The Application of the Principal Curve Analysis Technique to Smooth Beam Lines

Friedsam, H.; Oren, W.

doi:10.2172/878884

Cited by 11 publications

(12 citation statements)

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“…We will adapt a tool from statistics called principal curves, and its generalization, principal manifolds to describe the behavior of a dynamical system evolving in a high dimensions by a smooth manifold of much lower dimensionality, as suggested in Sec. V. While principal curves have found many data processing applications from speech recognition [22], medicine [24], and physics [23], we will develop this area for use in a scenario where dynamics evolve on a stable submanifold. Considering data points sampled from the invariant measure of a dynamical system as an orbit evolves, whose ω-limit set may not be smooth, we may nonetheless expect the data to reside stably near a smooth manifold.…”

Section: Principal Curves and Principal Manifoldsmentioning

confidence: 99%

A Low-Dimensional Principal Manifold as the "Attractor Backbone" of a Chaotic Beam System

Bollt

Skufca

2015

Int. J. Bifurcation Chaos

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We model an elastic beam subject to a contact load which displaces under a chaotic external forcing, motivated by application of a ship carrying either a crane, or fluids in internal tanks. This model not only has rich dynamics and relevance in its own right, it gives rise to a Partial Differential Equation (PDE) whose solutions are chaotic, with an attractor whose points lie "near" a low-dimensional curve. This form identifies a data-driven dimensionality reduction which encapsulates a Cartesian product, approximately, of a principal manifold, corresponding to spatial regularity, against a temporal complex dynamics of the intrinsic variable of the manifold. The principal manifold element serves to translate the complex information at one site to all other sites on the beam. Although points of the attractor do not lie on the principal manifold, they lie sufficiently close that we describe that manifold as a "backbone" running through the attractor, allowing the manifold to serve as a suitable space to approximate behaviors.

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Section: Principal Curves and Principal Manifoldsmentioning

confidence: 99%

A Low-Dimensional Principal Manifold as the "Attractor Backbone" of a Chaotic Beam System

Bollt

Skufca

2015

Int. J. Bifurcation Chaos

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“…Roughly, the purpose is to search for a curve passing through the middle of the observations, as illustrated in Figure 1. Principal curves have a broad range of applications in many different areas, such as physics (Hastie and Stuetzle [26], Friedsam and Oren [23]), character and speech recognition (Kégl and Krzyżak [29], Reinhard and Niranjan [39]), mapping and geology (Brunsdon [10], Stanford and Raftery [43], Banfield and Raftery [4], Einbeck, Tutz and Evers [20,21]), natural sciences (De'ath [14], Corkeron, Anthony and Martin [13], Einbeck, Tutz and Evers [20]) and medicine (Wong and Chung [46], Caffo, Crainiceanu, Deng and Hendrix [11]). …”

Section: Principal Curvesmentioning

confidence: 99%

Parameter Selection for Principal Curves

Biau

Fischer

2012

IEEE Trans. Inform. Theory

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-Principal curves are nonlinear generalizations of the notion of first principal component. Roughly, a principal curve is a parameterized curve in R d which passes through the "middle" of a data cloud drawn from some unknown probability distribution. Depending on the definition, a principal curve relies on some unknown parameters (number of segments, length, turn. . . ) which have to be properly chosen to recover the shape of the data without interpolating. In the present paper, we consider the principal curve problem from an empirical risk minimization perspective and address the parameter selection issue using the point of view of model selection via penalization. We offer oracle inequalities and implement the proposed approaches to recover the hidden structures in both simulated and real-life data.

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“…Principal curve algorithms have also been applied in other areas, including physics [11,15], natural language processing [16,17], geology [18,13,19,20], natural sciences [21,20] and bio-medical studies [22,14]. They are often useful over connected graphs, for example, in settings where the structure is not contiguous, there is noise and when sampling assumptions are needed.…”

Section: Introductionmentioning

confidence: 99%

A Length Penalized Probabilistic Principal Curve Algorithm With Applications To Handwritten Digits And Pharmacologic Colon Imaging

Chen

Weld

Hendrix

et al. 2020

Preprint

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The classical Principal Curve algorithm was developed as a nonlinear version of principal component analysis to model curves. However, existing principal curve algorithms with classical penalties, such as smoothness or ridge penalties, lack the ability to deal with complex curve shapes. In this manuscript, we introduce a robust and stable length penalty which solves issues of unnecessary curve complexity, such as the self-looping, that arise widely in principal curve algorithms. A novel probabilistic mixture regression model is formulated. A modified penalized EM(Expectation Maximization) Algorithm was applied to the model to obtain the penalized MLE. Two applications of the algorithm were performed. In the first, the algorithm was applied to the MNIST dataset of handwritten digits to find the centerline, not unlike defining a TrueType font. We demonstrate that the centerline can be recovered with this algorithm. In the second application, the algorithm was applied to construct a three dimensional centerline through single photon emission computed tomography images of the colon arising from the study of pre-exposure prophylaxis for HIV. The centerline in this application is crucial for understanding the distribution of the antiviral agents in the colon for HIV prevention. The new algorithms improves on previous applications of principal curves to this data.

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The Application of the Principal Curve Analysis Technique to Smooth Beam Lines

Cited by 11 publications

References 0 publications

A Low-Dimensional Principal Manifold as the "Attractor Backbone" of a Chaotic Beam System

A Low-Dimensional Principal Manifold as the "Attractor Backbone" of a Chaotic Beam System

Parameter Selection for Principal Curves

A Length Penalized Probabilistic Principal Curve Algorithm With Applications To Handwritten Digits And Pharmacologic Colon Imaging

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