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Time-frequency analysis (TFA) technology is an important tool for analyzing non-Gaussian mechanical fault vibration signals. In the complex background of infinite variance process noise and Gaussian colored noise, it is difficult for traditional methods to obtain the highly concentrated time-frequency representation (TFR) of fault vibration signals. Based on the insensitive property of fractional low-order statistics for infinite variance and Gaussian processes, robust fractional lower order adaptive linear chirplet transform (FLOACT) and fractional lower order adaptive scaling chirplet transform (FLOASCT) methods are proposed to suppress the mixed complex noise in this paper. The calculation steps and processes of the algorithms are summarized and deduced in detail. The experimental simulation results show that the improved FLOACT and FLOASCT methods have good effects on multi-component signals with short frequency intervals in the time-frequency domain and even cross-frequency trajectories in the strong impulse background noise environment. Finally, the proposed methods are applied to the feature analysis and extraction of the mechanical outer race fault vibration signals in complex background environments, and the results show that they have good estimation accuracy and effectiveness in lower MSNR, which indicate their robustness and adaptability.

Experiments in ecology are usually designed to provide tests of hypotheses on the influence of the mean intensity of causal processes, whereas the variance around mean effects has been largely overlooked as a causal force in biological assemblages. Repetition of experiments in space and time provides an estimate of this variability at specific scales, but does not explain how changes in variance generate structure in assemblages and the extent to which variance and mean intensity interact. This paper seeks to identify suitable procedures for empirical analyses on the influence of variance and mean intensity of predictor ecological variables on spatial and temporal patterns in natural populations. A survey of the ecological literature indicates that temporal variability in studies of disturbance and in analyses of consumer-resource interactions is generally expressed in terms of frequency of events. This is inappropriate, as frequency confounds the variance with the mean effect size of a process. A possible solution to the problem involves experimental designs in which levels of intensity and those of variability are chosen independently over explicit spatial or temporal scales and treated as fixed, orthogonal factors. Examples are offered for various scenarios of consumer-resource interactions along with indications for statistical tests of hypotheses. Such novel approaches have important ramifications for understanding variability in a wide range of ecological contexts and for predicting the response of assemblages to increased environmental fluctuations, including those expected under modified climate conditions.

Post-processing synchrosqueezing transform and synchroextracting transform methods can improve TFR resolution for fault diagnosis. The normal and fault signal can be described by infinite variance process, and 1 < α ≤ 2, even the background noise belongs to the process under complex conditions. The effect of traditional SST and SET methods is greatly reduced and even lost in infinite variance process environment. Several robust post-processing methods are proposed including FSET, FSSET, FSOSET and FMSST technology employing infinite variance process statistical model and FLOS, and their mathematical derivation are completed in this paper. The proposed methods are compared with the conventional methods, and the results show that the proposed methods achieve better results than the existing methods. In addition, the new methods are applied to diagnose the bearing outer race DE signals polluted by infinite variance process, the result demonstrates that they have performance advantages. Finally, the characteristics, shortcomings and application scenarios of the improved algorithms are summarized.

Despite decades of field research on greater sage-grouse, range-wide demographic data have yet to be synthesized into a sensitivity analysis to guide management actions. We reviewed range-wide demographic rates for greater sage-grouse from 1938 to 2011 and used data from 50 studies to parameterize a 2-stage, female-based population matrix model. We conducted life-stage simulation analyses to determine the proportion of variation in population growth rate (λ) accounted for by each vital rate, and we calculated analytical sensitivity, elasticity, and variance-stabilized sensitivity to identify the contribution of each vital rate to λ. As expected for an upland game bird, greater sage-grouse showed marked annual and geographic variation in several vital rates. Three rates were demonstrably important for population growth: female survival, chick survival, and nest success. Female survival and chick survival, in that order, had the most influence on λ per unit change in vital rates. However, nest success explained more of the variation in λ than did the survival rates. In lieu of quantitative data on specific mortality factors driving local populations, we recommend that management efforts for greater sage-grouse first focus on increasing female survival by restoring large, intact sagebrush-steppe landscapes, reducing persistent sources of human-caused mortality, and eliminating anthropogenic habitat features that subsidize species that prey on juvenile, yearling, and adult females. Our analysis also supports efforts to increase chick survival and nest success by eliminating anthropogenic habitat features that subsidize chick and nest predators, and by managing shrub, forb, and grass cover, height, and composition to meet local brood-rearing and nesting habitat guidelines. We caution that habitat management to increase chick survival and nest success should not reduce the cover or height of sagebrush below that required for female survival in other seasons (e.g., fall, winter). The success or failure of management actions for sage-grouse should be assessed by measuring changes in vital rates over long time periods to avoid confounding with natural, annual variation.

