Randomized singular spectrum analysis for long time series

ABSTRACT

Singular spectrum analysis (SSA) is a relatively new method for time series analysis and comes as a non-parametric alternative to the classical methods. This methodology has proven to be effective in analysing non-stationary and complex time series since it is a non-parametric method and do not require the classical assumptions over the stationarity or over the normality of the residuals. Although SSA have proved to provide advantages over traditional methods, the challenges that arise when long time series are considered, make the standard SSA very demanding computationally and often not suitable. In this paper we propose the randomized SSA which is an alternative to SSA for long time series without losing the quality of the analysis. The SSA and the randomized SSA are compared in terms of quality of the model fit and forecasting, and computational time. This is done by using Monte Carlo simulations and real data about the daily prices of five of the major world commodities.     

Common singular spectrum analysis of several time series

Singular spectrum analysis (SSA) is a non-parametric time series modelling technique where an observed time series is unfolded into the column vectors of a Hankel structured matrix, known as a trajectory matrix. For noise-free signals the column vectors of the trajectory matrix lie on a single R-flat. Singular value decomposition (SVD) can be used to find the orthonormal base vectors of the linear subspace parallel to this R-flat. SSA can essentially handle functions that are governed by a linear recurrent formula (LRF) and include the broad class of functions that was proposed by Buchstaber [1994. Time series analysis and Grassmannians. Amer. Math. Soc. Transl. 162 (2), 1–17]. SSA is useful to model time series with complex cyclical patterns that increase over time.Various methods have been studied to extend SSA for application to several time series, see Golyandina et al. [2003. Variants of the Caterpillar SSA-method for analysis of multidimensional time series (in Russian) hhttp://www.gistatgroup.com/cat/i]. Prior to that Von Storch and Zwiers (1999) and Allen and Robertson (1996) (see Ghil et al. [2002. Advanced spectral methods for climatic time series. Rev. Geophys. 40 (1), 3.1–3.41]) used multi-channel SSA (M-SSA), to apply SSA to “grand” block matrices. Our approach is different from all of these by using the common principal components approaches introduced by Flury [1988. Common Principal Components and Related Multivariate Models. Wiley, New York]. In this paper SSA is extended to several time series which are similar in some respects, like cointegrated, i.e. sharing a common R-flat. By using the common principal component (CPC) approach of Flury [1988. Common Principal Components and Related Multivariate Models. Wiley, New York] the SSA method is extended to common singular spectrum analysis (CSSA) where common features of several time series can be studied. CSSA decomposes the different original time series into the sum of a common small number of components which are related to common trend and oscillatory components and noise. The determination of the most likely dimension of the supporting linear subspace is studied using a heuristic approach and a hierarchical selection procedure.     

Understanding exponential smoothing via kernel regression

Exponential smoothing is the most common model-free means of forecasting a future realization of a time series. It requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one-step-ahead forecasts. In this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation. We also use theoretical developments from the kernel regression field to derive, for the first time, asymptotic properties of exponential smoothing forecasters.     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

Bayesian analysis of semiparametric Bernstein polynomial regression models for data with sample selection

In regression analysis, a sample selection scheme often applies to the response variable, which results in missing not at random observations on the variable. In this case, a regression analysis using only the selected cases would lead to biased results. This paper proposes a Bayesian methodology to correct this bias based on a semiparametric Bernstein polynomial regression model that incorporates the sample selection scheme into a stochastic monotone trend constraint, variable selection, and robustness against departures from the normality assumption. We present the basic theoretical properties of the proposed model that include its stochastic representation, sample selection bias quantification, and hierarchical model specification to deal with the stochastic monotone trend constraint in the nonparametric component, simple bias corrected estimation, and variable selection for the linear components. We then develop computationally feasible Markov chain Monte Carlo methods for semiparametric Bernstein polynomial functions with stochastically constrained parameter estimation and variable selection procedures. We demonstrate the finite-sample performance of the proposed model compared to existing methods using simulation studies and illustrate its use based on two real data applications.     

Forecasting before,during, and after recession with singular spectrum analysis

The aim of this research is to apply the singular spectrum analysis (SSA) technique, which is a relatively new and powerful technique in time series analysis and forecasting, to forecast the 2008 UK recession, using eight economic time series. These time series were selected as they represent the most important economic indicators in the UK. The ability to understand the underlying structure of these series and to quickly identify turning points such as the on-set of the recent recession is of key interest to users. In recent years, the SSA technique has been further developed and applied to many practical problems. Hence, these series will provide an ideal practical test of the potential benefits from SSA during one of the most challenging periods for econometric analyses of recent years. The results are compared with those obtained using the ARIMA and Holt–Winters models as these methods are currently used as standard forecasting methods in the Office for National Statistics in the UK.     

