Empirical Autocorrelation Research Articles

Diagnostic Checking in Multivariate ARMA Models With Dependent Errors Using Normalized Residual Autocorrelations

ABSTRACTIn this paper, we derive the asymptotic distribution of normalized residual empirical autocovariances and autocorrelations under weak assumptions on the noise. We propose new portmanteau statistics for vector autoregressive moving average models with uncorrelated but nonindependent innovations by using a self-normalization approach. We establish the asymptotic distribution of the proposed statistics. This asymptotic distribution is quite different from the usual chi-squared approximation used under the independent and identically distributed assumption on the noise, or the weighted sum of independent chi-squared random variables obtained under nonindependent innovations. A set of Monte Carlo experiments and an application to the daily returns of the CAC40 is presented. Supplementary materials for this article are available online.

Calendar Spread Exchange Options Pricing with Gaussian Random Fields

Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance introduces a bias in the valuation of calendar spread exchange options. As the payoff of these options depends on two asset values at different times, particular care must be taken for the modeling of covariance and autocorrelation. This article proposes a simple alternative model for asset prices with sub-exponential, exponential and hyper-exponential autocovariance structures. In the proposed approach, price processes are seen as conditional Gaussian fields indexed by the time. In general, this process is not a semi-martingale, and therefore, we cannot rely on stochastic differential calculus to evaluate options. However, option prices are still calculable by the technique of the change of numeraire. A numerical illustration confirms the important influence of the covariance structure in the valuation of calendar spread exchange options for Brent against WTI crude oil and for gold against silver.

Modeling long correlation times using additive binary Markov chains: Applications to wind generation time series

Wind power generation exhibits a strong temporal variability, which is crucial for system integration in highly renewable power systems. Different methods exist to simulate wind power generation but they often cannot represent the crucial temporal fluctuations properly. We apply the concept of additive binary Markov chains to model a wind generation time series consisting of two states: periods of high and low wind generation. The only input parameter for this model is the empirical autocorrelation function. The two-state model is readily extended to stochastically reproduce the actual generation per period. To evaluate the additive binary Markov chain method, we introduce a coarse model of the electric power system to derive backup and storage needs. We find that the temporal correlations of wind power generation, the backup need as a function of the storage capacity, and the resting time distribution of high and low wind events for different shares of wind generation can be reconstructed.

Continuous-Time Random Walk with multi-step memory: an application to market dynamics

An extended version of the Continuous-Time Random Walk (CTRW) model with memory is herein developed. This memory involves the dependence between arbitrary number of successive jumps of the process while waiting times between jumps are considered as i.i.d. random variables. This dependence was established analyzing empirical histograms for the stochastic process of a single share price on a market within the high frequency time scale. Then, it was justified theoretically by considering bid-ask bounce mechanism containing some delay characteristic for any double-auction market. Our model appeared exactly analytically solvable. Therefore, it enables a direct comparison of its predictions with their empirical counterparts, for instance, with empirical velocity autocorrelation function. Thus, the present research significantly extends capabilities of the CTRW formalism.

Spatial distribution of leprosy in Brazil: a literature review.

Leprosy remains a public health problem in developing countries. Among communicable diseases, it is one of the leading causes of permanent disability. Brazil had not reached the goal of reducing cases to less than 1 per 10,000 population. This study aimed to analyze the spatial distribution of leprosy cases in Brazil, using a literature review. The search strategy included the LILACS and MEDLINE databases with no language or period restriction. Ecological studies with spatial data analysis were considered as a criterion for the inclusion. We found 38 studies for review after the selection criteria. Among the epidemiological indicators of the disease, the most common was the new case detection rate. Several articles have explored the association between spatial distribution of leprosy and socioeconomic, demographic, and environmental factors. The most common unit of analysis was the municipality. The spatial distribution methods mostly used were: empirical Bayesian method, autocorrelation (Moran's I index) and Kernel estimates. The distribution of leprosy was very heterogeneous, independent of the unit of analysis. There was a decrease in the rate of detection and among under-15-year-olds, but some regions maintained high endemicity during the study period. The distribution and risk of illness were directly related to living conditions of the population. Improved access to health services was associated with increased detection rate in some regions. Spatial analysis seems to be a very useful tool to study leprosy and to guide interventions and surveillance.

General simulation algorithm for autocorrelated binary processes.

