Extrapolation With Gaussian Random Processes and Evolutionary Programming

In this contribution, we present the exact asymptotic behavior of the Parisian ruin probability for specific classes of conditionally Gaussian random processes, as the initial capital approaches infinity. We consider a risk process as a mixture of Gaussian and non-Gaussian random processes, where the aggregate claim process is modeled as a Gaussian process (stationary or non-stationary), while the premium income is represented by a different random process that is independent of the claim process.

The pharmacokinetic parameter estimation problem is reexamined within the framework of stochastic dynamic systems. Using this formalism, two sources of uncertainty are incorporated into the parameter estimation procedure: measurement error and process or model error. Consideration is given to linear dynamic models, with both model and measurement error terms modeled as Gaussian random processes. The maximum likelihood estimate of the parameters is obtained by using the Kaiman filter formulation of the model to compute the likelihood function which is then maximized by direct nonlinear optimization. This approach to maximum likelihood estimation, given process or model error as well as output error, is evaluated using several simulated pharmacokinetic parameter estimation problems.

Extreme value distributions for nonlinear transformations of vector Gaussian processes

Gaussian processes (GPs) are a well-known nonparametric Bayesian inference technique, but they suffer from scalability problems for large sample sizes, and their performance can degrade for non-stationary or spatially heterogeneous data. In this work, we seek to overcome these issues through (i) employing variational free energy approximations of GPs operating in tandem with online expectation propagation steps; and (ii) introducing a local splitting step which instantiates a new GP whenever the posterior distribution changes significantly as quantified by the Wasserstein metric over posterior distributions. Over time, then, this yields an ensemble of sparse GPs which may be updated incrementally, and adapts to locality, heterogeneity, and non-stationarity in training data. We provide a 1-dimensional example to illustrate the motivation behind our approach, and compare the performance of our approach to other Gaussian process methods across various data sets, which often achieves competitive, if not superior predictive performance, relative to other locality-based GP regression methods in which hyperparameters are learned in an online manner.

Chapter 2.3.6 - Scattering by Surfaces and Phase Screens

Optimum estimation (tracking) of the polarization plane of a linearly polarized electromagnetic wave is determined when the signal is a narrow-band Gaussian random process with a polarization plane angle which is also a Gaussian random process. This model is Compared to previous work and is applicable to space communication. The estimator performs a correlation operation similar to an amplitude -comparison monopulse angle tracker, giving the name correlation polarimeter. Under large signal-to-noise ratio (SNR), the estimator is causal. Performance of the causal correlation polarimeter is evaluated for arbitrary SNR. Optimum precorrelation filtering is determined. With low SNR, the performance of this system is far better than that of previously developed systems. Practical implementation is discussed. A scheme is given to reduce the effect of linearly polarized noise.

It is known that various types of location privacy attacks can be carried out using a personalized transition matrix that is learned for each target user, or a population transition matrix that is common to all target users. However, since many users disclose only a small amount of location information in their daily lives, the training data can be extremely sparse. The aim of this paper is to clarify the risk of location privacy attacks in this realistic situation. To achieve this aim, we propose a learning method that uses tensor factorization (or matrix factorization) to accurately estimate personalized transition matrices (or a population transition matrix) from a small amount of training data. To avoid the difficulty in directly factorizing the personalized transition matrices (or population transition matrix), our learning method first factorizes a transition count tensor (or matrix), whose elements are the number of transition counts that the user has made, and then normalizes counts to probabilities. We focus on a localization attack, which derives an actual location of a user at a given time instant from an obfuscated trace, and compare our learning method with the maximum likelihood (ML) estimation method in both the personalized matrix mode and the population matrix mode. The experimental results using four real data sets show that the ML estimation method performs only as well as a random guess in many cases, while our learning method significantly outperforms the ML estimation method in all of the four data sets.

The Maximum Likelihood (ML) and Best Linear Unbiased (BLU) estimators of the location and scale parameters of an extreme value distribution (Lawless [1982]) are compared under conditions of small sample sizes and Type I censorship. The comparisons were made in terms of the mean square error criterion. According to this criterion, the ML estimator of σ in the case of very small sample sizes (n < 10) and heavy censorship (low censoring time) proved to be more efficient than the corresponding BLU estimator. However, the BLU estimator for σ attains parity with the corresponding ML estimator when the censoring time increases even for sample sizes as low as 10. The BLU estimator of σ attains equivalence with the ML estimator when the sample size increases above 10, particularly when the censoring time is also increased. The situation is reversed when it came to estimating the location parameter μ, as the BLU estimator was found to be consistently more efficient than the ML estimator despite the improved performance of the ML estimator when the sample size increases. However, computational ease and convenience favor the ML estimators. Keywords: maximum likelihood estimatorsbest linear unbiased estimatorslife-testingreliability

With their greatly reduced sizes, low development cost, and rapid construction time, CubeSats have merged as a platform of considerable interest for a wide range of applications, including remote sensing. Many applications require the interpolation of sensor data into a regularly spaced grid for the development of downstream scientific products. This problem is complicated for CubeSat platforms due to potentially significant uncertainties associated with the spatial position of the satellite. In this paper, we present a probabilistic approach to the data interpolation problem in which we estimate both the platform location and data samples on a regular grid given observations corrupted by noise and location error. Our approach is based on a Gaussian process model to connect the measured data to the values on the grid. Two statistical models for positional uncertainties are considered, one based on an assumption of independent errors and another motivated by positional errors associated with a specific platform of interest, the MicroMAS radiometer. In each case, the maximum a posteriori estimate of the positions and the data is generated using an optimized Gaussian process regression (OGPR) method resulting in two algorithms: OGPR-IID and OGPR-PCA. The performance of this approach is tested on both simulated data and advanced technology microwave sounder data where significant improvements both qualitatively and quantitatively relative to traditional interpolation methods are observed.

