Multiple Time Scales Research Articles

When do plant hydraulics matter in terrestrial biosphere modelling?

The ascent of water from the soil to the leaves of vascular plants, described by the study of plant hydraulics, regulates ecosystem responses to environmental forcing and recovery from stress periods. Several approaches to model plant hydraulics have been proposed. In this study, we introduce four different versions of plant hydraulics representations in the terrestrial biosphere model T&C to understand the significance of plant hydraulics to ecosystem functioning. We tested representations of plant hydraulics, investigating plant water capacitance, and long-term xylem damages following drought. The four models we tested were a combination of representations including or neglecting capacitance and including or neglecting xylem damage legacies. Using the models at six case studies spanning semiarid to tropical ecosystems, we quantify how plant xylem flow, plant water storage and long-term xylem damage can modulate overall water and carbon dynamics across multiple time scales. We show that as drought develops, models with plant hydraulics predict a slower onset of plant water stress, and a diurnal variability of water and carbon fluxes closer to observations. Plant water storage was found to be particularly important for the diurnal dynamics of water and carbon fluxes, with models that include plant water capacitance yielding better results. Models including permanent damage to conducting plant tissues show an additional significant drought legacy effect, limiting plant productivity during the recovery phase following major droughts. However, when considering ecosystem responses to the observed climate variability, plant hydraulic modules alone cannot significantly improve the overall model performance, even though they reproduce more realistic water and carbon dynamics. This opens new avenues for model development, explicitly linking plant hydraulics with additional ecosystem processes, such as plant phenology and improved carbon allocation algorithms.

Toward the mechanism of jarastatin (rJast) inhibition of the integrin αVβ3

Disintegrins are a family of cysteine-rich small proteins that were first identified in snake venom. The high divergence of disintegrins gave rise to a plethora of functions, all related to the interaction with integrins. Disintegrins evolved to interact selectively with different integrins, eliciting many physiological outcomes and being promising candidates for the therapy of many pathologies. We used NMR to determine the structure and dynamics of the recombinant disintegrin jarastatin (rJast) and its interaction with the cancer-related integrin αVβ3. rJast displayed the canonical fold of a medium-sized disintegrin and showed complex dynamic in multiple timescales. We used NMR experiments to map the interaction of rJast with αVβ3, and molecular docking followed by molecular dynamics (MD) simulation to describe the first structural model of a disintegrin/integrin complex. We showed that not only the RGD loop participates in the interaction, but also the N-terminal domain. rJast plasticity was essential for the interaction with αVβ3 and correlated with the main modes of motion depicted in the MD trajectories. In summary, our study provides novel structural insights that enhance our comprehension of the mechanisms underlying disintegrin functionality.

Nonlinear vibration of microbeams subjected to a uniform magnetic field and rested on nonlinear elastic foundation

Abstract This study investigates the nonlinear vibration motions of the Euler–Bernoulli microbeam on a nonlinear elastic foundation in a uniform magnetic field based on Modified Couple Stress Theory (MCST). The effect of size, foundation, and magnetic field on the nonlinear vibration motion of microbeam has been examined. The governing equations related to the nonlinear vibration motions of the microbeam are obtained by using Hamilton’s Principle, and the Multiple Time Scale Method was used to obtain the solutions for the governing equations. The linear natural frequencies of microbeam are presented in the table according to nonlinear parameters and boundary conditions. The linear and nonlinear natural frequency ratio graphs are shown. The present study results are also compared with previous work for validation. It is observed that length scale parameters and magnetic force have a more significant effect on the natural frequency of microbeams. It is seen that when the linear elastic foundation coefficient, the Pasternak foundation and the magnetic force effects increase, the ratio of nonlinear and linear natural frequency decreases.

Meta-reinforcement learning via orbitofrontal cortex

The meta-reinforcement learning (meta-RL) framework, which involves RL over multiple timescales, has been successful in training deep RL models that generalize to new environments. It has been hypothesized that the prefrontal cortex may mediate meta-RL in the brain, but the evidence is scarce. Here we show that the orbitofrontal cortex (OFC) mediates meta-RL. We trained mice and deep RL models on a probabilistic reversal learning task across sessions during which they improved their trial-by-trial RL policy through meta-learning. Ca2+/calmodulin-dependent protein kinase II-dependent synaptic plasticity in OFC was necessary for this meta-learning but not for the within-session trial-by-trial RL in experts. After meta-learning, OFC activity robustly encoded value signals, and OFC inactivation impaired the RL behaviors. Longitudinal tracking of OFC activity revealed that meta-learning gradually shapes population value coding to guide the ongoing behavioral policy. Our results indicate that two distinct RL algorithms with distinct neural mechanisms and timescales coexist in OFC to support adaptive decision-making.

