We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

The Analysis of Time Series: An Introduction, 4th edn. By C. Chatfield. ISBN 0 412 31820 2. Chapman and Hall, London, 1989. 242 pp. £13.50.

The Analysis of Time Series: An Introduction

Thank you for reading stochastic processes and filtering theory. Maybe you have knowledge that, people have look numerous times for their favorite novels like this stochastic processes and filtering theory, but end up in harmful downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they are facing with some infectious bugs inside their computer. stochastic processes and filtering theory is available in our digital library an online access to it is set as public so you can download it instantly. Our digital library saves in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the stochastic processes and filtering theory is universally compatible with any devices to read.

Stochastic Processes And Filtering Theory

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

The vast quantity of information brought by big data as well as the evolving computer hardware encourages success stories in the machine learning community. In the meanwhile, it poses challenges for the Gaussian process regression (GPR), a well-known nonparametric, and interpretable Bayesian model, which suffers from cubic complexity to data size. To improve the scalability while retaining desirable prediction quality, a variety of scalable GPs have been presented. However, they have not yet been comprehensively reviewed and analyzed to be well understood by both academia and industry. The review of scalable GPs in the GP community is timely and important due to the explosion of data size. To this end, this article is devoted to reviewing state-of-the-art scalable GPs involving two main categories: global approximations that distillate the entire data and local approximations that divide the data for subspace learning. Particularly, for global approximations, we mainly focus on sparse approximations comprising prior approximations that modify the prior but perform exact inference, posterior approximations that retain exact prior but perform approximate inference, and structured sparse approximations that exploit specific structures in kernel matrix; for local approximations, we highlight the mixture/product of experts that conducts model averaging from multiple local experts to boost predictions. To present a complete review, recent advances for improving the scalability and capability of scalable GPs are reviewed. Finally, the extensions and open issues of scalable GPs in various scenarios are reviewed and discussed to inspire novel ideas for future research avenues.

/pdf/when-gaussian-process-meets-big-data-a-review-of-scalable-489iuo4vst.pdf

When Gaussian Process Meets Big Data: A Review of Scalable GPs

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.

/pdf/learning-stationary-time-series-using-gaussian-processes-9ppx2ktsfd.pdf

Learning stationary time series using Gaussian processes with nonparametric kernels

The multikernel least-mean-square algorithm is introduced for adaptive estimation of vector-valued nonlinear and nonstationary signals. This is achieved by mapping the multivariate input data to a Hilbert space of time-varying vector-valued functions, whose inner products (kernels) are combined in an online fashion. The proposed algorithm is equipped with novel adaptive sparsification criteria ensuring a finite dictionary, and is computationally efficient and suitable for nonstationary environments. We also show the ability of the proposed vector-valued reproducing kernel Hilbert space to serve as a feature space for the class of multikernel least-squares algorithms. The benefits of adaptive multikernel (MK) estimation algorithms are illuminated in the nonlinear multivariate adaptive prediction setting. Simulations on nonlinear inertial body sensor signals and nonstationary real-world wind signals of low, medium, and high dynamic regimes support the approach.

Multikernel Least Mean Square Algorithm

Early approaches to multiple-output Gaussian processes (MOGPs) relied on linear combinations of independent, latent, single-output Gaussian processes (GPs). This resulted in cross-covariance functions with limited parametric interpretation, thus conflicting with the ability of single-output GPs to understand lengthscales, frequencies and magnitudes to name a few. On the contrary, current approaches to MOGP are able to better interpret the relationship between different channels by directly modelling the cross-covariances as a spectral mixture kernel with a phase shift. We extend this rationale and propose a parametric family of complex-valued cross-spectral densities and then build on Cramer's Theorem (the multivariate version of Bochner's Theorem) to provide a principled approach to design multivariate covariance functions. The so-constructed kernels are able to model delays among channels in addition to phase differences and are thus more expressive than previous methods, while also providing full parametric interpretation of the relationship across channels. The proposed method is first validated on synthetic data and then compared to existing MOGP methods on two real-world examples.

/pdf/spectral-mixture-kernels-for-multi-output-gaussian-processes-wvloo62ba9.pdf

Spectral Mixture Kernels for Multi-Output Gaussian Processes

The existence and uniqueness conditions of quaternion reproducing kernel Hilbert spaces (QRKHS) are established in order to provide a mathematical foundation for the development of quaternion-valued kernel learning algorithms. This is achieved through a rigorous account of left quaternion Hilbert spaces, which makes it possible to generalise standard RKHS to quaternion RKHS. Quaternion versions of the Riesz representation and Moore-Aronszajn theorems are next introduced, thus underpinning kernel estimation algorithms operating on quaternion-valued feature spaces. The difference between the proposed quaternion kernel concept and the existing real and vector approaches is also established in terms of both theoretical advantages and computational complexity. The enhanced estimation ability of the so-introduced quaternion-valued kernels over their real- and vector-valued counterparts is validated through kernel ridge regression applications. Simulations on real world 3D inertial body sensor data and nonlinear channel equalisation using novel quaternion cubic and Gaussian kernels support the approach.

Quaternion Reproducing Kernel Hilbert Spaces: Existence and Uniqueness Conditions

Timely detection and accurate description of extreme events, such as natural disasters and other crisis situations, are crucial for emergency management and mitigation. Extreme-event detection is challenging, since one has to rely upon reports from human observers appointed to specific geographical areas, or on an expensive and sophisticated infrastructure. In the case of earthquakes, geographically dense sensor networks are expensive to deploy and maintain. Therefore, only some regions—or even countries—are able to acquire useful information about the effects of earthquakes in their own territory. An inexpensive and viable alternative to this problem is to detect extreme real-world events through people's reactions in online social networks. In particular, Twitter has gained popularity within the scientific community for providing access to real-time “citizen sensor” activity. Nevertheless, the massive amount of messages in the Twitter stream, along with the noise it contains, underpin a number of difficulties when it comes to Twitter-based event detection. We contribute to address these challenges by proposing an online method for detecting unusual bursts in discrete-time signals extracted from Twitter. This method only requires a one-off semisupervised initialization and can be scaled to track multiple signals in a robust manner. We also show empirically how our proposed approach, which was envisioned for generic event detection, can be adapted for worldwide earthquake detection, where we compare the proposed model to the state of the art for earthquake tracking using social media. Experimental results validate our approach as a competitive alternative in terms of precision and recall to leading solutions, with the advantage of implementation simplicity and worldwide scalability.

Felipe Tobar

Papers

Learning stationary time series using Gaussian processes with nonparametric kernels

Multikernel Least Mean Square Algorithm

Spectral Mixture Kernels for Multi-Output Gaussian Processes

Quaternion Reproducing Kernel Hilbert Spaces: Existence and Uniqueness Conditions

Robust Detection of Extreme Events Using Twitter: Worldwide Earthquake Monitoring