Empirical Autocorrelation Function Research Articles

Application of Generalized Cauchy Process on Modeling the Long-Range Dependence and Self-Similarity of Sea Surface Chlorophyll Using 23 years of Remote Sensing Data

Understanding the temporal characteristics of sea surface chlorophyll (SSC) is helpful for marine environmental management. This study chose 10 time series of remote daily sea surface chlorophyll products from the European Space Agency during the period from July 29, 1998 to December 31, 2020. A generalized Cauchy model was employed to capture the local and global behaviors of sea surface chlorophyll from a fractal perspective; the fractal dimension D measures the local similarity while the Hurst parameter H measures the global long-range dependence. The generalized Cauchy model was fitted to the empirical autocorrelation function values of each SSC series. The results showed that the sea surface chlorophyll was multi-fractal in both space and time with the D values ranging from 1.0000 to 1.7964 and H values ranging from 0.6757 to 0.8431. Specifically, regarding the local behavior, 9 of the 10 series had low D values (<1.5), representing weak self-similarity; on the other hand, regarding the global behavior, high H values represent strong long-range dependence that may be a general phenomenon of daily sea surface chlorophyll.

Comparison of Enhanced Noise Model Performance Based on Analysis of Civilian GPS Data.

We recorded the time series of location data from stationary, single-frequency (L1) GPS positioning systems at a variety of geographic locations. The empirical autocorrelation function of these data shows significant temporal correlations. The Gaussian white noise model, widely used in sensor-fusion algorithms, does not account for the observed autocorrelations and has an artificially large variance. Noise-model analysis—using Akaike’s Information Criterion—favours alternative models, such as an Ornstein–Uhlenbeck or an autoregressive process. We suggest that incorporating a suitable enhanced noise model into applications (e.g., Kalman Filters) that rely on GPS position estimates will improve performance. This provides an alternative to explicitly modelling possible sources of correlation (e.g., multipath, shadowing, or other second-order physical phenomena).Dataset License: BY-NC-ND

On the field validation of α-μ fading coefficients estimator based on the autocorrelation function for ionospheric amplitude scintillation

The ionosphere in low-latitude regions has intense dynamics with great variability – not only spatially but also temporally. The most critical effect found in the low-latitude region is the scintillation in phase and amplitude due to plasma bubble occurrence in the ionospheric layer. To augmentation systems like GBAS or SBAS, designed to provide Category-I (CAT I) precision approach service in low-visibility conditions, plasma bubbles can compromise the performance requirements for air navigation precision approach in a way that the service provided by the system is considered unviable. In practical terms, this is the main reason why the Brazilian Department of Airspace Control (DECEA) still has not implemented any augmentation system in Brazil. Scintillation may lead to loss of signal lock of the affected satellite. In this context, a better understanding of the scintillation pattern and its statistical properties gain particular relevance, once this is the most concerning issue for the performance of any GNSS receiver in low-latitude regions especially in the context of augmentation systems for air navigation. Previous works validated the use of the α-μ distribution to characterize amplitude scintillation. The present work is concerned with proposing a new method for the estimation of the α-μ coefficients based on the α-μ autocorrelation function. The method is validated using field data and comparing the results from empirical autocorrelation function with the results obtained from the moment-based estimator. Additionally, the efficiency of the method is proven by analyzing measurements of Level-Crossing Rate (LCR) and Average Fading Duration (AFD) with their respective theoretical formulation.

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.

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.

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.

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.

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.

An Efficient Approach to Spatiotemporal Analysis and Modeling of Air Pollution Data

A statistically efficient approach is adopted for modeling spatial time-series of large data sets. The estimation of the main diagnostic tool such as the likelihood function in Gaussian spatiotemporal models is a cumbersome task when using extended spatial time-series such as air pollution. Here, using the Innovation Algorithm, we manage to compute it for many spatiotemporal specifications. These specifications refer to the spatial periodic-trend, the spatial autoregressive moving average, the spatial autoregressive integrated and fractionally integrated moving average Gaussian models. Our method is applied to daily pollutants over a large metropolitan area like Athens. In the applied part of our paper, we first diagnose temporal and spatial structures of data using non-likelihood based criteria, such as the empirical autocorrelation and covariance functions. Second, we use likelihood and non-likelihood based criteria to select a spatiotemporal model among various specifications. Finally, using kriging we regionalize the resulting parameter estimates of the best-fitted model in space at any unmonitored location in the Athens region. The results show that a specific autoregressive integrated moving average spatiotemporal model can optimally perform in within and out of spatial sample estimation. Supplemental materials for this article are available from the journal website.