Climate influence on compound solar and wind droughts in Australia

Solar photovoltaic and wind power are central to Australia’s renewable energy future, implying an energy sector vulnerable to weather and climate variability. Alignment of weather systems and the influence of large-scale climate modes of variability risks widespread reductions in solar and wind resources, and could induce grid-wide impacts. We therefore systematically analyse the relationship between compound solar radiation and wind speed droughts with weather systems and climate modes of variability over multiple time scales. We find that compound solar and wind droughts occur most frequently in winter, affecting at least five significant energy-producing regions simultaneously on 10% of days. The associated weather systems vary by season and by drought type, although widespread cloud cover and anticyclonic circulation patterns are common features. Indices of major climate modes are not strong predictors of grid-wide droughts, and are typically within one standard deviation of the mean during seasons with the most widespread events. However, the spatial imprints of the teleconnections display strong regional variations, with drought frequencies varying by more than ten days per season between positive and negative phases of climate modes in some regions. The spatial variability of these teleconnection patterns suggests that droughts in one region may be offset by increased resource in another. Our work highlights the opportunity for minimising the impact of energy production variability by utilising weather and climate intelligence. Exploiting the spatial variability associated with daily weather systems and the seasonal influence of climate modes could help build a more climate-resilient renewables-dominated energy system.

Dynamic time scales ensemble framework for similarity-based remaining useful life prediction under multiple failure modes

In modern industry, the stochastic degradation of mechanical equipment typically involves multiple failure modes, which heavily affects the reliability of the remaining useful life (RUL) prediction. The similarity-based methods have been widely deployed in RUL prediction due to their flexibility, but it is still challenging to accurately identify similar degradation trajectories under varying failure modes. The obstacles lie in the interference of reference trajectories under different degradation states and the insufficiency of measuring trajectory trends. Therefore, this paper proposes a dynamic scales ensemble method based on the mean removal Canberra distance with failure identification (FI-MRC-DSE) for similarity-based prognosis. Firstly, a gated recurrent unit autoencoder network is employed to adaptively extract failure features from multi-dimensional monitoring data to support the targeted selection of reference trajectories. Then, the similarity matching is performed based on the proposed MRC distance instead of the commonly used Euclidean distance, enhancing the perception of degradation trends. Finally, the matching results across multiple time scales, which are dynamically determined by the instance's degradation state, are integrated to obtain the predicted RUL. It effectively overcomes the insufficient utilization of trajectory caused by the single time scale. In the experiments, the superiority of our developed similarity-based FI-MRC-DSE method is demonstrated by comparison with the state-of-the-art similarity-based methods. The effectiveness analyses and the ablation study show that all three key components contribute to accurate prognosis under multiple failure modes.

Spatial–Temporal Variation Characteristics and Driving Factors of Net Primary Production in the Yellow River Basin over Multiple Time Scales

Vegetation net primary productivity (NPP) serves as a crucial and intuitive indicator for assessing ecosystem health. However, the nonlinear dynamics and influencing factors operating at various time scales are not yet fully understood. Here, the ensemble empirical mode decomposition (EEMD) method was used to analyze the spatiotemporal patterns of NPP and its association with hydrothermal factors and anthropogenic activities across different temporal scales for the Yellow River Basin (YRB) from 2000 to 2020. The results indicate that: (1) the annual average NPP was 236.37 g C/m2 in the YRB and increased at rates of 4.64 g C/m2/a1 (R2 = 0.86, p < 0.01) during 2000 to 2020. Spatially, nonlinear analysis indicates that 72.77% of the study area exhibits a predominantly increasing trend in NPP, while 25.17% exhibits a reversing trend. (2) On a 3-year time scale, warming has resulted in an increase in NPP in the majority of areas of the study area (69.49%). As the time scale widens, the response of vegetation to climate change becomes more prominent; especially under the long-term trend, the percentage areas of the correlation between vegetation and precipitation and temperature increased with significance, reaching 48.21% and 11.57%, respectively. (3) Through comprehensive time analysis and multivariate regression analysis, it was confirmed that both human activities and climate factors had comparable impacts on vegetation growth. Among different vegetation types, climate was still the main factor affecting grassland NPP, and only 15.74% of grassland was affected by human activities. For shrubland, forest, and farmland, human activity was a dominating factor for vegetation NPP change. There are still few studies on vegetation change using nonlinear methods in the Yellow River Basin, and most studies have not considered the effect of time scale on vegetation evolution. The findings highlight the significance of multi-time scale analysis in understanding the vegetation dynamics and providing scientific guidance for future vegetation restoration and conservation efforts.

