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In this paper we use a rotation property of Wiener measure to define a very general multiple Fourier–Feynman transform on Wiener space. We then proceed to establish its many algebraic properties as well as to establish several relationships between this generalized multiple transform and the corresponding generalized convolution product.

We define two generalized sequential transforms on the function space $C_{a,b}[0,T]$. In order to find the essential structure of the generalized sequential transforms we first investigate the relationships between the generalized analytic ${{𝒵}}_k$-Feynman integral and the generalized ${{𝒵}}_k$-function space integral. We then establish the existences of our sequential transforms on $C_{a,b}[0,T]$. One of these sequential transforms identifies the generalized analytic ${{𝒵}}_k$-Fourier–Feynman transform and the other transform plays a role as an inverse transform of the generalized analytic ${{𝒵}}_k$-Fourier–Feynman transform.

Voice conversion (VC) is a technique that aims to transform the individuality of a source speech so as to mimic that of a target speech while keeping the message unaltered. In our previous work, Gaussian process (GP) was introduced into the literature of VC for the first time, for the sake of overcoming the “over-fitting” problem inherent in the state-of-the-art VC methods, which gives very promising results. However, standard GP usually acts as somewhat a smoothing device more than a universal approximator. In this paper, we further attempt to improve the flexibility of GP-based VC by resorting to the expressive kernels that are derived to model the spectral density with Gaussian mixture model (GMM). Our new method benefits from the expressiveness of the new kernel while the inference of GP remains simple and analytic as usual. Experiments demonstrate both objectively and subjectively that the individualities of the converted speech are much more closer to those of the target while speech quality obtained is comparable to the standard GP-based method.

In this paper, we propose a new approach for anomaly detection in video surveillance. This approach is based on a nonparametric Bayesian regression model built upon Gaussian process priors. It establishes a set of basic vectors describing motion patterns from low-level features via online clustering, and then constructs a Gaussian process regression model to approximate the distribution of motion patterns in kernel space. We analyze different anomaly measure criterions derived from Gaussian process regression model and compare their performances. To reduce false detections caused by crowd occlusion, we utilize supplement information from previous frames to assist in anomaly detection for current frame. In addition, we address the problem of hyperparameter tuning and discuss the method of efficient calculation to reduce computation overhead. The approach is verified on published anomaly detection datasets and compared with other existing methods. The experiment results demonstrate that it can detect various anomalies efficiently and accurately.

Understanding the expansion rate history is a primary goal in cosmology. There are various ways to measure the expansion rate at different redshifts and measuring it through the luminosity distance of SNIa is widely used in cosmology. In this work, we split the latest dataset of SNIa (the Pantheon+ sample) into several redshift bins and compute the expansion rate through a non-parametric technique. To do this, we use the Gaussian process to obtain the expansion rate considering data in different redshift bins. For the sake of comparison, we also consider the $\mathrm{\Lambda }$CDM model and find the expansion history through an MCMC analysis. Our results indicate that the degeneracy between free parameters prevents a good constraint using the low redshifts data but this is not the case in the GP scenario. Moreover, we study how high redshift SNIa can affect the current expansion rate and observe that data at $z>0.5$ has almost no impact on the current expansion rate in both approaches.

To further explore the research of financial time series prediction and broaden the application scope of Gaussian distribution probability density equation, based on the nonlinear differential equation of Gaussian distribution probability density, the semi-supervised Gaussian process model is taken as the research object to discuss the application of semi-supervised Gaussian process model in the stock market. Shanghai 180 Index, Shanghai (Securities) Composite Index and the yield of three stocks are studied in detail. The specific results are as follows. The prediction accuracy of Shanghai 180 Index is 83%, and the prediction accuracy of Shanghai (Securities) Composite Index is 78%. The stock yield series curves of the three stocks have the characteristics of peak and thick tail, which do not obey the normal distribution. Among the three models of semi-supervised Gaussian process model, initial Gaussian process model and SVM algorithm model, the prediction accuracy of semi-supervised Gaussian process model is the best, and the prediction accuracy of the yield of three stocks is 82.45%, 85.03% and 84.53%, respectively. The research results can fully prove that the semi-supervised Gaussian process model has good application effect in stock time series prediction. The research content can provide scientific and sufficient reference for the follow-up research of financial time series, and also has important significance for the research of Gaussian process model.