Pointwise Regularity Exponents and Well-Behaved Residuals in Stock Markets

The article deals with a class of stochastic processes, the Multifractional Processes with Random Exponent (MPRE), recently introduced to gain flexibility in modeling many complex phenomena. We claim that MPRE can capture in a very parsimonious way most of the well known financial stylized facts. In particular, we prove that the process unconditional distributions are fat-tailed and high-peaked and show that, as it occurs for asset returns, the empirical autocorrelation functions of the process increments are close to zero whereas significant values are exhibited by squared (or absolute) increments. Furthermore, we provide evidence that the sole knowledge of functional parameter of the MPRE allows to calculate residuals that perform much better than those obtained by other discrete models such as the GARCH family.

Modeling Spatiotemporal Forest Health Monitoring Data

Forest health monitoring schemes were set up across Europe in the 1980s in response to concerns about air pollution-related forest dieback (Waldsterben) and have continued since then. Recent threats to forest health are climatic extremes likely due to global climate change and increased ground ozone levels and nitrogen deposition. We model yearly data on tree crown defoliation, an indicator of tree health, from a monitoring survey carried out in Baden-Württemberg, Germany since 1983. On a changing irregular grid, defoliation and other site-specific variables are recorded. In Baden-Württemberg, the temporal trend of defoliation differs among areas because of site characteristics and pollution levels, making it necessary to allow for space–time interaction in the model. For this purpose, we propose using generalized additive mixed models (GAMMs) incorporating scale-invariant tensor product smooths of the space–time dimensions. The space–time smoother allows separate smoothing parameters and penalties for the space and time dimensions and thus avoids the need to make arbitrary or ad hoc choices about the relative scaling of space and time. The approach of using a space–time smoother has intuitive appeal, making it easy to explain and interpret when communicating the results to nonstatisticians, such as environmental policy makers. The model incorporates a nonlinear effect for mean tree age, the most important predictor, allowing the separation of trends in time, which may be pollution-related, from trends that relate purely to the aging of the survey population. In addition to a temporal trend due to site characteristics and other conditions modeled with the space–time smooth, we account for random temporal correlation at site level by an autoregressive moving average (ARMA) process. Model selection is carried out using the Bayes information criterion (BIC), and the adequacy of the assumed spatial and temporal error structure is investigated with the empirical semivariogram and the empirical autocorrelation function.

Theory of the in-circuit observation of VLSICs using autocorrelation functions

A method of formation of an autocorrelation function (ACF) as applied to the problem of observation of interior points of processor VLSICs is considered. As elementary events for the synthesis of empirical ACF values, the antecedently determined vectors z ξ,t = x η,t ∪ x ω,t and y λ,t + τ are used; the setting for them is performed by an integrated circuit control automat. It is shown that the principle of statistical control using ACF in asymptotics is more definite than linear signature analysis or condition vector counting.

How can we Define the Concept of Long Memory? An Econometric Survey

In this paper we discuss different aspects of long memory behavior and applicable parametric models. We discuss the confusion that can arise when the empirical autocorrelation function decreases in a hyperbolic way.

Parallelization algorithms for modeling ARM processes

AutoRegressive Modular (ARM) processes are a new class of nonlinear stochastic processes, which can accurately model a large class of stochastic processes, by capturing the empirical distribution and autocorrelation function simultaneously. Given an empirical sample path, the ARM modeling procedure consists of two steps: a global search for locating the minima of a nonlinear objective function over a large parametric space, and a local optimization of optimal or near optimal models found in the first step. In particular, since the first task calls for the evaluation of the objective function at each vector of the search space, the global search is a time consuming procedure. To speed up the computations, parallelization of the global search can be effectively used by partitioning the search space among multiple processors, since the requisite communication overhead is negligible.This paper describes two space-partitioning methods, called Interleaving and Segmentation, respectively. The speedups resulting from these methods are compared for their performance in modeling real-life data.

