Classifying Multivariate Signals in Rolling Bearing Fault Detection Using Adaptive Wide-Kernel CNNs

With the developments in improved computation power and the vast amount of (automatic) data collection, industry has become more data-driven. These data-driven approaches for monitoring processes and machinery require different modeling methods focusing on automated learning and deployment. In this context, deep learning provides possibilities for industrial diagnostics to achieve improved performance and efficiency. These deep learning applications can be used to automatically extract features during training, eliminating time-consuming feature engineering and prior understanding of sophisticated (signal) processing techniques. This paper extends on previous work, introducing one-dimensional (1D) CNN architectures that utilize an adaptive wide-kernel layer to improve classification of multivariate signals, e.g., time series classification in fault detection and condition monitoring context. We used multiple prominent benchmark datasets for rolling bearing fault detection to determine the performance of the proposed wide-kernel CNN architectures in different settings. For example, distinctive experimental conditions were tested with deviating amounts of training data. We shed light on the performance of these models compared to traditional machine learning applications and explain different approaches to handle multivariate signals with deep learning. Our proposed models show promising results for classifying different fault conditions of rolling bearing elements and their respective machine condition, while using a fairly straightforward 1D CNN architecture with minimal data preprocessing. Thus, using a 1D CNN with an adaptive wide-kernel layer seems well-suited for fault detection and condition monitoring. In addition, this paper clearly indicates the high potential performance of deep learning compared to traditional machine learning, particularly in complex multivariate and multi-class classification tasks.

Gaussian mixture model decomposition of multivariate signals

We propose a greedy variational method for decomposing a non-negative multivariate signal as a weighted sum of Gaussians, which, borrowing the terminology from statistics, we refer to as a Gaussian mixture model. Notably, our method has the following features: (1) It accepts multivariate signals, i.e., sampled multivariate functions, histograms, time series, images, etc., as input. (2) The method can handle general (i.e., ellipsoidal) Gaussians. (3) No prior assumption on the number of mixture components is needed. To the best of our knowledge, no previous method for Gaussian mixture model decomposition simultaneously enjoys all these features. We also prove an upper bound, which cannot be improved by a global constant, for the distance from any mode of a Gaussian mixture model to the set of corresponding means. For mixtures of spherical Gaussians with common variance $$\sigma ^2$$ σ 2 , the bound takes the simple form $$\sqrt{n}\sigma $$ n σ . We evaluate our method on one- and two-dimensional signals. Finally, we discuss the relation between clustering and signal decomposition, and compare our method to the baseline expectation maximization algorithm.

Sparse Representations of Random Signals

Sparse (fast) representations of deterministic signals have been well studied. Among other types there exists one called adaptive Fourier decomposition (AFD) for functions in analytic Hardy spaces. Through the Hardy space decomposition of the $L^2$ space the AFD algorithm also gives rise to sparse representations of signals of finite energy. To deal with multivariate signals the general Hilbert space context comes into play. The multivariate counterpart of AFD in general Hilbert spaces with a dictionary has been named pre-orthogonal AFD (POAFD). In the present study we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces and stochastic Hilbert spaces. To analyze random analytic signals we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) those expressible as the sum of a deterministic signal and an error term (SAFDI); and for (ii) those from different sources obeying certain distributive law (SAFDII). In the later part of the paper we drop off the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and SPOAFDII. Like AFDs and POAFDs for deterministic signals, the developed stochastic POAFD algorithms offer powerful tools to approximate and thus to analyze random signals.

An MEWMA-based segmental multivariate hidden Markov model for degradation assessment and prediction

Degradation is an unavoidable phenomenon in industrial systems. Hidden Markov models (HMMs) have been used for degradation modeling. In particular, segmental HMMs have been developed to model the explicit relationship between degradation signals and hidden states. However, existing segmental HMMs deal only with univariate cases, whereas in real systems, signals from various sensors are collected simultaneously, which makes it necessary to adapt the segmental HMMs to deal with multivariate processes. Also, to make full use of the information from the sensors, it is important to differentiate stable signals from deteriorating ones, but there is no good way for this, especially in multivariate processes. In this paper, the multivariate exponentially weighted moving average (MEWMA) control chart is employed to identify deteriorating multivariate signals. Specifically, the MEWMA statistic is used as a comprehensive indicator for differentiating multivariate observations. Likelihood Maximization is used to estimate the model parameters. To avoid underflow, the forward and backward probabilities are normalized. In order to assess degradation, joint probabilities are defined and derived. Further, the occurrence probability of each degradation state at the current time, as well as in the future, is derived. The Commercial Modular Aero-Propulsion System Simulation (C-MAPSS) dataset of NASA is employed for comparative analysis. In terms of degradation assessment and prediction, the proposed model performs very well in general. By sensitivity analysis, we show that in order to improve further the performance of the method, the weight of the chart should be set relatively small, whereas the method is not sensitive to the change of the in-control average run length (ARL).

Trait–fitness associations do not predict within-species phenotypic evolution over 2 million years

Long-term patterns of phenotypic change are the cumulative results of tens of thousands to millions of years of evolution. Yet, empirical and theoretical studies of phenotypic selection are largely based on contemporary populations. The challenges in studying phenotypic evolution, in particular trait–fitness associations in the deep past, are barriers to linking micro- and macroevolution. Here, we capitalize on the unique opportunity offered by a marine colonial organism commonly preserved in the fossil record to investigate trait–fitness associations over 2 Myr. We use the density of female polymorphs in colonies of Antartothoa tongima as a proxy for fecundity, a fitness component, and investigate multivariate signals of trait–fitness associations in six time intervals on the backdrop of Pleistocene climatic shifts. We detect negative trait–fitness associations for feeding polymorph (autozooid) sizes, positive associations for autozooid shape but no particular relationship between fecundity and brood chamber size. In addition, we demonstrate that long-term trait patterns are explained by palaeoclimate (as approximated by ∂ 18 O), and to a lesser extent by ecological interactions (i.e. overgrowth competition and substrate crowding). Our analyses show that macroevolutionary outcomes of trait evolution are not a simple scaling-up from the trait–fitness associations.

An Optimal Bayesian method for point analysis in discrete time

In this work we consider time series with a finite number of discrete point changes. We assume that the data in each segment follows a different probability density functions (pdf). We focus on the case where the data in all segments are modeled by Gaussian probability density functions with different means, variances and correlation lengths. We put a prior law on the change point instances (Poisson process) as well as on these different parameters(conjugate priors) and give the expression of the posterior probality distributions of these change points. The computations are done by using an appropriate Markov Chain Monte Carlo (MCMC) technique. The problem as we stated can also be considered as an unsupervised classification and/or segmentation of the time serie. This analogy gives us the possibility to propose alternative modeling and computation of change points, which are more appropriate for multivariate signals, for example in image processing.