlsanity.com/paper/1606.05336/) (2016)
- [Neural Autoregressive Flows](https://mlsanity.com/paper/1804.00779/) (2018)
- [The Power of Depth for Feedforward Neural Networks](https://mlsanity.com/paper/1512.03965/) (2015)
- [Riemannian Continuous Normalizing Flows](https://mlsanity.com/paper/2006.10605/) (2020)
- [Normalizing Flows on Tori and Spheres](https://mlsanity.com/paper/2002.02428/) (2020)
- [Neural Spatio-Temporal Point Processes](https://mlsanity.com/paper/2011.04583/) (2020)
- [Normalizing Flows on Riemannian Manifolds](https://mlsanity.com/paper/1611.02304/) (2016)
- [Sum-of-Squares Polynomial Flow](https://mlsanity.com/paper/1905.02325/) (2019)
- [Neural Networks for Parameter Estimation in Intractable Models](https://mlsanity.com/paper/2107.14346/) (2021)
- [Fast covariance parameter estimation of spatial Gaussian process models using neural networks](https://mlsanity.com/paper/2012.15339/) (2020)
- [DeepKriging: Spatially Dependent Deep Neural Networks for Spatial Prediction](https://mlsanity.com/paper/2007.11972/) (2020)
- [Deep Compositional Spatial Models](https://mlsanity.com/paper/1906.02840/) (2019)
- [Bayesian Recurrent Neural Network Models for Forecasting and Quantifying Uncertainty in Spatial-Temporal Data](https://mlsanity.com/paper/1711.00636/) (2017)
- [Deep Integro-Difference Equation Models for Spatio-Temporal Forecasting](https://mlsanity.com/paper/1910.13524/) (2019)
- [Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields](https://mlsanity.com/paper/2108.04211/) (2021)
- [Deep Gaussian Markov Random Fields](https://mlsanity.com/paper/2002.07467/) (2020)
- [Bayesian Inverse Reinforcement Learning for Collective Animal Movement](https://mlsanity.com/paper/2009.04003/) (2020)
- [Determinantal point process models on the sphere](https://mlsanity.com/paper/1607.03675/) (2016)
- [Non-Homogeneous Poisson Process Intensity Modeling and Estimation using Measure Transport](https://mlsanity.com/paper/2007.00248/) (2020)
- [Compositionally-warped additive mixed modeling for a wide variety of non-Gaussian spatial data](https://mlsanity.com/paper/2101.03684/) (2021)

## Built on by (most cited first, 1 of 1)

- [Statistical Deep Learning for Spatial and Spatio-Temporal Data](https://mlsanity.com/paper/2206.02218/) (2022)

## Abstract (arXiv)

Recent years have seen an increased interest in the application of methods and techniques commonly associated with machine learning and artificial intelligence to spatial statistics. Here, in a celebration of the ten-year anniversary of the journal Spatial Statistics, we bring together normalizing flows, commonly used for density function estimation in machine learning, and spherical point processes, a topic of particular interest to the journal's readership, to present a new approach for modeling non-homogeneous Poisson process intensity functions on the sphere. The central idea of this framework is to build, and estimate, a flexible bijective map that transforms the underlying intensity function of interest on the sphere into a simpler, reference, intensity function, also on the sphere. Map estimation can be done efficiently using automatic differentiation and stochastic gradient descent, and uncertainty quantification can be done straightforwardly via nonparametric bootstrap. We investigate the viability of the proposed method in a simulation study, and illustrate its use in a proof-of-concept study where we model the intensity of cyclone events in the North Pacific Ocean. Our experiments reveal that normalizing flows present a flexible and straightforward way to model intensity functions on spheres, but that their potential to yield a good fit depends on the architecture of the bijective map, which can be difficult to establish in practice.