Multivariate singular spectrum analysis for forecasting revisions to real-time data

Real-time data on national accounts statistics typically undergo an extensive revision process, leading to multiple vintages on the same generic variable. The time between the publication of the initial and final data is a lengthy one and raises the question of how to model and forecast the final vintage of data – an issue that dates from seminal articles by Mankiw et al. [51 Mankiw, N. G., Runkle, M. and Shapiro, M. D. 1984. Are preliminary announcements of the money stock rational forecasts?. J. Monetary Econ., 14: 15–27. [Crossref], [Web of Science ®] [Google Scholar]], Mankiw and Shapiro [52 Mankiw, N. G. and Shapiro, M. D. 1986. News or noise? An analysis of GNP revisions. Surv. Curr. Bus. May, : 20–25.  [Google Scholar]] and Nordhaus [57 Nordhaus, W. D. 1987. Forecasting efficiency: Concepts and applications. Rev. Econ. Stat., 4: 667–674.  [Google Scholar]]. To solve this problem, we develop the non-parametric method of multivariate singular spectrum analysis (MSSA) for multi-vintage data. MSSA is much more flexible than the standard methods of modelling that involve at least one of the restrictive assumptions of linearity, normality and stationarity. The benefits are illustrated with data on the UK index of industrial production: neither the preliminary vintages nor the competing models are as accurate as the forecasts using MSSA.     

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.     

Statistical dimension estimation in singular spectrum analysis

Summary Singular spectrum analysis has been proposed in the field of nonlinear dynamical systems as filtering method. In this paper a criterion to choose the number of components which leads to the best filtering is proposed. The selection is made by minimizing the prediction error.     

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.     

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.     

Tourist number prediction of historic buildings by singular spectrum analysis

A wooden historic building located in Tibet, China, experienced structural damage when subjected to tourists visit. This kind of ancient building attends to too many visitors every day because heritage sites never fail to attract tourists. There should be a balance between accepting the visitors and the protection of historic buildings considering the importance of the cultural relics. In this paper, the singular spectrum analysis (SSA) is used for forecasting the number of tourist for the building management to exercise maintenance measures to the structure. The analyzed results can be used to control the tourist flow to avoid excessive pedestrian loading on the structure. The relationship between the measured acceleration from the structure and the tourist number is firstly studied. The root-mean-square (RMS) value of the measured acceleration in the passage route of the tourist is selected for forecasting future tourist number. The forecasting results from different methods are compared. The SSA is found slightly outperforms the autoregressive integrated moving average model (ARIMA), the X-11-ARIMA model and the cubic spline extrapolation in terms of the RMS error, mean absolute error and mean absolute percentage error for long-term prediction, whereas the opposite is observed for short-term forecasting.     

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.     

Multivariate analysis in genstat

Genstat is a general statistical language for data analysis. The facilities for multivariate and cluster analysis within the language are described as well as the many vector and matrix operations which can be used to form multivariate analysis programs. The contents of the standard macro library relevant to multivariate analysis are also discussed.     

Multivariate analysis of within-group variability

This paper describes a consultancy problem where the question was to compare the variability of two groups of plants using several variables of widely differing types. The two-stage approach adopted is uncommon, and may be more generally useful to any analysis where variables are not commensurate. The Appendix describes the measures of similarity used within the paper, and the important result (equation 4) that gives the squared distances of a set of points from their centroid.     

Multivariate regression analysis and canonical variates

A method called FICYREG of estimating regression coefficients is introduced. This is a generalization to the multivariate regression problem of the James-Stein estimator. When suitably représentés FICYREG emerges as a rule in which the canonical variates and canonical correlations have an intrinsic role to play. By exploiting these objects FICYREG is able to achieve stability against the influence of the “noise” present in problems where the responses are correlated so that some of the response vector's canonical variates will be essentially independent of all others including the predictors. The least squares (LS) estimator is, by contrast, highly sensitive to this noise. The use of FICYREG is illustrated in terms of an example, and its peformance is compared to the LS estimator when a quadratic loss function is assumed. The cases of both fixed and random predictors are considered. Overall, FICYREG outperforms the LS estimator.     

Multivariate thin plate spline estimates for the posterior probabilities in the classification problem

A nonparametric estimate for the posterior probabilities in the classification problem using multivariate thin plate splines is proposed. This method presents a nonpararnetric alternative to logistic discrimination as well as to survival curve estimation. The degree of smoothness of the estimate is determined from the data using generalized crossvalidation.     

Multivariate convex risk statistics with scenario analysis

Abstract

In this article, we introduce three new classes of multivariate risk statistics, which can be considered as data-based versions of multivariate risk measures. These new classes are multivariate convex risk statistics, multivariate comonotonic convex risk statistics and multivariate empirical-law-invariant convex risk statistics, respectively. Representation results are provided. The arguments of proofs are mainly developed by ourselves. It turns out that all the relevant existing results in the literature are special cases of those obtained in this article.     

