In sensing applications, sensors cannot always measure the latent quantity of interest at the required resolution, sometimes they can only acquire a blurred version of it due the sensor's transfer function. To recover latent signals when only noisy mixed measurements of the signal are available, we propose the Gaussian process mixture of measurements (GPMM), which models the latent signal as a Gaussian process (GP) and allows us to perform Bayesian inference on such signal conditional to a set of noisy mixture of measurements. We describe how to train GPMM, that is, to find the hyperparameters of the GP and the mixing weights, and how to perform inference on the latent signal under GPMM; additionally, we identify the solution to the underdetermined linear system resulting from a sensing application as a particular case of GPMM. The proposed model is validated in the recovery of three signals: A smooth synthetic signal, a real-world heart-rate time series and a step function, where GPMM outperformed the standard GP in terms of estimation error, uncertainty representation, and recovery of the spectral content of the latent signal.
The GP framework is well suited to the sensing setting in fig. , since modelling latent signal as a GP results in the posterior distribution of the latent process (given the mixture of measurements) being Gaussian as well. Previous GP-based models for convolution processes @cite_3 @cite_12 @cite_9 @cite_11 model signals a convolution between a continuous-time filter and a white-noise process, which is unsuitable to represent the latent process in the sensing application where the spatial correlation of the process is fundamental. Conversely, @cite_8 allows GPs as latent functions but to address the multi-output case, where the aim is to perform inference on the outputs rather than the latent processes. Furthermore, these methods consider continuous-time convolution filters, which is computationally demanding and requires, e.g., variational approximations @cite_2 @cite_14 . Consequently, closed-form and computationally-efficient Bayesian reconstruction of the latent process is still an open problem in sensing applications.
@cite_14 @cite_8 @cite_9 @cite_3 @cite_2 @cite_12 @cite_11
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In this paper, we propose a Gaussian process (GP) model for analysis of nonlinear time series. Formulation of our model is based on the consideration that the observed data are functions of latent variables, with the associated mapping between observations and latent representations modeled through GP priors. In addition, to capture the temporal dynamics in the modeled data, we assume that subsequent latent representations depend on each other on the basis of a hidden Markov prior imposed over them. Derivation of our model is performed by marginalizing out the model parameters in closed form using GP priors for observation mappings, and appropriate stick-breaking priors for the latent variable (Markovian) dynamics. This way, we eventually obtain a nonparametric Bayesian model for dynamical systems that accounts for uncertainty in the modeled data. We provide efficient inference algorithms for our model on the basis of a truncated variational Bayesian approximation. We demonstrate the efficacy of our approach considering a number of applications dealing with real-world data, and compare it with the related state-of-the-art approaches. We present a sparse approximation approach for dependent output Gaussian processes (GP). Employing a latent function framework, we apply the convolution process formalism to establish dependencies between output variables, where each latent function is represented as a GP. Based on these latent functions, we establish an approximation scheme using a conditional independence assumption between the output processes, leading to an approximation of the full covariance which is determined by the locations at which the latent functions are evaluated. We show results of the proposed methodology for synthetic data and real world applications on pollution prediction and a sensor network. 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. We introduce the Gaussian Process Convolution Model (GPCM), a two-stage non-parametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametric-window moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational free-energy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals. Sparse Gaussian process methods that use inducing variables require the selection of the inducing inputs and the kernel hyperparameters. We introduce a variational formulation for sparse approximations that jointly infers the inducing inputs and the kernel hyperparameters by maximizing a lower bound of the true log marginal likelihood. The key property of this formulation is that the inducing inputs are defined to be variational parameters which are selected by minimizing the Kullback-Leibler divergence between the variational distribution and the exact posterior distribution over the latent function values. We apply this technique to regression and we compare it with other approaches in the literature. A continuous spatial model can be constructed by convolving a very simple, perhaps independent, process with a kernel or point spread function. This approach for constructing a spatial process offers a number of advantages over specification through a spatial covariogram. In particular, this process convolution specification leads to computational simplifications and easily extends beyond simple stationary models. This paper uses process convolution models to build space and space-time models that are flexible and able to accommodate large amounts of data. Data from environmental monitoring is considered.