Multiple output gaussian process regression (original) (raw)

Approximate Inference in Related Multi-output Gaussian Process Regression

Lecture Notes in Computer Science, 2017

In Gaussian Processes a multi-output kernel is a covariance function over correlated outputs. Using a prior known relation between outputs, joint auto-and cross-covariance functions can be constructed. Realizations from these joint-covariance functions give outputs that are consistent with the prior relation. One issue with gaussian process regression is efficient inference when scaling upto large datasets. In this paper we use approximate inference techniques upon multi-output kernels enforcing relationships between outputs. Results of the proposed methodology for theoretical data and real world applications are presented. The main contribution of this paper is the application and validation of our methodology on a dataset of real aircraft flight tests, while imposing knowledge of aircraft physics into the model.

Dependent Gaussian Processes

Advances in neural information processing systems

Gaussian processes are usually parameterised in terms of their covariance functions. However, this makes it difficult to deal with multiple outputs, because ensuring that the covariance matrix is positive definite is problematic. An alternative formulation is to treat Gaussian processes as white noise sources convolved with smoothing kernels, and to parameterise the kernel instead. Using this, we extend Gaussian processes to handle multiple, coupled outputs.

Sparse convolved multiple output gaussian processes

2009

Recently there has been an increasing interest in methods that deal with multiple outputs. This has been motivated partly by frameworks like multitask learning, multisensor networks or structured output data. From a Gaussian processes perspective, the problem reduces to specifying an appropriate covariance function that, whilst being positive semi-definite, captures the dependencies between all the data points and across all the outputs. One approach to account for non-trivial correlations between outputs employs convolution processes. Under a latent function interpretation of the convolution transform we establish dependencies between output variables. The main drawbacks of this approach are the associated computational and storage demands. In this paper we address these issues. We present different sparse approximations for dependent output Gaussian processes constructed through the convolution formalism. We exploit the conditional independencies present naturally in the model. This leads to a form of the covariance similar in spirit to the so called PITC and FITC approximations for a single output. We show experimental results with synthetic and real data, in particular, we show results in pollution prediction, school exams score prediction and gene expression data.

Scalable High-Order Gaussian Process Regression

2019

While most Gaussian processes (GP) work focus on learning single-output functions, many applications, such as physical simulations and gene expressions prediction, require estimations of functions with many outputs. The number of outputs can be much larger than or comparable to the size of training samples. Existing multi-output GP models either are limited to low-dimensional outputs and restricted kernel choices, or assume oversimplified low-rank structures within the outputs. To address these issues, we propose HOGPR, a High-Order Gaussian Process Regression model, which can flexibly capture complex correlations among the outputs and scale up to a large number of outputs. Specifically, we tensorize the high-dimensional outputs, introducing latent coordinate features to index each tensor element (i.e., output) and to capture their correlations. We then generalize a multilinear model to a hybrid of a GP and latent GP model. The model is endowed with a Kronecker product structure ove...

Computationally Efficient Convolved Multiple Output Gaussian Processes

2020

Online Sparse Multi-Output Gaussian Process Regression and Learning

IEEE Transactions on Signal and Information Processing over Networks

This paper proposes an approach for online training of a sparse multi-output Gaussian process (GP) model using sequentially obtained data. The considered model combines linearly multiple latent sparse GPs to produce correlated output variables. Each latent GP has its own set of inducing points to achieve sparsity. We show that given the model hyperparameters, the posterior over the inducing points is Gaussian under Gaussian noise since they are linearly related to the model outputs. However, the inducing points from different latent GPs would become correlated, leading to a full covariance matrix cumbersome to handle. Variational inference is thus applied and an approximate regression technique is obtained, with which the posteriors over different inducing point sets can always factorize. As the model outputs are non-linearly dependent on the hyperparameters, a novel marginalized particle filer (MPF)-based algorithm is proposed for the online inference of the inducing point values and hyperparameters. The approximate regression technique is incorporated in the MPF and its distributed realization is presented. Algorithm validation using synthetic and real data is conducted, and promising results are obtained.

Gaussian process regression with functional covariates and multivariate response

Chemometrics and Intelligent Laboratory Systems, 2017

Gaussian process regression (GPR) has been shown to be a powerful and effective nonparametric method for regression, classification and interpolation, due to many of its desirable properties. However, most GPR models consider univariate or multivariate covariates only. In this paper we extend the GPR models to cases where the covariates include both functional and multivariate variables and the response is multidimensional. The model naturally incorporates two different types of covariates: multivariate and functional, and the principal component analysis is used to de-correlate the multivariate response which avoids the widely recognised difficulty in the multi-output GPR models of formulating covariance functions which have to describe the correlations not only between data points but also between responses. The usefulness of the proposed method is demonstrated through a simulated example and two real data sets in chemometrics.

Gaussian Processes for Regression and Optimisation

Gaussian processes have proved to be useful and powerful constructs for the purposes of regression. The classical method proceeds by parameterising a covariance function, and then infers the parameters given the training data. In this thesis, the classical approach is augmented by interpreting Gaussian processes as the outputs of linear filters excited by white noise. This enables a straightforward definition of dependent Gaussian processes as the outputs of a multiple output linear filter excited by multiple noise sources. We show how dependent Gaussian processes defined in this way can also be used for the purposes of system identification.

A conditional one-output likelihood formulation for multitask Gaussian processes

Neurocomputing, 2022

Multitask Gaussian processes (MTGP) are the Gaussian process (GP) framework's solution for multioutput regression problems in which the T elements of the regressors cannot be considered conditionally independent given the observations. Standard MTGP models assume that there exist both a multitask covariance matrix as a function of an intertask matrix, and a noise covariance matrix. These matrices need to be approximated by a low rank simplification of order P in order to reduce the number of parameters to be learnt from T 2 to T P. Here we introduce a novel approach that simplifies the multitask learning by reducing it to a set of conditioned univariate GPs without the need for any low rank approximations, therefore completely eliminating the requirement to select an adequate value for hyperparameter P. At the same time, by extending this approach with both a hierarchical and an approximate model, the proposed extensions are capable of recovering the multitask covariance and noise matrices after learning only 2T parameters, avoiding the validation of any model hyperparameter and reducing the overall complexity of the model as well as the risk of overfitting. Experimental results over synthetic and real problems confirm the advantages of this inference approach in its ability to accurately recover the original noise and signal matrices, as well as the achieved performance improvement in comparison to other state of art MTGP approaches. We have also integrated the model with standard GP toolboxes, showing that it is computationally competitive with state of the art options.

Hierarchical gaussian process regression

2010

We address an approximation method for Gaussian process (GP) regression, where we approximate covariance by a block matrix such that diagonal blocks are calculated exactly while off-diagonal blocks are approximated. Partitioning input data points, we present a two-layer hierarchical model for GP regression, where prototypes of clusters in the upper layer are involved for coarse modeling by a GP and data points in each cluster in the lower layer are involved for fine modeling by an individual GP whose prior mean is given by the corresponding prototype and covariance is parameterized by data points in the partition. In this hierarchical model, integrating out latent variables in the upper layer leads to a block covariance matrix, where diagonal blocks contain similarities between data points in the same partition and off-diagonal blocks consist of approximate similarities calculated using prototypes. This particular structure of the covariance matrix divides the full GP into a pieces of manageable sub-problems whose complexity scales with the number of data points in a partition. In addition, our hierarchical GP regression (HGPR) is also useful for cases where partitions of data reveal different characteristics. Experiments on several benchmark datasets confirm the useful behavior of our method.

Multiple output gaussian process regression (original) (raw)

Related papers