Multivariate analysis of variance for functional data

Functional data are being observed frequently in many scientific fields, and therefore most of the standard statistical methods are being adapted for functional data. The multivariate analysis of variance problem for functional data is considered. It seems to be of practical interest similarly as the one-way analysis of variance for such data. For the MANOVA problem for multivariate functional data, we propose permutation tests based on a basis function representation and tests based on random projections. Their performance is examined in comprehensive simulation studies, which provide an idea of the size control and power of the tests and identify differences between them. The simulation experiments are based on artificial data and real labeled multivariate time series data found in the literature. The results suggest that the studied testing procedures can detect small differences between vectors of curves even with small sample sizes. Illustrative real data examples of the use of the proposed testing procedures in practice are also presented.     

Multivariate analysis with an autoregressive covariance model

This paper addresses the problem of testing the multivariate linear hypothesis when the errors follow an antedependence model (Gabriel, 1961, 1962). Antedependence can be formulated as a nonstationary autoregressive model of general order. Three test statistics are derived that provide analogs to three commonly used MANOVA statistics: Wilks' Lambda, the Lawley-Hotelling Trace, and Pillai's Trace. Formulas are given for each of these statistics that show how they can be obtained From any statistical computing package that calculates the usual MANOVA statistics. These antedependent statistics would be appropriate in analyzing certain multivariate data sets in which repeated measurements are taken on the same subjects over a period of time.     

Multivariate online regression analysis with heterogeneous streaming data

New data collection and storage technologies have given rise to a new field of streaming data analytics, called real-time statistical methodology for online data analyses. Most existing online learning methods are based on homogeneity assumptions, which require the samples in a sequence to be independent and identically distributed. However, inter-data batch correlation and dynamically evolving batch-specific effects are among the key defining features of real-world streaming data such as electronic health records and mobile health data. This article is built under a state-space mixed model framework in which the observed data stream is driven by a latent state process that follows a Markov process. In this setting, online maximum likelihood estimation is made challenging by high-dimensional integrals and complex covariance structures. In this article, we develop a real-time Kalman-filter-based regression analysis method that updates both point estimates and their standard errors for fixed population average effects while adjusting for dynamic hidden effects. Both theoretical justification and numerical experiments demonstrate that our proposed online method has statistical properties similar to those of its offline counterpart and enjoys great computational efficiency. We also apply this method to analyze an electronic health record dataset.     

Multivariate partially linear single-index models: Bayesian analysis

Partially linear single-index models play important roles in advanced non-/semi-parametric statistics due to their generality and flexibility. We generalise these models from univariate response to multivariate responses. A Bayesian method with free-knot spline is used to analyse the proposed models, including the estimation and the prediction, and a Metropolis-within-Gibbs sampler is provided for posterior exploration. We also utilise the partially collapsed idea in our algorithm to speed up the convergence. The proposed models and methods of analysis are demonstrated by simulation studies and are applied to a real data set.     

Multivariate analysis of a reference collection of elm leaves

SUMMARY Analysis of leaf measurements of a reference collection of elm leaves suggests that the leaves, collected from trees which were selected by R. H. Richens as spanning the range of elm in Europe, fall into two primary groups. The first of these groups is made up of the representatives of the Wych elm (Ulmus glabra), which is known to be a distinct species. The second, larger, group is made up of an assemblage of subgroups that represent English elm (Ulmus procera), Ulmus minor and hybrids between Ulmus minor and Ulmus glabra. A minimum variance cluster analysis suggests that these subgroups are reasonably distinct, and discriminant analysis, logistic regression and a genetic algorithm are used to help identify the subgroups.     

Multivariate analysis with a stationary time series covariance structure

When a covariance matrix has a pattern associated with a stationary time series on the errors, it is shown how certain hypothesis testing problems In multivariate analysis can be transformed into a product of two similar multivariate problems that each involve unpatterned covariance matrices.     

Multivariate survival mixed models for genetic analysis of longevity traits

A class of multivariate mixed survival models for continuous and discrete time with a complex covariance structure is introduced in a context of quantitative genetic applications. The methods introduced can be used in many applications in quantitative genetics although the discussion presented concentrates on longevity studies. The framework presented allows to combine models based on continuous time with models based on discrete time in a joint analysis. The continuous time models are approximations of the frailty model in which the baseline hazard function will be assumed to be piece-wise constant. The discrete time models used are multivariate variants of the discrete relative risk models. These models allow for regular parametric likelihood-based inference by exploring a coincidence of their likelihood functions and the likelihood functions of suitably defined multivariate generalized linear mixed models. The models include a dispersion parameter, which is essential for obtaining a decomposition of the variance of the trait of interest as a sum of parcels representing the additive genetic effects, environmental effects and unspecified sources of variability; as required in quantitative genetic applications. The methods presented are implemented in such a way that large and complex quantitative genetic data can be analyzed. Some key model control techniques are discussed in a supplementary online material.