SiZer for time series: A new approach to the analysis of trends
TL;DR: A new visualization is proposed, which shows the statistician the range of trade-offs that are available in SiZer, and demonstrates the effectiveness of the method.
Abstract: Smoothing methods and SiZer are a useful statistical tool for discovering statistically significant structure in data. Based on scale space ideas originally developed in the computer vision literature, SiZer (SIgnificant ZERo crossing of the derivatives) is a graphical device to assess which observed features are `really there' and which are just spurious sampling artifacts. In this paper, we develop SiZer like ideas in time series analysis to address the important issue of significance of trends. This is not a straightforward extension, since one data set does not contain the information needed to distinguish `trend' from `dependence'. A new visualization is proposed, which shows the statistician the range of trade-offs that are available. Simulation and real data results illustrate the effectiveness of the method.
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TL;DR: A simple factor method is proposed, based on bimodal kernels, to effectively deal with correlated data in the local polynomial regression framework and establish consistency of the estimator.
Abstract: We present a fully automated framework to estimate derivatives nonparametrically without estimating the regression function. Derivative estimation plays an important role in the exploration of structures in curves (jump detection and discontinuities), comparison of regression curves, analysis of human growth data, etc. Hence, the study of estimating derivatives is equally important as regression estimation itself. Via empirical derivatives we approximate the qth order derivative and create a new data set which can be smoothed by any nonparametric regression estimator. We derive L1 and L2 rates and establish consistency of the estimator. The new data sets created by this technique are no longer independent and identically distributed (i.i.d.) random variables anymore. As a consequence, automated model selection criteria (data-driven procedures) break down. Therefore, we propose a simple factor method, based on bimodal kernels, to effectively deal with correlated data in the local polynomial regression framework.
64 citations
Cites background or methods from "SiZer for time series: A new approa..."
...…1999; Gijbels and Goderniaux, 2004) (jump detection and discontinuities), inference of significant features in data, trend analysis in time series (Rondonotti et al., 2007), comparison of regression curves (Park and Kang, 2008), analysis of human growth data (Müller, 1988; Ramsay and Silverman,…...
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...Also, a brief summary of local polynomial regression is given....
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TL;DR: A graphical method based on SiZer (SIgnificant ZERo crossing of the differences) analysis, which is a scale-space visualization tool for statistical inferences, to find the differences between two curves that are present at each resolution level.
50 citations
Cites background from "SiZer for time series: A new approa..."
...For the one curve case, Rondonotti, Marron, and Park (2007) extended the original SiZer to SiZer for time series....
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...Some work was done in Rondonotti, Marron, and Park (2007), but we plan to improve their method and extend it to the comparison of several time series in the future....
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TL;DR: In this article, the authors investigated the Valanginian Nannoconid decline in two sections of the Vocontian Basin, La Charce and Vergol, which are biostratigraphically well-constrained and contain well-preserved calcareous nannofossils.
47 citations
Cites background from "SiZer for time series: A new approa..."
...A smoothing either computes a local average or locally fits a function indexed by a “smoothing parameter”, a “window width” or a “bandwidth” called h (Marron and Chaudhuri, 1998a,b; Rondonotti et al., 2007)....
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...Statistical smoothingmethods assist in the identification of important and non-obvious structure in complex datasets, but sometimes trends are generated that could be spurious sampling artifacts (Marron and Chaudhuri, 1998a; Rondonotti et al., 2007; Morton et al., 2009)....
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TL;DR: The experimental results strongly supported selection for early flowering in short season and selection for late flowering in long season conditions, and found support for the predicted asymmetry of the flowering time fitness function, including a ‘tail of zeros’ at later flowering dates.
Abstract: Selection gradient analysis examines the strength and direction of phenotypic selection as well as the curvature of fitness functions, allowing predictions on and insights into the process of evolution in natural populations. However, traditional linear and quadratic selection analyses are not capable of detecting other features of fitness functions, such as asymmetry or thresholds, which may be relevant for understanding key aspects of selection on many traits. In these cases, additional analyses are needed to test specific hypotheses about fitness functions. In this study we used several approaches to analyze selection on a major life-history trait—flowering time—in the annual plant Brassica rapa subjected to experimentally abbreviated and lengthened growing seasons. We used a model that incorporated a tradeoff between the time allocated to growth versus the time allocated to reproduction in order to predict fitness function shape. The model predicted that optimal flowering time shifts to earlier and later dates as the growing season contracts and expands. It also showed the flowering time fitness function to be asymmetrical: reproductive output increases modestly between the earliest and the optimal flowering date, but then falls sharply with later dates, truncating in a ‘tail of zeros’. Our experimental results strongly supported selection for early flowering in short season and selection for late flowering in long season conditions. We also found support for the predicted asymmetry of the flowering time fitness function, including a ‘tail of zeros’ at later flowering dates. The form of the fitness function revealed here has implications for interpreting estimates of selection on flowering time in natural populations and for refining predictions on evolutionary response to climate change. More generally, this study illustrates the value of diverse statistical approaches to understanding mechanisms of natural selection.
