Partial Autocorrelations Research Articles

Bayesian estimation in generalized linear models for longitudinal data with hyperspherical coordinates

Under the framework of generalized linear models (GLM), the generalized estimating equation (GEE) method is typically applied for longitudinal data analysis. However, there are a series of problems due to the misspecification of the within-subject correlation structure, especially in Bayesian estimation. To handle these difficulties, in this paper, we construct a class of generalized estimating equations for longitudinal data with hyperspherical coordinates (HPC) and propose a Bayesian approach established through empirical likelihood (EL). Additionally, an efficient Markov chain Monte Carlo (MCMC) procedure is developed for the required computation of the posterior distribution. As proved by the simulation studies and an application to a real longitudinal data set, our method not only performs better than traditional empirical likelihood estimation and Bayesian estimation with partial autocorrelations (PAC) but also is suitable for non-Gaussian data.

A Review of Seasonal Adjustment Diagnostics

SummarySeasonal adjustment methods are used to process and publish thousands of time series across the world each month, and judgement of the adequacy relies heavily upon seasonal adjustment diagnostics. This paper discusses tests for the adequacy of a seasonal adjustment, first reviewing the broader background on tests for seasonality and then proceeding to four tests that are appropriate for stationary forms of seasonality. We contrast time‐domain and frequency‐domain approaches, focusing upon four diagnostics available in the seasonal adjustment literature (and software packages) and applying the methods to a large collection of public use time series. Each of the four tests is designed around a distinct formulation of seasonality, and hence, empirical performances differ; we compare and contrast the methods and include discussion on how diagnostic results can be used to improve faulty seasonal adjustments. Directions for future research, involving inverse partial autocorrelations and polyspectra, are also discussed, and the methodologies are supported by theoretical and simulation results.

Fractional and fractal processes applied to cryptocurrencies price series

IntroductionCryptocurrencies have been attracting the attention from media, investors, regulators and academia during the last years. In spite of some scepticism in the financial area, cryptocurrencies are a relevant subject of academic research. ObjectivesIn this paper, several tools are adopted as an instrument that can help market agents and investors to more clearly assess the cryptocurrencies price dynamics and, thus, guide investment decisions more assertively while mitigating risks. MethodsWe consider three methods, namely the Auto-Regressive Integrated Moving Average (ARIMA), Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) and Detrended Fluctuation Analysis, and three indices given by the Hurst and Lyapunov exponents or the Fractal Dimension. This information allows assessing the behaviour of the time series, such as their persistence, randomness, predictability and chaoticity. ResultsThe results suggest that, except for the Bitcoin, the other cryptocurrencies exhibit the characteristic of mean reverting, showing a lower predictability when compared to the Bitcoin. The results for the Bitcoin also indicate a persistent behavior that is related to the long memory effect. ConclusionsThe ARFIMA reveals better predictive performance than the ARIMA for all cryptocurrencies. Indeed, the obtained residual values for the ARFIMA are smaller for the auto and partial auto correlations functions, as well as for confidence intervals.

