Mallows, C. (1973), `Some comments on c p ', Technometrics 15, 661--675.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....MSEP estimates often require the estimation of the error variance whereas the cross validation statistic does not. However, in general cross validation overestimates the MSEP for the linear regression case (Bunke Droge [4] which is a disadvantage compared to other estimates like C p (Mallows [28]) or those derived by the bootstrap (Efron [12] Efron [14] investigated the relations between cross validation, jackknife and some bootstrap estimates of the prediction error. He remarked that cross validation gives reasonable estimates for smooth predictors. Bunke Droge [4] showed that ....

....for model selection in general regression (see e.g. Cook Weisberg [8] But this procedure has some de ciencies. Shao [32] pointed out that in linear models, leave one out cross validation is asymptotically equivalent to the Akaike information criterion (AIC) Akaike [1] the C p (Mallows [28]) the jackknife, and the bootstrap (Efron [14] But these do not provide consistent model selection, meaning that they do not provide the best predictive model with probability 1, as n 1. It has been found by Stone [35] that the probability of choosing a good model converges to one, but this ....

Mallows, C. L. (1973). Some comments on C p . Technometrics 15, 661-675.

....of this quantity are strongly related with tuning parameters for different modeling techniques. This relation is reflected in a number of applications. Firstly, the variance of the noise plays an important role in various complexity criteria as Akaike s information criteria [1] and C p statistic [13] which can be used to select the appropriate model amongst a set (class) of models (see e.g. 20] and [19] Another point is that the presented approximator gives rise to good initial starting values of the tuning parameters in ridge regression, least squares) support vector machines, ....

....plays an important role for doing model selection and setting tuning parameters. Examples of such applications are given. 1. Well known complexity criteria (or model selection criteria) such as the Akaike Information Criterion [1] the Bayesian Information Criterion [18] and C p statistic [13] take the form of a prediction error which consist of the sum of a training set error (e.g. the residual sum of squares) and a complexity term. In general: J(S) f(x i ; S) # QN ( f) e , 8) see [6] The complexity term QN ( f) represents a penalty term which grows as ....

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C. Mallows, "Some comments on Cp," Technometrics, vol. 15, pp. 661--675, 1973.

.... select a model from a so that the generalization error JG is minimized: min , S#M JG [X , S] 2) In general, the model should be fixed for active learning [4, 7, 3, 6, 5, 14, 15, 18] and conversely the training examples gathered at fixed sample points are required for model selection [8, 1, 13, 12, 11, 2, 16, 17]. This implies that the problem of active learning with model selection can not be generally solved by simply combining existing active learning and model selection techniques. We call this the active learning model selection dilemma. In this paper, we suggest a basic strategy for solving this ....

..... Leave one out cross validation (CV) 9] Akaike s information criterion (AIC) 1] Corrected AIC (cAIC) 13] Bayesian information criterion (BIC) 12] Vapnik s measure (VM) 2] Note that for optimal sampling with A # n An = I Sn , SIC essentially agrees with Mallows s CL [8]. Figure 6 depicts the simulation results. The left column corresponds to (M, # ) 300, 0.04) and the right column corresponds to (M, # ) 100, 0.07) The top seven graphs show the values of the error and model selection criteria corresponding to the order n of the model Sn (see ....

C. L. Mallows. Some comments on C P . Technometrics, 15(4):661--675, 1973.

....values so that the criterion is optimized [3, 4, 5] The most crucial point in this approach is how well the alternative criterion approximates the original ESE. This topic is also a traditional concern in the communities of statistics and machine learning, and it has been extensively studied [8, 9, 10, 11, 12, 13]. Most of the methods proposed so far proved their usefulness in the asymptotic sense. However, in practice, we are interested in the case with finite samples. So far, an estimator of ESE called the subspace information criterion (SIC) was proposed [14, 15] Among several other interesting ....

....we use Daubechies s compactly supported wavelets with external phase and 4 vanishing moments [25] As mentioned above, there are a large number of parameter optimization methods for the regularization filter. Here we compare the proposed SIC with some of the representative methods: Mallows s C L [8], the leave one out cross validation (CV) 26] the generalized cross validation (GCV) 12] the network information criterion (NIC) 27] a Bayesian information criterion (ABIC) 28] and Vapnik s measure (VM) 29] We measure the actual error of the restored image f by the following mean ....

