Bootstrapping a powerful mixed portmanteau test for time series.

A new portmanteau test statistic is proposed for detecting nonlinearity in time series data. The new portmanteau statistic is calculated from the log of the determinant of a matrix comprised of the autocorrelations and cross-correlations of the residuals and squared residuals of a fitted time series. The asymptotic distribution of the proposed test statistic is derived as a linear combination of chi-square distributed random variables and can be approximated by a gamma distribution. A bootstrapping approach is shown to be robust when distributional assumptions are relaxed. The efficacy of the statistic is studied against linear and nonlinear dependency structures of some stationary time series models. It is shown that the new test can provide higher power than other tests in many situations. We demonstrate the advantages of the proposed test by investigating linear and nonlinear effects in an economic series and two environmental time series.

A novel comparative study of NNAR approach with linear stochastic time series models in predicting tennis player's performance.

BACKGROUND: Prediction models have gained immense importance in various fields for decision-making purposes. In the context of tennis, relying solely on the probability of winning a single match may not be sufficient for predicting a player's future performance or ranking. The performance of a tennis player is influenced by the timing of their matches throughout the year, necessitating the incorporation of time as a crucial factor. This study aims to focus on prediction models for performance indicators that can assist both tennis players and sports analysts in forecasting player standings in future matches. METHODOLOGY: To predict player performance, this study employs a dynamic technique that analyzes the structure of performance using both linear and nonlinear time series models. A novel approach has been taken, comparing the performance of the non-linear Neural Network Auto-Regressive (NNAR) model with conventional stochastic linear and nonlinear models such as Auto-Regressive Integrated Moving Average (ARIMA), Exponential Smoothing (ETS), and TBATS (Trigonometric Seasonal Decomposition Time Series). RESULTS: The study finds that the NNAR model outperforms all other competing models based on lower values of Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Percentage Error (MAPE). This superiority in performance metrics suggests that the NNAR model is the most appropriate approach for predicting player performance in tennis. Additionally, the prediction results obtained from the NNAR model demonstrate narrow 95% Confidence Intervals, indicating higher accuracy and reliability in the forecasts. CONCLUSION: In conclusion, this study highlights the significance of incorporating time as a factor when predicting player performance in tennis. It emphasizes the potential benefits of using the NNAR model for forecasting future player standings in matches. The findings suggest that the NNAR model is a recommended approach compared to conventional models like ARIMA, ETS, and TBATS. By considering time as a crucial factor and employing the NNAR model, both tennis players and sports analysts can make more accurate predictions about player performance.

Incorporating Russo-Ukrainian war in Brent crude oil price forecasting: A comparative analysis of ARIMA, TARMA and ENNReg models.

This article investigates the performance of three models - Autoregressive Integrated Moving Average (ARIMA), Threshold Autoregressive Moving Average (TARMA) and Evidential Neural Network for Regression (ENNReg) - in forecasting the Brent crude oil price, a crucial economic variable with a significant impact on the global economy. With the increasing complexity of the price dynamics due to geopolitical factors such as the Russo-Ukrainian war, we examine the impact of incorporating information on the war on the forecasting accuracy of these models. Our analysis shows that incorporating the impact of the war can significantly improve the forecasting accuracy of the models, and the ENNReg model with the inclusion of the dummy variable outperforms the other models during the war period. Including the war variable has enhanced the forecasting accuracy of the ENNReg model by 0.11%. These results carry significant implications regarding policymakers, investors, and researchers interested in developing accurate forecasting models in the presence of geopolitical events such as the Russo-Ukrainian war. The results can be used by the governments of oil-exporting countries for budget policies.

Motivation-related influences on fNIRS signals during walking exercise: a permutation entropy approach.

