Trend analysis in the presence of short- and long-range correlations with application to regional warming.

Many real-world time series exhibit both significant short- and long-range temporal correlations. Such correlations enhance the errors of linear trend analysis. In this paper, we provide a general framework for trend analysis under the consideration of such correlations. We propose a parsimonious model containing both a single short-range autoregressive parameter and long-range fractional parameter. We derive analytical closed-form results for the error bars of the least-squares estimate of the trend for such time series, highlighting the different effects of short- and of long-range correlations. We employ an ensemble method for the automated extraction of scaling regions to estimate the fractional parameter of the data model together with its error bar, and the Grünwald-Letnikov derivative for the identification of the autoregressive parameter. We apply this framework to the study of warming trends on gridded temperature data in central Europe. We make use of the redundancy of the trend signal in adjacent grid points using methods of spatial averaging and the first principal component of empirical orthogonal function analysis. We find good agreement between the results of these two methods. We find a statistically significant decadal warming trend in central Europe over the past 70 years, which shows a particularly dramatic increase over the past 20 years.

Controls that expedite first-passage times in disordered systems.

First-passage time statistics in disordered systems exhibiting scale invariance are studied widely. In particular, long trapping times in energy or entropic traps are fat-tailed distributed, which slow the overall transport process. We study the statistical properties of the first-passage time of biased processes in different models, and we employ the big-jump principle that shows the dominance of the maximum trapping time on the first-passage time. We demonstrate that the removal of this maximum significantly expedites transport. As the disorder increases, the system enters a phase where the removal shows a dramatic effect. Our results show how we may speed up transport in strongly disordered systems exploiting scale invariance. In contrast to the disordered systems studied here, the removal principle has essentially no effect in homogeneous systems; this indicates that improving the conductance of a poorly conducting system is, theoretically, relatively easy as compared to a homogeneous system.

Improved clinical outcomes in response to a 12-week blended digital and community-based long-COVID-19 rehabilitation programme.

Introduction: Two million people in the UK are experiencing long COVID (LC), which necessitates effective and scalable interventions to manage this condition. This study provides the first results from a scalable rehabilitation programme for participants presenting with LC. Methods: 601 adult participants with symptoms of LC completed the Nuffield Health COVID-19 Rehabilitation Programme between February 2021 and March 2022 and provided written informed consent for the inclusion of outcomes data in external publications. The 12-week programme included three exercise sessions per week consisting of aerobic and strength-based exercises, and stability and mobility activities. The first 6 weeks of the programme were conducted remotely, whereas the second 6 weeks incorporated face-to-face rehabilitation sessions in a community setting. A weekly telephone call with a rehabilitation specialist was also provided to support queries and advise on exercise selection, symptom management and emotional wellbeing. Results: The 12-week rehabilitation programme significantly improved Dyspnea-12 (D-12), Duke Activity Status Index (DASI), World Health Orginaisation-5 (WHO-5) and EQ-5D-5L utility scores (all p < 0.001), with the 95% confidence intervals (CI) for the improvement in each of these outcomes exceeding the minimum clinically important difference (MCID) for each measure (mean change [CI]: D-12: -3.4 [-3.9, -2.9]; DASI: 9.2 [8.2, 10.1]; WHO-5: 20.3 [18.6, 22.0]; EQ-5D-5L utility: 0.11 [0.10, 0.13]). Significant improvements exceeding the MCID were also observed for sit-to-stand test results (4.1 [3.5, 4.6]). On completion of the rehabilitation programme, participants also reported significantly fewer GP consultations (p < 0.001), sick days (p = 0.003) and outpatient visits (p = 0.007) during the previous 3 months compared with baseline. Discussion: The blended and community design of this rehabilitation model makes it scalable and meets the urgent need for an effective intervention to support patients experiencing LC. This rehabilitation model is well placed to support the NHS (and other healthcare systems worldwide) in its aim of controlling the impacts of COVID-19 and delivering on its long-term plan. Clinical trial registration: https://www.isrctn.com/ISRCTN14707226, identifier 14707226.

Scalable modEls of Community rehAbilitation for Individuals Recovering From COVID:19 reLated illnEss: A Longitudinal Service Evaluation Protocol-"SeaCole Cohort Evaluation".

Introduction: High levels of physical, cognitive, and psychosocial impairments are anticipated for those recovering from the COVID-19. In the UK, ~50% of survivors will require additional rehabilitation. Despite this, there is currently no evidence-based guideline available in England and Wales that addresses the identification, timing and nature of effective interventions to manage the morbidity associated following COVID-19. It is now timely to accelerate the development and evaluation of a rehabilitation service to support patients and healthcare services. Nuffield Health have responded by configuring a scalable rehabilitation pathway addressing the immediate requirements for those recovering from COVID-19 in the community. Methods and Analysis: This long-term evaluation will examine the effectiveness of a 12-week community rehabilitation programme for COVID-19 patients who have been discharged following in-patient treatment. Consisting of two distinct 6-week phases; Phase 1 is an entirely remote service, delivered via digital applications. Phase 2 sees the same patients transition into a gym-based setting for supervised group-based rehabilitation. Trained rehabilitation specialists will coach patients across areas such as goal setting, exercise prescription, symptom management and emotional well-being. Outcomes will be collected at 0, 6, and 12 weeks and at 6- and 12-months. Primary outcome measures will assess changes in health-related quality of life (HR-QOL) and COVID-19 symptoms using EuroQol Five Dimension Five Level Version (EQ-5D-5L) and Dyspnea-12, respectively. Secondary outcome measures of the Duke Activity Status Questionnaire (DASI), 30 s sit to stand test, General Anxiety Disorder-7 (GAD-7), Patient Health Questionnaire-9 (PHQ-9), Patient Experience Questionnaire (PEQ) and Quality Adjusted Life Years (QALY) will allow for the evaluation of outcomes, mediators and moderators of outcome, and cost-effectiveness of treatment. Discussion: This evaluation will investigate the immediate and long-term impact, as well as the cost effectiveness of a blended rehabilitation programme for COVID-19 survivors. This evaluation will provide a founding contribution to the literature, evaluating one of the first programmes of this type in the UK. The evaluation has international relevance, with the potential to show how a new model of service provision can support health services in the wake of COVID-19. Trial Registration: Current Trials ISRCTN ISRCTN14707226 Web: http://www.isrctn.com/ISRCTN14707226.