COVID-19: Fitting a ROR Prediction Model for Cuba as Vaccination Advance

Background: We are still in the midst of a deadly pandemic caused by the SARS-CoV-2 virus causing COVID-19, and at the moment there is a shortage of licensed vaccines. Several models and methodologies have been applied in the study, analysis, prediction and modeling of COVID-19 in the world, so it is important to estimate the trend in the behavior of the epidemiological curve of the COVID-19 pandemic worldwide.Methods: In the realization of the study were used the data of the pandemic of new cases to COVID-19 in Cuba, which were taken from the press conference offered by the Ministry of Public Health (MINSAP), by Dr. Francisco Durán García (National Director of Hygiene and Epidemiology of MINSAP), from Monday to Sunday, at 9.00am on national television and Cuban radio, since the pandemic began in March 2020 until July 15, 2021. The forecast was made using the methodology of Regressive Objective Regression (ROR), which has been implemented for different variables (viruses, parasites and bacteria that have circulated in Villa Clara province, Cuba) and even the SARS-CoV-2 virus causing COVID-19. A very long term forecast was made, up to November 1, 2021 with data taken up to July 15, 2021. The objective regressive modeling (ROR) is based on a combination of dummy variables with ARIMA modeling. In this methodology it is necessary to first create the dichotomous variables DS, DI and NoC (NoC: Number of cases of the base, DS represents a sawtooth function and DI this same function, but in inverted form), in such a way that the variable to be modeled is trapped between these parameters and a large amount of variance can be explained. Subsequently, the module corresponding to the Regression analysis of the statistical package SPSS version 19.0 is executed, specifically the ENTER method where the predicted variable and the ERROR are obtained. Then, the autocorrelograms of the ERROR variable will be obtained, paying attention to the maximums of the significant partial autocorrelations PACF. Finally, these regressed variables are included in the new regression in a process of successive approximations until a white noise in the regression errors is obtained.Objective: This research was undertaken to predict new COVID-19 cases, as well as the impact of vaccination in Cuba using the Regressive Objective Regression (ROR) methodology.Findings: The intensification and sustainability of hygienic and sanitary measures, as well as social distancing, and the massive, constant and sustained use of nasobuco, must be taken into account in the prognosis of new cases, since the number of new cases will be higher and may reach values higher than or close to 10,000 cases. As for the impact of vaccination in Cuba with the "Abdala" vaccine, progress is not yet as desired, since it will not be until October 2 of the current year that approximately 6.5 to 7 million people in Cuba will be vaccinated with the three doses, so that it will not be until October 22 that zero new cases could be reached, so that, from the first half of December of the current year, herd immunity could begin to appear in Cuba.Interpretation: Despite being a new disease in the world, COVID-19 can also be followed, modeled and predicted by means of ROR modeling, which allows reducing the number of deceased, severe and critical patients for a better management of the pandemic. On the other hand, the use of vaccines has shown that they can protect against COVID-19.Funding Information: None to declare. Declaration of Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Hierarchical Clustering for Smart Meter Electricity Loads Based on Quantile Autocovariances

In order to improve the efficiency and sustainability of electricity systems, most countries worldwide are deploying advanced metering infrastructures, and in particular household smart meters, in the residential sector. This technology is able to record electricity load time series at a very high frequency rates, information that can be exploited to develop new clustering models to group individual households by similar consumptions patterns. To this end, in this work we propose three hierarchical clustering methodologies that allow capturing different characteristics of the time series. These are based on a set of "dissimilarity" measures computed over different features: quantile auto-covariances, and simple and partial autocorrelations. The main advantage is that they allow summarizing each time series in a few representative features so that they are computationally efficient, robust against outliers, easy to automatize, and scalable to hundreds of thousands of smart meters series. We evaluate the performance of each clustering model in a real-world smart meter dataset with thousands of half-hourly time series. The results show how the obtained clusters identify relevant consumption behaviors of households and capture part of their geo-demographic segmentation. Moreover, we apply a supervised classification procedure to explore which features are more relevant to define each cluster.

Bayesian Estimation of Correlation Matrices of Longitudinal Data

Estimation of correlation matrices is a challenging problem due to the notorious positive-definiteness constraint and high-dimensionality. Reparameterizing Cholesky factors of correlation matrices in terms of angles or hyperspherical coordinates where the angles vary freely in the range [0,π) has become popular in the last two decades. However, it has not been used in Bayesian estimation of correlation matrices perhaps due to lack of clear statistical relevance and suitable priors for the angles. In this paper, we show for the first time that for longitudinal data these angles are the inverse cosine of the semi-partial correlations (SPCs). This simple connection makes it possible to introduce physically meaningful selection and shrinkage priors on the angles or correlation matrices with emphasis on selection (sparsity) and shrinking towards longitudinal structure. Our method deals effectively with the positive-definiteness constraint in posterior computation. We compare the performance of our Bayesian estimation based on angles with some recent methods based on partial autocorrelations through simulation and apply the method to a data related to clinical trial on smoking.

POISSON MODELS WITH DYNAMIC RANDOM EFFECTS AND NONNEGATIVE CREDIBILITIES PER PERIOD

AbstractThis paper provides a toolbox for the credibility analysis of frequency risks, with allowance for the seniority of claims and of risk exposure. We use Poisson models with dynamic and second-order stationary random effects that ensure nonnegative credibilities per period. We specify classes of autocovariance functions that are compatible with positive random effects and that entail nonnegative credibilities regardless of the risk exposure. Random effects with nonnegative generalized partial autocorrelations are shown to imply nonnegative credibilities. This holds for ARFIMA(0, d, 0) models. The AR(p) time series that ensure nonnegative credibilities are specified from their precision matrices. The compatibility of these semiparametric models with log-Gaussian random effects is verified. Gaussian sequences with ARFIMA(0, d, 0) specifications, which are then exponentiated entrywise, provide positive random effects that also imply nonnegative credibilities. Dynamic random effects applied to Poisson distributions are retained as products of two uncorrelated and positive components: the first is time-invariant, whereas the autocovariance function of the second vanishes at infinity and ensures nonnegative credibilities. The limit credibility is related to the three levels for the length of the memory in the random effects. The limit credibility is less than one in the short memory case, and a formula is provided.