Cortical activity is typically indexed by analyzing functional near-infrared spectroscopy (fNIRS) signals in terms of the mean (e.g., mean oxygenated hemoglobin; HbO). Entropy approaches have been proposed as useful complementary methods for analyzing fNIRS signals. Entropy methods consider the regularity of a time series, and in doing so, may provide additional insights into the underlying dynamics of brain activity. Recent research using fNIRS found that non-disabled adults exhibit widespread increases in cortical activity and walk faster when under "extra motivation" conditions (e.g., verbal encouragement, lap timer) compared to trials without such motivators ("standard motivation"). This ancillary analysis of that study aimed to assess the extent to which fNIRS permutation entropy (PE) was affected by motivational conditions and explained variance in self-reported motivation. No regional PE differences were found between different motivational conditions. However, a greater difference in PE between motivational conditions (higher in standard, lower in extra motivation) in the anterior prefrontal cortex (aPFC) was associated with greater self-determined motivation. PE was also higher (less regular) in the primary sensorimotor cortex lower limb area compared to all other cortical areas analyzed, except the dorsal premotor cortex, regardless of motivational condition. This study provides early evidence to suggest that while different motivational environments during walking activity influence the magnitude of fNIRS signals, they may not influence the regularity of cortical signals. However, the magnitude of PE difference between motivational conditions was related to self-determined motivation in the aPFC, and this is an area warranting further investigation.

Dynamics-based characterization and classification of biodiversity indicators.

Various biodiversity indicators, such as species richness, total abundance, and species diversity indices, have been developed to capture the state of ecological communities over space and time. As biodiversity is a multifaceted concept, it is important to understand the dimension of biodiversity reflected by each indicator for successful conservation and management. Here we utilized the responsiveness of biodiversity indicators' dynamics to environmental changes (i.e., environmental responsiveness) as a signature of the dimension of biodiversity. We present a method for characterizing and classifying biodiversity indicators according to environmental responsiveness and apply the methodology to monitoring data for a marine fish community under intermittent anthropogenic warm water discharge. Our analysis showed that 10 biodiversity indicators can be classified into three super-groups based on the dimension of biodiversity that is reflected. Group I (species richness and community mean of latitudinal center of distribution (cCOD)) showed the greatest robustness to temperature changes; Group II (species diversity and total abundance) showed an abrupt change in the middle of the monitoring period, presumably due to a change in temperature; Group III (species evenness) exhibited the highest sensitivity to environmental changes, including temperature. These results had several ecological implications. First, the responsiveness of species diversity and species evenness to temperature changes might be related to changes in the species abundance distribution. Second, the similar environmental responsiveness of species richness and cCOD implies that fish migration from lower latitudes is a major driver of species compositional changes. The study methodology may be useful in selecting appropriate indicators for efficient biodiversity monitoring.

Friction and Stiffness Dependent Dynamics of Accumulation Landslides with Delayed Failure.

We propose a new model for landslide dynamics under the assumption of a delay failure mechanism. Delay failure is simulated as a delayed interaction between adjacent blocks, which mimics the relationship between the accumulation and feeder part of the accumulation slope. The conducted research consisted of three phases. Firstly, the real observed movements of the landslide were examined to exclude the existence or the statistically significant presence of background noise. Secondly, we propose a new mechanical model of an accumulation landslide dynamics, with introduced delay failure, and with variable friction law. Results obtained indicate the onset of a transition from an equilibrium state to an oscillatory regime if delayed failure is assumed for different cases of slope stiffness and state of homogeneity/heterogeneity of the slope. At the end, we examine the influence of different frictional properties (along the sliding surface) on the conditions for the onset of instability. Results obtained indicate that the increase of friction parameters leads to stabilization of sliding for homogeneous geological environment. Moreover, increase of certain friction parameters leads to the occurrence of irregular aperiodic behavior, which could be ascribed to the regime of fast irregular sliding along the slope.

Nanophotonic reservoir computing for COVID-19 pandemic forecasting.

The coronavirus disease 2019 (COVID-19) has spread worldwide in unprecedented speed, and diverse negative impacts have seriously endangered human society. Accurately forecasting the number of COVID-19 cases can help governments and public health organizations develop the right prevention strategies in advance to contain outbreaks. In this work, a long-term 6-month COVID-19 pandemic forecast in second half of 2021 and a short-term 30-day daily ahead COVID-19 forecast in December 2021 are successfully implemented via a novel nanophotonic reservoir computing based on silicon optomechanical oscillators with photonic crystal cavities, benefitting from its simpler learning algorithm, abundant nonlinear characteristics, and some unique advantages such as CMOS compatibility, fabrication cost, and monolithic integration. In essence, the nonlinear time series related to COVID-19 are mapped to the high-dimensional nonlinear space by the optical nonlinear properties of nanophotonic reservoir computing. The testing-dataset forecast results of new cases, new deaths, cumulative cases, and cumulative deaths for six countries demonstrate that the forecasted blue curves are awfully close to the real red curves with exceedingly small forecast errors. Moreover, the forecast results commendably reflect the variations of the actual case data, revealing the different epidemic transmission laws in developed and developing countries. More importantly, the daily ahead forecast results during December 2021 of four kinds of cases for six countries illustrate that the daily forecasted values are highly coincident with the real values, while the relevant forecast errors are tiny enough to verify the good forecasting competence of COVID-19 pandemic dominated by Omicron strain. Therefore, the implemented nanophotonic reservoir computing can provide some foreknowledge on prevention strategy and healthcare management for COVID-19 pandemic.

