Tanimoto系数距离测量(Tanimoto coefficient distance measure)

Tanimoto系数距离测量(Tanimoto coefficient distance measure)

可以将两个对象具有相同的余弦和Tanimoto系数距离度量，其中

Tanimoto distance measure, d(x,y) = x.y / (|x|*|x|) + (|y|*|y|)- x*y

和

cosine measure, d(x,y) = x.y /(|x|* |x|) * (|y| *|y|)

Can two objects have identical cosine and Tanimoto coefficient distance measure, where

Tanimoto distance measure, d(x,y) = x.y / (|x|*|x|) + (|y|*|y|)- x*y

and

cosine measure, d(x,y) = x.y /(|x|* |x|) * (|y| *|y|)

最满意答案

Tanimoto相似系数 （ 不是真正的距离测量）由下式定义

d(x,y) = x.y / ((|x|*|x|) + (|y|*|y|)- x.y)

对于位向量x和y。

现在将其与余弦相似系数进行比较，

d(x,y) = x.y / (|x| * |y|)

分母因xy项而异。 如果xy为零，则Tanimoto和余弦相似系数将相同。

几何上，当且仅当x和y垂直时， xy为零。

由于x和y是位向量（即每个维度中的值只能是0或1），因此xy等于零意味着

x1*y1 + x2*y2 + ... + xn*yn = 0

如果xi * yi = 1 * 1 = 1，那么整个总和将为正。 要使整个总和为零，则没有术语 xi * yi等于1.它们必须全部等于0：

所以

x1*y1 = 0 x2*y2 = 0 ... xn*yn = 0

换句话说，如果xi是1，那么yi必须是0，反之亦然。

因此，有很多例子，Tanimoto相似度等于余弦相似度：

x = (0,1,0,1) y = (1,0,0,0)

例如。

The Tanimoto similarity coefficient (which is not a true distance measure) is defined by

d(x,y) = x.y / ((|x|*|x|) + (|y|*|y|)- x.y)

for bit vectors x and y.

Now compare that with the cosine similarity coefficent,

d(x,y) = x.y / (|x| * |y|)

The denominators differ by a x.y term. The Tanimoto and cosine similarity coefficients would be the same if x.y is zero.

Geometrically, x.y is zero if and only if x and y are perpendicular.

Since x and y are bit vectors (i.e. whose values in each dimension can only be 0 or 1), x.y equalling zero means

x1*y1 + x2*y2 + ... + xn*yn = 0

If xi*yi = 1*1 = 1, then the whole sum would be positive. For the whole sum to be zero, no term xi*yi can equal 1. They must all equal 0:

So

x1*y1 = 0 x2*y2 = 0 ... xn*yn = 0

In other words, if xi is 1, then yi must be 0, and vice versa.

So there are tons of examples where the Tanimoto similarity is equal to the cosine similarity:

x = (0,1,0,1) y = (1,0,0,0)

for instance.