Goodness‐of‐fit tests forβARMA hydrological time series modeling

AbstractWe address the issue of performing portmanteau testing inference using time series data that assume values in the standard unit interval. The motivation involves modeling the time series dynamics of the proportion of stocked hydroelectric energy in the South of Brazil. Our focus lies in the class of beta autoregressive moving average (βARMA) models. In particular, we wish to test the goodness‐of‐fit of such models. We consider several testing criteria that have been proposed for Gaussian time series models and introduce two new tests. We derive the asymptotic null distribution of the two proposed test statistics in two different scenarios, namely, when the tests are applied to an observed time series and when they are applied to the residuals from a fittedβARMA model. It is worth noticing that our results imply the asymptotic validity of standard portmanteau tests in the class ofβARMA models that are, under the null hypothesis, asymptotically equivalent to our test statistics. We use Monte Carlo simulation to assess the relative merits of the different portmanteau tests when used with fittedβARMA models. The simulation results we present show that the new tests are typically more powerful than a well‐known test whose test statistic is also based on residual partial autocorrelations. Overall, the tests we propose perform quite well. Finally, we model the dynamics of the proportion of stocked hydroelectric energy in Brazil. The results show that theβARMA model outperforms three alternative models and an exponential smoothing algorithm.

FORECASTING PRODUCTION VOLUME IN A PLASTICS ENTERPRISE

The functioning of production enterprises is based on satisfying the needs of customers through the timely manufacture of products in accordance with the demand existing on the market. The availability of the offered range of products is guaranteed by a correct preparation of forecasts of potential orders. This article presents a multiple-regression-method-based tool supporting the planning of production volumes in an enterprise depending on the calendar month. Reliability analysis of the developed model through the analysis of residuals and their autocorrelations and partial autocorrelations is also presented.
 JEL classification: C2, C22.

Generalised Linear Cepstral Models for the Spectrum of a Time Series

The paper introduces the class of generalised linear models with Box-Cox link for the spectrum of a time series. The Box-Cox transformation of the spectral density is represented as a finite Fourier polynomial, with coefficients, that we term generalised cepstral coefficients, providing a complete characterisation of the properties of the random process. The link function depends on a power transformation parameter and encompasses the exponential model (logarithmic link), the autoregressive model (inverse link), and the moving average model (identity link). One of the merits of this model class is the possibility of nesting alternative spectral estimation methods under the same likelihood-based framework, so that the selection of a particular parametric spectrum amounts to estimating the transformation parameter. We also show that the generalised cepstral coefficients are a one to one function of the inverse partial autocorrelations of the process, which can be used to evaluate the mutual information between the past and the future of the process.

Noise Created Through the Interactions of Arbitrageurs and Noise Traders. Estimation of an ARMA Model of Well – Known Closed – End Funds of Major Investment Banks

In this article, we have applied an autoregressive moving average, ARMA(2,2) model of order AR(1), AR(2), MA(1), MA(2) and SMA(12) to test the natural logarithmic monthly market returns of the of the closed-end funds of major investment banks such as Van Kampen income trust, Aberdeen Asia Pacific income fund, Credit Suisse asset management income, Scudder high income and Templeton global income. The purpose of this article is to eliminate the seasonality and to compare the forecasted with the actual expectations of the market prices of the closed-end funds that arises from the interaction of arbitrageurs and noise traders. The software that we have used is EViews 6. We have found that the natural logarithmic monthly market price returns of all closed-end funds are a stationary series. There is serial correlation in the natural logarithmic monthly market price returns of all closed-end funds before applying the ARMA model. The natural logarithmic monthly market price returns of all closed-end funds are not normally distributed. All equations except equation (8) show a covariance stationary and invertible ARMA process. The equation of Aberdeen Asia Pacific income fund shows a stationary ARMA, but not invertible. The ARMA model is properly specified as the difference between the actual and the estimated residuals of the autocorrelations and partial autocorrelations function of the correlogram are very small. After applying the ARMA(2,2) model in all closed-end funds, we have found that there is no serial correlation in the residuals of the estimated model. The residuals of all closed-end funds are homoskedastic and there is no additional ARCH effect. We have selected the model with the best forecasting ability in terms of the smallest mean absolute percentage error, the lowest value of the Akaike information criterion, Schwarz criterion, and covariance proportion. We have concluded that the best fit model is Templeton global income, (LNMPTGI) closed-end fund. The Akaike and Schwarz criterion are -4.015 and -3.89 respectively. The mean absolute percentage error is 73.78% and the covariance proportion is 0.37. The one – step ahead forecast of the natural logarithmic monthly price returns of all closed-end funds does not have power to forecast future values. It does not show unanticipated changes in the prices of the closed-end funds that could help the arbitrageurs to craft their investment strategy. The whole dataset is from 31/01/1990 to 31/12/2001. The total observations are 144 and the logarithmic monthly market returns observations are 143. The data was obtained from Thomson Financial Investment View database.

