D. Ormoneit and T. Hastie, "Optimal kernel shapes for local linear regression," in Advances in Neural Information Processing Systems 12, S. A. Solla, T. K. Leen, and K-R. Muller, Eds. 2000, pp. 540--546, The MIT Press.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with multivariate densities. Specifically, it is now possible to choose kernel shapes and sizes which have unequal width in various directions. Proposals for selecting kernel shape include using the covariance structure of the data [9] and iteratively searching over rotations and kernel sizes [19]. Certainly executing such a search can become computationally intensive, due to the coupled nature of all the quantities being estimated. Most of the e#cient univariate estimators (such as the plug in estimate [11, 22] do not appear to be easily extensible to the 2.5. Nonparametric Entropy ....

....multivariable search over the kernel size in each direction may be able to capture such shape related estimate improvements, but is a relatively computationally intensive task. It may also be that the optimal kernel shape is regionally dependent; but finding such regional shapes is an open problem [19]. As was mentioned in Section 2.4.1, another possibility is to use a fixed shape but vary size over the space. Using such a variable kernel, with each point a#ecting an area proportional to its neighborhood s density, can avoid oversmoothing which would otherwise adversely a#ect the estimate. ....

D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. Proc., Advances in Neural Information Processing Systems 12, pages 540--546, 2000.

....easy) to combine both approaches for improved performance when prior knowledge is available. Previous work on nearest neighbor variations based on other locally defined metrics can be found in [12, 9, 6, 7] and is very much related to the more general paradigm of Local Learning Algorithms [3, 1, 10]. We should also mention close similarities between our approach and the recently proposed Local Linear Embedding [11] method for dimensionality reduction. The idea of fantasizing points around the training points in order to define the decision surface is also very close to methods based on ....

D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. In S. A. Solla, T. K. Leen, and K-R. Mller, editors, Advances in Neural Information Processing Systems, volume 12. MIT Press, 2000.

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D. Ormoneit and T. Hastie, "Optimal kernel shapes for local linear regression," in Advances in Neural Information Processing Systems 12, S. A. Solla, T. K. Leen, and K-R. Muller, Eds. 2000, pp. 540--546, The MIT Press.

....the number of free parameters. As a minimization procedure, we use a variant of gradient descent that accounts for the entropy constraint. In particular, our algorithm relies on the fact that (7) is differentiable with respect to L. Due to space limitations, the interested reader is referred to [OH99] for a formal derivation of the involved gradients and for a detailed description of the optimization procedure. 5 Experiments In this section we will compare kernel shaping to standard local linear regression using a fixed spherical kernel in two examples. First, we evaluate the performance ....

D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. Tech. report, Department of Statistics, Stanford University, 1999. To appear.

.... in place of the local averaging rule (4) Practical applications of this idea are described in [25] Locally weighted regression can be shown to eliminate much of the bias at the boundaries of the state space and it is sometimes believed to lead to superior performance in regression problems [12, 18]. From a mathematical perspective, it is well known that locally weighted regression can be interpreted as a special case of local averaging using the notion of equivalent kernels [10] However, local regression estimates need to be suitably constrained in practice to guarantee the positivity of ....

D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. In S. A. Solla, T. K. Leen, and K-R. Muller, editors, Advances in Neural Information Processing Systems 12, pages 540-546. The MIT Press, 2000.

No context found.

D. Ormoneit and T. Hastie. Optimal kernel shapes for local linear regression. In S. A. Solla, T. K. Leen, and K-R. Mller, editors, Advances in Neural Information Processing Systems, volume 12. MIT Press, 2000.

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