Nonlinear Forced Vibration Analysis of Two-Directional Functionally Graded Truncated Conical Shells

It is crucial to comprehend the nonlinear vibration behavior of sophisticated engineering constructions to aid in the examination, creation and production procedures. The primary objective of this study, therefore, is to focus on exploring the nonlinear forced vibrational characteristics of two-directional functionally graded (TDFG) cone-shaped shell for the first time. To accomplish this, the motion equations of the TDFG cone-shaped shell are derived based upon the framework of first-order shear deformation theory (FSDT) in conjunction with von Kármán assumption. TDFG cone-shaped shell models have mechanical properties that can change smoothly and continuously across the thickness and length of the shell, following a power law distribution pattern. To discretize the partial differential equations and transform them into ordinary differential equations, the Galerkin approach was used and ultimately, these equations were solved using the multiple time scales method. The accuracy and effectiveness of this analytical model are indicated through comparison with existing solutions. Finally, some comprehensive parametric investigations are carried out to gain insight into the impacts of several factors on the nonlinear behavior of the shell, highlighting the significant influence of transverse and longitudinal gradient indices on the nonlinear frequency response, which is not considered before. The new insights gained from this research could serve as valuable benchmark outcomes as well as contribute to a more comprehensive understanding of the nonlinear vibrations in future analysis and design processes.

Digital Adaptive Behavioral Interventions to Improve HIV Prevention and Care: Innovations in Intervention Approach and Experimental Design.

Recent advances in digital technologies can be leveraged to adapt HIV prevention and treatment services to the rapidly changing needs of individuals in everyday life. However, to fully take advantage of these technologies, it is critical to effectively integrate them with human-delivered components. Here, we introduce a new experimental approach for optimizing the integration and adaptation of digital and human-delivered behavioral intervention components for HIV prevention and treatment. Typically, human-delivered components can be adapted on a relatively slow timescale (e.g., every few months or weeks), while digital components can be adapted much faster (e.g., every few days or hours). Thus, the systematic integration of these components requires an experimental approach that involves sequential randomizations on multiple timescales. Selecting an experimental approach should be motivated by the type of adaptive intervention investigators would like to develop, and the scientific questions they have about its construction.

Regional to global assessments of ocean transparency dynamics from 1997 to 2019

Water transparency, often quantified using the Secchi disk depth (Zsd), is a key parameter for assessing the water quality and ecological health in marine environments. However, a limited understanding of global Zsd dynamics over the past two decades exists. To assess the variations of ocean transparency over multiple timescales, the Ocean Colour Climate Change Initiative (OC-CCI) datasets are used to retrieve monthly global Zsd products during 1997–2019. On regional to global scales, significant Zsd variations on seasonal, interannual, and long-term timescales are identified. Seasonal and interannual variations of Zsd in most open oceans are found to be closely linked to phytoplankton dynamics and climate change. In the coastal zones, the Zsd variability is attributed to various physical and environmental factors, including seasonal monsoons, river outflow, sediment resuspension, eutrophication, etc. Moreover, the climatological seasonal Zsd variations of 22 typical ocean areas and the associated influencing factors are investigated on regional scales. The long-term trends of Zsd suggest the widespread expansion of oligotrophic waters within the North Pacific, North Atlantic, and South Indian Ocean gyres, indicating ongoing ocean desertification. The OC-CCI Zsd datasets can be used to build a merged centennial time series of Zsd in marine optics by linking to historical measurements. By relying on satellite-derived products, global and time-varying images offer a more comprehensive understanding of regional to global Zsd dynamics.