A Durbin–Levinson regularized estimator of high-dimensional autocovariance matrices

We consider the problem of estimating the high-dimensional autocovariance matrix of a stationary random process, with the purpose of out of sample prediction and feature extraction. This problem has received several solutions. In the nonparametric framework, the literature has concentrated on banding and tapering the sample autocovariance matrix. This paper proposes and evaluates an alternative approach, based on regularizing the sample partial autocorrelation function, via a modified Durbin-Levinson algorithm that receives as input the banded and tapered partial autocorrelations and returns a sample autocovariance sequence which is positive definite. We show that the regularized estimator of the autocovariance matrix is consistent and its convergence rates is established. We then focus on constructing the optimal linear predictor and we assess its properties. The computational complexity of the estimator is of the order of the square of the banding parameter, which renders our method scalable for high-dimensional time series. The performance of the autocovariance estimator and the corresponding linear predictor is evaluated by simulation and empirical applications.

Abrupt and gradual temperature changes influence on the climatic suitability of Northwestern Alpine grapevine-growing regions for the invasive grape leafhopper Scaphoideus titanus Ball (Hemiptera, Cicadellidae)

The paper aims to elucidate the influence of abrupt and gradual climate changes on the suitability for colonization of Scaphoideus titanus populations in grapevine-growing areas of the Northwestern Alpine region. This study spans several decades of temperature recordings and is carried out in ten grapevine-growing areas. A time-varying distributed delay with attrition model, linked to a grapevine phenology model, is used to simulate the development of S. titanus populations and produce an annual Climatic Suitability Index (CSI). Area-specific CSI time series were obtained. The Breusch-Godfrey test revealed few significant partial autocorrelations in nine areas and the occurrence of six consecutive - first decreasing and then increasing - partial correlations in one case only. The occurrence of abrupt and gradual changes of the index were studied via multiple least square regression analyses. In general, the climatic suitability of all areas tended to improve through time. However, gradual and abrupt temperature changes were not consistently reflected in gradual and abrupt CSI patterns: abrupt and gradual CSI changes were observed in two areas, abrupt changes were detected in three areas, and exclusively gradual changes in the remaining five. Pest control institutions of the region under study may deal with different scenarios of pest status such as long-time presence and increasing risks, high colonization risks or limited colonization risks for the foreseeable future. Institutions charged with pest control elsewhere are advised to use a mechanistic demographic model to study area-specific infestation patterns and colonization risks because the results obtained here cannot be transferred to other areas without site-specific evaluations.

A Durbin-Levinson Regularized Estimator of High Dimensional Autocovariance Matrices

We consider the problem of estimating the high-dimensional autocovariance matrix of a stationary random process, with the purpose of out of sample prediction and feature extraction. This problem has received several solutions. In the nonparametric framework, the literature has concentrated on banding and tapering the sample autocovariance matrix. This paper proposes and evaluates an alternative approach, based on regularizing the sample partial autocorrelation function, via a modified Durbin-Levinson algorithm that receives as input the banded and tapered partial autocorrelations and returns a sample autocovariance sequence which is positive definite. We show that the regularized estimator of the autocovariance matrix is consistent and its convergence rates is established. We then focus on constructing the optimal linear predictor and we assess its properties. The computational complexity of the estimator is of the order of the square of the banding parameter, which renders our method scalable for high-dimensional time series. The performance of the autocovariance estimator and the corresponding linear predictor is evaluated by simulation and empirical applications.