Current environmental changes may increase temporal variability of life history traits of species thus affecting their long‐term population growth rate and extinction risk. If there is a general relationship between environmental variances (EVs) and mean annual survival rates of species, that relationship could be used as a guideline for analyses of population growth and extinction risk for populations, where data on EVs are missing. For this purpose, we present a comprehensive compilation of 252 EV estimates from 89 species belonging to five vertebrate taxa (birds, mammals, reptiles, amphibians and fish) covering mean annual survival rates from 0.01 to 0.98. Since variances of survival rates are constrained by their means, particularly for low and high mean survival rates, we assessed whether any observed relationship persisted after applying two types of commonly used variance stabilizing transformations: relativized EVs (observed/mathematical maximum) and logit‐scaled EVs. With raw EVs at the arithmetic scale, mean–variance relationships of annual survival rates were hump‐shaped with small EVs at low and high mean survival rates and higher (and widely variable) EVs at intermediate mean survival rates. When mean annual survival rates were related to relativized EVs the hump‐shaped pattern was less distinct than for raw EVs. When transforming EVs to logit scale the relationship between mean annual survival rates and EVs largely disappeared. The within‐species juvenile‐adult slopes were mainly positive at low (<0.5) and negative at high (>0.5) mean survival rates for raw and relativized variances while these patterns disappeared when EVs were logit transformed. Uncertainties in how to interpret the results of relativized and logit‐scaled EVs, and the observed high variation in EV's for similar mean annual survival rates illustrates that extrapolations of observed EVs and tests of life history drivers of survival–EV relationships need to also acknowledge the large variation in these parameters. The relationship between mean survival rate and its temporal environmental (process) variance is dependent on the mathematical scaling.

Within statistical process control (SPC), normality is often assumed as the underlying probabilistic generator where the process variance is assumed equal for all rational subgroups. The parameters of the underlying process are usually assumed to be known—if this is not the case, some challenges arise in the estimation of unknown parameters in the SPC environment especially in the case of few observations. This paper proposes a bivariate beta type distribution to guide the user in the detection of a permanent upward or downward step shift in the process’ variance that does not directly rely on parameter estimates, and as such presents itself as an attractive and intuitive approach for not only potentially identifying the magnitude of the shift, but also the position in time where this shift is most likely to occur. Certain statistical properties of this distribution are derived and simulation illustrates the theoretical results. In particular, some insights are gained by comparing the newly proposed model’s performance with an existing approach. A multivariate extension is described, and useful relationships between the derived model and other bivariate beta distributions are also included.

The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties. Conditional moments of stochastic processes can be used to price financial derivatives whose payoffs depend on conditional moments of underlying assets. In general, the transition probability density function (PDF) of a stochastic process is often unavailable in closed form. Thus, the conditional moments, which can be directly computed by applying the transition PDFs, may be unavailable in closed form. In this work, we studied an inhomogeneous nonlinear drift constant elasticity of variance (IND-CEV) process, which is a class of diffusions that have time-dependent parameter functions; therefore, their sample paths are asymmetric. The closed-form formulas for conditional moments of the IND-CEV process were derived without having a condition on eigenfunctions or the transition PDF. The analytical results were examined through Monte Carlo simulations.

