Using difference-based methods for inference in nonparametric regression with time series errors

Summary. We show that difference-based methods can be used to construct simple and explicit estimators of error covariance and autoregressive parameters in nonparametric regression with time series errors. When the error process is Gaussian our estimators are efficient, but they are available well beyond the Gaussian case. As an illustration of their usefulness we show that difference-based estimators can be used to produce a simplified version of time series cross-validation. This new approach produces a bandwidth selector that is equivalent, to both first and second orders, to that given by the full time series cross-validation algorithm. Other applications of difference-based methods are to variance estimation and construction of confidence bands in nonparametric regression.     

Comments on “Canonical Correlation in Multivariate Time Series Analysis With an Application to One-Year-Ahead and Multiyear- Ahead Macroeconomic Forecasting,” by P. W. Otter

We examine the orthogonality assumption of seasonal and nonseasonal components for official quarterly unemployment figures in Germany and the United States. Although nonperiodic correlations do not seem to reject the orthogonality assumption, a periodic analysis based on correlation functions that vary with the seasons indicates the violation of orthogonality. We find that the unadjusted data can be described by periodic autoregressive models with a unit root. In simulations we replicate the empirical findings for the German data, where we use these simple models to generate artificial samples. Multiplicative seasonal adjustment leads to large periodic correlations. Additive adjustment leads to smaller ones.     

Ordering and selecting components in multivariate or functional data linear prediction

Summary.  The problem of component choice in regression-based prediction has a long history. The main cases where important choices must be made are functional data analysis, and problems in which the explanatory variables are relatively high dimensional vectors. Indeed, principal component analysis has become the basis for methods for functional linear regression. In this context the number of components can also be interpreted as a smoothing parameter, and so the viewpoint is a little different from that for standard linear regression. However, arguments for and against conventional component choice methods are relevant to both settings and have received significant recent attention. We give a theoretical argument, which is applicable in a wide variety of settings, justifying the conventional approach. Although our result is of minimax type, it is not asymptotic in nature; it holds for each sample size. Motivated by the insight that is gained from this analysis, we give theoretical and numerical justification for cross-validation choice of the number of components that is used for prediction. In particular we show that cross-validation leads to asymptotic minimization of mean summed squared error, in settings which include functional data analysis.     

Extraction of sea temperature in the Barents Sea by a scale space multiresolution method – prospects for Atlantic salmon

Variation of marine temperature at different time scales is a central environmental factor in the life cycle of marine organisms, and may have particular importance for various life stages of anadromous species, for example, Atlantic salmon. To understand the salient features of temperature variation we employ scale space multiresolution analysis, that uses differences of smooths of a time series to decompose it as a sum of scale-dependent components. The number of resolved components can be determined either automatically or by exploring a map that visualizes the structure of the time series. The statistical credibility of the features of the components is established with Bayesian inference. The method was applied to analyze a marine temperature time series measured from the Barents Sea and its correlation with the abundance of Atlantic salmon in three Barents Sea rivers. Besides the annual seasonal variation and a linear trend, the method revealed mid time-scale (～10 years) and long time-scale (～30 years) variation. The 10-year quasi-cyclical component of the temperature time series appears to be connected with a similar feature in Atlantic salmon abundance. These findings can provide information about the environmental factors affecting seasonal and periodic variation in survival and migrations of Atlantic salmon and other migratory fish.     

Nonparametric change point detection for periodic time series

We consider detection of multiple changes in the distribution of periodic and autocorrelated data with known period. To account for periodicity we transform the sequence of vector observations by arranging them in matrices and thereby producing a sequence of independently and identically distributed matrix observations. We propose methods of testing the equality of matrix distributions and present methods that can be applied to matrix observations using the E-divisive algorithm. We show that periodicity and autocorrelation degrade existing change detection methods because they blur the changes that these procedures aim to discover. Methods that ignore the periodicity have low power to detect changes in the mean and the variance of periodic time series when the periodic effects overwhelm the true changes, while the proposed methods detect such changes with high power. We illustrate the proposed methods by detecting changes in the water quality of Lake Kasumigaura in Japan. The Canadian Journal of Statistics 48: 518–534; 2020 © 2020 Statistical Society of Canada     

Multivariate Real-Time Signal Extraction by a Robust Adaptive Regression Filter

We propose a new regression-based filter for extracting signals online from multivariate high frequency time series. It separates relevant signals of several variables from noise and (multivariate) outliers.

