| 1. |
- Basna, Rani, 1981, et al.
(författare)
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Data driven orthogonal basis selection for functional data analysis
- 2022
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Ingår i: Journal of Multivariate Analysis. - : Elsevier BV. - 0047-259X. ; 189
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Tidskriftsartikel (refereegranskat)abstract
- Functional data analysis is typically performed in two steps: first, functionally representing discrete observations, and then applying functional methods, such as the functional principal component analysis, to the so-represented data. While the initial choice of a functional representation may have a significant impact on the second phase of the analysis, this issue has not gained much attention in the past. Typically, a rather ad hoc choice of some standard basis such as Fourier, wavelets, splines, etc. is used for the data transforming purpose. To address this important problem, we present its mathematical formulation, demonstrate its importance, and propose a data driven method of functionally representing observations. The method chooses an initial functional basis by an efficient placement of the knots. A simple machine learning style algorithm is utilized for the knot selection and recently introduced orthogonal spline bases splinets are eventually taken to represent the data. The benefits are illustrated by examples of analyses of sparse functional data. (C) 2021 The Author(s). Published by Elsevier Inc.
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| 2. |
- Djehiche, Boualem, 1962-, et al.
(författare)
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A functional Hodrick-Prescott filter
- 2017
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Ingår i: Journal of Inverse and Ill-Posed Problems. - New York : WALTER DE GRUYTER GMBH. - 0928-0219 .- 1569-3945. ; 25:2, s. 135-148
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Tidskriftsartikel (refereegranskat)abstract
- We propose a functional version of the Hodrick-Prescott filter for functional data which take values in an infinite-dimensional separable Hilbert space. We further characterize the associated optimal smoothing operator when the associated linear operator is compact and the underlying distribution of the data is Gaussian.
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| 3. |
- Djehiche, Boualem, et al.
(författare)
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On the Functional Hodrick-Prescott Filter with Non-compact Operators
- 2016
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Ingår i: Random Operators and Stochastic Equations. - : Walter de Gruyter. - 0926-6364 .- 1569-397X. ; 24:1, s. 33-42
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Tidskriftsartikel (refereegranskat)abstract
- We study a version of the functional Hodrick-Prescott filter where the associated operator is not necessarily compact, but merely closed and densely defined with closed range. We show that the associate doptimal smoothing operator preserves the structure obtained in the compact case, when the underlying distribution of the data is Gaussian.
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| 4. |
- Liu, Xijia, et al.
(författare)
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Dyadic diagonalization of positive definite band matrices and efficient B-spline orthogonalization
- 2022
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Ingår i: Journal of Computational and Applied Mathematics. - : Elsevier. - 0377-0427 .- 1879-1778. ; 414
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Tidskriftsartikel (refereegranskat)abstract
- A dyadic algorithm for diagonalizing an arbitrary positive definite band matrix, referred to as a band Gramian, is obtained to efficiently orthogonalize the B-splines. The algorithm can be also used as a fast inversion method for a band Gramian characterized by remarkable sparsity of the diagonalizing matrix. There are two versions of the algorithm: the first one is more efficient and is applicable to a Toeplitz band Gramian while the second one is more general, works with any Gramian matrix, but is more computationally intensive. In the context of the B-splines, these two cases result in new symmetric orthogonalization procedures and correspond to equally and arbitrarily spaced knots, respectively. In the algorithm, the sparsity of a band Gramian is utilized to produce a natural dyadic net of orthogonal splines, rather than a sequence of them. Such a net is thus naturally referred to as a splinet. The splinets exploit “near-orthogonalization” of the B-splines and feature locality expressed through a small size of the total support set and computational efficiency that is a result of a small number of inner product evaluations needed for their construction. These and other efficiencies are formally quantified by upper bounds and asymptotic rates with respect to the number of splines in a splinet. An additional assessment is provided through numerical experiments. They suggest that the theoretical bounds are rather conservative and the method is even more efficient than the bounds indicate. The dyadic net-like structures and the locality bear some resemblance to wavelets but in fact, the splinets are fundamentally different because they do not aim at capturing the resolution scales. The orthogonalization method together with efficient spline algebra and calculus has been implemented in R-package Splinets available on CRAN.
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| 5. |
- Nassar, Hiba
(författare)
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A consistent estimator of the smoothing operator in the functional Hodrick-Prescott filter
- 2018
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Ingår i: Communications in Statistics - Theory and Methods. - : Informa UK Limited. - 0361-0926 .- 1532-415X. ; 47:12, s. 3029-3042
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we consider a version of the functional Hodrick-Prescott ?filter for functional time series. We show that the associated optimal smoothing operator preserves the 'noise-to-signal' structure. Moreover, we propose a consistent estimator of this optimal smoothing operator.
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| 6. |
- Nassar, Hiba, et al.
(författare)
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Empirically Driven Orthonormal Bases for Functional Data Analysis
- 2021
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Ingår i: Numerical Mathematics and Advanced Applications, ENUMATH 2019 - European Conference. - Cham : Springer International Publishing. - 2197-7100 .- 1439-7358. - 9783030558741 - 9783030558734 ; 139, s. 773-783
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Konferensbidrag (refereegranskat)abstract
- In implementations of the functional data methods, the effect of the initial choice of an orthonormal basis has not been properly studied. Typically, several standard bases such as Fourier, wavelets, splines, etc. are considered to transform observed functional data and a choice is made without any formal criteria indicating which of the bases is preferable for the initial transformation of the data. In an attempt to address this issue, we propose a strictly data-driven method of orthonormal basis selection. The method uses B-splines and utilizes recently introduced efficient orthornormal bases called the splinets. The algorithm learns from the data in the machine learning style to efficiently place knots. The optimality criterion is based on the average (per functional data point) mean square error and is utilized both in the learning algorithms and in comparison studies. The latter indicate efficiency that could be used to analyze responses to a complex physical system.
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| 7. |
- Nassar, Hiba, 1982-
(författare)
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Functional Hodrick-Prescott Filter
- 2013
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Licentiatavhandling (övrigt vetenskapligt/konstnärligt)abstract
- The study of functional data analysis is motivated by their applications in various fields of statistical estimation and statistical inverse problems.In this thesis we propose a functional Hodrick-Prescott filter. This filter is applied to functional data which take values in an infinite dimensional separable Hilbert space. The filter depends on a smoothing parameter. In this study we characterize the associated optimal smoothing parameter when the underlying distribution of the data is Gaussian. Furthermore we extend this characterization to the case when the underlying distribution of the data is white noise.
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| 8. |
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