Stress measures how faithfully multidimensional scaling (MDS) preserves original pairwise dissimilarities in a lower-dimensional representation. It acts as the model’s cost or loss: smaller values indicate closer agreement between original distances and fitted distances. section 6.2.2
For points i and j, stress compares the original distance dij with the lower-dimensional distance ||xi − xj||. MDS searches for coordinates that minimize the resulting discrepancy. section 6.2.2
A practical MDS workflow uses stress to assess dimensionality:
- Scale the input data or distance matrix before fitting MDS. section 6.3
- Fit MDS repeatedly across candidate component counts and record each model’s stress_ value. section 6.3
- Plot stress against the number of components and inspect the kink or elbow to select a practical dimensionality. section 6.3
- Interpret lower stress as better fit, while recognizing that optimization can converge at a local rather than global minimum. section 6.2.2
Comparing candidate MDS dimensions
A five-city distance matrix is scaled, fitted with MDS for one through nine components, and evaluated through mds.stress_. Plotting the collected values reveals kinks around two and three components; the chosen embedding can then be plotted and its stress printed. section 6.3
Table 6.2 Stress values and their goodness of fit
|
Stress values
|
Goodness of fit
|
|---|---|
|
0.200
|
Poor
|
|
0.100
|
Fair
|
|
0.050
|
Good
|
|
0.025
|
Excellent
|
|
0.000
|
Perfect
|
Use the stress-to-fit table as a quick interpretive reference: 0.200 is poor, 0.100 fair, 0.050 good, 0.025 excellent, and 0.000 perfect. section 6.2.2