Rate-induced tipping in ecosystems and climate: the role of unstable states, basin boundaries and transient dynamics

Abstract. The climate system as well as ecosystems might undergo relatively sudden qualitative changes in the dynamics when environmental parameters or external forcings vary due to anthropogenic influences. The study of these qualitative changes, called tipping phenomena, requires the development of new methodological approaches that allow phenomena observed in nature to be modeled, analyzed and predicted, especially concerning the climate crisis and its consequences. Here we briefly review the mechanisms of classical tipping phenomena and investigate rate-dependent tipping phenomena which occur in non-autonomous systems characterized by multiple timescales in more detail. We focus on the mechanism of rate-induced tipping caused by basin boundary crossings. We unravel the mechanism of this transition and analyze, in particular, the role of such basin boundary crossings in non-autonomous systems when a parameter drift induces a saddle-node bifurcation in which new attractors and saddle points emerge, including their basins of attraction. Furthermore, we study the detectability of those bifurcations by monitoring single trajectories in state space and find that depending on the rate of environmental parameter drift, such saddle-node bifurcations might be masked or hidden, and they can only be detected if a critical rate of environmental drift is crossed. This analysis reveals that unstable states of saddle type are the organizing centers of the global dynamics in non-autonomous multistable systems and as such need much more attention in future studies.

Characterizing Performance of Freshwater Wetland Methane Models Across Time Scales at FLUXNET‐CH4 Sites Using Wavelet Analyses

AbstractProcess‐based land surface models are important tools for estimating global wetland methane (CH4) emissions and projecting their behavior across space and time. So far there are no performance assessments of model responses to drivers at multiple time scales. In this study, we apply wavelet analysis to identify the dominant time scales contributing to model uncertainty in the frequency domain. We evaluate seven wetland models at 23 eddy covariance tower sites. Our study first characterizes site‐level patterns of freshwater wetland CH4 fluxes (FCH4) at different time scales. A Monte Carlo approach was developed to incorporate flux observation error to avoid misidentification of the time scales that dominate model error. Our results suggest that (a) significant model‐observation disagreements are mainly at multi‐day time scales (<15 days); (b) most of the models can capture the CH4 variability at monthly and seasonal time scales (>32 days) for the boreal and Arctic tundra wetland sites but have significant bias in variability at seasonal time scales for temperate and tropical/subtropical sites; (c) model errors exhibit increasing power spectrum as time scale increases, indicating that biases at time scales <5 days could contribute to persistent systematic biases on longer time scales; and (d) differences in error pattern are related to model structure (e.g., proxy of CH4 production). Our evaluation suggests the need to accurately replicate FCH4 variability, especially at short time scales, in future wetland CH4 model developments.

Seasonal and Storm Event‐Based Dynamics of Dissolved Organic Carbon (DOC) Concentration in a Mediterranean Headwater Catchment

AbstractThis study investigates the spatial and temporal dynamics of Dissolved Organic carbon (DOC) concentration in a Mediterranean headwater catchment (Turbolo River catchment, southern Italy) equipped with two multi‐parameter sondes providing more than two‐year (May 2019–November 2021) continuous high‐frequency measurements of several DOC‐related parameters. The sondes were installed in two nested sections, a quasi‐pristine upstream sub‐catchment and a downstream outlet with anthropogenic water quality disturbances. DOC estimates were achieved by correcting the fluorescent dissolved organic matter—fDOM—values through an original procedure not requiring extensive laboratory measurements. Then, DOC dynamics at the seasonal and storm event scales were analyzed. At the seasonal scale, results confirmed the climate control on DOC production, with increasing background concentrations in hot and dry summer months. The hydrological regulation proved crucial for DOC mobilization and export, with the top 10th percentile of discharge associated with up to 79% of the total DOC yield. The analysis at the storm scale using flushing and hysteresis indices highlighted substantial differences between the two catchments. In the steeper upstream catchment, the limited capability of preserving hydraulic connection over time with DOC sources determined the prevalence of transport as the limiting factor to DOC export. In the downstream catchment, transport‐ and source‐limited processes were observed almost equally. The correlation between the hysteretic behavior and antecedent precipitation was not linear since the process reverted to transport‐limited for high accumulated rainfall values. Exploiting high‐resolution measurements, the study provided insights into DOC export dynamics in nested headwater catchments at multiple time scales.