The apparent ubiquity of binary random processes in physics and many other fields has attracted considerable attention from the modeling community. However, generation of binary sequences with prescribed autocorrelation is a challenging task owing to the discrete nature of the marginal distributions, which makes the application of classical spectral techniques problematic. We show that such methods can effectively be used if we focus on the parent continuous process of beta distributed transition probabilities rather than on the target binary process. This change of paradigm results in a simulation procedure effectively embedding a spectrum-based iterative amplitude-adjusted Fourier transform method devised for continuous processes. The proposed algorithm is fully general, requires minimal assumptions, and can easily simulate binary signals with power-law and exponentially decaying autocorrelation functions corresponding, for instance, to Hurst-Kolmogorov and Markov processes. An application to rainfall intermittency shows that the proposed algorithm can also simulate surrogate data preserving the empirical autocorrelation.

TEMPORAL SHAPING OF SIMULATED TIME SERIES WITH CYCLICAL SAMPLE PATHS

Temporal shaping of time series is the activity of deriving a time series model with a prescribed marginal distribution and some sample path characteristics. Starting with an empirical sample path, one often computes from it an empirical histogram (a step-function density) and empirical autocorrelation function. The corresponding cumulative distribution function is piecewise linear, and so is the inverse distribution function. The so-called inversion method uses the latter to generate the corresponding distribution from a uniform random variable on [0,1), histograms being a special case. This paper shows how to manipulate the inverse histogram and an underlying marginally uniform process, so as to “shape” the model sample paths in an attempt to match the qualitative nature of the empirical sample paths, while maintaining a guaranteed match of the empirical marginal distribution. It proposes a new approach to temporal shaping of time series and identifies a number of operations on a piecewise-linear inverse histogram function, which leave the marginal distribution invariant. For cyclical processes with a prescribed marginal distribution and a prescribed cycle profile, one can also use these transformations to generate sample paths which “conform” to the profile. This approach also improves the ability to approximate the empirical autocorrelation function.

New procedure for gear fault detection and diagnosis using instantaneous angular speed

Besides the extreme complexity of gear dynamics, the fault diagnosis results in terms of vibration signal are sometimes easily misled and even distorted by the interference of transmission channel or other components like bearings, bars. Recently, the research field of Instantaneous Angular Speed (IAS) has attracted significant attentions due to its own advantages over conventional vibration analysis. On the basis of IAS signal's advantages, this paper presents a new feature extraction method by combining the Empirical Mode Decomposition (EMD) and Autocorrelation Local Cepstrum (ALC) for fault diagnosis of sophisticated multistage gearbox. Firstly, as a pre-processing step, signal reconstruction is employed to address the oversampled issue caused by the high resolution of the angular sensor and the test speed. Then the adaptive EMD is used to acquire a number of Intrinsic Mode Functions (IMFs). Nevertheless, not all the IMFs are needed for the further analysis since different IMFs have different sensitivities to fault. Hence, the cosine similarity metric is introduced to select the most sensitive IMF. Even though, the sensitive IMF is still insufficient for the gear fault diagnosis due to the weakness of the fault component related to the gear fault. Therefore, as the final step, ALC is used for the purpose of signal de-noising and feature extraction. The effectiveness and robustness of the new approach has been validated experimentally on the basis of two gear test rigs with gears under different working conditions. Diagnosis results show that the new approach is capable of effectively handling the gear fault diagnosis i.e., the highlighted quefrency and its rahmonics corresponding to the rotary period and its multiple are displayed clearly in the cepstrum record of the proposed method.

The Method and Criterion for Quality Assessment of Random Number Sequences

Abstract. The authors analyze the dependence of the uniformity of distribution of signs of empirical autocorrelation function with respect to the number of overlapping symbols of intervals into which a sequence of random numbers is divided. A feasible "threshold" of overlap is established, below which the signs of the autocorrelation function are uniformly distributed. The concept of barrier function is defined and used to develop a criterion for the quality assessment of random number generators. The technique of its application and its implementation for several well-known generators are presented.

Vibration condition monitoring system for wind turbine bearings based on noise suppression with multi-point data fusion

A noise suppression method for feature frequency extraction that is supplemented with multi-point data fusion was investigated in consideration of issues involving wind turbine vibration signals subject to high noise disturbance. The difficulty of extracting early weak fault features was examined as well. First, a de-noising and feature extraction method that uses EMD-Correlation was developed by adding empirical mode decomposition (EMD) and autocorrelation de-noising to the wavelet package transform under the effects of white noise and short-term disturbance noise in wind turbine vibration signals. Second, an EMD-Correlation analysis model for feature frequency extraction supplemented with multi-point data fusion was established with reference to adaptive resonance theory-2 to highlight the feature frequency of a possible early weak fault. Third, the results obtained with the actual and simulated fault vibration signals of wind turbine bearing faults and the outcomes of comparing the different feature frequency extraction methods show that the proposed method of EMD-Correlation that is supplemented with multi-point data fusion can not only effectively reduce white noise and short-term disturbance noise but can also extract the feature frequency of early weak faults. Finally, prototype hardware and software were developed for a wind turbine condition monitoring system based on the aforementioned fault feature extraction algorithms and tested in an actual wind turbine generation system.