This paper develops finite-series solutions of a hybrid system of interconnected partial differential equations for computing the conditional moments of regime-switching extended constant elasticity of variance processes with generalized drift and diffusion coefficients. The regime-switching mechanism is modeled by a continuous-time, finite-state, irreducible Markov chain with m regimes, for any integer m≥1. For any real γ>0, we identify a tractable class of processes where the γth conditional moment admits an explicit finite power series representation in the initial state, arising from the polynomial structure. The analytical framework is derived via a Feynman–Kac representation adapted for regime-switching diffusions and validated for accuracy and efficiency using Monte Carlo simulations. In addition, we investigate the asymptotic behavior of the first conditional moment for a two-state regime-switching constant elasticity of variance process with nonlinear drift, emphasizing the effects of symmetry in the Markov intensity matrix and comparisons with the corresponding linear-drift case. Applications in futures pricing demonstrate the framework’s relevance for derivative pricing and risk management.

Population dynamic models combine density dependence and environmental effects. Ignoring sampling uncertainty might lead to biased estimation of the strength of density dependence. This is typically addressed using state‐space model approaches, which integrate sampling error and population process estimates. Such models seldom include an explicit link between the sampling procedures and the true abundance, which is common in capture–recapture settings. However, many of the models proposed to estimate abundance in the presence of capture heterogeneity lead to incomplete likelihood functions and cannot be straightforwardly included in state‐space models. We assessed the importance of estimating sampling error explicitly by taking an intermediate approach between ignoring uncertainty in abundance estimates and fully specified state‐space models for density‐dependence estimation based on autoregressive processes. First, we estimated individual capture probabilities based on a heterogeneity model for a closed population, using a conditional multinomial likelihood, followed by a Horvitz–Thompson estimate for abundance. Second, we estimated coefficients of autoregressive models for the log abundance. Inference was performed using the methodology of integrated nested Laplace approximation (INLA). We performed an extensive simulation study to compare our approach with estimates disregarding capture history information, and using R‐package VGAM, for different parameter specifications. The methods were then applied to a real data set of gray‐sided voles Myodes rufocanus from Northern Norway. We found that density‐dependence estimation was improved when explicitly modeling sampling error in scenarios with low process variances, in which differences in coverage reached up to 8% in estimating the coefficients of the autoregressive processes. In this case, the bias also increased assuming a Poisson distribution in the observational model. For high process variances, the differences between methods were small and it appeared less important to model heterogeneity. We assessed the importance of including capture history information in the estimation of density dependence from capture–recapture data. We performed an extensive simulation study, evaluating the performance of different methods in the estimation of different population processes. We found that disregarding sampling error can bias the density‐dependence estimates in low variance autoregressive population models.

Two multi-parameter distributions, namely the Pearson type IV and metalog distributions, are discussed and suggested as alternatives to the normal distribution for modelling path delay data that determines the maximum clock frequency (FMAX) of a microprocessor or other digital circuit. These distributions outperform the normal distribution in goodness-of-fit statistics for simulated path delay data derived from a fabricated microcontroller, with the six-term metalog distribution offering the best fit. Furthermore, 99.7% confidence intervals are calculated for some extreme quantiles on each dataset using the previous distributions. Considering the six-term metalog distribution estimates as the golden standard, the relative errors in single paths vary between 4 and 14% for the normal distribution. Finally, the within-die (WID) variation maximum critical path delay distribution for multiple critical paths is derived under the assumption of independence between the paths. Its density function is then used to compute different maximum delays for varying numbers of critical paths, assuming each path has one of the previous distributions with the metalog estimates as the golden standard. For 100 paths, the relative errors are at most 14% for the normal distribution. With 1000 and 10,000 paths, the corresponding errors extend up to 16 and 19%, respectively.