Photoplethysmogram Recording Length: Defining Minimal Length Requirement from Dynamical Characteristics.

Photoplethysmography is a widely used technique to noninvasively assess heart rate, blood pressure, and oxygen saturation. This technique has considerable potential for further applications-for example, in the field of physiological and mental health monitoring. However, advanced applications of photoplethysmography have been hampered by the lack of accurate and reliable methods to analyze the characteristics of the complex nonlinear dynamics of photoplethysmograms. Methods of nonlinear time series analysis may be used to estimate the dynamical characteristics of the photoplethysmogram, but they are highly influenced by the length of the time series, which is often limited in practical photoplethysmography applications. The aim of this study was to evaluate the error in the estimation of the dynamical characteristics of the photoplethysmogram associated with the limited length of the time series. The dynamical properties were evaluated using recurrence quantification analysis, and the estimation error was computed as a function of the length of the time series. Results demonstrated that properties such as determinism and entropy can be estimated with an error lower than 1% even for short photoplethysmogram recordings. Additionally, the lower limit for the time series length to estimate the average prediction time was computed.

Experimental Diagnosis on Combustion Characteristic of Shock Wave Focusing Initiation Engine.

A shock wave focusing initiation engine was assembled and tested in an experimental program. The effective pyrolysis rate of the pre-combustor was evaluated over a range of supplementary fuel ratio in this paper. Results highlight two operational modes of the resonant cavity: (1) pulsating combustion mode, (2) stable combustion mode. The appearance of the two combustion modes is jointly affected by the flow and the structural characteristic value of the combustion chamber. This paper uses images, time-frequency analysis, and nonlinear time series analysis methods to identify and distinguish these two combustion modes. It is believed that the interaction between the combustion chamber and the supply plenum is the probable reason for different combustion modes. The experiment has found that structural parameters and import flow parameters have an impact on the initiation of the combustion chamber. Increasing the injection pressure can appropriately broaden the fuel-rich boundary of initiation. Low equivalence ratio and high injection pressure can also appropriately increase the combustion working frequency in a small range. From the perspective of pressure utilization, under the premise of ensuring successful initiation, injection pressure should not be too high.

A Waveform-Independent Measure of Recurrent Neural Activity.

Rhythmic neural activity, so-called oscillations, plays a key role in neural information transmission, processing, and storage. Neural oscillations in distinct frequency bands are central to physiological brain function, and alterations thereof have been associated with several neurological and psychiatric disorders. The most common methods to analyze neural oscillations, e.g., short-time Fourier transform or wavelet analysis, assume that measured neural activity is composed of a series of symmetric prototypical waveforms, e.g., sinusoids. However, usually, the models generating the signal, including waveform shapes of experimentally measured neural activity are unknown. Decomposing asymmetric waveforms of nonlinear origin using these classic methods may result in spurious harmonics visible in the estimated frequency spectra. Here, we introduce a new method for capturing rhythmic brain activity based on recurrences of similar states in phase-space. This method allows for a time-resolved estimation of amplitude fluctuations of recurrent activity irrespective of or specific to waveform shapes. The algorithm is derived from the well-established field of recurrence analysis, which, in comparison to Fourier-based analysis, is still very uncommon in neuroscience. In this paper, we show its advantages and limitations in comparison to short-time Fourier transform and wavelet convolution using periodic signals of different waveform shapes. Furthermore, we demonstrate its application using experimental data, i.e., intracranial and noninvasive electrophysiological recordings from the human motor cortex of one epilepsy patient and one healthy adult, respectively.

A Nonlinear Time-Series Analysis to Identify the Thresholds in Relationships Between Antimicrobial Consumption and Resistance in a Chinese Tertiary Hospital.