Non-reversible Metropolis-Hastings

The classical Metropolis-Hastings (MH) algorithm can be extended to generate non-reversible Markov chains. This is achieved by means of a modification of the acceptance probability, using the notion of vorticity matrix. The resulting Markov chain is non-reversible. Results from the literature on asymptotic variance, large deviations theory and mixing time are mentioned, and in the case of a large deviations result, adapted, to explain how non-reversible Markov chains have favorable properties in these respects. We provide an application of NRMH in a continuous setting by developing the necessary theory and applying, as first examples, the theory to Gaussian distributions in three and nine dimensions. The empirical autocorrelation and estimated asymptotic variance for NRMH applied to these examples show significant improvement compared to MH with identical stepsize.

Correlation-Based Spectrum Sensing With Oversampling in Cognitive Radio

In wireless communication, the amplitude and phase of the transmitted signal have certain patterns during one symbol duration, which introduces high correlation among the samples obtained by oversampling at the receiver. In this work, we aim to explore such correlation information for cognitive radios to enhance the performance of spectrum sensing. By jointly considering the signal modulation, multipath fading, and oversampling rate, we derive the distribution of the empirical autocorrelation function for the obtained samples, on which we propose two efficient spectrum-sensing algorithms, and then analyze their performance. Our theoretical results reveal that the proposed algorithms with oversampling perform strictly better than the conventional energy detection scheme, while requiring the same level of prior information. Finally, we show through simulations that the derived statistical characteristics approximate the true statistical distribution of the autocorrelation function well, and the proposed sensing algorithms significantly improve the sensing performance compared to several existing sensing schemes.

The affinely invariant distance correlation

Sz\'{e}kely, Rizzo and Bakirov (Ann. Statist. 35 (2007) 2769-2794) and Sz\'{e}kely and Rizzo (Ann. Appl. Statist. 3 (2009) 1236-1265), in two seminal papers, introduced the powerful concept of distance correlation as a measure of dependence between sets of random variables. We study in this paper an affinely invariant version of the distance correlation and an empirical version of that distance correlation, and we establish the consistency of the empirical quantity. In the case of subvectors of a multivariate normally distributed random vector, we provide exact expressions for the affinely invariant distance correlation in both finite-dimensional and asymptotic settings, and in the finite-dimensional case we find that the affinely invariant distance correlation is a function of the canonical correlation coefficients. To illustrate our results, we consider time series of wind vectors at the Stateline wind energy center in Oregon and Washington, and we derive the empirical auto and cross distance correlation functions between wind vectors at distinct meteorological stations.

An Integrated Algorithm for Estimating Portfolio VaR

In this paper, we propose an integrated algorithm for estimating portfolio Value at Risk (VaR). The integrated algorithm contains three parts, return series simulation, portfolio optimization and portfolio VaR calculation. In order to obtain accurate return series being consistent with empirical cases, autocorrelation and heteroskedasticity along with volatility clustering are compensated by Autoregressive (AR) model and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, meantime, the tails are simulated by Extreme Value Theory (EVT) and the dependencies are described by the selected copula function. q-Gaussian distribution is also introduced into portfolio optimization to construct a q-Gaussian Mean-Variance model. With the simulated portfolio returns and the optimized portfolio weights, the portfolio VaR could be calculated. Finally, an empirical study of seven Asia composite indexes is performed and the result suggests that by using the proposed integrated algorithm, a better result with lower risk and higher expected returns is obtained compared with other methods.

Randomness, Symmetry, and Scaling of Mesoscale Eddy Life Cycles

Abstract It is shown that the life cycles of nonlinear mesoscale eddies, a major component of low-frequency ocean physical variability, have a characteristic structure that differs fundamentally from that which would be expected on the basis of classical interpretations of ocean eddy evolution in terms of mean flow instability and equilibration followed by frictional, radiative, or barotropic decay, or of vortex merger dynamics in quasigeostrophic turbulent cascades. Further, it is found that these life cycles can be accurately modeled in terms of the large-amplitude excursions of a stochastic process. These conclusions, which apply in the corresponding global-mean context, follow from the examination of ensemble-mean and standard deviation time series of normalized eddy amplitude from an automated eddy identification and tracking analysis of a nearly two decade–merged satellite altimeter record of global sea surface height (SSH). The resulting series are found to have several striking and unexpected characteristics, including time-reversal symmetry and approximate self-similarity. Consistent results are obtained from a similar analysis of a 7-yr record of global SSH from a numerical ocean circulation model. The basic qualitative and quantitative statistical properties of these series can be remarkably well reproduced with an extremely simple stochastic model, in which the SSH increments between successive time points are random numbers, and the eddy life cycles are represented by excursions exceeding a given threshold. The stochastic model is found also to predict accurately the empirical autocorrelation structure of the underlying observed SSH field itself, when the autocorrelations are computed along long planetary (Rossby) wave characteristics.

