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Large-Scale Cox Process Inference Using Variational Fourier Features

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Large-Scale Cox Process Inference Using Variational Fourier Features Large-Scale Cox Process Inference using Variational Fourier Features ST John 1 James Hensman 1 Abstract spatiotemporal domain, which cannot be computed in gen- eral. There are three potential remedies to this issue. First, Gaussian process modulated Poisson processes the classic approach is to discretize the input domain (see, provide a flexible framework for modeling spa- e.g., Taylor et al., 2015), assuming that the rate is constant tiotemporal point patterns. So far this was re- over each grid cell. However, this discretized structure stricted to one dimension, binning to a pre- scales poorly with the number of grid cells. Second, we determined grid, or small data sets of up to a can use a thinning strategy to construct an exact MCMC few thousand data points. Here we introduce Cox sampler (Adams et al., 2009; Gunter et al., 2014), which process inference based on Fourier features. This demands the inverse link function to be bounded, e.g., sig- sparse representation induces global rather than moidal. However, this is prohibitively expensive for more local constraints on the function space and is com- than a few thousand points. Finally, we can use a square- putationally efficient. This allows us to formulate root link function λ(·) = f(·)2, as proposed by Lloyd et al. a grid-free approximation that scales well with the (2015). Combined with variational inference for the latent number of data points and the size of the domain. function, this can make the integral tractable. Although their We demonstrate that this allows MCMC approxi- approach scales linearly with the number of events, it scales mations to the non-Gaussian posterior. In practice, poorly with the size of the domain, especially if the rate we find that Fourier features have more consistent has high variance. Flaxman et al. (2017) make use of the optimization behavior than previous approaches. same link function with a frequentist approach, and Walder Our approximate Bayesian method can fit over & Bishop (2017) use a Laplace approximation. 100 000 events with complex spatiotemporal pat- terns in three dimensions on a single GPU. In this work we build on Lloyd et al. (2015)’s proposal. We similarly make use of the square-root link function for tractability, but introduce several innovations that greatly 1. Introduction improve the practical applicability of the method. Modeling spatiotemporal point patterns is a common task First, we extend the derivation to be able to use the Fourier in geostatistics, for example in ecology and epidemiology representation of the GP proposed by Hensman et al. (2018). (Diggle et al., 2013; Vanhatalo & Vehtari, 2007). We are This allows for a choice of kernels in the Matern´ family. further motivated by modeling events occurring in a city, Additionally, we show how the posterior process can be e.g., taxi pickups or crime incidents (Flaxman et al., 2018). approximated using standard MCMC methods (Hensman et al., 2015). This allows for flexible representations of the Gaussian process (GP) modulated Poisson processes pro- posterior over functions f(·), and a Bayesian treatment of vide a flexible Bayesian model for such data. The model the hyperparameters, whereas Lloyd et al. (2015)’s approach includes a GP prior over a latent function f(·), which is was restricted to a Gaussian approximation for f(·) and related to the rate λ(·) of an inhomogeneous Poisson pro- point estimates of the hyperparameters. cess through a link function. This is usually taken to be f(·) = log(λ(·)), resulting in the Log Gaussian Cox pro- Second, we deliver insights and improvements to the model. cess (Møller et al., 1998). These models are computationally The use of a square-root link function can result in nodal challenging as they are doubly intractable (Murray, 2007). lines: where the latent function f(·) crosses zero, the rate The likelihood involves an integral of the process over the function λ(·) must approach zero on both sides, resulting in regions of the posterior that have unsatisfactory artifacts. 