Standard Time Series Methods Research Articles

Functional data analysis: An application to COVID‐19 data in the United States in 2020

BackgroundIn this paper, we conduct an analysis of the COVID‐19 data in the United States in 2020 via functional data analysis methods. Through this research, we investigate the effectiveness of the practice of public health measures, and assess the correlation between infections and deaths caused by the COVID‐19. Additionally, we look into the relationship between COVID‐19 spread and geographical locations, and propose a forecasting method to predict the total number of confirmed cases nationwide.MethodsThe functional data analysis methods include functional principal analysis methods, functional canonical correlation analysis methods, an expectation‐maximization (EM) based clustering algorithm and a functional time series model used for forecasting.ResultsIt is evident that the practice of public health measures helps to reduce the growth rate of the epidemic outbreak over the nation. We have observed a high canonical correlation between confirmed and death cases. States that are geographically close to the hot spots are likely to be clustered together, and population density appears to be a critical factor affecting the cluster structure. The proposed functional time series model gives more reliable and accurate predictions of the total number of confirmed cases than standard time series methods.ConclusionsThe results obtained by applying the functional data analysis methods provide new insights into the COVID‐19 data in the United States. With our results and recommendations, the health professionals can make better decisions to reduce the spread of the epidemic, and mitigate its negative effects to the national public health.

Non-Negative Time Series Reconstruction via Singular Spectrum Analysis: A Case Study of Precipitation Dynamics in Nigeria

A proper understanding and analysis of the processes involved in seasonal precipitation variability and dynamics is essential to provide reliable information about climate change and how it can affect matters of critical importance such as water availability and agricultural productivity in urban cities. Precipitation data, as many other time series data present only non-negative observations, are is not constrained by standard time series methods. In this paper, we propose a modified singular spectrum analysis (SSA) algorithm for decomposition and reconstruction of time series with non-negative values. Our approach uses a non-negative matrix factorization (NMF) instead of the singular value decomposition in the SSA algorithm. The new algorithm is compared with the classic SSA algorithm by considering a simulation study and observed data of monthly precipitation of four major cities in Nigeria (Lagos, Kano, Ibadan and Kaduna). Although in terms of mean stared errors both methods give similar results, the percentage of negative fitted values for reconstructions with the classical SSA algorithm reached more than [Formula: see text] in our real data application, which is inappropriate for non-negative time series. The proposed adaptation of the SSA algorithm for non-negative time series data provides an important development with applications in many fields where time series data has non-negative constraints.

The impact of delayed transfers of care on emergency departments: common sense arguments, evidence and confounding

ObjectivesThere have been claims that Delayed Transfers of Care (DTOCs) of inpatients to home or a less acute setting are related to Emergency Department (ED) crowding. In particular DTOCs were...

Discriminant Analysis for the Eigenvalues of Variance Covariance Matrix of FFT Scaling of DNA Sequences: An Empirical Study of Some Organisms

Many studies discussed different numerical representations of DNA sequences. One naive approach for exploring the nature of a DNA sequence is to assign numerical values (or scales) to the nucleotides and then proceed with standard time series methods. The analysis will depend actually on the particular assignment of numerical values.Discriminant analysis aims to examine the dependence of one qualitative (classification) variable from several quantitative variables according to number of variations of qualitative variable we can distinction. Actually, there is a discriminant analysis for two or more groups. The essential work of discriminant analysis is to get the optimal assigning rules that will minimize the likelihood of incorrect classification of elements. In this paper, we discussed the discriminant analysis of the first, second, third and fourth eigenvalues of variance covariance matrix of Fast Fourier Transform (FFT) for numerical values representation of DNA sequences of five organisms, Human, E. coli, Rat, Wheat and Grasshopper. The analysis is based on three methods (All Variables, Forward Selection and Backward Selection) of discrimination. Functions have been reached whereby discrimination is made among organisms under consideration. Empirical studies are conducted to show the value of our point of view and the applications based on. Therefore, we recommended that, other empirical studies should be done for other organisms and statistical methods by using the point of view adopted here. Also, aspects stated here must be used in an applied manner for DNA sequences discrimination.