Renewable Electric Energy System Planning Considering Seasonal Electricity Imbalance Risk

The growing renewable integration significantly enhances the seasonal electricity imbalance of the electric energy system. However, traditional power system planning methods usually take into account the hourly power balance within typical days but seldom consider the seasonal imbalance risk in the long-term timescale. Therefore, this paper proposes a generation-transmission-storage co-planning model considering the seasonal imbalance risk brought by the long-term uncertainty of renewable generation in the power system. The Conditional Value at Risk (CVaR) method is introduced to quantify the seasonal imbalance risk. The power system energy balance constraints are decoupled into short-term (i.e., hourly) and long-term (i.e., monthly) energy balance so that the operation of the power system can be considered within different timescales and the seasonal imbalance risk could be calculated. The validity of the proposed model is first investigated via a modified Garver's 6-bus system case study. Furthermore, the research on the HRP-38 system (HRP represents high renewable penetration) illustrates that our proposed model could effectively control the energy imbalance risk from multiple timescales. Finally, the proposed method is applied to the whole China power system at the province level, proving that the proposed planning method could effectively enhance the reliability of China's power system, decreasing both the average annual short-term and long-term energy imbalance by 30%.

Coordinated X-ray and UV absorption within the accretion disk wind of the active galactic nucleus PG 1126-041

Context. Accretion disk winds launched close to supermassive black holes (SMBHs) are a viable mechanism providing feedback between the SMBH and the host galaxy. Aims. We aim to characterize the X-ray properties of the inner accretion disk wind of the nearby active galactic nucleus PG 1126-041 and to study its connection with the UV-absorbing wind. Methods. We performed a spectroscopic analysis of eight XMM-Newton observations of PG 1126-041 taken between 2004 and 2015, using both phenomenological models and the most advanced accretion disk wind models available. For half of the data set, we were able to compare the X-ray analysis results with the results of quasi-simultaneous, high-resolution, spectroscopic UV observations taken with the Cosmic Origins Spectrograph on board the Hubble Space Telescope. Results. The X-ray spectra of PG 1126-041 are complex and absorbed by ionized material, which is highly variable on multiple timescales, sometimes as short as 11 days. Accretion disk wind models can account for most of the X-ray spectral complexity of PG 1126-041, with the addition of massive clumps, represented by a partially covering absorber. Variations in column density (NH ∼ 5 − 20 × 1022 cm−2) of the partially covering absorber drive the observed X-ray spectral variability of PG 1126-041. The absorption from the X-ray partially covering gas and from the blueshifted C IV troughs appear to vary in a coordinated way. Conclusions. The line of sight toward PG 1126-041 offers a privileged view through a highly dynamic nuclear wind originating on inner accretion disk scales, making the source a very promising candidate for future detailed studies of the physics of accretion disk winds around SMBHs.

Variability of temperature measurements recorded by a wearable device by biological sex

BackgroundFemales have been historically excluded from biomedical research due in part to the documented presumption that results with male subjects will generalize effectively to females. This has been justified in part by the assumption that ovarian rhythms will increase the overall variance of pooled random samples. But not all variance in samples is random. Human biometrics are continuously changing in response to stimuli and biological rhythms; single measurements taken sporadically do not easily support exploration of variance across time scales. Recently we reported that in mice, core body temperature measured longitudinally shows higher variance in males than cycling females, both within and across individuals at multiple time scales.MethodsHere, we explore longitudinal human distal body temperature, measured by a wearable sensor device (Oura Ring), for 6 months in females and males ranging in age from 20 to 79 years. In this study, we did not limit the comparisons to female versus male, but instead we developed a method for categorizing individuals as cyclic or acyclic depending on the presence of a roughly monthly pattern to their nightly temperature. We then compared structure and variance across time scales using multiple standard instruments.ResultsSex differences exist as expected, but across multiple statistical comparisons and timescales, there was no one group that consistently exceeded the others in variance. When variability was assessed across time, females, whether or not their temperature contained monthly cycles, did not significantly differ from males both on daily and monthly time scales.ConclusionsThese findings contradict the viewpoint that human females are too variable across menstrual cycles to include in biomedical research. Longitudinal temperature of females does not accumulate greater measurement error over time than do males and the majority of unexplained variance is within sex category, not between them.

