Does NMDS require raw feature vectors?

No. It can work from an appropriate pairwise dissimilarity matrix.

Does it preserve distance ratios?

Not generally. Preserving ordinal relationships does not preserve numerical ratios.

Is the two-dimensional solution unique?

No. Geometric symmetries and different optimization solutions can produce different coordinates.

AI Glossary / Probabilistic machine learning / Advanced

Nonmetric multidimensional scaling

Definition

Nonmetric multidimensional scaling (NMDS) represents objects as points in a low-dimensional space while trying to preserve the ordering of their pairwise dissimilarities. It fits a monotonic relationship between the supplied dissimilarities and distances on the map, rather than requiring the numerical dissimilarities themselves to become the plotted distances.

What it means

People may reliably judge that two products are more alike than another pair without making meaningful numerical statements about exactly how much more alike. NMDS uses this ordering information to construct a spatial representation.
Nearby points should generally correspond to relatively similar objects, subject to the quality of the fit. However, the axes do not automatically represent named attributes, and distances do not retain the input rating scale. A visually attractive map is an approximation that needs checking, not a discovery of uniquely determined coordinates.

How it works

Begin with pairwise dissimilarities, a chosen embedding dimension and initial coordinates. Calculate distances between the coordinates. Fit ordered target distances, called disparities, using a monotonic transformation of the input dissimilarities. Isotonic regression is one way to perform this step.
The coordinates and disparities are adjusted to reduce a stress criterion measuring their disagreement. Scale constraints or normalization prevent a meaningless collapsed solution. Different stress definitions and tie-handling conventions exist, so comparisons should identify the exact criterion used.
The optimization can depend on initialization. Repeated starts, inspection of the distance-versus-dissimilarity relationship and comparisons across dimensions help assess a result. Monotonicity is generally nondecreasing: fitted disparities may be tied even when some input dissimilarities differ. The aim is an ordinal representation, not guaranteed perfect preservation of every strict inequality.

Three practical examples

Product research: analysts map products using perceived dissimilarity judgments. They compare the map with independently recorded attributes before assigning business meaning to its axes.
Document exploration: researchers use a pairwise dissimilarity measure to inspect broad relationships among document groups. They check individual pair fits before interpreting isolated points.
Survey comparison: a team constructs maps for different respondent groups and aligns them before comparing patterns. A rotation or reflection alone is not evidence that the groups disagree.

Think of it this way

Imagine arranging places on a sketch using only statements such as “these two are closer than those two.” The sketch can respect those comparisons without matching actual kilometers. Turning the paper around changes the coordinates but not the relationships. Some collections of comparisons may be impossible to reproduce perfectly in the chosen number of dimensions.

Worked example

Consider an original example with three objects A, B and C. Their dissimilarities are AB = 1, BC = 2 and AC = 9. The relevant order is AB < BC < AC.
Place the objects on a line at A = 0, B = 1 and C = 3. Their distances are AB = 1, BC = 2 and AC = 3. A monotonic transformation that maps input values 1, 2 and 9 to disparities 1, 2 and 3 matches these distances exactly. This gives zero discrepancy for these pairs, although the map does not preserve the numerical value 9.
By contrast, exact numerical preservation of all three input dissimilarities as Euclidean distances is impossible: distance AC cannot exceed distance AB plus distance BC, yet 9 exceeds 1 + 2. The ordinal objective avoids requiring that impossible equality.
Now inspect a different fixed layout, A = 0, B = 2 and C = 3. In input-order sequence AB, BC, AC, its distances are 2, 1 and 3. The first two violate the desired ordering. Equal-weight isotonic least squares pools them to disparities 1.5, 1.5 and 3. The residual sum of squares is (2 - 1.5) squared + (1 - 1.5) squared + (3 - 3) squared = 0.5.
For this explicitly specified, unrescaled fixed-layout calculation, dividing by the sum of squared map distances gives 0.5 / (4 + 1 + 9) = 1/28. Taking the square root gives approximately 0.189. This illustrates a normalized discrepancy, not the reported output of a complete software run with its own normalization and iterative updates. Moving the coordinates to the first layout removes the discrepancy. No optimizer execution or unique solution is claimed.

Illustrative case study

Fictional company Aspen Products surveys how customers perceive similarities among several service packages. Its analysts construct a dissimilarity matrix, record how ties and missing responses are handled, and compare NMDS solutions from several starting layouts.
Before labeling the axes “convenience” and “cost,” they compare the coordinates with separate customer ratings on those attributes. They also inspect poorly fitted pairs and repeat the analysis across respondent subsets. The map becomes a discussion aid supported by the survey evidence. This is a fictional workflow and does not assert that a particular market structure or commercial improvement was discovered.

Common misconceptions and mistakes

Nonmetric does not mean that the plotted distances lack ordinary geometric properties. The output can use Euclidean distances; it is the relationship to the input dissimilarities that is ordinal. Low stress does not prove the dissimilarity measure is substantively appropriate.
A map can be translated, rotated or reflected without changing its distances. Consequently, raw coordinate signs are not intrinsic findings. A low-dimensional representation may also distort some pairs despite a reasonable overall score. Universal stress cutoffs are risky without considering dimension, data size, ties and the particular stress definition.

Business use

NMDS can help communicate perceived relationships when numerical rating intervals are difficult to justify. It is useful for exploration and discussion, provided the input judgments and map fit are explained. Decisions should rely on supporting evidence about customers, products or outcomes rather than treating visual clusters as automatically meaningful market segments.

Check your understanding

Why can the first toy layout fit ordinal information perfectly?

Answer: Its three distances have the same order as the input dissimilarities.

Why can no Euclidean layout reproduce 1, 2 and 9 exactly?

Answer: Those values violate the triangle inequality.

Why were 2 and 1 replaced with equal disparities?

Answer: Their order violated the monotonic constraint, so equal-weight isotonic regression pooled them.

Related terms

Multidimensional Scaling
Isotonic regression
Dimensionality reduction
Dissimilarity matrix
Stress
Ordinal data
Shepard diagram

Learn next

Plot input dissimilarities against map distances, compare monotonic fits and inspect which pairs contribute most to stress. Then compare solutions across dimensions and initializations.

Sources

Scikit-learn User Guide, Multidimensional Scaling, version 1.8, describes disparities, isotonic regression and stress: https://scikit-learn.org/1.8/modules/manifold.html#multidimensional-scaling . The three-object calculation and fictional case are original teaching illustrations. Official versioned NMDS discussion independently read; isotonic pooling and fixed-layout discrepancy recalculated.

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