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Distance metrics in pattern recognition

A metric, , is a global function on a domain which indicates a human notion of distance. In our case, the domain is the set of all possible data instances. A metric function must conform to the following five axioms
1. non-negativity,
2. symmetry,
3. identity,
4. definiteness,
5. triangle inequality,
These constraints are not counter-intuitive. The first and second constraints define a metric as a scalar and not a directional vector quantity. The third property reflects the axiomatic belief that the distance between two objects must be zero when they are the identical. The fourth property strengthens the previous one by enforcing the constraint that the distance metric must only be zero when two objects are identical, and for no other pairs. The final constraint states that the minimal distance metric between two points must be that along the most direct path between them. The ubiquitous Euclidean distance over the multidimensional space of continuous real numbers, ,
Where the points take the form and , satisfies these constraints and is therefore a metric. A variant of the Euclidean, the Squared Euclidean , is also used in



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