35 citations
Cites methods from "SiZer for time series: A new approa..."
...We assessed the robustness of trend line features against smoothing with the graphical procedure called ‘significant zero crossing of the derivatives’ (SiZer) (Rondonotti et al. 2007; Sonderegger et al. 2009)....
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TL;DR: A new time series analysis approach, which combines the wavelet analysis with the visualization tools SiZer and SiNos, and it is shown that this new methodology can reveal hidden local non-stationary behavior of time series, that are otherwise difficult to detect.
29 citations
Cites background or methods from "SiZer for time series: A new approa..."
...The details of the multiple comparison test procedure in dependent SiZer can be found in Park, Marron, and Rondonotti (2004) and Rondonotti, Marron, and Park (2004)....
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...Some work in this direction includes Teverovsky and Taqqu (1997), Dang and Molnár (1999), Mikosch and Stǎricǎ (2000), Rondonotti, Marron, and Park (2004), Giraitis, Kokoszka and Leipus (2001), Stoev et al. (2006), Veitch and Abry (2001), and Uhlig, Bonaventure, and Rapier (2003)....
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...Rondonotti, Marron, and Park (2004) extended SiZer to time series....
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...The thorough power studies of the original SiZer and its time series version have been well-addressed in Hannig and Marron (2006), and Rondonotti, Marron, and Park (2004), respectively....
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References
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Book•
19 Aug 2009
TL;DR: In this article, the mean and autocovariance functions of ARIMA models are estimated for multivariate time series and state-space models, and the spectral representation of the spectrum of a Stationary Process is inferred.
Abstract: 1 Stationary Time Series.- 2 Hilbert Spaces.- 3 Stationary ARMA Processes.- 4 The Spectral Representation of a Stationary Process.- 5 Prediction of Stationary Processes.- 6* Asymptotic Theory.- 7 Estimation of the Mean and the Autocovariance Function.- 8 Estimation for ARMA Models.- 9 Model Building and Forecasting with ARIMA Processes.- 10 Inference for the Spectrum of a Stationary Process.- 11 Multivariate Time Series.- 12 State-Space Models and the Kalman Recursions.- 13 Further Topics.- Appendix: Data Sets.
5,260 citations
TL;DR: A general approach to Time Series Modelling and ModeLLing with ARMA Processes, which describes the development of a Stationary Process in Terms of Infinitely Many Past Values and the Autocorrelation Function.
Abstract: Preface 1 INTRODUCTION 1.1 Examples of Time Series 1.2 Objectives of Time Series Analysis 1.3 Some Simple Time Series Models 1.3.3 A General Approach to Time Series Modelling 1.4 Stationary Models and the Autocorrelation Function 1.4.1 The Sample Autocorrelation Function 1.4.2 A Model for the Lake Huron Data 1.5 Estimation and Elimination of Trend and Seasonal Components 1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality 1.5.2 Estimation and Elimination of Both Trend and Seasonality 1.6 Testing the Estimated Noise Sequence 1.7 Problems 2 STATIONARY PROCESSES 2.1 Basic Properties 2.2 Linear Processes 2.3 Introduction to ARMA Processes 2.4 Properties of the Sample Mean and Autocorrelation Function 2.4.2 Estimation of $\gamma(\cdot)$ and $\rho(\cdot)$ 2.5 Forecasting Stationary Time Series 2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past Values 2.6 The Wold Decomposition 1.7 Problems 3 ARMA MODELS 3.1 ARMA($p,q$) Processes 3.2 The ACF and PACF of an ARMA$(p,q)$ Process 3.2.1 Calculation of the ACVF 3.2.2 The Autocorrelation Function 3.2.3 The Partial Autocorrelation Function 3.3 Forecasting ARMA Processes 1.7 Problems 4 SPECTRAL ANALYSIS 4.1 Spectral Densities 4.2 The Periodogram 4.3 Time-Invariant Linear Filters 4.4 The Spectral Density of an ARMA Process 1.7 Problems 5 MODELLING AND PREDICTION WITH ARMA PROCESSES 5.1 Preliminary Estimation 5.1.1 Yule-Walker Estimation 5.1.3 The Innovations Algorithm 5.1.4 The Hannan-Rissanen Algorithm 5.2 Maximum Likelihood Estimation 5.3 Diagnostic Checking 5.3.1 The Graph of $\t=1,\ldots,n\ 5.3.2 The Sample ACF of the Residuals
3,732 citations
"SiZer for time series: A new approa..." refers background or methods in this paper
...Figure 2 (a) shows the Strikes data from [4]....