INTRODUCTION: Balancing the benefits and risks of antimicrobials in health care requires an understanding of their effects on antimicrobial resistance at the population scale. Therefore, we aimed to investigate the association between the population antibiotics use and resistance rates and further identify their critical thresholds. METHODS: Data for monthly consumption of six antibiotics (daily defined doses [DDDs]/1000 inpatient-days) and the number of cases caused by five common drug-resistant bacteria (occupied bed days [OBDs]/10,000 inpatient-days) from inpatients during 2009-2020 were retrieved from the electronic prescription system at Nanjing Drum Tower Hospital, a tertiary hospital in Jiangsu Province, China. Then, a nonlinear time series analysis method, named generalized additive models (GAM), was applied to analyze the pairwise relationships and thresholds of these antibiotic consumption and resistance. RESULTS: The incidence densities of carbapenem-resistant Acinetobacter baumannii (CRAB), carbapenem-resistant Klebsiella pneumoniae (CRKP), and aminoglycoside-resistant Pseudomonas aeruginosa were all strongly synchronized with recent hospital use of carbapenems and glycopeptides. Besides, the prevalence of carbapenem-resistant Escherichia coli was also highly connected the consumption of carbapenems and fluoroquinolones. To lessen resistance, we determined a threshold for carbapenem and glycopeptide usage, where the maximum consumption should not exceed 31.042 and 25.152 DDDs per 1000 OBDs, respectively; however, the thresholds of fluoroquinolones, third-generation cephalosporin, aminoglycosides, and ß-lactams have not been identified. CONCLUSIONS: The inappropriate usage of carbapenems and glycopeptides was proved to drive the incidence of common drug-resistant bacteria in hospitals. Nonlinear time series analysis provided an efficient and simple way to determine the thresholds of these antibiotics, which could provide population-specific quantitative targets for antibiotic stewardship.

Reliability of Running Stability during Treadmill and Overground Running.

Running stability is the ability to withstand naturally occurring minor perturbations during running. It is susceptible to external and internal running conditions such as footwear or fatigue. However, both its reliable measurability and the extent to which laboratory measurements reflect outdoor running remain unclear. This study aimed to evaluate the intra- and inter-day reliability of the running stability as well as the comparability of different laboratory and outdoor conditions. Competitive runners completed runs on a motorized treadmill in a research laboratory and overground both indoors and outdoors. Running stability was determined as the maximum short-term divergence exponent from the raw gyroscope signals of wearable sensors mounted to four different body locations (sternum, sacrum, tibia, and foot). Sacrum sensor measurements demonstrated the highest reliabilities (good to excellent; ICC = 0.85 to 0.91), while those of the tibia measurements showed the lowest (moderate to good; ICC = 0.55 to 0.89). Treadmill measurements depicted systematically lower values than both overground conditions for all sensor locations (relative bias = -9.8% to -2.9%). The two overground conditions, however, showed high agreement (relative bias = -0.3% to 0.5%; relative limits of agreement = 9.2% to 15.4%). Our results imply moderate to excellent reliability for both overground and treadmill running, which is the foundation of further research on running stability.

Biometric Identification Based on Eye Movement Dynamic Features.

The paper presents studies on biometric identification methods based on the eye movement signal. New signal features were investigated for this purpose. They included its representation in the frequency domain and the largest Lyapunov exponent, which characterizes the dynamics of the eye movement signal seen as a nonlinear time series. These features, along with the velocities and accelerations used in the previously conducted works, were determined for 100-ms eye movement segments. 24 participants took part in the experiment, composed of two sessions. The users' task was to observe a point appearing on the screen in 29 locations. The eye movement recordings for each point were used to create a feature vector in two variants: one vector for one point and one vector including signal for three consecutive locations. Two approaches for defining the training and test sets were applied. In the first one, 75% of randomly selected vectors were used as the training set, under a condition of equal proportions for each participant in both sets and the disjointness of the training and test sets. Among four classifiers: kNN (k = 5), decision tree, naïve Bayes, and random forest, good classification performance was obtained for decision tree and random forest. The efficiency of the last method reached 100%. The outcomes were much worse in the second scenario when the training and testing sets when defined based on recordings from different sessions; the possible reasons are discussed in the paper.

