A badness-of-fit measure for the entire MDS representation.
Several measures exist.
Raw stress
σ_{r} (d′, X) = ∑_{i<j} w_{ij}(d′_{ij} – d_{ij}(X))^{2} |
= ∑_{i<j} w_{ij}d′_{ij}^{2} + ∑_{i<j} w_{ij}d_{ij}^{2}(X) – 2 ∑_{i<j} w_{ij}d′_{ij}d_{ij}(X) |
= η_{d′}^{2} + η^{2}(X) – 2ρ(d′, X) |
where the d′_{ij} are the disparities that are the result from the transformation of the dissimilarities, i.e., f(δ_{ij}). 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
σ_{n} = σ_{r} / η_{d′}^{2} |
This is the stress function that we minimize by iterative majorization. It goes back to De Leeuw (1977).
Kruskal's stress-1
σ_{1} = √ (∑_{i<j} w_{ij}(d′_{ij} – d_{ij}(X))^{2} / ∑_{i<j} w_{ij}d_{ij}^{2}(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
σ_{2} = √ (∑_{i<j} w_{ij}(d′_{ij} – d_{ij}(X))^{2} / ∑_{i<j} w_{ij}(d_{ij}(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 σ_{1} and σ_{n}
When we have calculated σ_{n} for Configuration X, disparities d′ and Weight W we cannot directly use X, d′ and W to calculate σ_{1} because the scale of X is not necessarily optimal for σ_{1}. We allow therefore a scale factor b > 0 and try to calculate σ_{1} (d′, b X). We minimize the resulting expression for b and substitute the result back into the formula for stress, i.e.,
σ_{1}^{2} (d′, b X) = (η_{d′}^{2} + b^{2} η^{2}(X) – 2 b ρ(d′, X)) / b^{2} η^{2}(X) |
dσ_{1}^{2} (b) / db == 0, gives |
σ_{1}^{2} = (1 - ρ^{2} / (η_{d′}^{2}·η^{2}(X))) |
This means that σ_{1} = √ σ_{n}.
Relation between σ_{2} and σ_{n}
We can do the same trick as before for σ_{2}:
σ_{2}^{2} (d′, b X) = (η_{d′}^{2} + b^{2} η^{2}(X) – 2 b ρ(d′, X)) / (b^{2} ∑_{i<j} w_{ij}(d_{ij}(X) - averageDistance)^{2}) |
From which we derive:
σ_{2} = √ ((η_{d′}^{2} · η^{2}(X) - ρ^{2}(d′, X)) / (η_{d′}^{2} · ∑_{i<j} w_{ij}(d_{ij}(X) - averageDistance)^{2})) |
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© djmw, January 8, 1998