Gaussian Process Product Models for Nonparametric Nonstationarity
Description
Stationarity is often an unrealistic prior assumption for Gaussian process regression. One solution is to predefine an explicit nonstationary covariance function, but such covariance functions can be difficult to specify and require detailed prior knowledge of the nonstationarity. We propose the Gaussian process product model (GPPM) which models data as the pointwise product of two latent Gaussian processes to nonparametrically infer nonstationary variations of amplitude. This approach differs from other nonparametric approaches to covariance function inference in that it operates on the outputs rather than the inputs, resulting in a significant reduction in computational cost and required data for inference, while improving scalability to high-dimensional input spaces. We present an approximate inference scheme using Expectation Propagation. This variational approximation yields convenient GP hyperparameter selection and compact approximate predictive distributions.
| Slides | |
| 0:00 | Gaussian Process Product Models for Nonparametric Nonstationarity |
| 0:30 | Gaussian Processes for Regression - 1 |
| 0:47 | Gaussian Processes for Regression - 2 |
| 1:06 | Gaussian Processes for Regression - 3 |
| 1:17 | Gaussian Processes for Regression - 4 |
| 1:41 | Gaussian Processes for Regression: Predictions and Hyperparameters - 1 |
| 2:21 | Gaussian Processes for Regression: Predictions and Hyperparameters - 2 |
| 2:36 | Gaussian Processes for Regression: The Covariance Function - 1 |
| 3:08 | Gaussian Processes for Regression: The Covariance Function - 2 |
| 3:23 | Gaussian Processes for Regression: Stationarity vs Nonstationarity - 1 |
| 3:44 | Gaussian Processes for Regression: Stationarity vs Nonstationarity - 2 |
| 3:53 | Gaussian Processes for Regression: Stationarity vs Nonstationarity - 3 |
| 4:08 | Gaussian Processes for Regression: Stationarity vs Nonstationarity - 4 |
| 4:17 | Gaussian Processes for Regression: Stationarity vs Nonstationarity - 5 |
| 4:31 | Latent Space Extensions of Stationary Covariances - 1 |
| 4:49 | Latent Space Extensions of Stationary Covariances - 2 |
| 5:07 | Latent Space Extensions of Stationary Covariances - 3 |
| 5:19 | Latent Space Extensions of Stationary Covariances - 4 |
| 5:43 | Outline |
| 5:46 | Outline - Gaussian Process Product Model |
| 5:59 | The Gaussian Process Product Model: Varying Amplitudes - 1 |
| 6:14 | The Gaussian Process Product Model: Varying Amplitudes - 2 |
| 6:19 | The Gaussian Process Product Model: Varying Amplitudes - 3 |
| 6:36 | The Gaussian Process Product Model |
| 7:27 | The Gaussian Process Product Model: Samples from the Model - 1 |
| 7:54 | The Gaussian Process Product Model: Samples from the Model - 2 |
| 8:25 | Outline - Inference |
| 8:34 | GPPM Posterior - 1 |
| 8:41 | GPPM Posterior - 2 |
| 8:47 | GPPM Posterior - 3 |
| 8:49 | GPPM Posterior - 4 |
| 8:57 | GPPM Posterior - 5 |
| 9:09 | Approximate Inference - 1 |
| 9:52 | Approximate Inference - 2 |
| 10:06 | EP in a Nutshell - 1 |
| 10:40 | EP in a Nutshell - 2 |
| 11:10 | EP in a Nutshell - 3 |
| 11:21 | EP in the GPPM Model - 1 |
| 11:36 | EP in the GPPM Model - 2 |
| 11:46 | EP in the GPPM Model - 3 |
| 12:15 | EP in the GPPM Model - 4 |
| 12:32 | EP in the GPPM Model - 5 |
| 13:14 | EP in the GPPM Model - 6 |
| 13:48 | EP in the GPPM Model: Optimizing Hyperparameters - 1 |
| 14:02 | EP in the GPPM Model: Optimizing Hyperparameters - 2 |
| 14:11 | EP in the GPPM Model: Optimizing Hyperparameters - 3 |
| 14:15 | EP in the GPPM Model: Optimizing Hyperparameters - 4 |
| 14:29 | EP in the GPPM Model: Optimizing Hyperparameters - 5 |
| 14:33 | EP in the GPPM Model: Optimizing Hyperparameters - 6 |
| 14:38 | EP in the GPPM Model: Optimizing Hyperparameters - 7 |
| 15:02 | EP in the GPPM Model: Making Predictions - 1 |
| 15:20 | EP in the GPPM Model: Making Predictions - 2 |
| 15:51 | EP in the GPPM Model: Making Predictions - 3 |
| 16:01 | Outline - Results |
| 16:05 | Results |
| 16:49 | Motorcycle Helmet Data - 1 |
| 17:14 | Motorcycle Helmet Data - 2 |
| 17:32 | SP500 Log Daily Returns - 1 |
| 17:52 | SP500 Log Daily Returns - 2 |
| 18:01 | Heart Rate Data - 1 |
| 18:20 | Heart Rate Data - 2 |
| 18:27 | Summary - 1 |
| 18:28 | Summary - 2 |
| 18:51 | - Questions |
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