Judicial Indicators and Business Cycles in Brazil

This paper presents an innovative data set of judicial indicators in Brazil, based on the contents of court digital diaries (Diario da Justica Eletronico) for the state of Sao Paulo. These daily, well-structured text records allow computing aggregate, highfrequency court data for new cases, resolved cases and case-activity in general, as well as case type-level data, for a broad range of legal subjects including eviction, foreclosures, debt instruments, bankruptcy, dissolution of partnership, tax, indemnities, alimony, among others. The dataset also includes business starts, extracted from the commercial registry diary. Text records also allow monitoring litigation activity of the financial sector, which is related to credit market conditions. Ease of computation and timeliness makes this dataset attractive as potential leading or coincident indicators of economic activity and therefore, suitable for nowcasting. The first part of the paper presents the methodology of this novel panel dataset. The second part presents, based on standard time-series methods, an assessment of the predictive power of proposed judicial indicators in anticipating the path of a vector of business cycle variables. Results suggest that the novel set of judicial indicators improves forecasting in the short-run up to three months, relatively to benchmark models as well as relatively to “market” forecasts of the “present”. Relationships between judicial and economic indicators are discussed, particularly in the case of the credit market-related indicators. Results are in line with Fama and French (1989), Stock and Watson (1989), Chen (1991) and Gilchrist and Zakrajˇsek (2011), which suggest that delinquency rates and market risk premium are important predictors of business conditions.

Quantifying uncertainty in predictions of groundwater levels using formal likelihood methods

Informal and formal likelihood methods can be used to quantify uncertainty in modelled predictions of groundwater levels (GWLs). Informal methods use a relatively subjective criterion to identify sets of plausible or behavioural parameters of the GWL models. In contrast, formal methods specify a statistical model for the residuals or errors of the GWL model. The formal uncertainty estimates are only reliable when the assumptions of the statistical model are appropriate.We apply the formal approach to historical reconstructions of GWL hydrographs from four UK boreholes. We test whether a model which assumes Gaussian and independent errors is sufficient to represent the residuals or whether a model which includes temporal autocorrelation and a general non-Gaussian distribution is required. Groundwater level hydrographs are often observed at irregular time intervals so we use geostatistical methods to quantify the temporal autocorrelation rather than more standard time series methods such as autoregressive models.According to the Akaike Information Criterion, the more general statistical model better represents the residuals of the GWL model. However, no substantial difference between the accuracy of the GWL predictions and the estimates of their uncertainty is observed when the two statistical models are compared. When the general model is applied, significant temporal correlation over periods ranging from 3 to 20months is evident for the different boreholes. When the GWL model parameters are sampled using a Markov Chain Monte Carlo approach the distributions based on the general statistical model differ from those of the Gaussian model, particularly for the boreholes with the most autocorrelation. These results suggest that the independent Gaussian model of residuals is sufficient to estimate the uncertainty of a GWL prediction on a single date. However, if realistically autocorrelated simulations of GWL hydrographs for multiple dates are required or if the distributions of the GWL model parameters are of interest, then the more general statistical model should be used.

Robust Statistical Error Estimation of Local Power Tallies in Monte Carlo Calculation of Light Water Reactor

The overlapping batch means method (OBM) has been investigated for robust statistical error estimation of local power tallies in Monte Carlo (MC) reactor core calculation. Originally, a nonoverlapping version was introduced in MC criticality calculation by Gelbard and Prael. However, the issue of batch size optimization was thought of as a lack of robustness. In this work, OBM with asymptotic bias correction was implemented with the batch size of the square root of the number of generations and compared with the orthonormally weighted standardized time series method (OWSTS). Numerical tests were conducted for various positions of the core of a pressurized water reactor. Results obtained indicate that neither OBM nor OWSTS consistently outperforms the other in terms of an overall performance measure incorporating bias and stability. Therefore, OBM with asymptotic bias correction can be an option to statistical error estimation in production MC criticality codes since OWSTS lacks an automated process to determine the number of weighting functions and can output the estimate only at the final generation. It is also shown that OBM with asymptotic bias correction performs equally regardless of the batch size.

