Sample Autocorrelation Function Research Articles

Exploring the Depths of the Autocorrelation Function: Its Departure from Normality

In this article, we study the autocorrelation function (ACF), which is a crucial element in time series analysis. We compare the distribution of the ACF, both from a theoretical and empirical point of view. We focus on white noise processes (WN), i.e., uncorrelated, centered, and identically distributed variables, whose ACFs are supposed to be asymptotically independent and converge towards the same normal distribution. But, the study of the sum of the sample ACF contradicts this property. Thus, our findings reveal a deviation of the sample ACF from normality beyond a specific lag. Note that this phenomenon is observed for white noise of varying lengths, and evenforn the residuals of an ARMA(p,q) model. This discovery challenges traditional assumptions of normality in time series modeling. Indeed, when modeling a time series, the crucial step is to validate the estimated model by checking that the associated residuals form white noise. In this study, we show that the widely used portmanteau tests are not completely accurate. Box–Pierce appears to be too conservative, whereas Ljung–Box is too liberal. We suggest an alternative method based on the ACF for establishing the reliability of the portmanteau test and the validity of the estimated model. We illustrate our methodology using money stock data in the USA.

Instability Feature Extraction Method of Compressor Based on Sample Autocorrelation Function

The stability of aero-engine compressor is related to flight safety. Severe instability may lead to engine surge, parking or mechanical damage and even cause flight accidents. Aiming at this problem, a compressor instability feature extraction method based on sample autocorrelation function is proposed, and an autocorrelation analysis model of fluctuating pressure is established. The model is verified and analyzed by using the flight test data of an engine, which shows the feasibility of this method. It can provide technical support for real-time monitoring and quantitative evaluation of aero-engine compressor stability and improve safe flight efficiency.

Two New Estimators for the Autocorrelation Function Through Singular Spectrum Analysis

It is around a century that sample autocorrelation function has been introduced and used as a standard tool in time series analysis. A vast literature can be found on the statistical properties of the sample autocorrelation function. However, it has been highlighted recently that the sum of the sample autocorrelation function over the lags 1 to [Formula: see text] is −0.5 for all time series of length T. This property produces a big concern for the cases in which all available sample autocorrelations are used in the inference. This paper provides two new alternative for estimating the autocorrelation function. These estimators come from the idea of singular spectrum analysis which is a non-parametric technique for time series analysis. The paper utilizes a simulation study to illustrate the performance of the new approach. The results suggest that further improvement to the sample autocorrelation is possible and the new methods provide an attractive alternative to the classical approach.

Time Series Model to Forecast Fresh Coconut Exports from Sri Lanka

Coconut accounts for approximately 12% of all agricultural produce in Sri Lanka with the total land area under cultivation covering 409, 244 hectares ranking second to rice production. The primary regions for coconut cultivation are the Puttalam and Kurunegala districts in North-Western Province and Gampaha district in the Western Province, forming what is known as the Coconut Triangle. This region accounts for 232,270 hectares (50.94%) of the overall coconut cultivation area. The remaining coconut cultivation areas are found in the Southern Province, specifically in the districts of Galle (13,833 hectares), Matara (14,946 hectares), and Hambantota (25,837 hectares), and in non-traditional regions of the Eastern and Northern provinces. The annual coconut production varies between 2,800 to 3,000 million nuts. Having advanced knowledge of exporting coconuts offers numerous advantages to Sri Lanka, particularly in terms of establishing forward contracts with other countries. Based on secondary data of annual fresh coconut exports from 1981 to 2020 obtained from the Coconut Development Authority (CDA) of Sri Lanka, the paper developed ARIMA (2,1,0) model to forecast export. The model was selected out of three parsimonious models which were identified from the Sample Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) of the stationary series and a comparison of significant parameters and lowest values of Akaike Information Criterion (AIC), Schwarz Bayesian Information Criterion (SBIC) and Hannan-Quinn Information Criterion (HQIC). The errors of the fitted model were found to be random and constant variance. The model was validated using 2021 and 2022 data. The percentage errors for 2021 and 2022 are 20.23% and -29.57% respectively. The predictions for 2023 and 2024 are 14696 and 15052 respectively. The model can be used effectively by the Coconut Development Authority for decision-making. However, it is suggested to develop the model further to reduce the percentage error. Keywords: ARIMA model, Forecast, Fresh Coconut Exports and Validate