1PROWLER.io, 66-68 Hills Road, Cambridge CB2 1LA, In addition, these nodal lines create multiple modes in the United Kingdom. Correspondence to: ST John <[email protected]>, posterior that are hard to sample and difficult to approximate < > James Hensman [email protected] . well with a Gaussian approach. We investigate these effects Proceedings of the 35 th International Conference on Machine and demonstrate how they can be mitigated by suitable Learning, Stockholm, Sweden, PMLR 80, 2018. Copyright 2018 specification of the prior parameters. by the author(s). Large-Scale Cox Process Inference using Variational Fourier Features Third, motivated by spatiotemporal problems where a peri- 0 for any value of f. Lloyd et al. (2015) used λ(·) = f(·)2, odic function is expected a priori, we construct a periodic which has the advantage of tractable analytic derivations kernel that can be combined with the Fourier features ap- and, with a sparse GP approximation, linear scaling in the proach to Matern´ kernels under our inference scheme. We number of observed events. However, the lack of injectivity also show how to obtain a closed-form confidence interval of f 2 can lead to nodal lines (see section 4.5). To mitigate for the rate λ under a Gaussian approximation. this disadvantage, we extend the link function to include a constant offset1 β, so that λ(·) = (f(·) + β)2. Defining Θ The key result of our work is that we can now scale Bayesian to contain β and the GP hyperparameters, our model is point process inference on a single GPU to hundreds of thou- sands of events from multiple observations with complex p(D; f(·); Θ)=p(D j λ=(f(·)+β)2)p(f(·) j Θ)p(Θ): (3) spatiotemporal patterns. We demonstrate this on the Porto taxi data set (Moreira-Matias et al., 2013). For tractable inference we approximate the exact GP pos- terior p(f(·); Θ j D) by q(f(·) j Θ)q(Θ), where the approx- 2. Cox Process Inference imate GP posterior is q(f(·) j Θ) = GP(f;µ ~(·); Σ(~ ·; ·)). [λ(·)] =µ ~(·)2 + Σ(~ ·) A Cox process is an inhomogeneous Poisson process, where Then Eq(f(·)) is the expected rate un- the rate λ is itself a stochastic process. The probability of der the posterior. We now derive the inference objective, N and later discuss our choices for q(f(·)). the data D = fxngn=1 under an inhomogeneous Poisson process with known rate function λ(·) is given by 2.3. Inference Objective N Z nx Y λ(xn) n p(D j λ(·)) = exp − λ(x) dx ; (1) We rederive the Evidence Lower Bound Objective (ELBO) n ! T n=1 xn from Lloyd et al. (2015) to account for our extended model specification. We optimize q(f(·)) by minimizing the KL where T is the domain of the observation and nxn is the divergence to the true posterior, where we leave the depen- multiplicity of events at location xn. In the following, we dence on the hyperparameters implicit:2 assume that all events are distinct and nxn = 1. We want to infer the posterior distribution of the rate func- K = KL[q(f(·)) k p(f(·) j D)] tion given an observation D, which is given by p(λ(·) j D) = h p(f(·); D)=p(D)i p(λ(·))p(D j λ(·)) = −Eq(f(·)) log R p(λ(·))p(D j λ(·)) dλ(·) . The marginal likelihood in the de- q(f(·)) nominator involves a double integral, which gives rise to the h p(f(·))i = − log − log p(D j f(·)) + log p(D) so-called “double intractability” (Murray, 2007). Eq q(f(·)) Eq = KL[q(f(·)) k p(f(·))] − LD + log p(D) 2.1. Multiple Observations = log p(D) − L; (4) We can consider multiple observations corresponding to sev- eral draws from the same distribution. For example, when where L is the ELBO from Lloyd et al. (2015). Minimiz- considering the rate of taxi pickups in a city, we can take into ing K with respect to a variational distribution q(f(·)) is (o) No account observations Do = fxn gn=1 from several days, equivalent to maximizing the ELBO L. We use the result o = 1;:::;O, each containing No events. Instead of model- from Matthews et al. (2016) that the KL between the ap- ing each day independently, we model the pickup rate of the proximate posterior and prior processes is a KL divergence average day, while making use of all available data. Assum- at the inducing points. For details, see Matthews (2016). ing that observations are i.i.d. with the same rate function For our Cox process model, the likelihood term L is λ(·), the likelihood factorizes across observations, D O O Q LD = Eq(f(·))[log p(fDogo=1g j f(·))] p(D1;:::; DO j λ(·)) = o=1 p(Do j λ(·)); (2) h Z i = −O λ(x) dx where p(Do j λ(·)) is given by eq. (1). Eq(f(·)) T O N X X o (o) + q(f(·))[log λ(xn )] 2.2. Cox Process Constructions with Gaussian o=1 n=1 E Processes = −OLfx + Lfn ; (5) We model the intensity function λ(·) using a GP, f(·) ∼ GP. 1This is equivalent to a prior mean on the GP. Our method While f(·) is unbounded between −∞ and +1, the rate could also be extended to include covariates whose coefficients λ(·) must be non-negative everywhere. This is achieved by can be learnt within our inference frameworks. an inverse link function ρ(·) that ensures λ(·) = ρ(f(·)) ≥ 2The full derivation is given in the Supplementary Material. Large-Scale Cox Process Inference using Variational Fourier Features where we allow for multiple observations, and the sum in where σ2 = k(x; x) is the variance of the kernel (assum- PO R Lf extends over the N = No events in all obser- ing stationarity), Ψ = ku(x)ku(x)| dx, and Φ = n o=1 R T vations.
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