Multi-Response Heteroscedastic Gaussian Process Models and Their Inference

Despite the widespread utilization of Gaussian process models for versatile nonparametric modeling, they exhibit limitations in effectively capturing abrupt changes in function smoothness and accommodating relationships with heteroscedastic errors. Addressing these shortcomings, the heteroscedastic Gaussian process (HeGP) regression seeks to introduce flexibility by acknowledging the variability of residual variances across covariates in the regression model. In this work, we extend the HeGP concept, expanding its scope beyond regression tasks to encompass classification and state-space models. To achieve this, we propose a novel framework where the Gaussian process is coupled with a covariate-induced precision matrix process, adopting a mixture formulation. This approach enables the modeling of heteroscedastic covariance functions across covariates. To mitigate the computational challenges posed by sampling, we employ variational inference to approximate the posterior and facilitate posterior predictive modeling. Additionally, our training process leverages an EM algorithm featuring closed-form M-step updates to efficiently evaluate the heteroscedastic covariance function. A notable feature of our model is its consistent performance on multivariate responses, accommodating various types (continuous or categorical) seamlessly. Through a combination of simulations and real-world applications in climatology, we illustrate the model's prowess and advantages. By overcoming the limitations of traditional Gaussian process models, our proposed framework offers a robust and versatile tool for a wide array of applications.

Heteroscedastic Gaussian Process Regression on the Alkenone over Sea Surface Temperatures

To restore the historical sea surface temperatures (SSTs) better, it is important to construct a good calibration model for the associated proxies. In this paper, we introduce a new model for alkenone (${\rm{U}}_{37}^{\rm{K}'}$) based on the heteroscedastic Gaussian process (GP) regression method. Our nonparametric approach not only deals with the variable pattern of noises over SSTs but also contains a Bayesian method of classifying potential outliers.

Dual Proxy Gaussian Process Stack: Integrating Benthic $δ^{18}{\rm{O}}$ and Radiocarbon Proxies for Inferring Ages on Ocean Sediment Cores

Ages in ocean sediment cores are often inferred using either benthic ${\delta}^{18}{\rm{O}}$ or planktonic ${}^{14}{\rm{C}}$ of foraminiferal calcite. Existing probabilistic dating methods infer ages in two distinct approaches: ages are either inferred directly using radionuclides, e.g. Bacon [Blaauw and Christen (2011)]; or indirectly based on the alignment of records, e.g. HMM-Match [Lin et al. (2014)]. In this paper, we introduce a novel algorithm for integrating these two approaches by constructing Dual Proxy Gaussian Process (DPGP) stacks, which represent a probabilistic model of benthic ${\delta}^{18}{\rm{O}}$ change (and its timing) based on a set of cores. While a previous stack construction algorithm, HMM-Match, uses a discrete age inference model based on Hidden Markov models (HMMs) [Durbin et al. (1998)] and requires a number of records enough to sufficiently cover all its ages, DPGP stacks with time-varying variances are constructed with continuous ages obtained by particle smoothing [Doucet et al. (2001); Klaas et al. (2006)] and Markov-chain Monte Carlo (MCMC) [Peters (2008)] algorithms, and can be derived from a small number of records by applying the Gaussian process regression [Rasmussen and Williams (2005)]. As an example of the stacking method, we construct a local stack from 6 cores in the deep northeastern Atlantic Ocean and compare it to a deterministically constructed ${\delta}^{18}{\rm{O}}$ stack of 58 cores from the deep North Atlantic [Lisiecki and Stern (2016)]. We also provide two examples of how dual proxy alignment ages can be inferred by aligning additional cores to the stack.