Shrunken Social Brains? A Minimal Model of the Role of Social Interaction in Neural Complexity.

The social brain hypothesis proposes that enlarged brains have evolved in response to the increasing cognitive demands that complex social life in larger groups places on primates and other mammals. However, this reasoning can be challenged by evidence that brain size has decreased in the evolutionary transitions from solitary to social larger groups in the case of Neolithic humans and some eusocial insects. Different hypotheses can be identified in the literature to explain this reduction in brain size. We evaluate some of them from the perspective of recent approaches to cognitive science, which support the idea that the basis of cognition can span over brain, body, and environment. Here we show through a minimal cognitive model using an evolutionary robotics methodology that the neural complexity, in terms of neural entropy and degrees of freedom of neural activity, of smaller-brained agents evolved in social interaction is comparable to the neural complexity of larger-brained agents evolved in solitary conditions. The nonlinear time series analysis of agents' neural activity reveals that the decoupled smaller neural network is intrinsically lower dimensional than the decoupled larger neural network. However, when smaller-brained agents are interacting, their actual neural complexity goes beyond its intrinsic limits achieving results comparable to those obtained by larger-brained solitary agents. This suggests that the smaller-brained agents are able to enhance their neural complexity through social interaction, thereby offsetting the reduced brain size.

Seasonal dynamics of terrestrially sourced nitrogen influenced Karenia brevis blooms off Florida's southern Gulf Coast.

Harmful algal blooms (HABs) threaten coastal ecological systems, public health, and local economies, but the complex physical, chemical, and biological processes that culminate in HABs vary by locale and are often poorly understood. Despite broad recognition that cultural eutrophication may exacerbate nearshore bloom events, the association is typically not linear and is often difficult to quantify. Off the Gulf Coast of Florida, Karenia brevis blooms initiate in the open waters of the Gulf of Mexico, and advection of cells supplies nearshore blooms. However, past work has struggled to describe the relationship between terrestrial nutrient runoff and bloom maintenance near the Gulf Coast. This study applied a novel nonlinear time series (NLTS) analytical framework to investigate whether nearshore bloom dynamics observed near Charlotte Harbor, FL were causally and systematically driven by terrestrially sourced inputs of nitrogen, phosphorus, and freshwater between 2012 and 2018. Singular spectrum analysis (SSA) isolated low-dimensional, deterministic signals in K. brevis log10-density dynamics and in the dynamics of nine of 10 candidate drivers. The predominantly seasonal K. brevis signal was strong, explaining 77.6% of the total variance in the observed time series. Causal tests with convergent cross-mapping provided evidence that nitrogen concentrations measured at the discharge point of the Caloosahatchee River systematically influenced K. brevis bloom dynamics. However, further causal testing failed to link these nitrogen dynamics to an upstream basin, possibly due to data limitations. The results support the hypothesis that anthropogenic nitrogen runoff facilitated the growth of K. brevis blooms near Charlotte Harbor and suggest that bloom events would be mitigated by nitrogen source and transport controls within the Caloosahatchee and/or Kissimmee River basins. More broadly, this work demonstrates that management-relevant causal inferences into the drivers of HABs may be drawn from available monitoring records.

Time series and fractal analyses of wheezing: a novel approach.

Since the outbreak of the pandemic Coronavirus Disease 2019, the world is in search of novel non-invasive methods for safer and early detection of lung diseases. The pulmonary pathological symptoms reflected through the lung sound opens a possibility of detection through auscultation and of employing spectral, fractal, nonlinear time series and principal component analyses. Thirty-five signals of vesicular and expiratory wheezing breath sound, subjected to spectral analyses shows a clear distinction in terms of time duration, intensity, and the number of frequency components. An investigation of the dynamics of air molecules during respiration using phase portrait, Lyapunov exponent, sample entropy, fractal dimension, and Hurst exponent helps in understanding the degree of complexity arising due to the presence of mucus secretions and constrictions in the respiratory airways. The feature extraction of the power spectral density data and the application of principal component analysis helps in distinguishing vesicular and expiratory wheezing and thereby, giving a ray of hope in accomplishing an early detection of pulmonary diseases through sound signal analysis.

Nonlinear time series and principal component analyses: Potential diagnostic tools for COVID-19 auscultation.

