C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the set S is often referred to as a training set. Several properties make the nearest neighbour decision rule quite attractive, including its intuitive simplicity and the theorem that the asymptotic error rate of the nearest neighbour rule is bounded from above by twice the Bayes error rate [6, 8, 15]. See [16] for an extensive survey of the nearestneighbour decision rule and its relatives. Furthermore, for point sets in small dimensions, there are efficient and practical algorithms for preprocessing a set S so that the nearest neighbour of a query point q can be found quickly. The ....

C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.

....it is natural to study L 2 ( convergence of the regression estimate m n to m. In particular, the estimator m n is called weakly (strongly) universally consistent if its excess error satis es km m n k 0 in probability (a.s. for all distributions of (X; Y ) with EjY j 1. Stone [18] rst pointed out that there exist weakly universally consistent estimators. He considered local averaging estimates, i.e. estimates of the form W ni (x; X 1 ; X n )Y i = W ni (x)Y i ; where W ni (x) are the data dependent weights governing the local averaging about x. ....

....estimate, W ni (x; X 1 ; X n ) is chosen to be 1=k if X i is one of the k nearest neighbors of x among X 1 ; X n , and zero otherwise. Note in particular that i=1 W ni = 1. If k n 1; k n =n 0 (3) then the consistency of the k nearest neighbor estimate was established by Stone [18] and by Devroye et al. 6] It is of great importance to be able to estimate the minimum mean squared error L accurately, even before one of the above regression estimates is applied: in a standard nonparametric regression design process, one considers a nite number of real valued features ....

Stone, C. J. (1977). Consistent nonparametric regression. Annals of Statistics, 5, pp. 595-645.

....above by twice the Bayes error (the error of the best possible rule) More precisely, and for the more general case of M pattern classes the bounds proved by Cover and Hart [21] are given by: P e P e [1 NN ] P e (2 MP e = M 1) where P e is the optimal Bayes probability of error. Stone [101] and Devroye [36] generalized these results by proving the bounds for all distributions. In other words, the nearest neighbor of Z contains at least half of the total discrimination information contained in an in nite size training set. Furthermore, a simple generalization of this rule called the ....

C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348{ 1360, 1977.

....the set S is often referred to as a training set. Several properties make the nearest neighbour decision rule quite attractive, including its intuitive simplicity and the theorem that the asymptotic error rate of the nearest neighbour rule is bounded from above by twice the Bayes error rate [6, 8, 15]. See [16] for an extensive survey of the nearestneighbour decision rule and its relatives. Furthermore, for point sets in small dimensions, there are efficient and practical algorithms for preprocessing a set S so that the nearest neighbour of a query point q can be found quickly. The ....

C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.

.... k should be large in order to use as much evidence as possible; 33 on the other hand, k should be no more than a small fraction of n in order to keep the k nearest samples in a relatively small feature space neighborhood of the data point to classify [28] Quantitatively, it was shown [25, 66] that if k satis es both of the following two conditions: k = 1 k n = 0 then the kNN classi er (for n 1) will give an optimal (minimum) error probability. In practice n is nite, so k has to be chosen more carefully; a commonly used value is k = p n (Enas [29] suggested values of n ....

C. Stone. Consistent nonparametric regression. Annals of Statistics, 5:595-645, 1977. (with discussion).

....can classify based on a feature value. Many results have been obtained for nonparametric classification in the literature (see Devroye, Gyorfi, and Lugosi [16] for a review) A surprising result is that universally consistent estimators of f and classifiers exist (see, e.g. 15] 17] 27] and [32]) However, the convergence could be arbitrarily slow( 16] Chapter 7) If one knows a priori that the target function f belongs to a nonparametric class of functions, uniform convergence rates are possible. A few methods have been shown to converge at certain rates when f is in some nonparametric ....

C.J. Stone, "Consistent nonparametric regression," Ann. Statist.,vol. 8, pp. 1348-1360, 1977.

....implies a trade off between bias (systematic error) and variance (random error) h is determined by trial and error cross validating on a separate test set. A more detailed analysis of the approach and its properties are beyond the scope of this present paper; the interested reader is referred to [30]. Back propagation rule is generally criticized on the grounds that learning takes many epochs over the training set and also that the performance depends on a good choice of H the number of hidden units. The method just explained belongs to the class of memory based reelhods where approximation ....

Stone, C.J., "Consistent Nonparametric Regression," The Annals of Statistics, 5 (1977) 595 645.

