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2.2 Feature Space Metrics 29

2.2 Feature Space Metrics

In the previous chapter we used a distance measure for assessing pattern similarity
in the feature space. A measuring rule d(x,y) for the distance between two vectors
x and y is considered a metric if it satisfies the following properties:

(2- 10a)
(2- lob)
(2- 1 OC)
(2- 1 Od)

If the metric has the property

d(ax, ay) = la1 d(x,y) with a€ 31, (2- 10e)
it is called a norm and denoted d(x, y)= Ilx - yll .
The most "natural" metric is the Euclidian norm when an adequate system of
coordinates is used. For some classification problems it may be convenient to
evaluate pattern similarity using norms other than the Euclidian. The four most
popular norms for evaluating the distance of a feature vector x from a prototype m,
all with do = 0, are:

Euclidian norm:

d
Squared Euclidian norm': Ilx -mils = (Y, - rn, >2
i= l
d
City-block norm: (Ix - mlIc = Ixi mi ( (2- 1 1 c)
,=I
Chebychev norm: IIx -mil, = maxi(Jxj - mil) (2- 1 1 d)

Compared with the usual Euclidian norm the squared Euclidian norm grows
faster with patterns that are further apart. The city-block norm dampens the effect
of single large differences in one dimension since they are not squared. The
Chebychev norm uses the biggest deviation in any of the dimensions to evaluate
the distance. It can be shown that all these norms are particular cases of the power
norm':

' Note that for IIx - mils and /r - mllp,r we will have in relax the norm definition. allowing
a scale factor for (a(.