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conceptual vector
distances

Generality

Many distances are possible (to be continued)

Angular distance

Let X and Y two conceptual vectors of dimension n. Let alpha, the angle between X and Y, we have :

cos(alpha) = X . Y
= (x₁y₁ + … + x_ny_n) / |X||Y|
= (x₁y₁ + … + x_ny_n) / (sqrt(x₁x₁ + … + x_nx_n) * sqrt(y₁y₁ + … + y_ny_n))

cos(alpha) is the dot product between X and Y. We call D_A(X,Y) Angular Distance between X and Y, the alpha value :

D_A(X,Y) = alpha = Acos(X . Y)

We have the following properties (some are common to any distance function) :

D_A(X,X) = 0

D_A(X,Y) = D_A(Y,X)

D_A(X,Y) + D_A(Y,Z) &Mac179; D_A(X,Z)

0 &Mac178; D_A(X,Y) &Mac178; pi

Intensity distance

Let X and Y two conceptual vectors. Let V be the intersection between X and Y such that:

V = Inter(X, Y) = sqrt²(X_Y)

hence each v_i = (x_iy_i)^1/2

We recall the mass of V :

Mass(V) = sqrt(v₁² + … + v_n²) / encoding-size
= sqrt(x₁y₁ + … + x_ny₁) / encoding-size

Note that in this case, we always have : 0 &Mac178; Mass(V) &Mac178; 1

D_I(X,Y) = Acos(Mass(V))

Canonical distance

Let X and Y two conceptual vectors. The Canonical distance (aka Euclidian) is :

D_C(X,Y) = srqt ((x₁-y₁)^2 + … + (x_i-y_i)^2 + … + (x_i-y_i)^2)

Playing field

Corrected Angular distance Get the between

between amour

and passion

with the lexicon as

The Angular Distance = Non Corrected Absoluted Synonymy.

Natural distances Get the between

between amour

and passion

with the lexicon as

Rationale

The Intensity Distance take into accont only the norm of the relative intersection between X and Y, which the Angular Distance cannot. If D_A(X,Y) is close to 0 (resp. close to pi/2), then D_I(X,Y) will be close to 0 (resp. close to pi/2). The D_I(X,Y) may be diffrent from the D_A(X,Y) for intermediate values depending on ow X and Y intersect.

For a given intermediate D_A(X,Y), we may have :

D_I(X,Y) closer to 0 as X and Y intersect on few high components

D_I(X,Y) closer to pi/2 as X and Y intersect on many low components

Any situation between these two extremes (in that case D_I(X,Y) is not particulary usefull).

last update17 april 2001*

mathieu lafourcade LIRMM - 161, rue ADA - 34392 Montpellier Cedex 5 - France - Tél : (33) 04 67 41 85 71 - Fax : (33) 04 67 41 85 00 - courriel : lafourca@lirmm.fr