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...As the Chocolate data set, the Deaths data set comes with the software companion to [4]....
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...This data set comes with the software companion to [4]....
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Book•
01 Jan 1996
TL;DR: In this paper, the authors present a general approach to time series analysis based on simple time series models and the Autocorrelation Function (AFF) and the Wold Decomposition.
Abstract: Preface 1 INTRODUCTION 1.1 Examples of Time Series 1.2 Objectives of Time Series Analysis 1.3 Some Simple Time Series Models 1.3.3 A General Approach to Time Series Modelling 1.4 Stationary Models and the Autocorrelation Function 1.4.1 The Sample Autocorrelation Function 1.4.2 A Model for the Lake Huron Data 1.5 Estimation and Elimination of Trend and Seasonal Components 1.5.1 Estimation and Elimination of Trend in the Absence of Seasonality 1.5.2 Estimation and Elimination of Both Trend and Seasonality 1.6 Testing the Estimated Noise Sequence 1.7 Problems 2 STATIONARY PROCESSES 2.1 Basic Properties 2.2 Linear Processes 2.3 Introduction to ARMA Processes 2.4 Properties of the Sample Mean and Autocorrelation Function 2.4.2 Estimation of $\gamma(\cdot)$ and $\rho(\cdot)$ 2.5 Forecasting Stationary Time Series 2.5.3 Prediction of a Stationary Process in Terms of Infinitely Many Past Values 2.6 The Wold Decomposition 1.7 Problems 3 ARMA MODELS 3.1 ARMA($p,q$) Processes 3.2 The ACF and PACF of an ARMA$(p,q)$ Process 3.2.1 Calculation of the ACVF 3.2.2 The Autocorrelation Function 3.2.3 The Partial Autocorrelation Function 3.3 Forecasting ARMA Processes 1.7 Problems 4 SPECTRAL ANALYSIS 4.1 Spectral Densities 4.2 The Periodogram 4.3 Time-Invariant Linear Filters 4.4 The Spectral Density of an ARMA Process 1.7 Problems 5 MODELLING AND PREDICTION WITH ARMA PROCESSES 5.1 Preliminary Estimation 5.1.1 Yule-Walker Estimation 5.1.3 The Innovations Algorithm 5.1.4 The Hannan-Rissanen Algorithm 5.2 Maximum Likelihood Estimation 5.3 Diagnostic Checking 5.3.1 The Graph of $\t=1,\ldots,n\ 5.3.2 The Sample ACF of the Residuals
3,126 citations
Book•
31 Dec 1993
TL;DR: A basic problem when deriving information from measured data, such as images, originates from the fact that objects in the world, and hence image structures, exist as meaningful entities only over measured data.
Abstract: A basic problem when deriving information from measured data, such as images, originates from the fact that objects in the world, and hence image structures, exist as meaningful entities only over ...
2,452 citations
"SiZer for time series: A new approa..." refers methods in this paper
...While classical kernel methods seek to find the “optimal” smoothing parameter, SiZer is based on scale-space ideas from computer vision, see [17]....
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01 Jan 1994
TL;DR: In this paper, the authors propose that having more aspects to know and understand will lead to becoming a more precious person, and becoming more precious can be situated with the presentation of how your knowledge much.
Abstract: Of course, from childhood to forever, we are always thought to love reading. It is not only reading the lesson book but also reading everything good is the choice of getting new inspirations. Religion, sciences, politics, social, literature, and fictions will enrich you for not only one aspect. Having more aspects to know and understand will lead you become someone more precious. Yea, becoming precious can be situated with the presentation of how your knowledge much.
1,900 citations
"SiZer for time series: A new approa..." refers background in this paper
...Many monographs on smoothing are available in the statistical literature which in the last years include [2, 9, 11, 15, 22, 23]....
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