Unlike parallel univariate filters, the new procedure takes into account the local covariance structure between the single time series components. It is based on high-breakdown estimates, which makes it robust against (patches of) outliers in one or several of the components as well as against outliers with respect to the multivariate covariance structure. Moreover, the trade-off problem between bias and variance for the optimal choice of the window width is approached by choosing the size of the window adaptively, depending on the current data situation.

Furthermore, we present an advanced algorithm of our filtering procedure that includes the replacement of missing observations in real time. Thus, the new procedure can be applied in online-monitoring practice. Applications to physiological time series from intensive care show the practical effect of the proposed filtering technique.     

A comment on the orthogonalization of B-spline basis functions and their derivatives

Through the use of a matrix representation for B-splines presented by Qin (Vis. Comput. 16:177–186, 2000) we are able to reexamine calculus operations on B-spline basis functions. In this matrix framework the problem associated with generating orthogonal splines is reexamined, and we show that this approach can simplify the operations involved to linear matrix operations. We apply these results to a recent paper (Zhou et al. in Biometrika 95:601–619, 2008) on hierarchical functional data analysis using a principal components approach, where a numerical integration scheme was used to orthogonalize a set of B-spline basis functions. These orthogonalized basis functions, along with their estimated derivatives, are then used to construct estimates of mean functions and functional principal components. By applying the methods presented here such algorithms can benefit from increased speed and precision. An R package is available to do the computations.     

Bayesian analysis of two-component mixture distributions applied to estimating malaria attributable fractions

Malaria illness can be diagnosed by the presence of fever and parasitaemia. However, in highly endemic areas the diagnosis of clinical malaria can be difficult since children may tolerate parasites without fever and may have fever due to other causes. We propose a novel, simulation-based Bayesian approach for obtaining precise estimates of the probabilities of children with different levels of parasitaemia having fever due to malaria, by formulating the problem as a mixture of distributions. The methodology suggested is a general methodology for decomposing any two-component mixture distribution nonparametrically, when an independent training sample is available from one of the components. It is based on the assumption that one of the component distributions lies on the left of the other but there is some overlap between the distributions.     

The Performance of Periodic Autoregressive Models in Forecasting Seasonal U.K. Consumption

The parameters of a periodic model are allowed to vary according to the time at which observations are made. Periodic autoregressive models are fitted to the quarterly values of seasonally unadjusted real nondurable consumers' expenditure for the United Kingdom and its components. The periodic model offers no improvement over conventional specifications if the aggregate is modeled directly. On the other hand, periodic models generally perform well for the components, which contain additional seasonal information. The choice between a periodic or nonperiodic specification is also shown to have an important influence on the resulting dynamic properties.     

Generalizing the information content for stepped wedge designs: A marginal modeling approach

Stepped wedge trials are increasingly adopted because practical constraints necessitate staggered roll-out. While a complete design requires clusters to collect data in all periods, resource and patient-centered considerations may call for an incomplete stepped wedge design to minimize data collection burden. To study incomplete designs, we expand the metric of information content to discrete outcomes. We operate under a marginal model with general link and variance functions, and derive information content expressions when data elements (cells, sequences, periods) are omitted. We show that the centrosymmetric patterns of information content can hold for discrete outcomes with the variance-stabilizing link function. We perform numerical studies under the canonical link function, and find that while the patterns of information content for cells are approximately centrosymmetric for all examined underlying secular trends, the patterns of information content for sequences or periods are more sensitive to the secular trend, and may be far from centrosymmetric.     