TESTING FOR ZERO AUTOCORRELATION WHEN THE INNOVATIONS BELONG TO THE NORMAL DOMAIN OF ATTRACTION OF A CAUCHY LAW

We consider the asymptotic null distribution of the empirical autocorrelation function when the innovations of a moving average process belong to the normal domain of attraction of a Cauchy law. A series expansion for the density of the limiting null distribution is developed, and some critical values of the tests are computed numerically.

Arm processes and modeling methodology

ARM (Auto-Regressive Modular) processes constitute a broad class of nonlinear autoregressive schemes with modulo-1 reduction and additional transformations. Unlike their TES (Transform-Expand-Sample) precursors, which only admit iid innovation sequences, ARM processes admit dependent innovation sequences as well, so long as they are independent of the initial ARM variate. As such, the class of ARM processes constitutes a considerable generalization of the TES class, endowed with enhanced modeling flexibility. For example, a Markovian innovation sequence can model burstiness in traffic processes far better by making use of the Markovian state to capture the structure of bursts. This paper introduces ARM processes and derives their fundamental properties in terms of marginal distributions and autocorrelation functions. It defines several useful subclasses that illustrate the modeling flexibility of ARM classes. Finally, it outlines a modeling methodology of empirical time series that can simultaneously fit the empirical marginal distribution (histogram) and empirical autocorrelation function.

Modeling full-length VBR video using Markov-renewal-modulated TES models

This paper describes a general methodology for constructing accurate source models of long sequences (full-length movies) of full-motion compressed VBR (variable bit rate) video, directly from empirical data sets of observed bit rate (frame size) records. The main idea is to first cluster scenes into classes and use TES (transform-expand-sample) processes to carefully model their duration and bit rate processes, and then modulate these scene class models by a Markov-renewal process that governs scene transitions. The resulting model is a Markov-renewal-modulated TES (MRMT) process. The fidelity criteria imposed are relatively stringent in that they call for the simultaneous capture of empirical histograms and autocorrelation functions, as well as the qualitative appearance of the empirical data, including burstiness. To illustrate the methodology, the paper provides a detailed description of the modeling procedure, using a JPEG-like coding trace of the Star Wars movie as a working example. The resulting MRMT model was validated by comparing its simulation-based statistics to their empirical counterparts, and by a performance study of the corresponding loss rates for various buffer sizes. The results support the efficacy of the modeling methodology and suggest that suites of realistic and compact source models can be developed for other types of movies under a variety of coding schemes. These may be used to drive realistic Monte Carlo simulations of broadband telecommunications networks.

The empirical TES methodology: modeling empirical time series

TES (Transform-Expand-Sample) is a versatile class of stochastic sequences defined via an autoregressive scheme with modulo-1 reduction and additional transformations. The scope of TES encompasses a wide variety of sample path behaviors, which in turn give rise to autocorrelation functions with diverse functional forms - monotone, oscillatory, alternating, and others. TES sequences are readily generated on a computer, and their autocorrelation functions can be numerically computed from accurate analytical formulas at a modest computational cost.This paper presents the empirical TES modeling methodology which uses TES process theory to model empirical records. The novel feature of the TES methodology is that it expressly aims to simultaneously capture the empirical marginal distribution (histogram) and autocorrelation function. We draw attention to the non-parametric nature of TES modeling in that it always guarantees an exact match to the empirical marginal distribution. However, fitting the corresponding autocorrelation function calls for a heuristic search for a TES model over a large parametric space. Consequently, practical TES modeling of empirical records must currently rely on software assistance. A visual interactive software environment, called TEStool, has been designed and implemented to support TES modeling. The paper describes the empirical TES modeling methodology as implemented in TEStool and provides numerically-computable formulas for TES autocorrelations. Two examples illustrate the efficacy of the TES modeling approach. These examples serve to highlight the ability of TES models to capture first-order and second-order properties of empirical sample paths and to mimic their qualitative appearance.