Spatial prediction with mobile sensor networks using Gaussian processes with built-in Gaussian Markov random fields

In this paper, a new class of Gaussian processes is proposed for resource-constrained mobile sensor networks. Such a Gaussian process builds on a GMRF with respect to a proximity graph over a surveillance region. The main advantages of using this class of Gaussian processes over standard Gaussian processes defined by mean and covariance functions are its numerical efficiency and scalability due to its built-in GMRF and its capability of representing a wide range of non-stationary physical processes. The formulas for predictive statistics such as predictive mean and variance are derived and a sequential field prediction algorithm is provided for sequentially sampled observations. For a special case using compactly supported weighting functions, we propose a distributed algorithm to implement field prediction by correctly fusing all observations. Simulation and experimental results illustrate the effectiveness of our approach.     

Skew Gaussian processes for classification

Machine Learning - Gaussian processes (GPs) are distributions over functions, which provide a Bayesian nonparametric approach to regression and classification. In spite of their success, GPs have...     

Singular ergodic control for multidimensional Gaussian processes

A multidimensional Wiener process is controlled by an additive process of bounded variation. A convex nonnegative function measures the cost associated with the position of the state process, and the cost of controlling is proportional to the displacement induced. We minimize a limiting time-average expected (ergodic) criterion. Under reasonable assumptions, we prove that the optimal discounted cost converges to the optimal ergodic cost. Moreover, under some additional conditions there exists a convex Lipschitz continuous function solution to the corresponding Hamilton-Jacobi-Bellman equation which provides an optimal stationary feedback control.Research supported in part by NSF Grant DMS-8702236.Research supported in part by Grant AFOSR-88-D183.     

Gaussian processes for classification: mean-field algorithms

We derive a mean-field algorithm for binary classification with gaussian processes that is based on the TAP approach originally proposed in statistical physics of disordered systems. The theory also yields an approximate leave-one-out estimator for the generalization error, which is computed with no extra computational cost. We show that from the TAP approach, it is possible to derive both a simpler "naive" mean-field theory and support vector machines (SVMs) as limiting cases. For both mean-field algorithms and support vector machines, simulation results for three small benchmark data sets are presented. They show that one may get state-of-the-art performance by using the leave-one-out estimator for model selection and the built-in leave-one-out estimators are extremely precise when compared to the exact leave-one-out estimate. The second result is taken as strong support for the internal consistency of the mean-field approach.     

Gaussian sum approximation for non-linear fixed-point prediction

Fixed-point prediction is the estimation of the state of a system at a future fixed time based on a noisy measurement with sequence length that increases with current time. A recursive algorithm for generating fixed-point prediction is given using the integrated form of the chain rule. For non-linear systems no general filter solution exists ; thus a gaussian sum approximation is developed. The method provides a numerical approximation for the time-dependant a posteriori density from which a filter can be generated.     

Approximation problems with the divergence criterion for Gaussian variables and Gaussian processes

System identification for stationary Gaussian processes includes an approximation problem. Currently, the subspace algorithm for this problem enjoys much attention. This algorithm is based on a transformation of a finite time series to canonical variable form followed by a truncation. There is no proof that this algorithm is the optimal solution to an approximation problem with a specific criterion. In this paper it is shown that the optimal solution to an approximation problem for Gaussian random variables with the divergence criterion is identical to the main step of the subspace algorithm. An approximation problem for stationary Gaussian processes with the divergence criterion is formulated.     

Rényi entropy rate for Gaussian processes

In this paper, we introduce the definition of the conditional Rényi entropy for continuous random variables and show that the so-called chain rule holds. Then, we use this rule to obtain another relation for getting the rate of Rényi entropy. Using this relation and properties of the Rényi entropy we obtain the Rényi entropy rate for stationary Gaussian processes. Finally, we show that the bound for the Rényi entropy rate is simply the Shannon entropy rate and that the Rényi entropy rate reduces to the Shannon entropy rate as α→1.     