Percentile-t Bootstrap Confidence Intervals for Period Using Periodogram

The magnitude of light intensity of many stars varies over time in a periodic way. Therefore, estimation of period and making inference about this parameter are of great interest in astronomy. The periodogram can be used to estimate period, properly. Bootstrap confidence intervals for period suggested here, are based on using the periodogram and constructed by percentile-t methods. We prove that the equal-tailed percentile-t bootstrap confidence intervals for period have an error of order n ^?1. We also show that the symmetric percentile-t bootstrap confidence intervals reduce the error to order n ^?2, and hence have a better performance. Finally, we assess the theoretical results by conducting a simulation study, compare the results with the coverages of percentile bootstrap confidence intervals for period and then analyze a real data set related to the eclipsing system R Canis Majoris collected by Shiraz Biruni Observatory.     

Issues Involved With the Seasonal Adjustment of Economic Time Series

In the first part of this article, we briefly review the history of seasonal adjustment and statistical time series analysis in order to understand why seasonal adjustment methods have evolved into their present form. This review provides insight into some of the problems that must be addressed by seasonal adjustment procedures and points out that advances in modem time series analysis raise the question of whether seasonal adjustment should be performed at all. This in turn leads to a discussion in the second part of issues involved in seasonal adjustment. We state our opinions about the issues raised and review some of the work of other authors. First, we comment on reasons that have been given for doing seasonal adjustment and suggest a new possible justification. We then emphasize the need to define precisely the seasonal and nonseasonal components and offer our definitions. Finally, we discuss criteria for evaluating seasonal adjustments. We contend that proposed criteria based on empirical comparisons of estimated components are of little value and suggest that seasonal adjustment methods should be evaluated based on whether they are consistent with the information in the observed data. This idea is illustrated with an example.     

Issues Involved With the Seasonal Adjustment of Economic Time Series

In the first part of this article, we briefly review the history of seasonal adjustment and statistical time series analysis in order to understand why seasonal adjustment methods have evolved into their present form. This review provides insight into some of the problems that must be addressed by seasonal adjustment procedures and points out that advances in modern time series analysis raise the question of whether seasonal adjustment should be performed at all. This in turn leads to a discussion in the second part of issues invloved in seasonal adjustment. We state our opinions about the issues raised and renew some of the work of our authors. First, we comment on reasons that have been given for doing seasonal adjustment and suggest a new possible justification. We then emphasize the need to define precisely the seasonal and nonseasonal components and offer our definitions. Finally, we discuss our criteria for evaluating seasonal adjustments. We contend that proposed criteria based on empirical comparisons of estimated components are of little value and suggest that seasonal adjustment methods should be evaluated based on whether they are consistent with the information in the observed data. This idea is illustrated with an example.     

Negative binomial time series models based on expectation thinning operators

The study of count data time series has been active in the past decade, mainly in theory and model construction. There are different ways to construct time series models with a geometric autocorrelation function, and a given univariate margin such as negative binomial. In this paper, we investigate negative binomial time series models based on the binomial thinning and two other expectation thinning operators, and show how they differ in conditional variance or heteroscedasticity. Since the model construction is in terms of probability generating functions, typically, the relevant conditional probability mass functions do not have explicit forms. In order to do simulations, likelihood inference, graphical diagnostics and prediction, we use a numerical method for inversion of characteristic functions. We illustrate the numerical methods and compare the various negative binomial time series models for a real data example.     

An analysis of pulsation periods of long period variable stars

Summary.  We report the results of a period change analysis of time series observations for 378 pulsating variable stars. The null hypothesis of no trend in expected periods is tested for each of the stars. The tests are non-parametric in that potential trends are estimated by local linear smoothers. Our testing methodology has some novel features. First, the null distribution of a test statistic is defined to be the distribution that results in repeated sampling from a population of stars. This distribution is estimated by means of a bootstrap algorithm that resamples from the collection of 378 stars. Bootstrapping in this way obviates the problem that the conditional sampling distribution of a statistic, given a particular star, may depend on unknown parameters of that star. Another novel feature of our test statistics is that one-sided cross-validation is used to choose the smoothing parameters of the local linear estimators on which they are based. It is shown that doing so results in tests that are tremendously more powerful than analogous tests that are based on the usual version of cross-validation. The positive false discovery rate method of Storey is used to account for the fact that we simultaneously test 378 hypotheses. We ultimately find that 56 of the 378 stars have changes in mean pulsation period that are significant when controlling the positive false discovery rate at the 5% level.     