C. L. Mallows, Some comments on C P , Technometrics 15 (4) (1973) 661--675.

....these potential problems. The proper choice of the ridge parameter greatly affects the performance of ridge regression. Several methods of choosing a valid ridge parameter have found their way into engineering practice. The most common methods are the Discrepancy Principle (DP) 6] Mallows [7] CL, Generalized Cross Validation (GCV) 8] and the L curve method [9] Unfortunately, every parameter choice rule has its pitfalls. The high sensitivity of CL and DP to an underestimation of the noise level has limited their application to cases in which the noise level can be estimated with ....

....method of optimizing the vector of local ridge parameters has not been found to be practical. This paper presents an Evolutionary Algorithm method for optimizing the local ridge parameters to minimize Mallows CL. CL was chosen because it has proven to be an unbiased estimate of prediction error [7]. The methodology section derives the local ridge solution and describes the evolutionary programming strategy. The developed methodology is then applied to the development of two predictive models. These two examples show the advantages of local ridge to pass components with small variance and ....

C.L. Mallows, Some comments on CP", Technometrics, 15, No. 4, pp. 661-675, (1973)

....maximum likelihood function, where the size of the penalty depends on the number of nits required to encode the parameters and on prior knowledge about the parameters. The AIC is AIC = #) k (6.15) where k is the number of independent parameters. This is equivalent to Mallow s C p criterion [92], proposed in the linear regression context. I note that Shibata [136] and Li [79] show that AIC is asymptotically optimal (in the sense of minimising risk in selecting model dimension) provided the true distribution was not in the finite dimensional family considered. This optimality property of ....

C.L. Mallows. Some comments on c p . Technometrics, 15:661--675, 1973.

....multidimensional optimization with a vector of i s being optimized. 2.2 Objective function As mentioned earlier, the MPE can be estimated using the observed data. Therefore, an estimator of the MPE is a plausible choice of the objective function. To approximate the MPE one can use Mallows [7] CL ( H b X y CL n n i i trace 2 = 15) where H is the hat matrix defined as ( X V V X X X H diag = CL is an unbiased estimate of the MPE. However, it works reliably only for white Gaussian noise and correctly specified models. 2.3 Optimization of ....

C.L. Mallows, Some comments on CP", Technometrics, 15, No. 4, pp. 661675, (1973).

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Mallows, C. (1973), `Some comments on c p ', Technometrics 15, 661--675.

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Mallows, C.L. (1973). Some comments on C p . Technometrics 15, 661-675.

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Mallows, C. L. (1973). Some comments on C p . Technometrics, 15, 661--675.

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C. L. Mallows. Some comments on C P . Technometrics, 15(4):661--675, 1973.

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C. L. Mallows. Some comments on Cp . Technometrics, 15(4):661--675, November 1973.

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C.L. Mallows, "Some comments on CP ," Technometrics, vol. 15, no. 4, pp. 661--675, 1973.

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C.L. Mallows, "Some comments on C P ," Technometrics, vol. 15, no. 4, pp. 661--675, 1973.

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Mallows C.L. (1973). Some comments on Cp . Technometrics, 15, 661-675.

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C. Mallows, " Some comments on Cp ", " Technometrics ", 15, pp. 661 - 675, ( 1973 ).

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C.L. Mallows. Some comments on Cp. Technometrics, 40, 661-675, 1973.

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Mallows C.L. (1973) Some comments on Cp . Technometrics, 15, 661-675.

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Mallows, C.L. (1973). Some comments on C p , Technometrics 15, 661-675.

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Mallows, C.L. (1973) Some comments on C p . Technometrics, 15, 661-675.

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C. L. Mallows, Some Comments on CP , Technometrics, vol. 15, no. 4, pp. 661--675, 1973.

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C.L. Mallows, "Some comments on C P ," Technometrics, vol. 15, no. 4, pp. 661--675, 1973.

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C. L. Mallows, "Some comments on C P ", Technometrics, vol. 15, no. 4, pp. 661--675, 1973.

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Mallows, C.L. Some comments on Cp . Technometrics, 15:661-676, 1973.

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Mallows, C.L. Some comments on Cp . Technometrics, 15:661-676, 1973.

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