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Time continuous analyses, such as statistical parametric mapping (SPM), have been increasingly used in biomechanics research to determine differences between populations, interventions and methodologies. Currently, it is not known how sensitive time-continuous analyses are to timing variability that occur in gait data. We evaluated this sensitivity by examining the frequency of significant SPM outcomes between two walking speeds when lower limb kinematics and kinetics were segmented and aligned based on 40 repeatable gait events. These events, defined in the supplementary material, include a commonly used event like foot contact and other events that have been previously demonstrated to be repeatable. Repeatable gait events were determined from joint and segment kinematics, joint kinetics as well as ground reaction forces. We examined the frequency of statistical outcomes for a single subject with different numbers of strides analyzed and for a cohort of 10 subjects. Our findings demonstrate that gait interventions, such as changes in walking speed, can induce temporal shifts that affect time-continuous outcomes for both cohort- and subject-level analyses. As both timing and magnitude are important in gait data, researchers are encouraged to perform additional analyses to understand how both of these variables affect time-continuous analysis outcomes. Finally, we demonstrate that multiple SPM tests can be performed to determine if statistical outcomes are due to temporal shifting or differences in magnitude. It is important to understand how both timing and magnitude of biomechanical data influences time continuous analyses as these analyses inform injury prevention, device development and basic understanding of biomechanics.

For time-continuous analysis of gait, the problem of variations in cycle durations is resolved by normalizing to the gait cycle, but results depend on the definition of the cycle start. Gait cycle normalization ignores variations in gait phase durations, which results in averaging and comparing data across different phases. We propose gait phase normalization as part of a comprehensive method for independently analyzing magnitude and timing differences. First, gait phases are identified and differences in absolute and/or relative timing of phase durations or any point of interest between conditions or groups are analyzed using standard statistics. Next, time-continuous gait data is normalized to gait phases, and statistical parametric mapping (SPM) is used to assess magnitude differences in gait data. This approach is demonstrated on data recorded from ten young healthy adults walking on a treadmill at five different speeds. Sagittal knee angle was normalized to gait cycle or gait phase using five different gait cycle start events. Walking at different speeds resulted in significant changes in gait phase durations, highlighting a problem ignored by gait cycle normalization. SPM results for knee angle normalized to gait cycle varied from normalization to gait phases. Gait phase normalized SPM results were robust to the definition of the cycle start, in contrast to gait cycle normalized data. The approach of analyzing phase durations and normalizing data to gait phases overcomes previous limitations and enables a comprehensive analysis of magnitude and timing differences in time-continuous gait data and could be readily adapted to other tasks.

Parametric continuous-time analysis often entails derivations of continuous-time models from predefined discrete formulations. However, undetermined convergence rates of frequency-dependent parameters can result in ill-defined continuous-time limits, leading to modeling discrepancy, which impairs the reliability of fitting and forecasting. To circumvent this issue, we propose a simple solution based on functional data analysis (FDA) and truncated Taylor series expansions. It is demonstrated through a simulation study that our proposed method is superior—compared with misspecified parametric methods—in fitting and forecasting continuous-time stochastic processes, while the parametric method slightly dominates under correct specification, with comparable forecast errors to the FDA-based method. Due to its generally consistent and more robust performance against possible misspecification, the proposed FDA-based method is recommended in the presence of modeling discrepancy. Further, we apply the proposed method to predict the future return of the S&P 500, utilizing observations extracted from a latent continuous-time process, and show the practical efficacy of our approach in accurately discerning the underlying dynamics.