....model is known. However, when the latter is unavailable, nonparametric density estimation methods must be employed. In this paper we discuss some traditional ways for doing this and then propose a computationally attractive method that is based on nearest neighbor conditional density estimation [5]. We show how the auxiliary particle lter, which has a built in capacity to properly handle outliers, can be fully adapted to the proposed observation model, leading to an optimal lter. The proposed algorithm is very simple to implement and exhibits a high degree of robustness in practice, as it ....

....observations with PCA to a subspace of moderately low dimen sion, e.g. 10 D. Then, instead of modeling the density p(yjx) we invert it using the Bayes rule f(xjy)f(y) f(x) 10) and then model the density f(xjy) nonparametrically using nearest neighbor conditional density estimation [5]. This is feasible because the dimensionality of the state space is low (e.g. q = 3, if the state combines position and orientation of a robot on the plane) The robot states fx k g in the training set are assumed uniformly sampled over the state space and therefore the denominator in the above ....

[Article contains additional citation context not shown here]

C. J. Stone, \Consistent nonparametric regression (with discussion)," Ann. Statist., vol. 5, pp. 595-645, 1977.

....[3] provides a very elegant solution when the observation model is known. However, when the latter is unavailable, nonparametric estimation methods must be employed. We discuss some traditional ways for doing this and then propose a model based on nearest neighbor conditional density estimation [4]. We show how the auxiliary particle lter can be fully adapted to the proposed observation model, leading to an optimal lter. The proposed method is simple to implement and works e ectively even with high dimensional observations. We have successfully applied it on a Nomad robot equipped with ....

....with PCA to a subspace of moderately low dimension, e.g. 10 D. 3 Then, instead of modeling the density p(yjx) we invert it using the Bayes rule p(yjx) f(xjy)f(y) f(x) 9) and then model the density f(xjy) nonparametrically using nearest neighbor based conditional density estimation [4]. This is feasible because the dimensionality of the state space is low (e.g. q = 3, if the state combines position and orientation of the robot) The robot states fx k g in the training set are assumed uniformly sampled over the state space and therefore the denominator in the above formula can ....

[Article contains additional citation context not shown here]

C. J. Stone, \Consistent nonparametric regression (with discussion)," Ann. Statist., vol. 5, pp. 595-645, 1977.

....10] Particularly widely used have been kernel estimation based methods (e.g. 7, 9, 18] whose popularity is largely based on their simple, intuitively appealing de nitions that allow straightforward implementation and relatively transparent mathematical analysis. In local polynomial regression [1, 5, 7, 14, 16] kernel weighted least squares regression is used to t a low degree polynomial to the data that lies near the estimation point and an estimate for the regression function value is obtained by evaluating the t at the estimation point. The classical Nadaraya Watson kernel regression estimator [15] ....

C.J. Stone. Consistent nonparametric regression. Annals of Statistics, 5:595{ 620, 1977.

....from that of f(X i ; Z i )g. 1.2 Connection to Previous Work As noted above, the problem considered here is a special case of regression estimation from ergodic processes. The existence of regression estimates that are weakly L 2 consistent for any i.i.d. process was rst established by Stone [34] using nearest neighbor methods. Beginning with the papers [32, 33, 30] there has been a great deal of work on regression estimation from stationary, weakly dependent processes satisfying , and related mixing conditions. The majority of this work is devoted to central limit theorems and ....

Stone, C. (1977) Consistent nonparametric regression. Ann. Stat., 5 595-620.

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C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.

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C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348{ 1360, 1977.

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C. Stone, Consistent nonparametric regression. Annals of Statistics, Vol. 14, pp. 1348--1360, 1977

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Stone, C. J. (1977). Consistent nonparametric regression. The Annals of Statistics, 5, 595-645.

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C.J. Stone. Consistent nonparametric regression. The annals of statistics, 5:595--620, 1977.

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C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.

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Stone, C.J. Consistent nonparametric regression (with discussions), Ann. Statist. 5, pp. 595-645, 1977.

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C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.

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C. J. Stone, "Consistent nonparametric regression," Annals of Statistics, vol. 5, no. 4, pp. 595--645, 1977.

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C. Stone. Consistent nonparametric regression. Annals of Statistics, 8:1348--1360, 1977.

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Stone, C. J. (1977). Consistent nonparametric regression. Ann. Statist., 5, 595-645.

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C. J. Stone, Consistent nonparametric regression, Ann. Statist. 5 (1977), 595{ 645.

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Stone, C.J. (1977). Consistent nonparametric regression. Ann. Statist., 8, 1348-1360.

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Stone, C.S. (1977) Consistent nonparametric regression, Annals of Statistics, 5, 595-620.

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