Large deviations of the ballistic Lévy walk model.

We study the ballistic Lévy walk stemming from an infinite mean traveling time between collision events. Our study focuses on the density of spreading particles all starting from a common origin, which is limited by a "light" cone -v_{0}t

Extreme value theory for constrained physical systems.

We investigate extreme value theory for physical systems with a global conservation law which describes renewal processes, mass transport models, and long-range interacting spin models. As shown previously, a special feature is that the distribution of the extreme value exhibits a nonanalytical point in the middle of the support. We expose exact relationships between constrained extreme value theory and well-known quantities of the underlying stochastic dynamics, all valid beyond the midpoint in general, i.e., even far from the thermodynamic limit. For example, for renewal processes the distribution of the maximum time between two renewal events is exactly related to the mean number of these events. In the thermodynamic limit, we show how our theory is suitable to describe typical and rare events which deviate from classical extreme value theory. For example, for the renewal process we unravel dual scaling of the extreme value distribution, pointing out two types of limiting laws: a normalizable scaling function for the typical statistics and a non-normalized state describing the rare events.

Probabilistic properties of detrended fluctuation analysis for Gaussian processes.

Detrended fluctuation analysis (DFA) is one of the most widely used tools for the detection of long-range dependence in time series. Although DFA has found many interesting applications and has been shown to be one of the best performing detrending methods, its probabilistic foundations are still unclear. In this paper, we study probabilistic properties of DFA for Gaussian processes. Our main attention is paid to the distribution of the squared error sum of the detrended process. We use a probabilistic approach to derive general formulas for the expected value and the variance of the squared fluctuation function of DFA for Gaussian processes. We also get analytical results for the expected value of the squared fluctuation function for particular examples of Gaussian processes, such as Gaussian white noise, fractional Gaussian noise, ordinary Brownian motion, and fractional Brownian motion. Our analytical formulas are supported by numerical simulations. The results obtained can serve as a starting point for analyzing the statistical properties of DFA-based estimators for the fluctuation function and long-memory parameter.

Theoretical foundation of detrending methods for fluctuation analysis such as detrended fluctuation analysis and detrending moving average.

We present a bottom-up derivation of fluctuation analysis with detrending for the detection of long-range correlations in the presence of additive trends or intrinsic nonstationarities. While the well-known detrended fluctuation analysis (DFA) and detrending moving average (DMA) were introduced ad hoc, we claim basic principles for such methods where DFA and DMA are then shown to be specific realizations. The mean-squared displacement of the summed time series contains the same information about long-range correlations as the autocorrelation function but has much better statistical properties for large time lags. However, the scaling exponent of its estimator on a single time series is affected not only by trends on the data but also by intrinsic nonstationarities. We therefore define the fluctuation function as mean-squared displacement with weighting kernel. We require that its estimator be unbiased and exhibit the correct scaling behavior for the random component of a signal, which is only achieved if the weighting kernel implies detrending. We show how DFA and DMA satisfy these requirements and we extract their kernel weights.

Detrended fluctuation analysis and the difference between external drifts and intrinsic diffusionlike nonstationarity.

Detrended fluctuation analysis (DFA) has been shown to be an effective method to study long-range correlation of nonstationary series. In principle, DFA considers F_{DFA}^{2}(s), the mean of variance around the local polynomial fit in segments with length s, and then estimates the scaling exponent α_{DFA} in F_{DFA}(s)∼s^{α_{DFA}} with varying s. Usually, the methodological studies of DFA focus on its effect on removing the drift due to the external trends. Only few paid attention to nonstationary series without drift, such as fractional Brownian motion (FBM) with nonstationarity due to its intrinsic dynamics. Both of these types of nonstationarity can shift the local mean by drift or diffusion and can be treated as the additive nonstationarity eliminable by the additive decomposition. In this study, we limit our discussion to such additive nonstationarity and furthermore specifically distinguish these two types of nonstationarity, namely the drift and the intrinsic diffusionlike nonstationarity. To understand how DFA works for the intrinsic diffusionlike nonstationarity, we take FBM as the example and seek for the answers to two fundamental questions: (1) what DFA removes from FBM; and (2) why DFA can handle such intrinsic diffusionlike nonstationarity, in contrast to methods only applicable to stationary series such as the fluctuation analysis. A crucial condition, i.e., statistical equivalence among all segments, is proposed and checked in the fluctuation analysis and DFA. As shown, the crucial condition is a natural requirement for the connection between DFA and autocorrelation function. With the help of the crucial condition, our study analytically and numerically demonstrates for the intrinsic diffusionlike nonstationary series that (1) rather than the nonstationarity as thought, DFA actually removes the difference among all segments; (2) the detrended segments fulfill the crucial condition so that the average over segments becomes equivalent to the ensemble average over realizations. These answers are also true for series with a drift. Thus, we provide a unified perspective to refresh the understanding of how DFA works on nonstationarity and underpin the mathematical ground of DFA.