Deep recurrent Gaussian processes for outlier-robust system identification

Gaussian Processes (GP) comprise a powerful kernel-based machine learning paradigm which has recently attracted the attention of the nonlinear system identification community, specially due to its inherent Bayesian-style treatment of the uncertainty. However, since standard GP models assume a Gaussian distribution for the observation noise, i.e., a Gaussian likelihood, the learning and predictive capabilities of such models can be severely degraded when outliers are present in the data. In this paper, motivated by our previous work on GP learning with data containing outliers and recent advances in hierarchical (deep GPs) and recurrent GP (RGP) approaches, we introduce an outlier-robust recurrent GP model, the RGP-t. Our approach explicitly models the observation layer, which includes a heavy-tailed Student-t likelihood, and allows for a hierarchy of multiple transition layers to learn the system dynamics directly from estimation data contaminated by outliers. In addition, we modify the original variational framework of standard RGP in order to perform inference with the new RGP-t model. The proposed approach is comprehensively evaluated using six artificial benchmarks, within several outlier contamination levels, and two datasets related to process industry systems (pH neutralization and heat exchanger), whose estimation data undergo large contamination rates. The simulation results obtained by the RGP-t model indicates an impressive resilience to outliers and a superior capability to learn nonlinear dynamics directly from highly outlier-contaminated data in comparison to existing GP models.     

Learning curves for LMS and regular Gaussian processes

Uses methods due to Guo, Ljung, and Wang (1997) to obtain explicit bounds on the error of the LMS algorithm used in a linear prediction of a signal using previous values of that signal. The signal is assumed to be a mean-zero Gaussian regular stationary random process. The bounds are then used to construct learning curves for the LMS algorithm in situations where the statistics of the process are only partially known     

Gaussian processes and neuronal modeling

This work is a contribution towards the understanding of certain features of mathematical models of single neurons. Emphasis is set on neuronal firing, for which the first passage time (FPT) problem bears a fundamental relevance. We focus the attention on modeling the change of the neuron membrane potential between two consecutive spikes by Gaussian stochastic processes, both of Markov and of non-Markov types. Methods to solve the FPT problems, both of a theoretical and of a computational nature, are sketched, including the case of random initial values. Significant similarities or diversities between computational and theoretical results are pointed out, disclosing the role played by the correlation time that has been used to characterize the neuronal activity. It is highlighted that any conclusion on this matter is strongly model-dependent. In conclusion, an outline of the asymptotic behavior of FPT densities is provided, which is particularly useful to discuss neuronal firing under certain slow activity conditions.     

Data fusion with Gaussian processes

This paper addresses the problem of fusing multiple sets of heterogeneous sensor data using Gaussian processes (GPs). Experiments on large scale terrain modeling in mining automation are presented. Three techniques in increasing order of model complexity are discussed. The first is based on adding data to an existing GP model. The second approach treats data from different sources as different noisy samples of a common underlying terrain and fusion is performed using heteroscedastic GPs. The final approach, based on dependent GPs, models each data set by a separate GP and learns spatial correlations between data sets through auto and cross covariances. The paper presents a unifying view of approaches to data fusion using GPs, a statistical evaluation that compares these approaches and multiple previously untested variants of them and an insight into the effect of model complexity on data fusion. Experiments suggest that in situations where data being fused is not rich enough to require a complex GP data fusion model or when computational resources are limited, the use of simpler GP data fusion techniques, which are constrained versions of the more generic models, reduces optimization complexity and consequently can enable superior learning of hyperparameters, resulting in a performance gain.     

Graph kernels and Gaussian processes for relational reinforcement learning

RRL is a relational reinforcement learning system based on Q-learning in relational state-action spaces. It aims to enable agents to learn how to act in an environment that has no natural representation as a tuple of constants. For relational reinforcement learning, the learning algorithm used to approximate the mapping between state-action pairs and their so called Q(uality)-value has to be very reliable, and it has to be able to handle the relational representation of state-action pairs. In this paper we investigate the use of Gaussian processes to approximate the Q-values of state-action pairs. In order to employ Gaussian processes in a relational setting we propose graph kernels as a covariance function between state-action pairs. The standard prediction mechanism for Gaussian processes requires a matrix inversion which can become unstable when the kernel matrix has low rank. These instabilities can be avoided by employing QR-factorization. This leads to better and more stable performance of the algorithm and a more efficient incremental update mechanism. Experiments conducted in the blocks world and with the Tetris game show that Gaussian processes with graph kernels can compete with, and often improve on, regression trees and instance based regression as a generalization algorithm for RRL. Editors: David Page and Akihiro Yamamoto     

Gaussian processes and limiting linear models

Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the Gaussian processes which encode the linear model either have probability of nearly zero or are otherwise unattainable without the explicit construction of a prior with the limiting linear model in mind. We develop such a prior, and show that its practical benefits extend well beyond the computational and conceptual simplicity of the linear model. For example, linearity can be extracted on a per-dimension basis, or can be combined with treed partition models to yield a highly efficient nonstationary model. Our approach is demonstrated on synthetic and real datasets of varying linearity and dimensionality.     