In this paper, for sparse multi-label data, based on inter-instance relations and inter-label correlation, a Sparse Multi-Label Kernel Gaussian Neural Network (SMLKGNN) framework is proposed. Double insurance for the sparse multi-label datasets is constructed with bidirectional relations such as inter-instance and inter-label. When instance features or label sets are too sparse to be extracted effectively, we argument that the inter-instance relations and inter-label correlation can supplement and deduce the relevant information. Meanwhile, to enhance the explainable of neural network, Gaussian process is adopted to simulate the real underlying distribution of multi-label dataset. Besides, this paper also considers that contributions of different features have different effects on the experimental results, thus self-attention is leveraged to balance various features. Finally, the applicability of the algorithm is verified in three sparse datasets, and the generalization performance is also validated in three groups of benchmark datasets.

While the proportional hazard model is recognized to be statistically meaningful for analyzing and estimating financial event risks, the existing literature that analytically deals with the valuation problems is very limited. In this paper, adopting the proportional hazard model in continuous time setting, we provide an analytical treatment for the valuation problems. The derived formulas, which are based on the generalized Edgeworth expansion and give approximate solutions to the valuation problems, are widely useful for evaluating a variety of financial products such as corporate bonds, credit derivatives, mortgage-backed securities, saving accounts and time deposits. Furthermore, the formulas are applicable to the proportional hazard model having not only continuous processes (e.g., Gaussian, affine, and quadratic Gaussian processes) but also discontinuous processes (e.g., Lévy and time-changed Lévy processes) as stochastic covariates. Through numerical examples, it is demonstrated that very accurate values can be quickly obtained by the formulas such as a closed-form formula.

There has been growing recent interest in probabilistic interpretations of kernel-based methods as well as learning in Banach spaces. The absence of a useful Lebesgue measure on an infinite-dimensional reproducing kernel Hilbert space is a serious obstacle for such stochastic models. We propose an estimation model for the ridge regression problem within the framework of abstract Wiener spaces and show how the support vector machine solution to such problems can be interpreted in terms of the Gaussian Radon transform.

Structural health monitoring data has been widely acknowledged as a significant source for evaluating the performance and health conditions of structures. However, a holistic framework that efficiently incorporates monitored data into structural identification and, in turn, provides a realistic life-cycle performance assessment of structures is yet to be established. There are different sources of uncertainty, such as structural parameters, computer model bias and measurement errors. Neglecting to account for these factors results in unreliable structural identifications, consequent financial losses, and a threat to the safety of structures and human lives. This paper proposes a new framework for structural performance assessment that integrates a comprehensive probabilistic finite element model updating approach, which deals with various structural identification uncertainties and structural reliability analysis. In this framework, Gaussian process surrogate models are replaced with a finite element model and its associate discrepancy function to provide a computationally efficient and all-round uncertainty quantification. Herein, the structural parameters that are most sensitive to measured structural dynamic characteristics are investigated and used to update the numerical model. Sequentially, the updated model is applied to compute the structural capacity with respect to loading demand to evaluate its as-is performance. The proposed framework’s feasibility is investigated and validated on a large lab-scale box girder bridge in two different health states, undamaged and damaged, with the latter state representing changes in structural parameters resulted from overloading actions. The results from the box girder bridge indicate a reduced structural performance evidenced by a significant drop in the structural reliability index and an increased probability of failure in the damaged state. The results also demonstrate that the proposed methodology contributes to more reliable judgment about structural safety, which in turn enables more informed maintenance decisions to be made.

