Some results concerning off-training-set and IID error for the Gibbs and the Bayes optimal generalizers

In this paper we analyse the average behaviour of the Bayes-optimal and Gibbs learning algorithms. We do this both for off-training-set error and conventional IID (independent identically distributed) error (for which test sets overlap with training sets). For the IID case we provide a major extension to one of the better known results. We also show that expected IID test set error is a non-increasing function of training set size for either algorithm. On the other hand, as we show, the expected off-training-set error for both learning algorithms can increase with training set size, for non-uniform sampling distributions. We characterize the relationship the sampling distribution must have with the prior for such an increase. We show in particular that for uniform sampling distributions and either algorithm, the expected off-training-set error is a non-increasing function of training set size. For uniform sampling distributions, we also characterize the priors for which the expected error of the Bayes-optimal algorithm stays constant. In addition we show that for the Bayes-optimal algorithm, expected off-training-set error can increase with training set size when the target function is fixed, but if and only if the expected error averaged over all targets decreases with training set size. Our results hold for arbitrary noise and arbitrary loss functions.     

国内外能源价格的非线性特征研究——以原油和天然气为例

基于BDS法和替代数据法,对布伦特原油现货价、大庆原油现货价以及NYMEX天然气期货价和亨利中心天然气现货价的日数据进行了非线性特征检验。研究表明,上述能源价格收益率不服从正态分布,存在长相关特征并具有一定的非线性结构。在替代数据法检验时发现,嵌入维数小于6时,上述能源价格的原始数据和替代数据没有显著性差异,表明在维数低于6时,可以用少于6个经济变量进行线性模拟,以对相关能源价格进行短期预测,以便为中国能源政策制定及能源行业的科学发展提供有益启示。