T. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....To my parents and Agn es Abstract Principal Curves: Learning, Design, and Applications Balazs Kegl, Ph.D. Concordia University, 2000 The subjects of this thesis are unsupervised learning in general, and principal curves in particular. Principal curves were originally defined by Hastie [Has84] and Hastie and Stuetzle [HS89] hereafter HS) to formally capture the notion of a smooth curve passing through the middle of a d dimensional probability distribution or data cloud. Based on the definition, HS also developed an algorithm for constructing principal curves of distributions and ....

....algorithm proceeds by minimizing G n (M ) over all admissible manifolds. In Chapter 3, we present several methods that follow this scheme. 1.2 Principal Curves The main subject of this thesis is the analysis and applications of principal curves. Principal curves were originally defined by Hastie [Has84] and Hastie and Stuetzle [HS89] hereafter HS) to formally capture the notion of a smooth curve passing through the middle of a d dimensional probability distribution or data cloud (to form an intuitive image, see Figure 1 on page 2) The original HS definition of principal curves is based on ....

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T. Hastie. Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....the result is the skeleton of the character. In this paper we propose another approach to skeletonization. The development of the method was inspired by the apparent similarity between the definitions of principal curves and the medial axis. Principal curves were defined by Hastie and Stuetzle [3, 4] (hereafter HS) as self consistent smooth curves which pass through the middle of a d dimensional probability distribution or data cloud, whereas the medial axis is a set of smooth curves that go equidistantly from the contours of a character. Therefore, by representing the black pixels of a ....

T. Hastie. Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....X, and the projection of X to this line gives the best linear summary of the data. For elliptical distributions the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [1] and Hastie and Stuetzle [2] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f(t) f 1 (t) f d (t) be a smooth (infinitely differentiable) curve in R parametrized by t 2 R, and for any x 2 R let t ....

T. Hastie, Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....principal curves from training data. In this paper we propose a practical construction based on the new definition. Simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity. 1 Introduction Hastie [2] and Hastie and Stuetzle [3] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Consider a d dimensional random vector X = X ; X ) with finite second moments, and let f(t) f 1 (t) f d (t) be a ....

T. Hastie, Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....rules that govern the deletion of black pixels. In this paper we propose another approach to skeletonization. The development of the method was inspired by the apparent similarity between the definition of principal curves and the medial axis. Principal curves were defined by Hastie and Stuetzle [11, 12] (hereafter HS) as self consistent smooth curves which pass through the middle of a d dimensional probability distribution or data cloud, whereas the medial axis is a set of smooth curves that go equidistantly from the contours of a character. Therefore, by representing the black pixels of a ....

T. Hastie, Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....X and the projection of X to this line gives the best linear summary of the data. For elliptical distributions the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [2] and Hastie and Stuetzle [3] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f(t) f 1 (t) f d (t) be a smooth curve in R parametrized by t 2 R, and for any x 2 R let t f (x) denote the parameter ....

T. Hastie, "Principal curves and surfaces." PhD Thesis, Stanford University, 1984.

....modeling the observed data as a linear mixture of (unknown) independent sources. ICA s proficiency in blind source separation [15] has found a particular niche in the analysis of EEG [18] and fMRI [21] signals of the brain. Nonlinear PCA (NLPCA) 16, 8] and nonlinear Principal Surfaces [9, 10] are extensions of these linear techniques. In the following section we will briefly review these principal manifolds, their derivation and subsequent statistical properties. In Section 3, an alternative technique using subspace densities and Bayesian similarity is presented and in Section 4 its ....

....the data points and their projections on that surface [10] Note that this is essentially a nonlinear regression on the data. Furthermore, the NLPCA computed by a multi layer sigmoidal network is equivalent with certain exceptions 2 to a principal surface under the more general definition [9, 10]. To summarize, the main properties of NLPCA are: y = f(x) x g(y) P (y) 3) corresponding to nonlinear projection, approximate reconstruction and (almost always) no prior knowledge or certainty regarding the joint distribution of the components, respectively. An example of a principal ....

T. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.

....X and the projection of X to this line gives the best linear summary of the data. For elliptical distributions, the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [1] and Hastie and Stuetzle [2] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f t f 1 t; f d t be a smooth (infinitely differentiable) curve in IR d parametrized by t 2 IR and, for any x 2 IR d ....

T. Hastie, Principal Curves and Surfaces, PhD thesis, Stanford Univ., 1984.

....and this parameterization is unique to choice of origin and sign flips, but defining parameterizations for higher dimensional surfaces is much more complicated. The unit speed property can be generalized for surfaces in terms of areas and volumes, but these parameterizations are not unique. See [8] for further discussion. 2.1 Principal Components Analysis Principal components analysis is a well established feature extraction method that assumes f in (1) is linear. Let the spectral decomposition of X 0 X be X 0 X = UU 0 , where U(p Theta p) is an orthogonal matrix whose column ....