Tourism specialization and economic output in small islands

Purpose – This study aims to investigate growth differentials among small islands and the impact of tourism specialization on the growth and the economic performance of small islands. Design/methodology/approach – The study is based on trade theory and uses data from a panel of small islands for 1995-2007. It applies panel regression and standard time series methods combined with a qualitative approach. Findings – Small islands experienced stronger basic patterns of growth than many developed countries, especially where economies of scale are not an issue. The findings further suggest that tourism specialization is not harmful to growth, and, in lieu of technological gaps and resource limitations, tourism specialization is a sound option. Size, a lack of complete sovereignty or independence and export orientation do not seem to affect the variance in the real per capita GDP at a greater degree. Finally, small islands may leverage returns to scale in global markets. Research limitations/implications – While tourism specialization is assumed to enhance growth, in the case of small island destinations, the study did not formally test whether increased terms-of-trade may be perpetually improved. Practical implications – The study prompted four policy suggestions: small island economies should engage in tourism specialization; small island economies should allocate more resources to the tourism industry than other economic sectors; the success of tourism specialization does not depend exclusively on comparative advantage; and institutional realities and path dependence may play a role in economic performance. Originality/value – The originality of this study lies in the detection of a paradox in mainstream economics that indicates that small islands may not enjoy sustained economic growth. The detection led to a surprising discovery that tourism specialization may propel growth. The value of the study is twofold: theoretical value is added by suggesting a reconceptualization of the construct capital; and, practical value is strengthened in the sense that tourism specialization may only work under a condition where upon tourism offerings command higher prices than other commodities.

Validation and forecasting accuracy in models of climate change

Abstract Forecasting researchers, with few exceptions, have ignored the current major forecasting controversy: global warming and the role of climate modelling in resolving this challenging topic. In this paper, we take a forecaster’s perspective in reviewing established principles for validating the atmospheric-ocean general circulation models (AOGCMs) used in most climate forecasting, and in particular by the Intergovernmental Panel on Climate Change (IPCC). Such models should reproduce the behaviours characterising key model outputs, such as global and regional temperature changes. We develop various time series models and compare them with forecasts based on one well-established AOGCM from the UK Hadley Centre. Time series models perform strongly, and structural deficiencies in the AOGCM forecasts are identified using encompassing tests. Regional forecasts from various GCMs had even more deficiencies. We conclude that combining standard time series methods with the structure of AOGCMs may result in a higher forecasting accuracy. The methodology described here has implications for improving AOGCMs and for the effectiveness of environmental control policies which are focussed on carbon dioxide emissions alone. Critically, the forecast accuracy in decadal prediction has important consequences for environmental planning, so its improvement through this multiple modelling approach should be a priority.

USING EXCURSIONS TO ANALYZE SIMULATION OUTPUT

We consider the steady-state simulation output analysis problem for a process that satisfies a functional central limit theorem. We construct an estimator for the time-average variance constant that is based on excursions of a process above the minimum. The resulting estimator does not require a fixed run length, and the memory requirement can be dynamically bounded. Standardized time series methods based on excursions are also described.

Overlapping Variance Estimators for Simulation

To estimate the variance parameter (i.e., the sum of covariances at all lags) for a steady-state simulation output process, we formulate certain statistics that are computed from overlapping batches separately and then averaged over all such batches. We form overlapping versions of the area and Cramér–von Mises estimators using the method of standardized time series. For these estimators, we establish (i) their limiting distributions as the sample size increases while the ratio of the sample size to the batch size remains fixed; and (ii) their mean-square convergence to the variance parameter as both the batch size and the ratio of the sample size to the batch size increase. Compared with their counterparts computed from nonoverlapping batches, the estimators computed from overlapping batches asymptotically achieve reduced variance while maintaining the same bias as the sample size increases; moreover, the new variance estimators usually achieve similar improvements compared with the conventional variance estimators based on nonoverlapping or overlapping batch means. In follow-up work, we present several analytical and Monte Carlo examples, and we formulate efficient procedures for computing the overlapping estimators with only order-of-sample-size effort.

