Nonparametric estimation of a periodic function

Hall, Peter; Reimann, J; Rice, John A.

doi:10.1093/biomet/87.3.545

Cited by 58 publications

(83 citation statements)

References 26 publications

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“…It is of interest in some areas of the physical sciences to accurately detect and resolve frequency components; two examples are variable stars (Pojmanski, 2002) and detection of gravitational waves (Cornish and Crowder, 2005;Umstätter et al, 2005). A non-parametric approach is often most suitable for fitting of the involved periodic functions (Hall et al, 2000). However, we assume here for simplicity that the observations Y = (Y 1 , .…”

Section: Numerical Illustration: Frequency Detectionmentioning

confidence: 99%

Lasso-type recovery of sparse representations for high-dimensional data

Meinshausen¹,

Yu²

2006

151

253

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The Lasso (Tibshirani, 1996) is an attractive technique for regularization and variable selection for high-dimensional data, where the number of predictor variables p is potentially much larger than the number of samples n. However, it was recently discovered (Zhao and Yu, 2006;Zou, 2005;Meinshausen and Bühlmann, 2006) that the sparsity pattern of the Lasso estimator can only be asymptotically identical to the true sparsity pattern if the design matrix satisfies the so-called irrepresentable condition. The latter condition can easily be violated in applications due to the presence of highly correlated variables.Here we examine the behavior of the Lasso estimators if the irrepresentable condition is relaxed. Even though the Lasso cannot recover the correct sparsity pattern, we show that the estimator is still consistent in the 2 -norm sense for fixed designs under conditions on (a) the number s n of non-zero components of the vector β n and (b) the minimal singular values of the design matrices that are induced by selecting of order s n variables. The results are extended to vectors β in weak q -balls with 0 < q < 1. Our results imply that, with high probability, all important variables are selected. The set of selected variables is a useful (meaningful) reduction on the original set of variables (p n > n). Finally, our results are illustrated with the detection of closely adjacent frequencies, a problem encountered in astrophysics. * Acknowledgments We would like to thank Noureddine El Karoui and Debashis Paul for pointing out interesting connections to Random Matrix theory. Some results of this manuscript have been presented at the Oberwolfach workshop "Qualitative Assumptions and Regularization for High-Dimensional Data". Nicolai Meinshausen is supported by DFG (Deutsche Forschungsgemeinschaft) and Bin Yu is partially supported by a Guggenheim fellowship and grants NSF DMS-0605165 (06-08), NSF DMS-03036508 (03-05) and ARO W911NF-05-1-0104 (05-07). Part of this work has been presented at Oberwolfach workshop 0645, "Qualitative Assumptions and Regularization in High-Dimensional Statistics". 1 Report Documentation PageForm Public reporting burden for the collection of information is estimated to average 1 hour per response, including the time for reviewing instructions, searching existing data sources, gathering and maintaining the data needed, and completing and reviewing the collection of information. Send comments regarding this burden estimate or any other aspect of this collection of information, including suggestions for reducing this burden, to Washington Headquarters Services, Directorate for Information Operations and Reports, 1215 Jefferson Davis Highway, Suite 1204, Arlington VA 22202-4302. Respondents should be aware that notwithstanding any other provision of law, no person shall be subject to a penalty for failing to comply with a collection of information if it does not display a currently valid OMB control number. REPORT DATE DEC 2006

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Section: Numerical Illustration: Frequency Detectionmentioning

confidence: 99%

Lasso-type recovery of sparse representations for high-dimensional data

Meinshausen¹,

Yu²

2006

151

253

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“…Previous approaches for automated alignment can be found, for example, in [46]. Based on this segmentation the individual walking "cycles" can be modeled using various statistical techniques, e.g.…”

Section: Previous Workmentioning

confidence: 99%

Representing cyclic human motion using functional analysis

Ormoneit¹,

Black

Hastie

et al. 2005

Image and Vision Computing

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We present a robust automatic method for modeling cyclic 3D human motion such as walking using motion-capture data. The pose of the body is represented by a time series of joint angles which are automatically segmented into a sequence of motion cycles. The mean and the principal components of these cycles are computed using a new algorithm that enforces smooth transitions between the cycles by operating in the Fourier domain. Key to this method is its ability to automatically deal with noise and missing data. A learned walking model is then exploited for Bayesian tracking of 3D human motion.

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“…, X p (t), and either a functional or scalar response, has re-cently received a great deal of attention. A few examples include Hastie and Mallows (1993); Hall et al (2000); Alter et al (2000); Hall et al (2001); James (2002); Cardot et al (2003); Ferraty and Vieu (2003); James and Silverman (2005); Muller and Stadtmuller (2005) ;Chen et al (2011), and Jiang and Wang (2011). See Chapter 15 of Ramsay and Silverman (2005) for a thorough discussion of the issues involved with fitting such data.…”

Section: Introductionmentioning

confidence: 99%

Index Models for Sparsely Sampled Functional Data

Radchenko

Qiao

James

2015

Journal of the American Statistical Association

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The regression problem involving functional predictors has many important applications and a number of functional regression methods have been developed. However, a common complication in functional data analysis is one of sparsely observed curves, that is predictors that are observed, with error, on a small subset of the possible time points. Such sparsely observed data induces an errors-in-variables model, where one must account for measurement error in the functional predictors. Faced with sparsely observed data, most current functional regression methods simply estimate the unobserved predictors and treat them as fully observed; thus failing to account for the extra uncertainty from the measurement error. We propose a new functional errors-invariables approach, Sparse Index Model Functional Estimation (SIMFE), which uses a functional index model formulation to deal with sparsely observed predictors. SIMFE has several advantages over more traditional methods. First, the index model implements a non-linear regression and uses an accurate supervised method to estimate the lower dimensional space into which the predictors should be projected. Second, SIMFE can be applied to both scalar and functional responses and multiple predictors. Finally, SIMFE uses a mixed effects model to effectively deal with very sparsely observed functional predictors and to correctly model the measurement error.

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Nonparametric estimation of a periodic function

Cited by 58 publications

References 26 publications

Lasso-type recovery of sparse representations for high-dimensional data

Lasso-type recovery of sparse representations for high-dimensional data

Representing cyclic human motion using functional analysis

Index Models for Sparsely Sampled Functional Data

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