Maritime abnormality detection using Gaussian processes

Novelty, or abnormality, detection aims to identify patterns within data streams that do not conform to expected behaviour. This paper introduces novelty detection techniques using a combination of Gaussian processes, extreme value theory and divergence measurement to identify anomalous behaviour in both streaming and batch data. The approach is tested on both synthetic and real data, showing itself to be effective in our primary application of maritime vessel track analysis.     

Application of evolutionary Gaussian processes regression by particle swarm optimization for prediction of dew point pressure in gas condensate reservoirs

One of the most critical quantities for characterizing a gas condensate reservoir is dew point pressure. But, accurate determination of dew point pressure is a very challengeable task in reservoir development. Experimental measurement of dew point pressure in PVT (Pressure, Volume, Temperature) cell is often difficult, especially in the case of lean retrograde gas condensate. So, different empirical correlations and equations of state are developed by researchers to calculate this property. Empirical correlations do not have ability to reliably duplicate the temperature behavior of constant composition fluids, and equations of state have convergence problem and need to be tuned against some experimental data. In addition, these approaches are not generalizable to unseen data, and they usually memorize the data used to develop them. In this paper, we develop an intelligent model to predict dew point pressure of gas condensate reservoirs using Gaussian processes optimized by particle swarm optimization. The developed model is generalizable and can estimate unseen data with the same distribution of training data accurately. Results show that the proposed method in this paper outperforms the previous published models and correlations.     

Multiple instance learning via Gaussian processes

Multiple instance learning (MIL) is a binary classification problem with loosely supervised data where a class label is assigned only to a bag of instances indicating presence/absence of positive instances. In this paper we introduce a novel MIL algorithm using Gaussian processes (GP). The bag labeling protocol of the MIL can be effectively modeled by the sigmoid likelihood through the max function over GP latent variables. As the non-continuous max function makes exact GP inference and learning infeasible, we propose two approximations: the soft-max approximation and the introduction of witness indicator variables. Compared to the state-of-the-art MIL approaches, especially those based on the Support Vector Machine, our model enjoys two most crucial benefits: (i) the kernel parameters can be learned in a principled manner, thus avoiding grid search and being able to exploit a variety of kernel families with complex forms, and (ii) the efficient gradient search for kernel parameter learning effectively leads to feature selection to extract most relevant features while discarding noise. We demonstrate that our approaches attain superior or comparable performance to existing methods on several real-world MIL datasets including large-scale content-based image retrieval problems.     

Gaussian processes and limiting linear models

Gaussian processes retain the linear model either as a special case, or in the limit. We show how this relationship can be exploited when the data are at least partially linear. However from the perspective of the Bayesian posterior, the Gaussian processes which encode the linear model either have probability of nearly zero or are otherwise unattainable without the explicit construction of a prior with the limiting linear model in mind. We develop such a prior, and show that its practical benefits extend well beyond the computational and conceptual simplicity of the linear model. For example, linearity can be extracted on a per-dimension basis, or can be combined with treed partition models to yield a highly efficient nonstationary model. Our approach is demonstrated on synthetic and real datasets of varying linearity and dimensionality.     

空间太阳能电站太阳能接收器二维展开过程的保结构分析

针对传统数值方法求解微分-代数方程过程中经常遇到的违约问题,本文以空间太阳能电站太阳能接收器的简化二维模型为例,采用辛算法模拟了简化模型的展开过程,研究了辛算法在求解过程中约束违约问题.首先,基于Hamilton变分原理,将描述简化二维模型展开过程的Euler-Lagrange方程导入Hamilton体系,建立其Hamilton正则方程;随后,采用s级PRK离散方法离散正则方程,得到其辛格式;最后,采用辛PRK格式模拟太阳能接收器的二维展开过程.模拟结果显示:本文构造的辛PRK格式能够很好地满足系统的位移约束.     

Stochastic variational hierarchical mixture of sparse Gaussian processes for regression

In this article, we propose a scalable Gaussian process (GP) regression method that combines the advantages of both global and local GP approximations through a two-layer hierarchical model using a variational inference framework. The upper layer consists of a global sparse GP to coarsely model the entire data set, whereas the lower layer comprises a mixture of sparse GP experts which exploit local information to learn a fine-grained model. A two-step variational inference algorithm is developed to learn the global GP, the GP experts and the gating network simultaneously. Stochastic optimization can be employed to allow the application of the model to large-scale problems. Experiments on a wide range of benchmark data sets demonstrate the flexibility, scalability and predictive power of the proposed method.