Penalised Complexity Priors for Stationary Autoregressive Processes

The autoregressive (AR) process of orderp(AR(p)) is a central model in time series analysis. A Bayesian approach requires the user to define a prior distribution for the coefficients of the AR(p) model. Although it is easy to write down some prior, it is not at all obvious how to understand and interpret the prior distribution, to ensure that it behaves according to the users' prior knowledge. In this article, we approach this problem using the recently developed ideas of penalised complexity (PC) priors. These prior have important properties like robustness and invariance to reparameterisations, as well as a clear interpretation. A PC prior is computed based on specific principles, where model component complexity is penalised in terms of deviation from simple base model formulations. In the AR(1) case, we discuss two natural base model choices, corresponding to either independence in time or no change in time. The latter case is illustrated in a survival model with possible time‐dependent frailty. For higher‐order processes, we propose a sequential approach, where the base model for AR(p) is the corresponding AR(p−1) model expressed using the partial autocorrelations. The properties of the new prior distribution are compared with the reference prior in a simulation study.

A Gini-based time series analysis and test for reversibility

Time reversibility is a fundamental hypothesis in time series. In this paper, Gini-based equivalents for time series concepts that enable to construct a Gini-based test for time reversibility under merely first-order moment assumptions are developed. The key idea is that the relationship between two variables using Gini (as measured by Gini autocorrelations and partial autocorrelations) can be measured in two directions, which are not necessarily equal. This implies a built-in capability to discriminate between looking at forward and backward directions in time series. The Gini creates two bi-directional Gini autocorrelations (and partial autocorrelations), looking forward and backward in time, which are not necessarily equal. The difference between them may assist in identifying models with underlying heavy-tailed and non-normal innovations. Gini-based test and Gini-based correlograms, which serve as visual tools to examine departures from the symmetry assumption, are constructed. Simulations are used to illustrate the suggested Gini-based framework and to validate the statistical test. An application to a real data set is presented.

Generalised Partial Autocorrelations and the Mutual Information between Past and Future

The paper introduces the generalised partial autocorrelation (GPAC) coefficients of a stationary stochastic process. The latter are related to the generalised autocovariances, the inverse Fourier transform coefficients of a power transformation of the spectral density function. By interpreting the generalised partial autocorrelations as the partial autocorrelation coefficients of an auxiliary process, we derive their properties and relate them to essential features of the original process. Based on a parameterisation suggested by Barndorff-Nielsen and Schou (1973) and on Whittle likelihood, we develop an estimation strategy for the GPAC coefficients. We further prove that the GPAC coefficients can be used to estimate the mutual information between the past and the future of a time series.

Sparsity Inducing Prior Distributions for Correlation Matrices of Longitudinal Data

For longitudinal data, the modeling of a correlation matrix R can be a difficult statistical task due to both the positive definite and the unit diagonal constraints. Because the number of parameters increases quadratically in the dimension, it is often useful to consider a sparse parameterization. We introduce a pair of prior distributions on the set of correlation matrices for longitudinal data through the partial autocorrelations (PACs), which vary independently over (−1,1). The first prior shrinks each of the PACs toward zero with increasingly aggressive shrinkage in lag. The second prior (a selection prior) is a mixture of a zero point mass and a continuous component for each PAC, allowing for a sparse representation. The structure implied under our priors is readily interpretable for time-ordered responses because each zero PAC implies a conditional independence relationship in the distribution of the data. Selection priors on the PACs provide a computationally attractive alternative to selection on the elements of R or R− 1 for ordered data. These priors allow for data-dependent shrinkage/selection under an intuitive parameterization in an unconstrained setting. The proposed priors are compared to standard methods through a simulation study and illustrated using a multivariate probit data example. Supplemental materials for this article (appendix, data, and R code) are available online.

ON WEIGHTED PORTMANTEAU TESTS FOR TIME‐SERIES GOODNESS‐OF‐FIT

Recent work in the literature has shown weighted variants of the classic portmanteau test for time series can be more powerful in many situations. In this article, we study the asymptotic distribution of weighted sums of the squared residual autocorrelations where both the sample size n and maximum lag of the statistic m grow large. Several weighting schemes are introduced, including a data‐adaptive statistic in which the weights are determined by a function of the sample partial autocorrelations. These statistics can provide more power than other portmanteau tests found in the literature and are much less sensitive to the choice of the maximum correlation lag. The efficacy of the proposed methods is further demonstrated through an analysis of Australian red wine sales.

On Outliers and Interventions in Count Time Series following GLMs

We discuss the analysis of count time series following generalised linear models in the presence of outliers and intervention effects. Different modifications of such models are formulated which allow to incorporate, detect and to a certain degree distinguish extraordinary events (interventions) of different types in count time series retrospectively. An outlook on extensions to the problem of robust parameter estimation, identification of the model orders by robust estimation of autocorrelations and partial autocorrelations, and online surveillance by sequential testing for outlyingness is provided.