A badness-of-fit measure for the entire MDS representation.

Several measures exist.

Raw stress

σr (d′, X) = ∑i<j wij(dijdij(X))2
= ∑i<j wijdij2 + ∑i<j wijdij2(X) – 2 ∑i<j wijdijdij(X)
= ηd2 + η2(X) – 2ρ(d′, X)

where the dij are the disparities that are the result from the transformation of the dissimilarities, i.e., f(δij). Raw stress can be misleading because it is dependent on the normalization of the disparities. The following measure tries to circumvent this inconvenience.

Normalized stress

σn = σr / ηd2

This is the stress function that we minimize by iterative majorization. It goes back to De Leeuw (1977).

Kruskal's stress-1

σ1 = √ (∑i<j wij(dijdij(X))2 / ∑i<j wijdij2(X))1/2

In this measure, which is due to Kruskal (1964), stress is expressed in relation to the size of X.

Kruskal's stress-2

σ2 = √ (∑i<j wij(dijdij(X))2 / ∑i<j wij(dij(X) - averageDistance)2)1/2.

In general, this measure results in a stress value that is approximately twice the value for stress-1.

Relation between σ1 and σn

When we have calculated σn for Configuration X, disparities d′ and Weight W we cannot directly use X, d′ and W to calculate σ1 because the scale of X is not necessarily optimal for σ1. We allow therefore a scale factor b > 0 and try to calculate σ1 (d′, b X). We minimize the resulting expression for b and substitute the result back into the formula for stress, i.e.,

σ12 (d′, b X) = (ηd2 + b2 η2(X) – 2 b ρ(d′, X)) / b2 η2(X)
dσ12 (b) / db == 0, gives
b = ηd2 / ρ
σ12 = (1 - ρ2 / (ηd2·η2(X)))

This means that σ1 = √ σn.

Relation between σ2 and σn

We can do the same trick as before for σ2:

σ22 (d′, b X) = (ηd2 + b2 η2(X) – 2 b ρ(d′, X)) / (b2i<j wij(dij(X) - averageDistance)2)

From which we derive:

σ2 = √ ((ηd2 · η2(X) - ρ2(d′, X)) / (ηd2 · ∑i<j wij(dij(X) - averageDistance)2))

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© djmw, January 8, 1998