估计模型维度的双评分准则及其应用
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摘要
在预测问题中,如经济预测,天气预报,地震预报案,用户要求预报准确;尤其是所谓“大趋势”或“大方向”一定要报对,如明年是丰收还是歉收,股票未来升值还是贬值,汛期洪峰高还是低。如果趋势报反了,模型就失去了价值。基于这样的思想,我们从权衡预测模型的数量误差(精评分S_1)和趋势误差(粗评分S_2)同时达到最小来确定模型维度,故称为双评分准则(CSC);其原理既不同于传统的统计检验也不同于Akaike信息量(AIC)。文中提出了一种新的CSC表达式。在线性统计模型中,S_1取为与均方根误差等价的量,即(N-k)R~2,N为样本量,k为模型线度,R为复相关系数,S_2取为Kullback最小判别信息统计量。证明了S_1中包含了信息量准则(AIC,BIC,HIC)所考虑的因素:殊差平方和小和维度k少;推导了假定S_1与S_2独立时CSC的分布为一渐近X~2分布。对两个不同试验区小麦产量(1855-1884年)之差的著名例子进行了计算,所选2阶多项式回归与运用AIC所得结果和Fisher-Yates判断一致。另给出了预报旬降水量的气象实例
In respect of prediction issues, such as economic forecast, weather forecast, seismic prediction and so on, a user requires correct forecast , in particular , a so-called "grand trend" or " grand direction " must be forecasted correctly, e.g. being rich harvest or poor harvest in the next year , a stock being risen or fallen in the future , flood peak being high or low in the following summer. If the trend forecast is missed , a model will be unvalued. Along with this idea , weighting both the quantitative error (fine score S1) and the qualitative trend error (rough score S2 ) and minimizing S1 and S2 simultaneously , a model dimension can be determined , so it is called the couple score criterion (CSC) . The CSC principle is not only different from a traditional statistical test but also from the Akaike information citerion (AIC). We propose a new CSC expression here . In the linear statistical model , let S1 be a quantity which corresponds to a root mean square error , i.e. (N-k)R2, where N is a sample size , k the model dimension , R multiple correlation coefficient , and let S2 be the Kullback minimum discriminant information statistic, It is proved that S1 contains both factors; small residence square sum and less dimension k, which are accounted for by the information ctierion (AIC, BIC, HIC); It is derived that the distribution of CSC is a asymtotic X2 statistical distribution if S1 being independent from S2 is supposed. A famous example of wheat production for 1855 to 1884 in two experimental areas is computed with CSC and the selected polynomial regression with 2 orders is coincident with that obtained by the use of AIC and also in agreement with the inference by Fisher-Yates. In addition , a meteorological case which predicts total precipitation for ten days is given.
In respect of prediction issues, such as economic forecast, weather forecast, seismic prediction and so on, a user requires correct forecast , in particular , a so-called "grand trend" or " grand direction " must be forecasted correctly, e.g. being rich harvest or poor harvest in the next year , a stock being risen or fallen in the future , flood peak being high or low in the following summer. If the trend forecast is missed , a model will be unvalued. Along with this idea , weighting both the quantitative error (fine score S1) and the qualitative trend error (rough score S2 ) and minimizing S1 and S2 simultaneously , a model dimension can be determined , so it is called the couple score criterion (CSC) . The CSC principle is not only different from a traditional statistical test but also from the Akaike information citerion (AIC). We propose a new CSC expression here . In the linear statistical model , let S1 be a quantity which corresponds to a root mean square error , i.e. (N-k)R2, where N is a sample size , k the model dimension , R multiple correlation coefficient , and let S2 be the Kullback minimum discriminant information statistic, It is proved that S1 contains both factors; small residence square sum and less dimension k, which are accounted for by the information ctierion (AIC, BIC, HIC); It is derived that the distribution of CSC is a asymtotic X2 statistical distribution if S1 being independent from S2 is supposed. A famous example of wheat production for 1855 to 1884 in two experimental areas is computed with CSC and the selected polynomial regression with 2 orders is coincident with that obtained by the use of AIC and also in agreement with the inference by Fisher-Yates. In addition , a meteorological case which predicts total precipitation for ten days is given.
引文
[1] Akaike.H., A new look at the statistical model identification, IEEE Transaction on Automatic Control, Vol.19, 716-723．
[2] 曹鸿兴,牛保山,张承先,统计模型选择的双评分准则及其在气象、水文预报中的应用,数理统计与应用概率,Vol.4 No.1,5-10,1989．
[3] Kullback, S., Information Theory and Statistics, Dover Publications, New York, 1959, 155-159
[4] 顾震潮,从信息论看天气预报的评分和使用,气象学报,Vol.28,1957,256-263．
[5] Kendall,S.M. and A. Stuart, The Advanced Theory of Statistcs, Vol. 2, Charles Griffin & Company London, 1979, 361-366．
[6] 中山大学概率论与数理统计编写组,概率论及数理统计,高等教育出版社,1980,上册139-151．
[7] 刘章温,赤池信息量准则AIC及其意义,数学的实践与认识,Vol.3,1980,64-71．
[2] 曹鸿兴,牛保山,张承先,统计模型选择的双评分准则及其在气象、水文预报中的应用,数理统计与应用概率,Vol.4 No.1,5-10,1989．
[3] Kullback, S., Information Theory and Statistics, Dover Publications, New York, 1959, 155-159
[4] 顾震潮,从信息论看天气预报的评分和使用,气象学报,Vol.28,1957,256-263．
[5] Kendall,S.M. and A. Stuart, The Advanced Theory of Statistcs, Vol. 2, Charles Griffin & Company London, 1979, 361-366．
[6] 中山大学概率论与数理统计编写组,概率论及数理统计,高等教育出版社,1980,上册139-151．
[7] 刘章温,赤池信息量准则AIC及其意义,数学的实践与认识,Vol.3,1980,64-71．