Prediction and Classification Models of Wordle Data Set Based On ARMA, XGBoost

This study intends to use machine learning method to solve the problem of predicting the number of word guessing and the difficulty of word guessing in natural language. Firstly, this study smoothed the data according to the COF method. According to the images of the sample autocorrelation function and the sample partial autocorrelation function, the ARMA model is constructed and the coefficient of the AMRA model is calculated, and the number of word contest people on March 1, 2023 is predicted to drop to [18002,20622]. Second, based on the seven attributes of the word, the XGBoost classification model is used to predict the value of the first attempt, and then the XGBoost prediction model is used to predict the results of the other attempts. Eerie distribution of different attempts to {0,8.890, 17.650, 28.191, 32.451, 11.349, 1.483}. The average accuracy of the model on the test set is 79.4%.

Bayesian structural identification using Gaussian Process discrepancy models

Bayesian model updating based on Gaussian Process (GP) models has received attention in recent years, which incorporates kernel-based GPs to provide enhanced fidelity response predictions. Although most kernel functions provide high fitting accuracy in the training data set, their out-of-sample predictions can be highly inaccurate. This paper investigates this problem by reformulating the problem on a consistent probabilistic foundation, reviewing common choices of kernel covariance functions, and proposing a new Bayesian model selection for kernel function selection, aiming to create a balance between fitting accuracy, generalizability, and model parsimony. Computational aspects are addressed via Laplace approximation and sampling techniques, providing detailed algorithms and strategies. Numerical and experimental examples are included to demonstrate the accuracy and robustness of the proposed framework. As a result, an exponential-trigonometric covariance function is characterized and justified based on the Bayesian model selection approach and observations of the sample autocorrelation function of the response discrepancies.

Uncovering Hidden Insights with Long-Memory Process Detection: An In-Depth Overview

Long-memory models are frequently used in finance and other fields to capture long-range dependence in time series data. However, correctly identifying whether a process has long memory is crucial. This paper highlights a significant limitation in using the sample autocorrelation function (ACF) to identify long-memory processes. While the ACF establishes the theoretical definition of a long-memory process, it is not possible to determine long memory by summing the sample ACFs. Hassani’s −12 theorem demonstrates that the sum of the sample ACF is always −12 for any stationary time series with any length, rendering any diagnostic or analysis procedures that include this sum open to criticism. The paper presents several cases where discrepancies between the empirical and theoretical use of a long-memory process are evident, based on real and simulated time series. It is critical to be aware of this limitation when developing models and forecasting. Accurately identifying long-memory processes is essential in producing reliable predictions and avoiding incorrect model specification.

Stochastic ACF-ZCL model of time-series signals with an application to restfulness assessment

In this paper, a stochastic model of a time-series signal has been derived using the first zero-crossing lag (ZCL) of the sample autocorrelation function (ACF). For this, the order of a power law series is obtained deterministically from its ZCL. This is extended to polynomial regression models and stochastic models of time-series signals. The residuals of both the stochastic ACF-ZCL model and the conventional PACF model represent the phasic component and noise but the former follow the beta distribution more consistently. The proposed model is applied for restfulness assessment of 10-minutes long differential dermal potentials (DDP) acquired simultaneously from all 4 limbs of subjects using a specific 4-channel data acquisition system. The findings clearly show the manner in which stages of restfulness affect the features of the deterministic (or tonic) and the stochastic (or phasic) components of the stochastic ACF-ZCL model.

