Finite-time and spectral-finite methods of optimal filtering of discrete signals

На основе содержания теоремы ортогонального проецирования излагаются методы оптимальных, линейных рекуррентных оценок, в общем случае, не марковских, сигналов, на фоне произвольных помех. Предлагаемые алгоритмы оптимальной обработки дискретных сигналов являются альтернативными методу фильтрации Калмана, не отличающимися заметно от них по точности обработки и являющимися более универсальными и простыми при их реализации. Универсальность исследуемых методов определяется применимостью их к широкому классу моделей сигналов, не требующих марковского свойства оцениваемого сигнала и изменения структуры алгоритма оценки в зависимости от моделей помех измерения в виде случайного коррелированного процесса или белого шума. Более простые структуры алгоритмов рассматриваемых методов по отношению к фильтрации Калмана объясняются отсутствием необходимости представления модели в пространстве состояний и требования решать нелинейное уравнение Риккати для реализации алгоритма. Спектрально-финитный алгоритм оптимальной оценки сигнала осуществляет сжатие информации в спектральном аспекте на основе использования метода нахождения собственных чисел и векторов и позволяет осуществить понижение размерности векторов результатов измерений вплоть до скалярных величин без заметной потери точности оценки. В качестве исходной информации необходимо знание корреляционной функции и математического ожидания оцениваемого дискретного сигнала и дисперсии и математического ожидания дискретной помехи. Based on the content of the orthogonal projection theorem, methods of optimal, linear recurrent estimates of, in general, non-Markov signals, against the background of arbitrary interference, are presented. The proposed algorithms for optimal processing of discrete signals are alternative to the Kalman filtering method, which do not differ significantly from them in terms of processing accuracy and are more universal and simple to implement. The universality of the studied methods is determined by their applicability to a wide class of signal models that do not require the Markov property of the estimated signal and changes in the structure of the estimation algorithm depending on the measurement interference models in the form of a random correlated process or white noise. The simpler structures of the algorithms of the methods under consideration in relation to Kalman filtering are explained by the absence of the need to represent the model in the state space and the requirement to solve the nonlinear Riccati equation for the implementation of the algorithm.

TrackFit: Uncertainty Quantification, Optimal Filtering and Interpolation of Tracks for Time-Resolved Lagrangian Particle Tracking

Advanced Lagrangian Particle Tracking methods (such as the STB algorithm (Schanz et al. 2016)) are a very useful tool for uncovering properties of flow. As a measurement technique, the results of such methods are perturbed by different sources of errors and noise. This work addresses the problem of optimal filtering of particle tracks as well as estimating uncertainties of derived quantities such as location, velocity and acceleration of observed particles. The behavior and performance of this new filtering method (“TrackFit”), first introduced at Gesemann et al. (2016) is analyzed and compared to the Savitzky–Golay filter (Savitzky and Golay (1964)) which is commonly used for these purposes. The optimal choice of parameters of this filtering method as well as the uncertainty quantification of the reconstructed tracks can be extracted from a spectral analysis of the recorded raw particle tracking data. This is in contrast to a Savitzky–Golay filter where the choice of parameters might often be driven by experience and gut feeling. Estimating the power spectral density (PSD) of the particle trajectory signals for the purpose of optimal filtering parameter selection represents a challenge due to possibly short trajectory signals. In the following work we will present a method for PSD estimation that is applicable in this scenario. In addition, we show that regardless of the choice of Savitzky–Golay filter parameters, the resulting filter will not approximate the ideal noise reduction filter well unlike the “TrackFit” described in this work.

Monitoring of Industrial Machine Using a Novel Blind Feature Extraction Approach

Machinery with several rotating and stationary components tends to produce non-stationary and random vibration signatures due to the fluctuations in the input loads and process defects due to long hours of operation. Traditional heuristics methods are suitable for the detection of fault signatures, however, they become more complicated when the level of uncertainty or randomness exceeds beyond control. A novel methodology to identify these fault signatures using optimal filtering of vibration data is proposed to eliminate any false alarms and is expected to provide a higher probability of correct diagnosis. In this paper, a detailed pipeline of the algorithms are presented along with the results of the investigation that was carried out. These investigations are performed using open-source vibration data published by the NASA prognostics centre. The performance of these algorithms are evaluated based on the ground truth results published by NASA researchers. Based on the performance of these algorithms several parameters are fine-tuned to ensure generalisation and reliable performance.

Applying Wavelet Filters in Wind Forecasting Methods

Wind is a physical phenomenon with uncertainties in several temporal scales, in addition, measured wind time series have noise superimposed on them. These time series are the basis for forecasting methods. This paper studied the application of the wavelet transform to three forecasting methods, namely, stochastic, neural network, and fuzzy, and six wavelet families. Wind speed time series were first filtered to eliminate the high-frequency component using wavelet filters and then the different forecasting methods were applied to the filtered time series. All methods showed important improvements when the wavelet filter was applied. It is important to note that the application of the wavelet technique requires a deep study of the time series in order to select the appropriate family and filter level. The best results were obtained with an optimal filtering level and improper selection may significantly affect the accuracy of the results.