The development of novel digital auscultation techniques has become highly significant in the context of the outburst of the pandemic COVID 19. The present work reports the spectral, nonlinear time series, fractal, and complexity analysis of vesicular (VB) and bronchial (BB) breath signals. The analysis is carried out with 37 breath sound signals. The spectral analysis brings out the signatures of VB and BB through the power spectral density plot and wavelet scalogram. The dynamics of airflow through the respiratory tract during VB and BB are investigated using the nonlinear time series and complexity analyses in terms of the phase portrait, fractal dimension, Hurst exponent, and sample entropy. The higher degree of chaoticity in BB relative to VB is unwrapped through the maximal Lyapunov exponent. The principal component analysis helps in classifying VB and BB sound signals through the feature extraction from the power spectral density data. The method proposed in the present work is simple, cost-effective, and sensitive, with a far-reaching potential of addressing and diagnosing the current issue of COVID 19 through lung auscultation.

Bayesian estimation of subset threshold autoregressions: short-term forecasting of traffic occupancy.

Traffic management authorities in metropolitan areas use real-time systems that analyze high-frequency measurements from fixed sensors, to perform short-term forecasting and incident detection for various locations of a road network. Published research over the last 20 years focused primarily on modeling and forecasting of traffic volumes and speeds. Traffic occupancy approximates vehicular density through the percentage of time a sensor detects a vehicle within a pre-specified time interval. It exhibits weekly periodic patterns and heteroskedasticity and has been used as a metric for characterizing traffic regimes (e.g. free flow, congestion). This article presents a Bayesian three-step model building procedure for parsimonious estimation of Threshold-Autoregressive (TAR) models, designed for location- day- and horizon-specific forecasting of traffic occupancy. In the first step, multiple regime TAR models reformulated as high-dimensional linear regressions are estimated using Bayesian horseshoe priors. Next, significant regimes are identified through a forward selection algorithm based on Kullback-Leibler (KL) distances between the posterior predictive distribution of the full reference model and TAR models with fewer regimes. Given the regimes, the forward selection algorithm can be implemented again to select significant autoregressive terms. In addition to forecasting, the proposed specification and model-building scheme, may assist in determining location-specific congestion thresholds and associations between traffic dynamics observed in different regions of a network. Empirical results applied to data from a traffic forecasting competition, illustrate the efficacy of the proposed procedures in obtaining interpretable models and in producing satisfactory point and density forecasts at multiple horizons.

Sample entropy and surrogate data analysis for Alzheimer's disease.

Alzheimer's disease (AD) is a neurological degenerative disease, which is mainly char-acterized by the memory loss. As electroencephalogram (EEG) device is relatively cheap, portable and non-invasive, it has been widely used in AD-related studies. We proposed a method to detect the differences between healthy subjects and AD patients, which combines classical sample entropy (Sam-pEn) and surrogate data method. EEGs from 14 AD patients and 20 healthy subjects were analyzed. The results based on the original data showed that the SampEn of AD patients was significantly de-creased (p < 0.01) at electrodes c3, f3, o2 and p4, which confirmed that AD could cause complexity loss. However, using original data could be subject to human judgement, so we generated a series of surrogate data. We found that, there were significant difference of SampEn between the original time series and their surrogate data at c3 and o2 electrodes and the differences between healthy subjects and AD patients can be verified. Our method is capable of distinguishing AD patients from healthy subjects, which is consistent with the concept of physiologic complexity, and providing insights for understanding of AD.

Nonlinear autoregressive sieve bootstrap based on extreme learning machines.

The aim of the paper is to propose and discuss a sieve bootstrap scheme based on Extreme Learning Machines for non linear time series. The procedure is fully nonparametric in its spirit and retains the conceptual simplicity of the residual bootstrap. Using Extreme Learning Machines in the resampling scheme can dramatically reduce the computational burden of the bootstrap procedure, with performances comparable to the NN-Sieve bootstrap and computing time similar to the ARSieve bootstrap. A Monte Carlo simulation experiment has been implemented, in order to evaluate the performance of the proposed procedure and to compare it with the NN-Sieve bootstrap. The distributions of the bootstrap variance estimators appear to be consistent, delivering good results both in terms of accuracy and bias, for either linear and nonlinear statistics (such as the mean and the median) and smooth functions of means (such as the variance and the covariance).