New practice for investors in Chinese stock market: From perspective of fractionally integrated realized GARCH model

In this paper, based on the Realized GARCH model, the fractional integration Realized GARCH model is proposed by combining long memory parameters with conditional variance and replacing the original realized measure with the realized measure obtained after daily, weekly and monthly weighting. Based on the 5-min high-frequency data of the SSE index, the fractional integration Realized GARCH model, Realized HAR GARCH model and Realized GARCH model are investigated for their fitting effect and predictive ability on market volatility, and Monte Carlo simulations are conducted from the error terms obeying normal distribution, t-distribution and chi-square distribution so as to compare the RMSE and MAE of the three types of models with respect to conditional variance. The empirical results show that the fractionally integrated Realized GARCH model is found to better capture the long-run correlation in volatility in certain intervals by comparing the theoretical and sample auto-correlation functions, while the overall predictive power of the model is better than the other two models. Finally, it provides technical support and suggestions for investors' risk control.

On the Measurement of Cointegration Relations Between the Indicators of the Time Series of the Current Account of the Balance of Payments and GDP (On the Example of the Republic of Azerbaijan)

The article solves the problem of constructing an econometric model (based on time series of macroeconomic indicators) that reflects the relationship between changes in the current account of the balance of payments and GDP dynamics (on the example of the Republic of Azerbaijan).In order to assess the correctness of the application of mathematical and statistical methods for analyzing the impact of foreign economic activity on the final results of the country's economic activity, the constructed model was tested. So, to test the hypotheses about the normal distribution of the series, the Jarque–Bera test, the augmented Dickey–Fuller test (ADF), the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test were used, and histograms and correlograms were constructed.In the process of studying the time series of the indicators under consideration, their non-stationarity was revealed both according to the results of descriptive statistics, the Jarque–Bera test, histograms, plots of the sample autocorrelation function (ACF) and partial autocorrelation function (PACF), and according to the Dickey–Fuller test. The transition to absolute increments, that is, to the first differences, was applied. In addition, the Johansen test was performed to identify cointegration relationships. Based on its results, when comparing linear and quadratic trends with similar characteristics, a linear one was chosen, indicating the presence of a cointegration relationship. After carrying out the Trace and Maximum Eigenvalue tests, an alternative hypothesis about the presence of one cointegration vector was accepted.The results obtained allow us to conclude that the relationship between the indicators of the current account of the balance of payments and Azerbaijan's GDP is of a long-term nature. They can also serve as a basis for building an error correction model.

A cubic regression model to measure the impact of accidents on the number of incidence in the State of Kuwait

Theories of regression analysis play an important role in safety traffic applications. The purpose of this paper is to develop a methodology capable of fitting the yearly number of incidences based on annual traffic accidents in Kuwait. More specifically, a cubic regression model was specified to fit the annual traffic incidents in Kuwait during the period 2002-2017. Estimation results using the maximum likelihood estimation and the goodness of fit using the sample autocorrelation function clearly demonstrated the appropriateness of the estimated cubic regression model.

SACF of the Errors of Stationary Time Series Models in the Presence of a Large Additive Outlier

In this paper, the limiting behaviour of the Sample Autocorrelation Function(SACF) of the errors {et} of First-Order Autoregressive (AR(1)), First-Order Moving Average (MA(1)) and First Order Autoregressive First-Order Moving Average (ARMA(1,1)) stationary time series models in the presence of a large Additive Outlier(AO) is discussed. It is found that the errors which are supposed to be uncorrelated due to either white noise process or normally distributed process are not so in the presence of a large additive outlier. The SACF of the errors follows a particular pattern based on the time series model. In the case of AR(1) model, at lag 1, the contaminated errors {et} are correlated, whereas at higher lags, they are uncorrelated. But in the MA(1) and ARMA(1,1) models, the contaminated errors {et} are correlated at all the lags. Furthermore it is observed that the intensity of correlations depends on the parameters of the respective models.

Trajectory Extrapolation for Manual Robot Remote Welding

This article describes an algorithm for the online extrapolation of hand-motion during remote welding. The aim is to overcome the spatial limitations of the human welder’s arms in order to cover a larger workspace with a continuous weld seam and to substantially relieve the welder from strain and fatigue. Depending on the sampled hand-motion data, an extrapolation of the given motion patterns is achieved by decomposing the input signals in a linear direction and a periodic motion component. An approach to efficiently determine the periodicity using a sampled autocorrelation function and the subsequent application of parameter identification using a spline function are presented in this paper. The proposed approach is able to resemble all practically relevant motion patterns and has been validated successfully on a remote welding system with limited input space and audio-visual feedback by an experienced welder.