Various Approaches for Predicting Land Cover in Mountain Areas

Using former maps, geographers intend to study the evolution of the land cover in order to have a prospective approach on the future landscape; predictions of the future land cover, by the use of older maps and environmental variables, are usually done through the GIS (Geographic Information System). We propose here to confront this classical geographical approach with statistical approaches: a linear parametric model (polychotomous regression modeling) and a nonparametric one (multilayer perceptron). These methodologies have been tested on two real areas on which the land cover is known at various dates; this allows us to emphasize the benefit of these two statistical approaches compared to GIS and to discuss the way GIS could be improved by the use of statistical models.     

Estimating nonlinear additive models with nonstationarities and correlated errors

In this paper, we study a nonparametric additive regression model suitable for a wide range of time series applications. Our model includes a periodic component, a deterministic time trend, various component functions of stochastic explanatory variables, and an AR(p) error process that accounts for serial correlation in the regression error. We propose an estimation procedure for the nonparametric component functions and the parameters of the error process based on smooth backfitting and quasimaximum likelihood methods. Our theory establishes convergence rates and the asymptotic normality of our estimators. Moreover, we are able to derive an oracle‐type result for the estimators of the AR parameters: Under fairly mild conditions, the limiting distribution of our parameter estimators is the same as when the nonparametric component functions are known. Finally, we illustrate our estimation procedure by applying it to a sample of climate and ozone data collected on the Antarctic Peninsula.     

Bayesian Analysis of Latent Threshold Dynamic Models

We discuss a general approach to dynamic sparsity modeling in multivariate time series analysis. Time-varying parameters are linked to latent processes that are thresholded to induce zero values adaptively, providing natural mechanisms for dynamic variable inclusion/selection. We discuss Bayesian model specification, analysis and prediction in dynamic regressions, time-varying vector autoregressions, and multivariate volatility models using latent thresholding. Application to a topical macroeconomic time series problem illustrates some of the benefits of the approach in terms of statistical and economic interpretations as well as improved predictions. Supplementary materials for this article are available online.     

A comparison of stepwise common singular spectrum analysis and horizontal multi-channel singular spectrum analysis

The problem of forecasting a time series by using information provided by a second time series is considered. Two multivariate extensions of Singular Spectrum Analysis (SSA) are compared in terms of forecast error: Horizontal Multi-channel SSA and Stepwise Common SSA. Different signal structures, defined in terms of trend, period, amplitude and phase, are investigated. In broad terms we find that neither Horizontal Multichannel SSA nor Stepwise Common SSA is best in all cases. Horizontal MSSA is outperformed particularly in cases where different trends are considered.     

Dimension reduction for the conditional mean and variance functions in time series

This paper deals with the nonparametric estimation of the mean and variance functions of univariate time series data. We propose a nonparametric dimension reduction technique for both mean and variance functions of time series. This method does not require any model specification and instead we seek directions in both the mean and variance functions such that the conditional distribution of the current observation given the vector of past observations is the same as that of the current observation given a few linear combinations of the past observations without loss of inferential information. The directions of the mean and variance functions are estimated by maximizing the Kullback–Leibler distance function. The consistency of the proposed estimators is established. A computational procedure is introduced to detect lags of the conditional mean and variance functions in practice. Numerical examples and simulation studies are performed to illustrate and evaluate the performance of the proposed estimators.     

A semiparametric approach for modelling multivariate nonlinear time series

In this article, a semiparametric time‐varying nonlinear vector autoregressive (NVAR) model is proposed to model nonlinear vector time series data. We consider a combination of parametric and nonparametric estimation approaches to estimate the NVAR function for both independent and dependent errors. We use the multivariate Taylor series expansion of the link function up to the second order which has a parametric framework as a representation of the nonlinear vector regression function. After the unknown parameters are estimated by the maximum likelihood estimation procedure, the obtained NVAR function is adjusted by a nonparametric diagonal matrix, where the proposed adjusted matrix is estimated by the nonparametric kernel estimator. The asymptotic consistency properties of the proposed estimators are established. Simulation studies are conducted to evaluate the performance of the proposed semiparametric method. A real data example on short‐run interest rates and long‐run interest rates of United States Treasury securities is analyzed to demonstrate the application of the proposed approach. The Canadian Journal of Statistics 47: 668–687; 2019 © 2019 Statistical Society of Canada     

On the Discretization of Continuous-Time Filters for Nonstationary Stock and Flow Time Series

This article discusses the discretization of continuous-time filters for application to discrete time series sampled at any fixed frequency. In this approach, the filter is first set up directly in continuous-time; since the filter is expressed over a continuous range of lags, we also refer to them as continuous-lag filters. The second step is to discretize the filter itself. This approach applies to different problems in signal extraction, including trend or business cycle analysis, and the method allows for coherent design of discrete filters for observed data sampled as a stock or a flow, for nonstationary data with stochastic trend, and for different sampling frequencies. We derive explicit formulas for the mean squared error (MSE) optimal discretization filters. We also discuss the problem of optimal interpolation for nonstationary processes – namely, how to estimate the values of a process and its components at arbitrary times in-between the sampling times. A number of illustrations of discrete filter coefficient calculations are provided, including the local level model (LLM) trend filter, the smooth trend model (STM) trend filter, and the Band Pass (BP) filter. The essential methodology can be applied to other kinds of trend extraction problems. Finally, we provide an extended demonstration of the method on CPI flow data measured at monthly and annual sampling frequencies.     