Music listening is one of the most enigmatic of human mental phenomena; it not only triggers emotions but also changes our behavior. During the music session many people are observed to exhibit varying emotional response, which can be influenced by diverse factors such as music genre and instrument as well as the personal attributes of audiences. In this study, we assume that there is an intrinsic, complex and implicit relationship between the basic sound features of music and human emotional response to the music. The response levels of 12 individuals to a representative repertoire of 36 classical/popular Chinese traditional music (CTM) are systematically analyzed using the chills as a quantitative indicator, totally resulting in 432 ($12\times 36$) CTM–individual pairs that define a systematic individual-to-music response profile (SPTMRP). Gaussian process (GP) is then employed to model the multivariate correlation of SPTMRP profile with 15 sound features (including 5 Timbres, 4 Rhythms and 6 Pitchs) and 5 individual features in a supervised manner, which is also improved by genetic algorithm (GA) feature selection and compared with other machine learning methods. It is shown that the built GP regression model possesses a strong internal fitting ability ($r_F^2=0.786$) and a good external predictive power ($r_P^2=0.593$), which performed much better than linear PLS and nonlinear SVM and RF, confirming that the human emotional response to music can be quantitatively explained by GP methodology. Statistical examination of the GP model reveals that the sound features contribute more significantly to emotional response than individual features; their importance increases in the order: $Pitch<Timbre<Rhythm$, in which the spectral centroid (SC), relative amplitude of salient peaks (RASP), ratio of peak amplitudes (RPA), sum of all rhythm histograms (SARH) and period of unfolded maximum peak (PUMP) as well as gender are primarily responsible for the response.

Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP $Tu$ that is the image of another GP $u$ under a linear transformation $T$ acting on the sample paths of $u$ are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when $T$ is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator $T$ acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille’s theorem for the Bochner integral of a Banach-valued random variable.

This paper presents a non-parametric Bayesian approach for modeling multiple response variables using a Gaussian process regression (GPR) model. The response functions are modeled using a dependent Gaussian process (GP) prior, and the estimation, prediction, and inference issues are discussed within this framework. To establish the information consistency of the dependent GPs prediction strategy, a stretching-restriction method is proposed. The covariance structure is constructed using convolved Gaussian processes (CGPs) to illustrate the results. Simulations and real data analyses show that the proposed dependent GPR yields reasonably good prediction accuracy.

Time course expression analysis constitutes a large portion of applications of microarray experiments. One primary goal of such experiments is to detect genes with the temporal changes over a period of time or at some interested time points. Difficulties arising from data with small number of replicates over only a few unaligned time points in multiple groups pose challenges for efficient statistical analysis. Some known methods are limited by the unverifiable assumptions or by the scope of applications for only two groups. We present a new method for detecting differentially expressed genes under nonhomogeneous time course experiments in multiple groups. The new method first models the time course curve of one gene by a Gaussian process to align the nonhomogeneous time course data and to compute the gradient of the time course curve as well, the latter of which is used as directional information to enhance the sensitivity of detection for temporal changes. Second, we adopt a nonparametric method to test a surrogate hypothesis based on the augmented data from the Gaussian process model. The proposed method is robust in terms of model fitting and testing. It does not require any distributional assumption for the observations or the test statistic and the method works for the case with as few as triplicate samples over four or five time points under multiple groups. We show the effectiveness and superiority of the new method in comparison with some existing methods using simulated models and two real data sets.

Large amounts of research efforts have been focused on learning gene regulatory networks (GRNs) based on gene expression data to understand the functional basis of a living organism. Under the assumption that the joint distribution of the gene expressions of interest is a multivariate normal distribution, such networks can be constructed by assessing the nonzero elements of the inverse covariance matrix, the so-called precision matrix or concentration matrix. This may not reflect the true connectivity between genes by considering just pairwise linear correlations. To relax this limitative constraint, we employ Gaussian process (GP) model which is well known as computationally efficient non-parametric Bayesian machine learning technique. GPs are among a class of methods known as kernel machines which can be used to approximate complex problems by tuning their hyperparameters. In fact, GP creates the ability to use the capacity and potential of different kernels in constructing precision matrix and GRNs. In this paper, in the first step, we choose the GP with appropriate kernel to learn the considered GRNs from the observed genetic data, and then we estimate kernel hyperparameters using rule-of-thumb technique. Using these hyperparameters, we can also control the degree of sparseness in the precision matrix. Then we obtain kernel-based precision matrix similar to GLASSO to construct kernel-based GRN. The findings of our research are used to construct GRNs with high performance, for different species of Drosophila fly rather than simply using the assumption of multivariate normal distribution, and the GPs, despite the use of the kernels capacity, have a much better performance than the multivariate Gaussian distribution assumption.

