多维标度

多维标度(英語:multidimensional scaling,缩写:MDS),又译多维尺度,又稱相似度结构分析similarity structure analysis),属于多重变量分析的方法之一,是社会学数量心理学市场营销统计实证分析的常用方法。

假設

  • 有許多特徵是互相關聯的,而受測者原本並不知道其特徵為何。
  • 存在著這樣一個空間:它的正交軸是欲尋找的特徵。
  • 这个特征空间满足这个要求:相似的对象能以相对较小的距离描摹出来

目的

多维标度是一个探索性的过程方法

  • 减少(观察)项目
  • 如果可能,在数据中揭示现有结构
  • 揭示相关特征
  • 寻找尽可能低维度的空间(“最小化条件”)
  • 空间必须满足“单调条件”
  • 解释空间的轴,依照假设提供关于感知和评判过程的信息

应用领域

用于评判和感知:

与其他多变量分析方法的比较

  • 相同:通过归因于少数几个不相關的特征来减少数据
  • 不同:多维标度仅仅需要相似性或者距离,而不需要相关性因子分析需要相关性
  • 如果仅仅对因子值感兴趣,可以用作因子分析的替代方法
  • 相同:把对象分组
  • 不同:聚类分析把观测到的特征当作分组标准,而多维标度仅仅取用感知到的差异
  • 为划分类别提供实际的支持

所使用的标量类型

  • 序数标量
  • 区隔标量
  • 比率标量

相似(度)矩阵

红色 橙色 黄色 绿色 蓝色 紫色
红色 -
橙色 6 -
黄色 8 0 -
绿色 10 8 9 -
蓝色 10 10 10 6 -
紫色 0 7 10 9 7 -

相似度矩阵举例(数字越小表示越相似)

例如,10个对象,2维空间,坐标个数则为10×2=20,“相似度”的个数为C102=45,数据压缩系数=相似度的个数÷坐标个数=45÷20=2.25(数据压缩系数要大于等于2才可接受,否则不能做多维标度分析)

数据采集的困难和问题

间接(数据)采集方法

完全排序法

Cn2对“相似度”进行排序,最相似的一对得到序数1,最不相似的一对得到序数Cn2

锚点法

评级法(Rating)

与“完全排序法”不同的是,虽然最相似的一对得到序数1,但是可以有多于一对得到相同的序数,最不相似的一对也不一定会依序得到Cn2

各种多维标度

简单多维标度

迭代多维标度

加权多维标度

对各维度进行不同的加权

多维展开(Multidimensional unfolding

多维标度方法

(古典)公制(多维)标度

  • 处理区隔标量和比率标量
  • 一定是采用欧氏距离

非公制多维标度

  • 处理序数标量
  • 不一定采用欧氏距离

评价

参见

外部链接

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