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.     

Extreme quantile estimation based on financial time series

Estimation of market risk is an important problem in finance. Two well-known risk measures, viz., value at risk and median shortfall, turn out to be extreme quantiles of the marginal distribution of asset return. Time series on asset returns are known to exhibit certain stylized facts, such as heavy tails, skewness, volatility clustering, etc. Therefore, estimation of extreme quantiles in the presence of such features in the data seems to be of natural interest. It is difficult to capture most of these stylized facts using one specific time series model. This motivates nonparametric and extreme value theory-based estimation of extreme quantiles that do not require exact specification of the asset return model. We review these quantile estimators and compare their known properties. Their finite sample performance are compared using Monte Carlo simulation. We propose a new estimator that exhibits encouraging finite sample performance while estimating extreme quantile in the right tail region.     

检验式设定错误对时间序列单位根检验小样本性质的影响

首先对单位根检验的两类常见的数据生成系统进行比较，然后利用蒙特卡洛实验研究了时间序列单位根检验式的设定问题。研究发现在利用DF检验和DF-GLS检验进行时间序列的单位根检验时，检验式设定错误直接影响着检验结果，尤其在推断时间序列是趋势平稳过程还是有时间趋势项的随机游走过程或有二阶时间趋势多项式的随机游走过程时，检验式的错误设定很容易将趋势平稳过程误判为非平稳过程。     

Bayesian nonparametric survival analysis using mixture of Burr XII distributions

ABSTRACT

Recently, the Bayesian nonparametric approaches in survival studies attract much more attentions. Because of multimodality in survival data, the mixture models are very common. We introduce a Bayesian nonparametric mixture model with Burr distribution (Burr type XII) as the kernel. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and real failure time datasets, we show the preference of this model and compare it with Dirichlet process mixture models with different kernels. The Markov chain Monte Carlo (MCMC) simulation methods to calculate the posterior distribution are used.     

随机效应半参数logit模型的惩罚似然估计研究

 在颇具争议的收入差距和健康关系研究中,为了降低可能存在的模型设定和遗漏变量偏误,本文提出了随机效应半参数logit模型,其中非参数的设定还可用于数据的初探性分析。随后本文提出了模型非参数和参数部分的估计方法。这里涉及的难点是随机效应的存在导致似然函数中的积分没有解析式,而非参数的存在更加大了估计难度。本文基于惩罚样条非参数估计方法和四阶Laplace近似方法建立了惩罚对数似然函数,其最大化采用了Newton_Raphson近似方法。文章还建立了惩罚样条中重要光滑参数的选取准则。模型在收入差距和健康实例中的估计结果表明数据支持收入差距弱假说,且非参数估计结果表明其具有U型形式,与实例估计结果的比较指出本文提出的估计方法是较准确的。     

On Robustness of Model-Based Bootstrap Schemes in Nonparametric Time Series Analysis

Theory in time series analysis is often developed under the assumption of finite-dimensional models for the data generating process. Whereas corresponding estimators such as those of a conditional mean function are reasonable even if the true dependence mechanism is more complex, it is usually necessary to capture the whole dependence structure asymptotically for the bootstrap to be valid. In contrast, we show that certain simplified bootstrap schemes which imitate only some aspects of the time series are consistent for quantities arising in nonparametric statistics. To this end, we generalize the well-known "whitening by windowing" principle to joint distributions of nonparametric estimators of the autoregression function. Consequently, we obtain that model-based nonparametric bootstrap schemes remain valid for supremum-type functionals as long as they mimic those finite-dimensional joint distributions consistently which determine the quantity of interest. As an application, we show that simple regression-type bootstrap schemes can be applied for the determination of critical values for nonparametric tests of parametric or semiparametric hypotheses on the autoregression function in the context of a general process.     

On the estimation problem of periodic autoregressive time series: symmetric and asymmetric innovations

Periodic autoregressive (PAR) models with symmetric innovations are widely used on time series analysis, whereas its asymmetric counterpart inference remains a challenge, because of a number of problems related to the existing computational methods. In this paper, we use an interesting relationship between periodic autoregressive and vector autoregressive (VAR) models to study maximum likelihood and Bayesian approaches to the inference of a PAR model with normal and skew-normal innovations, where different kinds of estimation methods for the unknown parameters are examined. Several technical difficulties which are usually complicated to handle are reported. Results are compared with the existing classical solutions and the practical implementations of the proposed algorithms are illustrated via comprehensive simulation studies. The methods developed in the study are applied and illustrate a real-time series. The Bayes factor is also used to compare the multivariate normal model versus the multivariate skew-normal model.