....the objective function given in (8) There are several methods that generalize (7) including [6] 7] 10] and [15] Also see Section 10 in [12] for additional discussion and references on these other approaches. 2. 2 Principal Curves and Surfaces Principal curves (PC) were first proposed in [8] and [9] PCA finds a unitlength vector u that satisfies the minimum distance property in (8) PC extends PCA by fitting a unit speed curve f under a similar objective function. The lengths of the projections of X onto PCA vector u were given in (4) the principal curves method generalizes this ....

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T. Hastie, Principal Curves and Surfaces, Ph.D. thesis, Stanford University, November 1984.

....X, and the projection of X to this line gives the best linear summary of the data. For elliptical distributions the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [1] and Hastie and Stuetzle [2] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f(t) f 1 (t) f d (t) be a smooth (infinitely differentiable) curve in R d parametrized by t 2 R, and for any x 2 R d ....

T. Hastie, Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....X, and the projection of X to this line gives the best linear summary of the data. For elliptical distributions the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [1] and Hastie and Stuetzle [2] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f(t) f 1 (t) f d (t) be a smooth (infinitely differentiable) curve in R d parametrized by t 2 R, and for any x 2 R d ....

T. Hastie, Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....principal curves from training data. In this paper we propose a practical construction based on the new definition. Simulation results demonstrate that the new algorithm compares favorably with previous methods both in terms of performance and computational complexity. 1 Introduction Hastie [2] and Hastie and Stuetzle [3] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Consider a d dimensional random vector X = X (1) X (d) with finite second moments, and let f(t) f 1 (t) f d ....

T. Hastie, Principal curves and surfaces. PhD thesis, Stanford University, 1984.

....X and the projection of X to this line gives the best linear summary of the data. For elliptical distributions the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [2] and Hastie and Stuetzle [3] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f(t) f 1 (t) f d (t) be a smooth curve in R d parametrized by t 2 R, and for any x 2 R d let t f (x) denote the ....

T. Hastie, "Principal curves and surfaces." PhD Thesis, Stanford University, 1984.

....X, and the projection of X to this line gives the best linear summary of the data. For elliptical distributions the first principal component is also self consistent, i.e. any point of the line is the conditional expectation of X over those points of the space which project to this point. Hastie [1] and Hastie and Stuetzle [2] hereafter HS) generalized the self consistency property of principal components and introduced the notion of principal curves. Let f(t) f 1 (t) f d (t) be a smooth (infinitely differentiable) curve in R d parametrized by t 2 R, and for any x 2 R d ....

T. Hastie, "Principal curves and surfaces." PhD Thesis, Stanford University, 1984.

....Estimation is carried out through an EM algorithm. Some comparisons are made to the HastieStuetzle definition. 1 Introduction Suppose we have a random vector Y = Y 1 ; Y 2 ; Y p ) with density gY (y) How can we draw a smooth curve f (s) through the middle of the distribution of Y Hastie (1984) and Hastie and Stuetzle (1989) hereafter HS) proposed a generalization of linear principal components known as principal curves. Let f (s) f 1 (s) f p (s) be a curve in R p parametrized by a real argument s and define the projection index s f (y) to be the value of s corresponding ....

Hastie, T. (1984). Principal curves and surfaces. Technical report, Stanford University.

....and shows that sequential NLPCA has several limitations. We summarize our conclusions regarding NLPCA and PC S in Section 7.1. 30 4.1 Curves and Surfaces 4.1.1 Definition The definitions of curves and surfaces, which generalize lines and planes, are given and discussed in this section. See Hastie (1984), Struik (1961) or Thorpe (1979) for more discussion of these topics. A (globally) parameterized r surface in R p is a vector of smooth functions f : A B, A R r , B R p , x = f(s) 0 B B B B f 1 (s) f 2 (s) f p (s) 1 C C C C A = 0 B B B B f 1 (s 1 ; s r ) ....

....and this parameterization is unique to choice of origin and sign flips, but defining parameterizations for higher dimensional surfaces is much more complicated. The unit speed property can be generalized for surfaces in terms of areas and volumes, but these parameterizations are not unique. See Hastie (1984) for further discussion. 4.2 Principal Curves and Surfaces Sections 2.1 and 3.3.3 introduced PCA as a method for estimating lines planes through the middle of a set of observations, i.e. so that sum of squared distances between the observations and their projections onto the line plane is ....

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Hastie, T. (1984). Principal Curves and Surfaces. PhD thesis, Stanford University.

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T. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.

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T. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.

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T. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.

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T.J. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.

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T.J. Hastie. Principal Curves and Surfaces. PhD thesis, Stanford University, 1984.

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