B. Chapmann and R. Tibshirani. An Introduction to the Bootstrap. Monograph on Statistics and Applied Probability. Chapman and Hall, 1993.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....m # (log s) d 1 s; which is a significant reduction compared to TPSFEM, especially for large s. TPSFEM: 16641 vars HISURF: 833 vars ADDFIT: 388 vars Figure 2: Modelling 2D aeromagnetic data with 735,700 records. 4. 3 ADDFIT We have adopted (generalised) additive models (Hastie and Tibshirani [7]) to deal with very high dimensional data. Figure 1 clearly shows its superiority with respect to complexity. However, this comes at a cost as the approximation power of additive models is much poorer as can be seen for the magnetisation example in Figure 2. Both advantages and disadvantages ....

T.J. Hastie and R.J. Tibshirani, Generalized additive models, Monographs on statistics and applied probability, vol. 43, Chapman and Hall, 1990.

....distributions. Hence, the model can be estimated as t = log oe t t ; 14) with t j log j t j 0:635, t j log j t j 0:635 and E( t ) 0. For the nonparametric estimation of the functions g j ( Delta) basically two methods are available: the backfitting procedure of Hastie and Tibshirani (1990), and the estimation based on marginal integration, as in Linton and Nielsen (1995) The integration estimator has the advantage of a closed form expression, which implies that it is non iterative. Also, confidence intervals can be constructed. On the other hand, for large dimensions q and large ....

....expression, which implies that it is non iterative. Also, confidence intervals can be constructed. On the other hand, for large dimensions q and large data sets, the estimator is computationally expensive and the choice of the bandwidths is a delicate problem. We used the backfitting procedure of Hastie and Tibshirani (1990), because efficient implementations in standard packages exist 3 . The backfitting procedure is iterative with the following steps: 1. Initialization: log g j ( t Gammaj ) 0; j = 1; q 2. Repeat for j = 1; q; 1; q; log g j ( t Gammaj ) S j t Gamma ....

Hastie, T. J., and R. J. Tibshirani (1990): Generalized Additive Models, vol. 43 of Monographs on Statistics and Applied Probability. Chapman and Hall, London.

....splits x i . A terminal node in a trained tree is labeled according to the majority class in the associated hyper rectangle. The particular tree classifier tested here uses a likelihood function to select the optimal splits and it is available as part of the S Plus statistical software package [5]. MARS estimates an unknown function r as r(x) a 0 P M k=1 a k B k (x) where the functions B k are multivariate splines. The algorithm begins with a forward stepwise phase which adds basis functions in a deliberate attempt to overfit the data. A backward model pruning stage then follows. ....

J. Chambers and T. Hastie, editors. Statistical Models in S. Chapman and Hall, New York, 1992.

....number of training vectors in the associated hyper rectangle. The tree classifier therefore uses the Bayes rule with the class posterior probabilities estimated by locally constant functions. The particular tree classifier used here is available as part of the S Plus statistical software package [2, 8, 106]. This implementation uses a likelihood function to select the optimal splits [10] Pruning was performed by the minimal cost complexity method. The cost of a subtree T is taken to be R ff (T ) R(T ) ff Delta size(T ) 22) where R(T ) is an estimate of the classification error of T , size ....

J. Chambers and T. Hastie, editors. Statistical Models in S. Chapman and Hall, New York, 1992.

....achieve a 10 times decrease in the current approximation error, one requires a 10 d=2 increase in the number of function evaluations. This yields an explicit characterization of the mentioned Curse of Dimensionality . Note. In other contexts, such as data smoothing (cf. Hastie and Tibshirani [4]) the increase in function evaluations associated with the increase in dimension is also viewed as a curse . FOR NUMERICAL DIFFERENTIATION, DIMENSIONALITY CAN BE A BLESSING 3 In numerical differentiation, there is no curse. This is a direct consequence of the local nature of differentiation. ....

T.J. Hastie and R.J. Tibshirani, Generalized additive models, Monographs on statistics and applied probability, vol. 43, Chapman and Hall, 1990.

....[95] which are also non parametric, as smoothers. They can give accurate derivative calculation and smooth interpolation, but still require the use of a smoother matrix. Moreover, the generalized cross validation statistic, which is used to select the degree of smoothness, tends to under smooth [38]. Consequently, heuristics are required to remedy this problem. On the computational aspect, smoothing splines are usually more computationally intensive. The universal approximation capability of PPR networks, with respect to the L 2 norm, follows directly from its convergence property, which ....

T.J. Hastie and R.J. Tibshirani. Generalized Additive Models. Monographs on Statistics and Applied Probability 43. Chapman and Hall, 1st edition, 1990.

....used smoothing splines as the smoothers, which give accurate derivative calculation and smooth interpolation. However, this still requires the use of a smoother matrix. Moreover, the generalized cross validation statistic, which is used to select the degree of smoothness, tends to under smooth [29]. Consequently, heuristics are required to remedy this problem. Besides, smoothing splines are usually more computationally intensive. On the other hand, the smoothers in PPL are parametric. They are represented as linear combinations of Hermite functions [30] of the form g(z) R X r=1 c r h ....

T.J. Hastie and R.J. Tibshirani, Generalized Additive Models, Monographs on Statistics and Applied Probability 43. Chapman and Hall, 1st edition, 1990.

....to using an identity link # k (#) # and # j ( of the form (1. 4) Alternative forms for # k (x) are used, for example, in generalized linear models (McCullagh and Nelder [17] natural thin plate splines (O Sullivan, Yandell, and Raynor [19] Gu [8] additive models (Hastie and Tibshirani [10]) and smoothing spline ANOVA (Wahba et al. 26] Let w denote those # kj and those parameters defining the functions # j for which estimates are required (any remaining parameters will have to be specified beforehand) Estimation of w and f 1 , f p is considered next. 2. A roughness ....

.... Wang [27] However, this method is very general and requires O(n 3 ) operations, and, in addition, one would have to perform the expensive computation of R(#) Since # # is a vector of m weighted additive fits, we may instead resort to the following approximation used by Hastie and Tibshirani [10] in the BRUTO algorithm and further developed for PPR by Roosen and Hastie [22] in their ASP algorithm, tr R(#) m p # j=1 df j tr S j . 6.2) Here, df j accounts for the degrees of freedom due to # j ( and # 1j , #mj . Roosen and Hastie consider linear # j ( and suggest df ....

<F3.799e+05> T.J. Hastie and R.J.<F3.89e+05> Tibshirani,<F3.829e+05> Generalized Additive<F3.89e+05> Models, Chapman and Hall, London, 1990.

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B. Chapmann and R. Tibshirani. An Introduction to the Bootstrap. Monograph on Statistics and Applied Probability. Chapman and Hall, 1993.

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B. Chapmann and R. Tibshirani. An Introduction to the Bootstrap. Monograph on Statistics and Applied Probability. Chapman and Hall, 1993.

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T. Hastie and R. Tibshirani. Generalized Additive Models, volume43ofMonographs on Statistics and Applied Probability. Chapman and Hall, London, 1990.

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T. Hastie and R. Tibshirani. GeneralizedAdditive Models,volume 43 of Monographs on Statistics and AppliedProbability. Chapman and Hall, London, 1990.

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T.J. Hastie, and R.J. Tibshirani, Generalized Additive Models, vol. 43 of Monographs on Statistics and Applied Probability, Chapman and Hall, London, 1990.

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