Efficient Computation of Overlapping Variance Estimators for Simulation

For a steady-state simulation output process, we formulate efficient algorithms to compute certain estimators of the process variance parameter (i.e., the sum of covariances at all lags), where the estimators are derived in principle from overlapping batches separately and then averaged over all such batches. The algorithms require order-of-sample-size work to evaluate overlapping versions of the area and Cramér–von Mises estimators arising in the method of standardized time series. Recently, Alexopoulos et al. showed that, compared with estimators based on nonoverlapping batches, the estimators based on overlapping batches achieve reduced variance while maintaining similar bias asymptotically as the batch size increases. We provide illustrative analytical and Monte Carlo results for M/M/1 queue waiting times and for a first-order autoregressive process. We also present evidence that the asymptotic distribution of each overlapping variance estimator can be closely approximated using an appropriately rescaled chi-squared random variable with matching mean and variance.

Permuted Standardized Time Series for Steady-State Simulations

We describe an extension procedure for constructing new standardized time series procedures from existing ones. The approach is based on averaging over sample paths obtained by permuting path segments. Analytical and empirical results indicate that permuting improves standardized time series methods. We compare permuting to an alternative extension procedure known as batching. We demonstrate the permuting method by applying it to estimators based on the maximum and the area of a normalized path.

Impact of Federal drug law enforcement on the supply of heroin in Australia

To conduct an empirical investigation of the efficacy of law enforcement in reducing heroin supply in Australia. Specifically, this paper addresses the question of whether heroin purity levels in the Australian Capital Territory (ACT) could be predicted by heroin seizures at the national level by the Australian Federal Police (AFP) in the preceding year. We considered two forms of evidence. First, a Bayesian Markov Chain Monte Carlo (MCMC) change-point model was used to discover (a) if there was a substantial increase in heroin seizures by the AFP, (b) when the increase began and (c) whether it occurred after increased funding to the Australian Federal Police for the purpose of drug law enforcement. Second, standard time-series methods were used to ascertain whether fluctuations in heroin seizure weights or the frequency of large-scale seizures after the aforementioned changes in seizure levels predicted fluctuations in heroin purity levels in the ACT after autocorrelation had been removed from the purity series. A Bayesian MCMC change-point model supported the hypothesis that heroin seizures rapidly increased about a year before the estimated decline in heroin purity and after the increased funding of AFP. The autoregression models suggested that 10-20% of the variance in the residuals of the heroin purity series was predicted by appropriately lagged residuals of the seizure-number and log-weight series, after autocorrelation had been removed. The overall results are consistent with the hypothesis that large-scale heroin seizures by the AFP reduce street-level heroin supply a year or so later, although the short-term dynamics suggest an 'opponent' response to residual fluctuations in seizures. To our knowledge, this is first time a connection has been identified between large-scale heroin seizures and street-level supply.

Time Series Modelling of Daily Tax Revenues

We provide a detailed discussion of time series modelling of daily data in general and daily tax revenues in particular. The main feature of the daily tax revenue series is the pattern within calendar months. Standard time series methods for seasonal adjustment and forecasting cannot be used since the number of banking days per calendar month varies and because there are two levels of seasonality: between months and within months. We propose a daily time series model based on unobserved components that allows for the classic decomposition into trend, seasonal plus irregular, but it also includes components for intra‐monthly, trading‐day and length‐of‐month effects. Such components typically rely on stochastic cubic spline, polynomial and dummy variable functions. State space techniques are used for the recursive computation of the likelihood and forecasts functions with special allowance for irregular spacing. The model is operational for daily forecasting at the Dutch Ministry of Finance. We present the model specification and discuss estimation and forecasting results up to December 1999. A comparative forecast evaluation is also presented.