Fast Channel Estimation in the Transformed Spatial Domain for Analog Millimeter Wave Systems

Fast channel estimation in millimeter-wave (mmWave) systems is a fundamental enabler of high-gain beamforming, which boosts coverage and capacity. The channel estimation stage typically involves an initial beam training process where a subset of the possible beam directions at the transmitter and receiver is scanned along a predefined codebook. Unfortunately, the high number of transmit and receive antennas deployed in mmWave systems increase the complexity of the beam selection and channel estimation tasks. In this work, we tackle the channel estimation problem in analog systems from a different perspective than used by previous works. In particular, we propose to move the channel estimation problem from the angular domain into the transformed spatial domain, in which estimating the angles of arrivals and departures corresponds to estimating the angular frequencies of paths constituting the mmWave channel. The proposed approach, referred to as transformed spatial domain channel estimation (TSDCE) algorithm, exhibits robustness to additive white Gaussian noise by combining low-rank approximations and sample autocorrelation functions for each path in the transformed spatial domain. Numerical results evaluate the mean square error of the channel estimation and the direction of arrival estimation capability. TSDCE significantly reduces the first, while exhibiting a remarkably low computational complexity compared with well-known benchmarking schemes.

Simulation of the Localized Modulus of Elasticity of Hardwood Boards by Means of an Autoregressive Model

An autoregressive (AR) model for the simulation of modulus of elasticity (MOE) fluctuations along hardwood boards without pronounced knot periodicity is presented. The model is calibrated based on contiguous, localized (100 mm gauge length) empirical MOEs from European white oak boards. The calibration involves the computation of the sample autocorrelation function (SACF). For this computation, the stationary data resulting from a two-step transformation method are used: (1) first, board-internal MOE normalization suppresses the interboard bias; and (2) the rather left-skewed normalized data are described by a log-gamma distribution, which is then mapped to N(0,1) to obtain the needed stationary data. A first-order AR process suffices to accurately describe the data and is thus used as a basis to simulate MOE profiles, followed by the sketched transformation procedure in reverse order. The quality of the simulated MOE profiles and of the methodological approach is shown by comparison with the experimental MOE profiles. It was found that the simulation model correctly reproduces the statistical information of the MOE variations while also producing very plausible profiles, with similar characteristics as the empirical MOE profiles. This model can be used to better study the size effect of glued-laminated timber by means of numerical simulations.

Analysis of autocorrelation function of stochastic processes by F-transform of higher degree

The autocorrelation function of a stochastic process is one of the essential mathematical tools in the description of variability that is successfully applied in many scientific fields such as signal processing or financial time series analysis and forecasting. The aim of the paper is to provide an analysis of fuzzy transform of higher degree applied to stochastic processes with a focus on its autocorrelation function. We introduce a higher-degree fuzzy transform of a stochastic process and investigate its basic properties. Further, we introduce a higher-degree fuzzy transform of bivariate functions in the tensor product of polynomial spaces and demonstrate its approximation ability. In addition, we show that the bivariate higher-degree fuzzy transform of multiplicative separable functions is a product of univariate fuzzy transform of the respective functions, which is a desirable property in the processing of functions of higher dimensions. The obtained results are used to demonstrate an interesting identity between the autocorrelation function of the fuzzy transform of a stochastic process and the bivariate fuzzy transform of the autocorrelation function of the stochastic process. This identity provides two ways to determine the autocorrelation function of the fuzzy transform of a stochastic process, which can be useful, especially in a situation where its direct calculation becomes complex or even impossible, for example, the calculation of a sample autocorrelation function.

