(New page: Metric Space (X,d) <math>d:X \times X \rightarrow \Re ^{+}</math> X is set, not necessarily vector space <math>x, y, z \in X</math> 1. <math>d(x,y)=d(y,x)</math> 2. <math>d(x,z)\leq d(...) |
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<math>x, y, z \in X</math> | <math>x, y, z \in X</math> | ||
| − | + | #<math>d(x,y)=d(y,x)</math> | |
| + | #<math>d(x,z)\leq d(x,y)+d(y,z)</math> | ||
| + | #<math>d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y)</math> | ||
| − | + | If X is vector space, metric can be induced by the norm <math>||\cdot||</math>. | |
| − | + | <math>d(x,y)=||y-x||</math> | |
| + | |||
| + | Norm is defined as follows | ||
| + | |||
| + | <math>||\cdot||: X \rightarrow \Re ^{+}</math> | ||
| + | |||
| + | #<math>|x| \geq 0, ||x||=0 \Leftrightarrow x=0</math> | ||
| + | #<math>||\alpha x||=|\alpha | ||x||</math> | ||
| + | #<math>||x+y|| \leq ||x|| + ||y||</math> | ||
| + | |||
| + | [[Category:ECE662]] | ||
| + | Defining metric, we can measure similarity of elements of set X. | ||
| + | |||
| + | Example of metric | ||
| + | #Minkowski Metric <math> \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p}</math> | ||
| + | #Riemannian Metric <math>D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})}</math> | ||
| + | #Tanimoto metric <math>D(S_1, S_2) = \frac {|S_1|+|S_2|-2|S_1 \bigcap S_2| }{|S_1|+|S_2|-|S_1 \bigcap S_2|} </math> | ||
| + | #Procrustes metric <math>D(p,\bar p)= min_{R,T} \sum_{i=1}^n | ||
| + | {\begin{Vmatrix} Rp_i+T-\bar p_i \end{Vmatrix}} _{L^2} </math>, R: Rotation, T: Translation | ||
Latest revision as of 08:34, 10 April 2008
Metric Space (X,d) $ d:X \times X \rightarrow \Re ^{+} $
X is set, not necessarily vector space
$ x, y, z \in X $
- $ d(x,y)=d(y,x) $
- $ d(x,z)\leq d(x,y)+d(y,z) $
- $ d(x,y) \geq 0, d(x,y)=0 \Leftrightarrow x=y) $
If X is vector space, metric can be induced by the norm $ ||\cdot|| $.
$ d(x,y)=||y-x|| $
Norm is defined as follows
$ ||\cdot||: X \rightarrow \Re ^{+} $
- $ |x| \geq 0, ||x||=0 \Leftrightarrow x=0 $
- $ ||\alpha x||=|\alpha | ||x|| $
- $ ||x+y|| \leq ||x|| + ||y|| $
Defining metric, we can measure similarity of elements of set X.
Example of metric
- Minkowski Metric $ \left( \sum_{i=1}^n \left| x_i - y_i \right|^p \right)^{1/p} $
- Riemannian Metric $ D(\vec{x_1},\vec{x_2})=\sqrt{(\vec{x_1}-\vec{x_2})^\top \mathbb{M}(\vec{x_1}-\vec{x_2})} $
- Tanimoto metric $ D(S_1, S_2) = \frac {|S_1|+|S_2|-2|S_1 \bigcap S_2| }{|S_1|+|S_2|-|S_1 \bigcap S_2|} $
- Procrustes metric $ D(p,\bar p)= min_{R,T} \sum_{i=1}^n {\begin{Vmatrix} Rp_i+T-\bar p_i \end{Vmatrix}} _{L^2} $, R: Rotation, T: Translation