Housing price time series is worth studying as it is closely related to the well-being of society. In the Hong Kong housing market from 1992 to 2010, signs of quasi-periodicity in housing price and transaction volume can be observed. We find that there is an overall periodicity of approximately 30 months in housing price changes and a strong lead–lag relationship between housing price and transaction volume. Analysis of the cross-covariance of the housing price, transaction volume and prime lending rate reveals that this quasi-periodicity is potentially driven by prime lending rates. Incorporation of quasi-periodicity into the kernel of Gaussian processes further enables us to construct a predictive model of the Hong Kong housing price trends that outperforms other traditional kernel functions.

Acoustic metamaterials are engineered microstructures with special mechanical and acoustic properties enabling exotic effects such as wave steering, focusing and cloaking. In this research, we develop a new machine learning framework for predicting optimal metastructures such as planar configurations of scatterers with specific functionalities. Specifically, we implement this framework by combining probabilistic generative modeling with deep learning and propose two models: a conditional variational autoencoder (CVAE) and a supervised variational autoencoder (SVAE) model. As an application of the method, here we design an acoustic cloak considering a minimization of total scattering cross-section (TSCS) for a set of cylindrical obstacles. We work with the sets of cylindrical objects confined in a region of space and streamline the design of configurations with minimal TSCS, demonstrating broadband cloaking effect at discrete sets of wavenumbers. After establishing the artificial neural networks that are capable of learning the TSCS based on the location of cylinders, we discuss our inverse design algorithms, combining variational autoencoders and the Gaussian process, for predicting optimal arrangements of scatterers given the TSCS. We show results for up to eight cylinders and discuss the efficiency and other advantages of the machine learning approach.

The control performance of model-based control approaches depends on the accuracy of the system dynamics. However, due to unknown environmental disturbances, quadrotor dynamics are time-varying during real-world flight, and real-time updating of the dynamics is required to avoid degradation of control performance due to model errors. To address this challenging issue, this paper proposes a hybrid controller that combines model predictive control (MPC) with model-based reinforcement learning (MBRL) to improve quadrotor trajectory tracking accuracy in unknown wind conditions. The MPC controller uses simple dynamics identified from offline flight data. The MBRL controller learns MPC-augmented dynamics represented by the Gaussian process (GP) and the residual policy from online flight data. The RL controller outputs additional actions to compensate for MPC control errors, while MPC can ensure the safety of RL exploration. The computational cost of GP when dealing with a large dataset affects the real-time performance. We design a priority data selection criterion to maintain a small dataset, balancing model accuracy and computational time. The accurate tracking results for different reference trajectories under various wind conditions demonstrate the robustness of the proposed approach. Comparative results with nominal MPC and related baselines show the advantages of the proposed approach in terms of control and real-time performance.

Bayesian optimisation, known for its minimal requirement of design parameter evaluations, is vital for global industrial process optimisation. It typically employs Gaussian processes as surrogate models for specific objectives. Traditional Bayesian optimisation is directly applied to the actual system. However, numerous industrial applications rely on simulation models. Although these models fail to represent the real system fully, they offer valuable insights to improve optimisation algorithms. A model that relies on the transfer of physical knowledge from the simulations to a Gaussian process model of the actual system is called a physics-informed Gaussian process model. Inspired by this, this chapter proposes a novel approach called simulation-informed Gaussian process. This approach constructs a Gaussian process kernel from simulation results to better capture design parameter-objective function correlations. This results in an accelerated Bayesian optimisation convergence of the actual system. We show this by comparing our method to conventional and physics-informed Bayesian optimisation. In addition, we offer insights into the consequences of integrating potentially misleading information into the Gaussian process framework.

Suppose that (X_i, Y_i,), i = 1, 2, … , n, are iid. random vectors with uniform marginals and a certain joint distribution F_ρ, where ρ is a parameter with ρ = ρ_o corresponds to the independence case. However, the X's and Y's are observed separately so that the pairing information is missing. Can ρ be consistently estimated? This is an extension of a problem considered in DeGroot and Goel (1980) which focused on the bivariate normal distribution with ρ being the correlation. In this paper we show that consistent discrimination between two distinct parameter values ρ₁ and ρ₂ is impossible if the density f_ρ of F_ρ is square integrable and the second largest singular value of the linear operator , h ∈ L²[0, 1], is strictly less than 1 for ρ = ρ₁ and ρ₂. We also consider this result from the perspective of a bivariate empirical process which contains information equivalent to that of the broken sample.