Two-stage multiple-comparison procedures for steady-state simulations

Procedures for multiple comparisons with the best are investigated in the context of steady-state simulation, whereby a number k of different systems (stochastic processes) are compared based upon their (asymptotic) means μ i ( i = 1,2,…, k ). The variances of these (asymptotically stationary) processes are assumed to be unknown and possibly unequal. We consider the problem of constructing simultaneous confidence intervals for μ i -max j≠i μ j ( i=1,2,…,k) which is known as multiple comparisons with the best (MCB). Our intervals are constrained to contain 0, and so are called constrained MCB intervals. In particular, two-stage procedures for construction of absolute- and relative-width confidence intervals are presented. Their validity is addressed by showing that the confidence intervals cover the parameters with probability of at least some user-specified threshold value, as the confidence intervals' width parameter shrinks to 0. The general assumption about the processes is that they satisfy a functional central limit theorem. The simulation output analysis procedures are based on the method of standardized time series (the batch means method is a special case). The techniques developed here extend to other multiple-comparison procedures such as unconstrained MCB, multiple comparisons with a control, and all-pairwise comparisons. Although simulation is the context in this paper, the results naturally apply to (asymptotically) stationary time series.

Multiple-comparison procedures for steady-state simulations

Suppose that there are $k \geq 2$ different systems (i.e., stochastic processes), where each system has an unknown steady-state mean performance and unknown asymptotic variance. We allow for the asymptotic variances to be unequal and for the distributions of the k systems to be different. We consider the problem of running independent, single-stage simulations to make multiple comparisons of the steady-state means of the different systems. We derive asymptotically valid (as the run lengths of the simulations of the systems tend to infinity) simultaneous confidence intervals for each of the following problems: all pairwise comparisons of means, all contrasts, multiple comparisons with a control and multiple comparisons with the best. Our confidence intervals are based on standardized time series methods, and we establish the asymptotic validity of each under the sole assumption that the stochastic processes representing the simulation output of the different systems satisfy a functional central limit theorem. Although simulation is the context of this paper, the results naturally apply to (asymptotically) stationary time series.

Two-Stage Stopping Procedures Based on Standardized Time Series

We propose some new two-stage stopping procedures to construct absolute-width and relative-width confidence intervals for a simulation estimator of the steady-state mean of a stochastic process. The procedures are based on the method of standardized time series proposed by Schruben and on Stein's two-stage sampling scheme. We prove that our two-stage procedures give rise to asymptotically valid confidence intervals (as the prescribed length of the confidence interval approaches zero and the size of the first stage grows to infinity). The sole assumption required is that the stochastic process satisfy a functional central limit theorem.

Notes: Conditions for the Applicability of the Regenerative Method

The regenerative method for estimating steady-state parameters is one of the basic methods in simulation output analysis. This method depends on central limit theorems for regenerative processes and weakly consistent estimates for the variance constants arising in the central limit theorems. A weak sufficient condition for both the central limit theorems and consistent estimates is given. Previous authors have implicitly made stronger moment assumptions which have led to strongly consistent variance estimates, more than is needed for the regenerative method to hold. The relationship between conditions for the validity of the regenerative method and those for the validity of standardized time series methods is also discussed.

Simulation Output Analysis Using Standardized Time Series

The method of standardized time series (STS) was proposed by Schruben as an approach for constructing asymptotic confidence intervals for the steady-state mean from a single simulation run. The STS method “cancels out” the variance constant while other methods attempt to consistently estimate the variance constant. Our goal in this paper is to generalize the STS method and to study some of its basic properties. Starting from a functional central limit theorem (FCLT) for the sample mean of the simulated process, a class of mappings of C[0,1] to ℝ is identified, each of which leads to a STS confidence interval. One of these mappings leads to the batch means method. A lower bound is obtained for the expected length of the asymptotic (as the run size becomes large) STS confidence intervals. This lower bound is not attained, but can be approached arbitrarily closely, by STS confidence intervals. Methods that consistently estimate the variance constant do realize this lower bound. The variance of the length of a STS confidence interval is of larger order (in the run length) than is that for the regenerative method.