A multiple time scale approach for anisotropic inertial wave turbulence

Wave turbulence is the study of the long-time statistical behaviour of equations describing a set of weakly nonlinear interacting waves. Such a theory, which has a natural asymptotic closure, allows us to probe the nature of turbulence more deeply than the exact Kolmogorov laws by rigorously proving the direction of the cascade and the existence of an inertial range, predicting stationary spectra for conserved quantities, or evaluating the Kolmogorov constant. An emblematic example is given by fast rotating fluids for which a wave turbulence theory has been derived by Galtier (Phys. Rev. E, vol. 68, issue 1, 2003, p. 015301). This work involves non-trivial analytical developments for a problem that is anisotropic by nature. We propose here a new path for the derivation of the kinetic equation by using the anisotropy at the beginning of the analysis. We show that the helicity basis is not necessary to obtain the wave amplitude equation for the canonical variables that involve a combination of poloidal and toroidal fields. The multiple time scale method adapted to this anisotropic problem is then used to derive the kinetic equation that is the same as the original work when anisotropy is eventually taken into account. This result proves the commutativity between asymptotic closure and anisotropy. In addition, the multiple time scale method informs us that the kinetic equation can be derived without imposing restrictions on the probability distribution of the wave amplitude such as quasi-Gaussianity, or on the phase such as random phase approximation that naturally occurs dynamically.

Spoken Word Recognition: A Focus on Plasticity

Psycholinguists define spoken word recognition (SWR) as, roughly, the processes intervening between speech perception and sentence processing, whereby a sequence of speech elements is mapped to a phonological wordform. After reviewing points of consensus and contention in SWR, we turn to the focus of this review: considering the limitations of theoretical views that implicitly assume an idealized (neurotypical, monolingual adult) and static perceiver. In contrast to this assumption, we review evidence that SWR is plastic throughout the life span and changes as a function of cognitive and sensory changes, modulated by the language(s) someone knows. In highlighting instances of plasticity at multiple timescales, we are confronted with the question of whether these effects reflect changes in content or in processes, and we consider the possibility that the two are inseparable. We close with a brief discussion of the challenges that plasticity poses for developing comprehensive theories of spoken language processing.

Formal theories clarify the complex: Generalizing a neural process account of the interaction of visual exploration and word learning in infancy.

The interaction of visual exploration and auditory processing is central to early cognitive development, supporting object discrimination, categorization, and word learning. Research has shown visual-auditory interactions to be complex, created from multiple processes and changing over multiple timescales. To better understand these interactions, we generalize a formal neural process model of early word learning to two studies examining how words impact 9- to 22-month-olds' attention to novelty. These simulations clarify the origin and nature of previously demonstrated effects of labels on visual exploration and the basis of mutual exclusivity effects in word learning. We use our findings to discuss key questions for this special section: what makes a good theory and how should formal theories interface with empirical paradigms and findings?

Refined Composite Multiscale Fuzzy Dispersion Entropy and Its Applications to Bearing Fault Diagnosis.

Rotary machines often exhibit nonlinear behavior due to factors such as nonlinear stiffness, damping, friction, coupling effects, and defects. Consequently, their vibration signals display nonlinear characteristics. Entropy techniques prove to be effective in detecting these nonlinear dynamic characteristics. Recently, an approach called fuzzy dispersion entropy (DE-FDE) was introduced to quantify the uncertainty of time series. FDE, rooted in dispersion patterns and fuzzy set theory, addresses the sensitivity of DE to its parameters. However, FDE does not adequately account for the presence of multiple time scales inherent in signals. To address this limitation, the concept of multiscale fuzzy dispersion entropy (MFDE) was developed to capture the dynamical variability of time series across various scales of complexity. Compared to multiscale DE (MDE), MFDE exhibits reduced sensitivity to noise and higher stability. In order to enhance the stability of MFDE, we propose a refined composite MFDE (RCMFDE). In comparison with MFDE, MDE, and RCMDE, RCMFDE's performance is assessed using synthetic signals and three real bearing datasets. The results consistently demonstrate the superiority of RCMFDE in detecting various patterns within synthetic and real bearing fault data. Importantly, classifiers built upon RCMFDE achieve notably high accuracy values for bearing fault diagnosis applications, outperforming classifiers based on refined composite multiscale dispersion and sample entropy methods.