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<>三点解释
* 比较的对象一般要远大于两个。(例如比较一个班级的成绩)
* 比较的指标也往往不只是一个方面的,例如成绩、工时数、课外竞赛得分等。
* 有很多指标不存在理论上的最大值和最小值,例如衡量经济增长水平的指标:GDP增速。
<>指标分类
* 极大型指标(效益型指标):越高(大)越好,如:成绩
* 极小型指标(成本型指标):越少(越小)越好,如坏品率
* 中间型指标:越接近某个值越好,如水质量评估时的PH
* 区间型指标:落在某个区间最好,如体温,水中植物性营养物量
<>建模步骤
1.将原始矩阵正向化
*
极小型指标 -> 极大型指标
max -x, 如果所有的元素均为正数,那么也可以使用1/x
*
中间型指标 -> 极大型指标
如:
*
区间型指标 -> 极大型指标
例如人的体温在36°~37°这个区间比较好
2.将正向化矩阵标准化
* 标准化的目的:消除不同指标量纲的影响
*
注意:标准化的方法有很多种,其主要目的就是去除量纲的影响,如:(x‐x的均值)/x的标准差,具体选用哪一种标准化的方法在多数情况下并没有很大的限制,这里我们采用的是前人的论文中用的比较多的一种标准化方法。
3.计算得分并归一化
有权重时计算得分:
只需在没有权重时的基础上:
*
注意:我们也可以先对标准化矩阵中的每个元素计算权重,然后直接用带权重的标准化矩阵来计算得分,这样得到的结果和上面在计算距离时引入权重得到的结果是几乎相同的。
* 可以用层次分析法给这些评价指标确定权重,当然,层次分析法的主观性太强了,更推荐使用熵权法来进行客观赋值。