Bootstrapping continuous-time autoregressive processes

We develop a bootstrap procedure for Levy-driven continuous-time autoregressive (CAR) processes observed at discrete regularly-spaced times. It is well known that a regularly sampled stationary Ornstein–Uhlenbeck process [i.e. a CAR(1) process] has a discrete-time autoregressive representation with i.i.d. noise. Based on this representation a simple bootstrap procedure can be found. Since regularly sampled CAR processes of higher order satisfy ARMA equations with uncorrelated (but in general dependent) noise, a more general bootstrap procedure is needed for such processes. We consider statistics depending on observations of the CAR process at the uniformly-spaced times, together with auxiliary observations on a finer grid, which give approximations to the derivatives of the continuous time process. This enables us to approximate the state-vector of the CAR process which is a vector-valued CAR(1) process, and whose sampled version, on the uniformly-spaced grid, is a multivariate AR(1) process with i.i.d. noise. This leads to a valid residual-based bootstrap which allows replication of CAR $$(p)$$ processes on the underlying discrete time grid. We show that this approach is consistent for empirical autocovariances and autocorrelations.

Regime Changes and Financial Markets

Regime-switching models can match the tendency of financial markets to often change their behavior abruptly and the phenomenon that the new behavior of financial variables often persists for several periods after such a change. Although the regimes captured by regime-switching models are identified by an econometric procedure, they often correspond to different periods in regulation, policy, and other secular changes. In empirical estimates, the means, volatilities, autocorrelations, and cross-covariances of asset returns often differ across regimes in a manner that allows regime-switching models to capture the stylized behavior of many financial series including fat tails, heteroskedasticity, skewness, and time-varying correlations. In equilibrium models, regimes in fundamental processes, such as consumption or dividend growth, strongly affect the dynamic properties of equilibrium asset prices and can induce nonlinear risk-return trade-offs. Regime switches also lead to potentially large consequences for investors' optimal portfolio choice.

Frequency detection in vortex flowmeter for low Reynolds number using piezoelectric sensor and installation effects

Piezoelectric sensors are one of the most widely used sensors for vortex flowmeter application due to their low cost. Various researchers have employed piezoelectric sensor for this application. However, the location of the sensor and the performance of vortex flowmeter under disturbed conditions are seldom reported. In the present study, experimental investigations are conducted with water as the working medium in a circular pipe of diameter 52.5mm. The optimum position of the piezoelectric sensor behind the trapezoidal bluff body is found out be 0.85 times the width of the bluff body. A new algorithm based on empirical mode decomposition and autocorrelation decay rate is suggested to identify the vortex shedding frequency under low Reynolds numbers flow condition. The performance of the flowmeter is also evaluated under different disturbed flow conditions to quantify the sensitivity of the flowmeter. The disturbances studied are single 90° bend, gate valve, globe valve, and two 90° out of plane bends. The overall uncertainty in the Strouhal number is within ±1.71%.

Structural Change and Spurious Persistence in Stochastic Volatility

We extend the well established link between structural change and estimated persistence from GARCH to stochastic volatility (SV) models. Whenever structural changes in some model parameters increase the empirical autocorrelations of the squares of the underlying time series, the persistence in volatility implied by the estimated model parameters follows suit. This explains why stochastic volatility often appears to be more persistent when estimated from a larger sample as then the likelihood increases that there might have been some structural change in between.

MARM Processes Part II: The Empirically-Based Subclass

MARM (Multivariate Autoregressive Modular) processes constitute a versatile class of multidimensional stochastic sequences which can exactly fit arbitrary multi-dimensional empirical histograms and approximately fit the leading empirical autocorrelations and cross-correlations. A companion paper (Part I) presented the general theory of MARM processes. This paper (Part II) proposes practical MARM modeling and forecasting methodologies of considerable generality, suitable for implementation on a computer. The purpose of Part II is twofold: (1) to specialize the general class of MARM processes to a practical subclass, called Empirically-Based MARM (EB-MARM) processes, suitable for modeling of empirical vector-valued time series, and devise the corresponding fitting and forecasting algorithms; and (2) to illustrate the efficacy of the EB-MARM fitting and forecasting algorithms. Specifically, we shall consider MARM processes with iid step-function innovation densities and distortions based on an empirical multi-dimensional histogram, as well as empirical autocorrelation and cross-correlation functions. Finally, we illustrate the efficacy of these methodologies with an example of a three-dimension time series vector, using a software environment, called MultiArmLab, which supports MARM modeling and forecasting.