予測結果の評価に関する一考察

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Many methods of evaluation of accuracy of prediction are proposed. The percentage of verification, the coefficient of association, X^2-test, RMS error, a value derived from the theory of entropy I, Theil's inequality coefficient are the examples of them. In this paper, I proposed tentatively U^* which is derived from Theil's inequality coefficient, and is defined by [numerical formula] (4.13) where π_i is the ith prediction P_i or the logarithm of it, log P_i, α_i is the ith actual value A_i or the logarithm of it, log A_i. When the π_i is P_i and the α_i is A_i, the coefficient U^* is the coefficient of inequality of difference (which is the coefficient of inequality measured by the difference between P_i and A_i) and is equivalent to the Theil's inequality coefficient U. And when π_i is log P_i and α_i is log A_i the coefficient U^* become the coefficient of inequality of proportion U' (which is the coefficient of inequality measured by the proportion of P_i to A_i). Therefore, the coefficient U^* generates systematically the coefficient U (Theil's inequality coefficient), that is the coefficient of inequality of difference, and the coefficient U', that is the coefficient of inequality of proportion (When we consider the π_i and α_i in the formula of U^* as P_i and A_i, respectively, U^* becomes the U, and when we consider them as log P_i and log A_i, U^* becomes the U'). If we can have an assumption that P_i=a+bA_i+ε_i (4.4) or (7.1) (where a and b are parameters and ε_i is the difference between P_i and a + bA_i) in other words, that the condition : [numerical formula] (7.2) is accepted (because, if [numerical formula], then we can get the relation expressed by equation (7.2)), and if we consider the difference between P_i and A_i, it would be better that the coefficient U is used. On the other hand, if we can have an assumption that log P_i=a+b log A_i+εi (4.10) or (7.3) or P_i = (kA^b)e_i (4.11) or (7.4) (where a, b, and k are parameters, ε_i is the difference between log p_i and a+b log A_i, and log e_i is ε_i) in other words, that the condition : P_i>0 (7.5) is accepted (because, even if [numerical formula], P_i is always positive), and if we consider the proportion of P_i to A_i or the value log P_i-log A_i, it would be better to calculate the coefficient U'. But, even if we have the condition [numerical formula] or P_i>0, the both U and U' may be used in each case, when we want only to measure the goodness of prediction by a measure of difference or a measure of proportion having no connection with the conditions. Furthermore, the measures s^<(+)*> (which may be called "positive standard deviation") and s^<(-)*> (which may be called "negative standard deviation") defined by equations (5.1) and (5.2) in this paper may be utilized when we want to know the distribution of the difference P_i-A_i or log P_i-log A_i. And even if the type of the distribution (probability density function) of the predictions are not strictly known, we can quantitatively show or discript the range of the distribution of predictions around the actual values by means of s^<(+)*> and s^<(-)*>. When we express the range of the existence of prediction by s^<(+)*> and s^<(-)*>, the range is considerably wide. But, it is thought that it is very difficult to express the range by narrower interval since, in general, the type of the distribution of prediction is not specified.
流通経済大学の論文

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鈴木啓祐
流通経済大学

予測結果の評価に関する一考察

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