Random label noise (or observational noise) widely exists in practical machine learning settings. While previous studies primarily focused on the effects of label noise to the performance of learning, our work intends to investigate the implicit regularization effects of label noise, under mini-batch sampling settings of stochastic gradient descent (SGD), with the assumption that label noise is unbiased. Specifically, we analyze the learning dynamics of SGD over the quadratic loss with unbiased label noise (ULN), where we model the dynamics of SGD as a stochastic differentiable equation with two diffusion terms (namely a doubly stochastic model). While the first diffusion term is caused by mini-batch sampling over the (label-noiseless) loss gradients, as in many other works on SGD (Zhu et al 2019 ICML 7654–63; Wu et al 2020 Int. Conf. on Machine Learning (PMLR) pp 10367–76), our model investigates the second noise term of SGD dynamics, which is caused by mini-batch sampling over the label noise, as an implicit regularizer. Our theoretical analysis finds such an implicit regularizer would favor some convergence points that could stabilize model outputs against perturbations of parameters (namely inference stability ). Though similar phenomenon have been investigated by Blanc et al (2020 Conf. on Learning Theory (PMLR) pp 483–513), our work does not assume SGD as an Ornstein–Uhlenbeck-like process and achieves a more generalizable result with convergence of the approximation proved. To validate our analysis, we design two sets of empirical studies to analyze the implicit regularizer of SGD with unbiased random label noise for deep neural network training and linear regression. Our first experiment studies the noisy self-distillation tricks for deep learning, where student networks are trained using the outputs from well-trained teachers with additive unbiased random label noise. Our experiment shows that the implicit regularizer caused by the label noise tends to select models with improved inference stability. We also carry out experiments on SGD-based linear regression with ULN, where we plot the trajectories of parameters learned in every step and visualize the effects of implicit regularization. The results back up our theoretical findings.

The electromagnetic transient (EMT) response of a power system can be determined either by time-domain (TD) or by frequency-domain (FD) methods. This chapter deals with those aspects of frequency domain analysis and of digital signal processing that have become essential for the analysis of transients in modern power systems. It provides a brief review of basic concepts of FD methods. Continuous-time Fourier analysis is introduced as an extension of phasor analysis which is more familiar to power engineers. The chapter presents the basic differences between continuous-time and discrete-time Fourier analysis. Of special interest in the chapter are: (1) the effect of aliasing, (2) the sampling theorem and (3) the principle of conservation of information. The chapter also provides a brief overview of multirate transient analysis methods based on the discrete Fourier transform (DFT) and the numerical Laplace transform (NLT).

We introduce a general framework that guides the management scientist's formulation of deterministic models of production processes. Using the framework, we reformulate the constraints of familiar linear programming-based planning models to specifically treat components of production lead time, thereby realizing a more accurate representation of the production process. In addition, the reformulation accommodates noninteger values for lead times as well as unequal-length planning periods. Manufacturing Resources Planning (MRP) and the Critical Path Method (CPM) are recast in terms of the framework, revealing opportunities for model generalization and extension, and their relationship to linear programming models.

The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.

Purpose Experimental animal models of acute respiratory distress syndrome (ARDS) have shown that the updated airway pressure release ventilation (APRV) methodologies may significantly improve oxygenation, maximize lung recruitment, and attenuate lung injury, without circulatory depression. This led us to hypothesize that early application of APRV in patients with ARDS would allow pulmonary function to recover faster and would reduce the duration of mechanical ventilation as compared with low tidal volume lung protective ventilation (LTV). Methods A total of 138 patients with ARDS who received mechanical ventilation for <48 h between May 2015 to October 2016 while in the critical care medicine unit (ICU) of the West China Hospital of Sichuan University were enrolled in the study. Patients were randomly assigned to receive APRV ( n  = 71) or LTV ( n  = 67). The settings for APRV were: high airway pressure (P high ) set at the last plateau airway pressure (P plat ), not to exceed 30 cmH 2 O) and low airway pressure ( P low ) set at 5 cmH 2 O; the release phase (T low ) setting adjusted to terminate the peak expiratory flow rate to ≥ 50%; release frequency of 10–14 cycles/min. The settings for LTV were: target tidal volume of 6 mL/kg of predicted body weight; P plat not exceeding 30 cmH 2 O; positive end-expiratory pressure (PEEP) guided by the PEEP–FiO 2 table according to the ARDSnet protocol. The primary outcome was the number of days without mechanical ventilation from enrollment to day 28. The secondary endpoints included oxygenation, P plat , respiratory system compliance, and patient outcomes. Results Compared with the LTV group, patients in the APRV group had a higher median number of ventilator-free days {19 [interquartile range (IQR) 8–22] vs. 2 (IQR 0–15); P  < 0.001}. This finding was independent of the coexisting differences in chronic disease. The APRV group had a shorter stay in the ICU ( P  = 0.003). The ICU mortality rate was 19.7% in the APRV group versus 34.3% in the LTV group ( P  = 0.053) and was associated with better oxygenation and respiratory system compliance, lower P plat , and less sedation requirement during the first week following enrollment ( P  < 0.05, repeated-measures analysis of variance). Conclusions Compared with LTV, early application of APRV in patients with ARDS improved oxygenation and respiratory system compliance, decreased P plat and reduced the duration of both mechanical ventilation and ICU stay.