Gaussian processes (GPs) are ubiquitously used in sciences and engineering as metamodels. Standard GPs, however, can only handle numerical or quantitative variables. In this paper, we introduce latent map Gaussian processes (LMGPs) that inherit the attractive properties of GPs and are also applicable to mixed data which have both quantitative and qualitative inputs. The core idea behind LMGPs is to learn a continuous, low-dimensional latent space or manifold which encodes all qualitative inputs. To learn this manifold, we first assign a unique prior vector representation to each combination of qualitative inputs. We then use a low-rank linear map to project these priors on a manifold that characterizes the posterior representations. As the posteriors are quantitative, they can be directly used in any standard correlation function such as the Gaussian or Matern. Hence, the optimal map and the corresponding manifold, along with other hyperparameters of the correlation function, can be systematically learned via maximum likelihood estimation. Through a wide range of analytic and real-world examples, we demonstrate the advantages of LMGPs over state-of-the-art methods in terms of accuracy and versatility. In particular, we show that LMGPs can handle variable-length inputs, have an explainable neural network interpretation, and provide insights into how qualitative inputs affect the response or interact with each other. We also employ LMGPs in Bayesian optimization and illustrate that they can discover optimal compound compositions more efficiently than conventional methods that convert compositions to qualitative variables via manual featurization.

The degradation of rolling-element bearings is mainly stochastic due to unforeseeable influences like short term overstraining, which hampers the prediction of the remaining useful lifetime. This stochastic behaviour is hardly describable with parametric degradation models, as it has been done in the past. Therefore, the two prognostic concepts presented and examined in this paper introduce a nonparametric approach by the application of a dynamic Gaussian Process (GP). The GP offers the opportunity to reproduce a damage course according to a set of training data and thereby also estimates the uncertainties of this approach by means of the GP’s covariance. The training data is generated by a stochastic degradation model that simulates the aforementioned highly stochastic degradation of a bearing fault. For prediction and state estimation of the feature, the trained dynamic GP is combined with the Unscented Kalman Filter (UKF) and evaluatedin the context of a case study. Since this prognostic approach has shown drawbacks during the evaluation, a multiple model approach based on GP-UKF is introduced and evaluated. It is shown that this combination offers an increased prognostic performance for bearing fault prediction.

In this paper, we propose a novel method of online learning of sparse pseudo-data, representative of the whole training data, for Gaussian Process (GP) regressions. We call the proposed method Incremental Sparse Pseudo-input Gaussian Process (ISPGP) regression. The proposed ISPGP algorithm allows for training from either a huge amount of training data by scanning through it only once or an online incremental training dataset. Thanks to the nature of the incremental learning algorithm, the proposed ISPGP algorithm can theoretically work with infinite data to which the conventional GP or SPGP algorithm is not applicable. From our experimental results on the KIN40K dataset, we can see that the proposed ISPGP algorithm is comparable to the conventional GP algorithm using the same number of training data. Although the proposed ISPGP algorithm performs slightly worse than Snelson and Ghahramani's SPGP algorithm, the level of performance degradation is acceptable.

A number of properties of separable covariance matrices are summarized. Expressions for the divergence of the corresponding two-dimensional Gaussian random processes are given in terms of row and column covariance matrices, and in terms of linear prediction parameters and maximum likelihood spectral estimates. Such time and frequency domain expressions are not widely known, even for one-dimensional random processes.

For a balanced two‐way mixed model, the maximum likelihood (ML) and restricted ML (REML) estimators of the variance components were obtained and compared under the non‐negativity requirements of the variance components by Lee and Kapadia (1984). In this note, for a mixed (random blocks) incomplete block model, explicit forms for the REML estimators of variance components are obtained. They are always non‐negative and have smaller mean squared error (MSE) than the analysis of variance (AOV) estimators. The asymptotic sampling variances of the maximum likelihood (ML) estimators and the REML estimators are compared and the balanced incomplete block design (BIBD) is considered as a special case. The ML estimators are shown to have smaller asymptotic variances than the REML estimators, but a numerical result in the randomized complete block design (RCBD) demonstrated that the performances of the REML and ML estimators are not much different in the MSE sense.

Over-the-air computation (OAC) is a promising technique to realize fast model aggregation in the uplink of federated edge learning (FEEL). OAC, however, hinges on accurate channel-gain precoding and strict synchronization among edge devices, which are challenging in practice. As such, how to design the maximum likelihood (ML) estimator in the presence of residual channel-gain mismatch and asynchronies is an open problem. To fill this gap, this paper formulates the problem of misaligned OAC for FEEL and puts forth a whitened matched filtering and sampling scheme to obtain oversampled, but independent samples from the misaligned and overlapped signals. Given the whitened samples, a sum-product ML (SP-ML) estimator and an aligned-sample estimator are devised to estimate the arithmetic sum of the transmitted symbols. In particular, the computational complexity of our SP-ML estimator is linear in the packet length, and hence is significantly lower than the conventional ML estimator. Extensive simulations on the test accuracy versus the average received energy per symbol to noise power spectral density ratio (EsN0) yield two main results: 1) In the low EsN0 regime, the aligned-sample estimator can achieve superior test accuracy provided that the phase misalignment is not severe. In contrast, the ML estimator does not work well due to the error propagation and noise enhancement in the estimation process. 2) In the high EsN0 regime, the ML estimator attains the optimal learning performance regardless of the severity of phase misalignment. On the other hand, the aligned-sample estimator suffers from a test-accuracy loss caused by phase misalignment.