Mixed Spectra for Stable Signals from Discrete Observations

This paper concerns the continuous-time stable alpha symmetric processes which are inivitable in the modeling of certain signals with indefinitely increasing variance. Particularly the case where the spectral measurement is mixed: sum of a continuous measurement and a discrete measurement. Our goal is to estimate the spectral density of the continuous part by observing the signal in a discrete way. For that, we propose a method which consists in sampling the signal at periodic instants. We use Jackson's polynomial kernel to build a periodogram which we then smooth by two spectral windows taking into account the width of the interval where the spectral density is non-zero. Thus, we bypass the phenomenon of aliasing often encountered in the case of estimation from discrete observations of a continuous time process.

Aliasing Free For Mixed Spectra for Stable Processes

This work focuses on the symmetric alpha stable processes with continuous time frequently used in modeling the signal with indefinitely growing variance when the spectral measure is mixed: sum of a continuous meseare and discrete measure. The objective of this paper is to estimate the spectral density of the continuous part from discrete observations of the signal. For that, we propose a method based on a sample of the signal at a periodic instant. The Jackson polynomial kernel is used for construct a periodogram. We smooth this periodogram by two spectral windows taking into account the width of the interval where the spectral density is nonzero. This technique allows to circumvent the phenomenon of aliasing often encountered in the estimation from the discrete observations of a process with a continuous time.

Remaining useful life estimation based on the joint use of an observer and a hidden Markov model

We propose an approach for failure prognosis based on the estimation of the Remaining Useful Life (RUL) of a system in a situation in which monitoring signals providing information about its degradation evolution are not measured and no operating model of the system is available. These conditions are of practical interest for industrial applications such as mechanical (e.g. rolling bearing) or electrical (e.g. wind turbine) devices or equipment-critical components (e.g. batteries) in which the addition of sensors to the system is not feasible (e.g. space limitations for sensors, cost, etc.). The approach is based on an estimation of the system degradation using residual generation (where the difference between the system and the observer outputs is processed) and Hidden Markov Models with discrete observations. The prediction of the system RUL is given by the Markov property concerning the mean time before absorption. The approach comprises two phases: a training phase to model the degradation behavior and an “on-line” use phase to estimate the remaining life of the system. Two case studies were conducted for RUL prediction to verify the effectiveness of the proposed approach.

Data-driven deep density estimation

Density estimation plays a crucial role in many data analysis tasks, as it infers a continuous probability density function (PDF) from discrete samples. Thus, it is used in tasks as diverse as analyzing population data, spatial locations in 2D sensor readings, or reconstructing scenes from 3D scans. In this paper, we introduce a learned, data-driven deep density estimation (DDE) to infer PDFs in an accurate and efficient manner, while being independent of domain dimensionality or sample size. Furthermore, we do not require access to the original PDF during estimation, neither in parametric form, nor as priors, or in the form of many samples. This is enabled by training an unstructured convolutional neural network on an infinite stream of synthetic PDFs, as unbound amounts of synthetic training data generalize better across a deck of natural PDFs than any natural finite training data will do. Thus, we hope that our publicly available DDE method will be beneficial in many areas of data analysis, where continuous models are to be estimated from discrete observations.

Least squares estimator for stochastic differential equations driven by small fractional Lévy noises from discrete observations

In this paper, we consider the problem of parameter estimation for stochastic differential equations with small fractional Lévy noises, based on discrete observations. Under certain regularity conditions on drift function, the consistency of least squares estimation has been established as a small dispersion coefficient [Formula: see text] and the number of discrete points [Formula: see text] simultaneously. We also obtain the asymptotic behavior of the estimator.

Parameter estimation and empirical analysis of fractional O-U process based on three machine learning algorithms

Fractional O-U process is a very classical stochastic process which is used to describe the time series of financial volatility. Three parameters need to be estimated in this process, and the estimation method based on discrete observations can be realized by machine learning optimization algorithm. In this study, the parameter estimation method of fractional O-U process is briefly described, and three optimization algorithms, Newton method, quasi Newton method and genetic algorithm, are used to estimate the parameters. The comparison shows that genetic algorithm is relatively accurate and efficient. Finally, the minute data of stock index futures are estimated based on fractional O-U process. The results show that the estimation of theta and Hurst index is relatively accurate, and the estimation error of volatility is large.