Distance measure - cosine distance

Python Learning Series ： Catalog

One 、 summary

Trigonometric functions , I believe everyone has learned in junior high school , And what we're talking about here Cosine distance （Cosine Distance） The calculation formula is similar to that learned in high school .

In Geometry , The cosine of the included angle can be used to measure two directions （ vector ） The difference of ; So it can be extended to machine learning , To measure the difference between sample vectors .

therefore , Our formula has to be changed slightly , So that it can be expressed as a vector .

Two 、 Calculation formula

① Cosine distance in two-dimensional plane

hypothesis Two dimensional plane There are two vectors inside ：$A(x_1,y_1)$ And $B(x_2,y_2)$

Then the $A$、$B$ The cosine distance formula of two vectors is ：

$$cos(\theta )=\frac{a\cdot b}{|a||b|}$$

$$\begin{array}{rl}cos(\theta )& =\frac{a\cdot b}{|a||b|}\\ & =\frac{x_1{x}_2+y_1{y}_2}{\sqrt{x_1^2+y_1^2}\sqrt{x_2^2+y_2^2}}\end{array}$$

② n Cosine distance in dimensional space

Generalized to n Two vectors of dimensional space $A(x_{11},x_{12},...,x_{1n})$ And $B(x_{21},x_{22},...,x_{2n})$, Then the cosine distance is ：

$$\begin{array}{rl}cos(\theta )& =\frac{a\cdot b}{|a||b|}\\ & =\frac{{\sum }_{k=1}^n{x}_{1k}{x}_{2k}}{\sqrt{{\sum }_{k=1}^n{x}_{1k}^2}\sqrt{{\sum }_{k=1}^n{x}_{2k}^2}}\end{array}$$

③ Be careful

• The value range of cosine distance is $[-1,1]$.
• The larger the cosine, the smaller the angle between the two vectors , The smaller the cosine, the larger the angle between the two vectors .
• When the directions of two vectors coincide, the cosine takes the maximum $1$, When two vectors are in opposite directions, the cosine is the minimum $-1$.