Real-time deterministic prediction of wave-induced ship responses based on short-time measurements

This paper studies real-time deterministic prediction of wave-induced ship motions using the autocorrelation functions (ACFs) from short-time measurements, namely the instantaneous ACFs. The Prolate Spheroidal Wave Functions (PSWF) are introduced to correct the large lag time errors in the instantaneous sample ACF, together with a modification of the autocorrelation (AC) matrix for ensuring its positive definiteness. The validity of the PSWF-based ACFs is first examined by using the ship motion measurements from model experiment under stationary wave excitations. It is shown that the use of PSWF-based ACFs leads to better prediction accuracy than direct use of sample ACFs. The validation is then extended to ship motion prediction using in-service data from a container ship, and an improvement of the prediction accuracy by PSWF-based ACFs is again found. Finally, the effectiveness of use of the instantaneous ACFs for non-stationary wave-induced responses is highlighted by comparing with the prediction results based on the ACFs from long-time measurements.

Estimation of autocorrelation function and spectrum density of wave-induced responses using prolate spheroidal wave functions

Predicting the wave-induced response in the near-future is of importance to ensure safety of ships. To achieve this target, a possible method for deterministic and conditional prediction of future responses utilizing measured data from the most recent past has been developed. Herein, accurate derivation of the autocorrelation function (ACF) is required. In this study, a new approach for deriving ACFs from measurements is proposed by introducing the Prolate Spheroidal Wave Functions (PSWF). PSWF can be used in two ways: fitting the measured response itself or fitting the sample ACF from the measurements. The paper contains various numerical demonstrations, using a stationary heave motion time series of a containership, and the effectiveness of the present approach is demonstrated by comparing with both a non-parametric and a parametric spectrum estimation method; in this case, Fast Fourier Transformation (FFT) and an Auto-Regressive (AR) model, respectively. The present PSWF-based approach leads to two important properties: (1) a smoothed ACF from the measurements, including an expression of the memory time, (2) a high frequency resolution in power spectrum densities (PSDs). Finally, the paper demonstrates that a fitting of the ACF using PSWF can be applied for deterministic motion predictions ahead of current time.

Global autocorrelation test based on the Monte Carlo method and impacts of eliminating nonstationary components on the global autocorrelation test

Autocorrelation and non-stationarity are always concerned in analysis of meteorological and hydrological time series. Current commonly used methods, such as the Box-Pierce (BP) test and Ljung-Box (LB) test, always preset the maximum order for the autocorrelation significance test without considering the existence of high-order autocorrelation coefficient(s), and also neglect a fact that the sum of sample autocorrelation function is a constant value. Moreover, the impacts of autocorrelation on the significance test of nonstationary components of sample time series have drawn much attention, but less attention is paid to the impacts of eliminating nonstationary components on the global autocorrelation significance test. These issues are addressed in the paper. Based on the Monte Carlo method, a global autocorrelation test method, the quadratic sum (QS) test, is presented for judging the existence of high-order autocorrelation coefficient(s) of a sample time series. Besides, two nonparametric trend estimators (nonlinear and linear trend estimators) are employed to investigate the impacts of eliminating nonstationary components on the global autocorrelation test. The results show that (i) the QS test method is more robust than the BP test and LB test in verifying the existence of significant high-order autocorrelation coefficient(s); and (ii) eliminating a linear trend has less damage on the original global autocorrelation structure of sample time series by comparing with eliminating a nonlinear trend. Therefore, it is recommended to initially eliminate the linear trend from a sample time series, and then judge the existence of high-order autocorrelation coefficients of the time series based on the QS test.

Goodness-of-fit testing for time series models via distance covariance

In many statistical modeling frameworks, goodness-of-fit tests are typically administered to the estimated residuals. In the time series setting, whiteness of the residuals is assessed using the sample autocorrelation function. For many time series models, especially those used for financial time series, the key assumption on the residuals is that they are in fact independent and not just uncorrelated. In this paper, we apply the auto-distance covariance function (ADCV) to evaluate the serial dependence of the estimated residuals. Distance covariance can discriminate between dependence and independence of two random vectors. The limit behavior of the test statistic based on the ADCV is derived for a general class of time series models. One of the key aspects in this theory is adjusting for the dependence that arises due to parameter estimation. This adjustment has essentially the same form regardless of the model specification. We illustrate the results in simulated examples.