Principal Curves

  title={Principal Curves},
  author={Trevor J. Hastie and Werner Stuetzle},
Principal curves are smooth one-dimensional curves that pass through the middle of a p-dimensional data set, providing a nonlinear summary of the data. They are nonparametric, and their shape is suggested by the data. The algorithm for constructing principal curves starts with some prior summary, such as the usual principal-component li e. The curve in each successive iteration is a smooth or local average of the p-dimensional points, where the definition of local is based on the distance in… 

Locally Defined Principal Curves and Surfaces

A novel theoretical understanding of principal curves and surfaces, practical algorithms as general purpose machine learning tools, and applications of these algorithms to several practical problems are presented.

On principal curves with a length constraint

Principal curves are defined as parametric curves passing through the ``middle'' of a probability distribution in R^d. In addition to the original definition based on self-consistency, several points

Adaptive smooth principal curves design

  • Limei ZhangZhongxuan Luo
  • Mathematics
    International Conference on Computer Graphics, Imaging and Visualization (CGIV'05)
  • 2005
A new principal curves design is presented which is based on a new subdivision scheme and generated in two steps: forming the fold line by joining up the nodes in order, generating scheduled smooth curves by iteratively subdividing the fold lines.

Research on the Construction Algorithm of Principal Curves

A new approach by defining principal curves as continuous curves based on the local tangent space in the sense of limit proves that this new principal curves not only satisfy the self-consistent property, but also are the unique existence for any given open covering.

Selecting the length of a principal curve within a Gaussian model

Principal curves are parameterized curves passing "through the middle" of a data cloud. These objects constitute a way of generalization of the notion of first principal component in Principal

Principal graphs and piecewise linear subspace constrained mean-shift

This work proposes a nonparametric principal graph algorithm, and applies it to optical character recognition, where handling the above mentioned irregularities like loops and self-intersections is a serious problem that appear in many characters.

Self-Consistent Locally Defined Principal Surfaces

  • Deniz ErdoğmuşU. Ozertem
  • Mathematics
    2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07
  • 2007
The concept of principal sets is introduced, which are the union of all principal surfaces with a particular dimensionality with rigorous conditions for a point to satisfy that can be evaluated using only the gradient and Hessian of the probability density at the point of interest.

Average curve of n smooth planar curves

Regularization-free principal curve estimation

This paper introduces a new objective function, facilitated through a modification of the principal curve estimation approach, for which all critical points are principal curves and minima, and removes the fundamental issue for model selection in principal curves estimation.

Parameter Selection for Principal Curves

  • G. BiauA. Fischer
  • Mathematics, Computer Science
    IEEE Transactions on Information Theory
  • 2012
This paper considers the principal curve problem from an empirical risk minimization perspective and addresses the parameter selection issue using the point of view of model selection via penalization and offers oracle inequalities and implements the proposed approach to recover the hidden structures in both simulated and real-life data.



Principal Curves and Surfaces

Principal curves are smooth one dimensional curves that pass through the middle of a p dimensional data set. They minimise the distance from the points, and provide a non-linear summary of the data.

Elementary Topics in Differential Geometry

Contents: Graphs and Level Sets.- Vector Fields.- The Tangent Space.- Surfaces.- Vector Fields on Surfaces Orientation.- The Gauss Map.- Geodesics.- Parallel Transport.- The Weingarten Map.-

Robust Locally Weighted Regression and Smoothing Scatterplots

Abstract The visual information on a scatterplot can be greatly enhanced, with little additional cost, by computing and plotting smoothed points. Robust locally weighted regression is a method for

Kernel Regression Estimation Using Repeated Measurements Data

Abstract The estimation of growth curves has been studied extensively in parametric situations. Here we consider the nonparametric estimation of an average growth curve. Suppose that there are

Topology from the differentiable viewpoint

Preface1Smooth manifolds and smooth maps1Tangent spaces and derivatives2Regular values7The fundamental theorem of algebra82The theorem of Sard and Brown10Manifolds with boundary12The Brouwer fixed

A completely automatic french curve: fitting spline functions by cross validation

The cross validation mean square error technique is used to determine the correct degree of smoothing, in fitting smoothing solines to discrete, noisy observations from some unknown smooth function.

A second generation nonlinear factor analysis

Nonlinear common factor models with polynomial regression functions, including interaction terms, are fitted by simultaneously estimating the factor loadings and common factor scores, using

Cross‐Validatory Choice and Assessment of Statistical Predictions

SUMMARY A generalized form of the cross-validation criterion is applied to the choice and assessment of prediction using the data-analytic concept of a prescription. The examples used to illustrate

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Topology from the Differentiable Vie'Wpoint,University Press of Virginia, Charlottesville

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