【摘要】 CRM(Center-Range Method,中点半径法)是求解区间数据回归模型的常用方法,其通过分别拟合区间中点和半径进行求解。研究了CRM方法的不足,当区间的中点波动比较大,而区间半径相对较小时,拟合中点和半径可能会导致许多预测区间与样本区间没有任何重叠。针对该问题,在CRM方法基础上,考虑在中点及半径误差平方和最小化的同时,增加一些约束条件,提出一种改进的区间回归方法。对区间回归分析的评价指标进行了研究,主要有均方根误差、决定系数、比率三个方面,选取这些指标用于本文所提出的约束方法的评价。通过蒙特卡洛模拟对约束方法进行评价,并选取2013年5月至2013年6月的沪深300指数和华夏上证50ETF构建区间样本数据,进行实证分析。模拟实验和实证分析都表明约束方法能够有效的减少预测区间和样本区间无任何重复的样本数目,在平均准确率和观测区间包含的预测区间的平均比率方面也具有明显的优势。更多还原

【Abstract】 Regression analysis is a statistical process for determining the relationships among variables. Traditional regression analysis takes point data as the research object. However, there are a large number of data which can’t be observed directly even though their variation intervals are available. For example, the stock index of a day is not a fixed data, because it always changes over time. A variety of methods to estimate the coefficients of interval regression models are studied in fuzzy theory, symbolic data analysis(SDA) as well as computer science. This paper also studies the estimation method for interval data. Symbolic data analysis is a theory of extracting systematic knowledge from huge data sets. In the framework of SDA, many regression methods have been developed. The Centre method(CM) assumes that the lower and upper bounds of the interval have the same coefficients, and the coefficients can be obtained by minimizing the sum of the square of the lower and upper bound errors. The Min Max method(Min Max) assumes that the coefficients of the lower and upper bounds are different, and they can be estimated separately by applying the Least Square method. The Center and Range method(CRM) uses the mid-points and ranges of the intervals to represent the intervals. Two linear regression relationships are constructed with the center and radius series. The coefficients can be calculated using the Least Square method. CRM performs the best among all these methods. One of the shortcomings of CRM is that the predicted interval may be meaningless, because the predicted radius is less than zero sometimes. The Constrained Center and Ranger method(CCRM) is proposed to solve this problem. In the CCRM, all the coefficients of the radius relationship are non-negative, which can ensure the forecast radius being non-negative. Another disadvantage of CRM is that it only fits the mid-points and the ranges of the intervals. In addition, it pays no attention to guarantee the prediction interval having overlaps with the sample interval. When the center series errors vary in a large range, there may be many predicted intervals which have no overlaps with the samples. The object of this paper is to solve this problem. A new constrained center and a range of methods are developed by adding some constrains to the CRM. The constraints ensure that the predicted intervals have some overlaps with the sample intervals. The constrained method can be expressed by a nonlinear programming. It is proved that the nonlinear programming is a convex programming. Thus, it can be solved through the K-T conditions. To evaluate the method, this paper studies the current main evaluating indicators. We summarize three kinds of measures, which are the Root Mean Square Error, the Coefficient of Determination and the Ratio. Some indicators are used in the paper, which are the root mean square error of the interval(IRMSE), the average accuracy rate(AR), the average percentage of predicted intervals contained in the observed intervals(PCO), as well as the average number of forecasts with 0% accuracy(0N). Both Monte Carlo simulations and empirical analysis are used to evaluate our method. In the Monte Carlo simulation experiments, both simple regression(p=1) and multivariate regression(p=3) are considered. In each circumstance, three conditions are considered. The main difference of these conditions is the ranges of the errors. The results show that the larger the range of the errors, the better the new method performs in the measures AR, PCO and 0N. The results of simple regression are different from the results of multivariate regression. The linear relationship between CSI 300 and 50 ETF is studied in the paper. The results indicate that the new method outperforms CRM in all the measures except the IRMSE. In a word, The Monte Carlo simulations and the empirical analysis show that the constrained method performs better or the same in the aspects of AR, PCO and 0N. Sometimes the constrained method has better results than the CRM method in all the measures. When there are some outliers in the samples, the new constrained method may not perform well. Thus, how to identify outliers and remove them are important research topics. Besides, how to achieve a balance between guaranteeing the overlaps and obtaining a lower IRMSE is another topic worth studying.更多还原