Copula Density Estimation Using Multiwavelets Based on the Multiresolution Analysis

Recently, wavelet has been used for copula density estimation. A known characteristic of wavelet functions is that they cannot be symmetric, orthogonal, and compact support at the same time while multiwavelets overcome this disadvantage. This article highlights the usefulness of the multiwavelet in order to approximate copula density functions. Possessing three appropriate properties at the same time, high smoothness, and high approximation order properties, multiwavelet can be more precise in copula density approximation. We make this approximation method more accurate by using multiresolution analysis. Finally, we apply our proposed method to approximate the copula density in actuarial data.     

Fuzzy Autoregressive Rules: Towards Linguistic Time Series Modeling

Fuzzy rule–based models, a key element in soft computing (SC), have arisen as an alternative for time series analysis and modeling. One difference with preexisting models is their interpretability in terms of human language. Their interactions with other components have also contributed to a huge development in their identification and estimation procedures. In this article, we present fuzzy rule–based models, their links with some regime-switching autoregressive models, and how the use of soft computing concepts can help the practitioner to solve and gain a deeper insight into a given problem. An example on a realized volatility series is presented to show the forecasting abilities of a fuzzy rule–based model.     

Market-Based Credit Ratings

We present a methodology for rating in real-time the creditworthiness of public companies in the U.S. from the prices of traded assets. Our approach uses asset pricing data to impute a term structure of risk neutral survival functions or default probabilities. Firms are then clustered into ratings categories based on their survival functions using a functional clustering algorithm. This allows all public firms whose assets are traded to be directly rated by market participants. For firms whose assets are not traded, we show how they can be indirectly rated by matching them to firms that are traded based on observable characteristics. We also show how the resulting ratings can be used to construct loss distributions for portfolios of bonds. Finally, we compare our ratings to Standard & Poors and find that, over the period 2005 to 2011, our ratings lead theirs for firms that ultimately default.     

Estimation and Variable Selection for Semiparametric Additive Partial Linear Models (SS-09-140)

Semiparametric additive partial linear models, containing both linear and nonlinear additive components, are more flexible compared to linear models, and they are more efficient compared to general nonparametric regression models because they reduce the problem known as "curse of dimensionality". In this paper, we propose a new estimation approach for these models, in which we use polynomial splines to approximate the additive nonparametric components and we derive the asymptotic normality for the resulting estimators of the parameters. We also develop a variable selection procedure to identify significant linear components using the smoothly clipped absolute deviation penalty (SCAD), and we show that the SCAD-based estimators of non-zero linear components have an oracle property. Simulations are performed to examine the performance of our approach as compared to several other variable selection methods such as the Bayesian Information Criterion and Least Absolute Shrinkage and Selection Operator (LASSO). The proposed approach is also applied to real data from a nutritional epidemiology study, in which we explore the relationship between plasma beta-carotene levels and personal characteristics (e.g., age, gender, body mass index (BMI), etc.) as well as dietary factors (e.g., alcohol consumption, smoking status, intake of cholesterol, etc.).     

Wavelet designs for estimating nonparametric curves with heteroscedastic error

In this paper, we discuss the problem of constructing designs in order to maximize the accuracy of nonparametric curve estimation in the possible presence of heteroscedastic errors. Our approach is to exploit the flexibility of wavelet approximations to approximate the unknown response curve by its wavelet expansion thereby eliminating the mathematical difficulty associated with the unknown structure. It is expected that only finitely many parameters in the resulting wavelet response can be estimated by weighted least squares. The bias arising from this, compounds the natural variation of the estimates. Robust minimax designs and weights are then constructed to minimize mean-squared-error-based loss functions of the estimates. We find the periodic and symmetric properties of the Euclidean norm of the multiwavelet system useful in eliminating some of the mathematical difficulties involved. These properties lead us to restrict the search for robust minimax designs to a specific class of symmetric designs. We also construct minimum variance unbiased designs and weights which minimize the loss functions subject to a side condition of unbiasedness. We discuss an example from the nonparametric literature.