Predictors of a multiple linear regression equation selected by GCV (Generalized Cross Validation) may contain undesirable predictors with no linear functional relationship with the target variable, but are chosen onl...Predictors of a multiple linear regression equation selected by GCV (Generalized Cross Validation) may contain undesirable predictors with no linear functional relationship with the target variable, but are chosen only by accident. This is because GCV estimates prediction error, but does not control the probability of selecting irrelevant predictors of the target variable. To take this possibility into account, a new statistics “GCVf” (“f”stands for “flexible”) is suggested. The rigidness in accepting predictors by GCVf is adjustable;GCVf is a natural generalization of GCV. For example, GCVf is designed so that the possibility of erroneous identification of linear relationships is 5 percent when all predictors have no linear relationships with the target variable. Predictors of the multiple linear regression equation by this method are highly likely to have linear relationships with the target variable.展开更多
Conventional Akaike’s Information Criterion (AIC) for normal error models uses the maximum-likelihood estimator of error variance. Other estimators of error variance, however, can be employed for defining AIC for nor...Conventional Akaike’s Information Criterion (AIC) for normal error models uses the maximum-likelihood estimator of error variance. Other estimators of error variance, however, can be employed for defining AIC for normal error models. The maximization of the log-likelihood using an adjustable error variance in light of future data yields a revised version of AIC for normal error models. It also gives a new estimator of error variance, which will be called the “third variance”. If the model is described as a constant plus normal error, which is equivalent to fitting a normal distribution to one-dimensional data, the approximated value of the third variance is obtained by replacing (n-1) (n is the number of data) of the unbiased estimator of error variance with (n-4). The existence of the third variance is confirmed by a simple numerical simulation.展开更多
The use of prediction error to optimize the number of splitting rules in a tree model does not control the probability of the emergence of splitting rules with a predictor that has no functional relationship with the ...The use of prediction error to optimize the number of splitting rules in a tree model does not control the probability of the emergence of splitting rules with a predictor that has no functional relationship with the target variable. To solve this problem, a new optimization method is proposed. Using this method, the probability that the predictors used in splitting rules in the optimized tree model have no functional relationships with the target variable is confined to less than 0.05. It is fairly convincing that the tree model given by the new method represents knowledge contained in the data.展开更多
文摘Predictors of a multiple linear regression equation selected by GCV (Generalized Cross Validation) may contain undesirable predictors with no linear functional relationship with the target variable, but are chosen only by accident. This is because GCV estimates prediction error, but does not control the probability of selecting irrelevant predictors of the target variable. To take this possibility into account, a new statistics “GCVf” (“f”stands for “flexible”) is suggested. The rigidness in accepting predictors by GCVf is adjustable;GCVf is a natural generalization of GCV. For example, GCVf is designed so that the possibility of erroneous identification of linear relationships is 5 percent when all predictors have no linear relationships with the target variable. Predictors of the multiple linear regression equation by this method are highly likely to have linear relationships with the target variable.
文摘Conventional Akaike’s Information Criterion (AIC) for normal error models uses the maximum-likelihood estimator of error variance. Other estimators of error variance, however, can be employed for defining AIC for normal error models. The maximization of the log-likelihood using an adjustable error variance in light of future data yields a revised version of AIC for normal error models. It also gives a new estimator of error variance, which will be called the “third variance”. If the model is described as a constant plus normal error, which is equivalent to fitting a normal distribution to one-dimensional data, the approximated value of the third variance is obtained by replacing (n-1) (n is the number of data) of the unbiased estimator of error variance with (n-4). The existence of the third variance is confirmed by a simple numerical simulation.
文摘The use of prediction error to optimize the number of splitting rules in a tree model does not control the probability of the emergence of splitting rules with a predictor that has no functional relationship with the target variable. To solve this problem, a new optimization method is proposed. Using this method, the probability that the predictors used in splitting rules in the optimized tree model have no functional relationships with the target variable is confined to less than 0.05. It is fairly convincing that the tree model given by the new method represents knowledge contained in the data.
