Early Fault Diagnosis Technology for Bearings Based on Quantile Multiscale Permutation Entropy

Early fault diagnosis of bearings is the basis of condition-based maintenance. To overcome the difficulty of early fault diagnosis for the mechanical system, a new conception named quantile multiscale permutation entropy (QMPE) is defined, and a new feature extraction method based on QMPE is proposed. On the basis of the multiscale entropy, the multiscale permutation entropy for the gathered vibration signal of equipment is obtained, and the sample quantile is calculated, which is employed to analyze the weak change of the variation signal. The proposed method is verified with the full lifetime datasets of a certain bearing, which proves that signal features extracted by the QMPE method can not only truly express the bearing detailed condition changing from normal to fault but also duly detect the early fault of the bearing. Comparing with other methods for early fault diagnosis, the proposed method can advance the finding time of the early fault obviously.

Nonparametric Limits of Agreement for Small to Moderate Sample Sizes: A Simulation Study

The assessment of agreement in method comparison and observer variability analysis of quantitative measurements is usually done by the Bland–Altman Limits of Agreement, where the paired differences are implicitly assumed to follow a normal distribution. Whenever this assumption does not hold, the 2.5% and 97.5% percentiles are obtained by quantile estimation. In the literature, empirical quantiles have been used for this purpose. In this simulation study, we applied both sample, subsampling, and kernel quantile estimators, as well as other methods for quantile estimation to sample sizes between 30 and 150 and different distributions of the paired differences. The performance of 15 estimators in generating prediction intervals was measured by their respective coverage probability for one newly generated observation. Our results indicated that sample quantile estimators based on one or two order statistics outperformed all of the other estimators and they can be used for deriving nonparametric Limits of Agreement. For sample sizes exceeding 80 observations, more advanced quantile estimators, such as the Harrell–Davis and estimators of Sfakianakis–Verginis type, which use all of the observed differences, performed likewise well, but may be considered intuitively more appealing than simple sample quantile estimators that are based on only two observations per quantile.

Moderate and Large Deviations for the Smoothed Estimate of Sample Quantiles

We derive the moderate and large deviations principle for the smoothed sample quantile from a sequence of independent and identically distributed samples of sizen.

Defining Sample Quantiles by the True Rank Probability

Many definitions exist for sample quantiles and are included in statistical software. The need to adopt a standard definition of sample quantiles has been recognized and different definitions have been compared in terms of satisfying some desirable properties, but no consensus has been found. We outline here that comparisons of the sample quantile definitions are irrelevant because the probabilities associated with order-ranked sample values are known exactly. Accordingly, the standard definition for sample quantiles should be based on the true rank probabilities. We show that this allows more accurate inference of the tails of the distribution, and thus improves estimation of the probability of extreme events.