Background The tip design and length of catheter impact catheter function. Two types of catheters with different tips, side-hole catheters and step-tip catheters, are commonly used during continuous renal replacement therapy (CRRT). However, there is insufficient evidence comparing their efficacy and safety in CRRT. In addition, whether the insertion of a longer catheter could enhance catheter function remains poorly studied and controversial. Methods In this open-label, three-arm, randomized trial, critically ill patients receiving CRRT were randomized to three groups. Group A received 20 cm side-hole catheters (GDHK‐1120), group B received 20 cm step-tip catheters (GDHK‐1320) and group C received 25 cm step-tip catheters (GDHK‐1325). The primary outcomes were the incidence of catheter dysfunction and catheter survival time. Results A total of 351 patients were enrolled, with 116 in group A, 117 in group B, and 118 in group C. The incidence of catheter dysfunction in group A (35.7%, 51/143) was significantly higher than that in group B (17.7%, 22/124) ( P  = 0.001). However, there was no difference between group B and group C (15.6%, 23/147) ( P  = 0.744). The catheter survival time was comparable between group A (5.5 days, IQR 2.5–9.3) and group B (5.0 days, IQR 3.0–10.0) ( P  = 0.626). In contrast, group C (6.4 days, IQR 3.9–12.0) demonstrated a significantly longer catheter survival time compared to group B ( P  = 0.019). Cox regression analysis identified BMI (HR 1.052, 95% CI 1.003–1.103, P  = 0.036) as an independent risk factor for catheter dysfunction. Results were not consistent across BMI tertiles, with similar results observed only in patients with a lower BMI (BMI < 24.2) (chi-square 13.65, P  = 0.001). There was also a trend that patients in group C have a longer filter lifespan (36.5 h, IQR 16.9–68.1, P  = 0.001) and a lower incidence of catheter-related thrombosis (10.40 per 1000 catheter-days, 95% CI 5.93, 17.83, P  = 0.019). Other secondary outcomes were not significantly different among groups. Conclusions Step-tip catheters may be preferable for CRRT, particularly for patients in the lower BMI tercile. Longer femoral vein catheterization demonstrated enhanced benefits in CRRT, especially among obese patients. Further high-quality, multicenter RCTs are essential to strengthen the evidence guiding catheter selection during CRRT. Trial registration : ChiCTR2300075107. Registered 25 August 2023.

This paper uncover a new perception of the dynamic interconnection between CO 2 emission and economic growth, renewable energy use, trade openness, and technological innovation in the Portuguese economy utilizing innovative Morlet wavelet analysis. The research applied continuous wavelet transform, wavelet correlation, the multiple and partial wavelet coherence, and frequency domain causality analyses are applied on variables of investigation using dataset between 1980 and 2019. The result of these analyses disclosed that the interconnection among the indicators progresses over time and frequency. The present analysis finds notable wavelet coherence and significant lead and lag interconnections in the frequency domain, while conflicting relationships among the variables are found in the time domain. The wavelet analysis according to economic viewpoint affirms that renewable energy consumption helps to curb CO 2 while trade openness, technological innovation, and economic growth contribute to CO 2 . The outcomes also proposed that renewable energy consumption decreases CO 2 in medium and long run in Portugal. Therefore, policymakers in Portugal should stimulate investment in renewable energy sources, establish restrictive laws, and enhance energy innovation.