Review: Some distance properties of latent root and vector methods used in multivariate analysis¶
Citation
- Gower, J. C. (1966). Some distance properties of latent root and vector methods used in multivariate analysis. Biometrika, 53(3-4), 325–338.
- DOI
Summary¶
This paper is concerned with the representation of a multivariate sample of size n as points P₁, P₂, …, Pₙ in a Euclidean space. The interpretation of the distance Δ(Pᵢ, Pⱼ) between the ith and jth members of the sample is discussed for some commonly used types of analysis, including both Q and R techniques. When all the distances between n points are known a method is derived which finds their co-ordinates referred to principal axes. A set of necessary and sufficient conditions for a solution to exist in real Euclidean space is found. Q and R techniques are defined as being dual to one another when they both lead to a set of n points with the same inter-point distances. Pairs of dual techniques are derived. In factor analysis the distances between points whose co-ordinates are the estimated factor scores can be interpreted as D² with a singular dispersion matrix.
Gower (1966) derives the construction Hifuku uses for classical multidimensional scaling, also called principal coordinates analysis, and states the condition under which it yields a real configuration. Section 3 contains both results.
Recovering coordinates from distances¶
Given a symmetric association matrix \( \mathbf{A} \) with latent roots \( \lambda_1 \ldots \lambda_n \) and vectors normalized so that the sum of squares of each vector equals its root, Gower shows (equation 4) that the distance between the points formed from the rows satisfies
The step that turns this into a method appears immediately afterward. Setting
gives \( \Delta(Q_i, Q_j) = d_{ij} \), which the paper describes as a direct method of finding the coordinates of a set of points from their interdistances alone. This is the origin of the familiar \( -\tfrac{1}{2}\mathbf{D}^2 \) construction.
Because adding a constant to every element of \( \mathbf{A} \) leaves the distances unchanged, the matrix may be centered without loss. Equation 7 gives the double-centering
and equation 8 confirms that it preserves the distance property. The rows of \( \alpha \) sum to zero, so \( \alpha \) has a zero root and the resulting coordinates have their centroid at the origin.
The computational procedure is set out in five steps at the end of Section 3: form \( \mathbf{A} \), transform to \( \alpha \) by equation 7, take the latent roots and vectors of \( \alpha \) scaling each vector so its sum of squares equals its root, read the coordinates from the rows, and obtain the residual sum of squares as the difference between the trace of \( \alpha \) and the sum of the \( k \) largest roots retained.
That last step is what makes the eigenvalue spectrum interpretable as retained and discarded variance, which is how Hifuku reports the dimensionality of a set of anchor distances.
The condition for a real configuration¶
Section 3 also gives the result that matters most for Hifuku's metric. A real configuration of points reproducing the distances exists if and only if \( \alpha \) is positive semi-definite. Gower notes that it is sufficient for \( \mathbf{A} \) to be positive semi-definite, since then all roots are non-negative.
The paper is explicit about the failure mode. Where two or more roots are negative, points cannot be found in real space with the required distances. A configuration may still be constructed, but it involves imaginary coordinate axes, and the consequences are geometrically incoherent: Gower gives the example of points \( Q_1(1, i) \) and \( Q_3(-1, -i) \), which are at zero distance from one another yet appear distant in any real model. Where the negative roots are small in modulus the practical effect is slight. A large negative root is not recoverable.
This is the reason Hifuku reports non-Euclidean mass at all. The tree metric
combines a squared branch-score term with a Robinson-Foulds term, and both are
squared Euclidean distances, so their non-negative combination is squared Euclidean
for any set of trees. That property is verified independently in
docs/theory/algebra/kuhner1994-robinson1981-euclidean-embedding.py. By Gower's
condition the centered matrix is therefore positive semi-definite by construction,
and the negative eigenvalue mass must be zero.
A measured value of zero consequently confirms the metric and the implementation. It is not a finding about the data, and it should not be reported as one. What the spectrum does measure is dimension: how many axes the distances occupy, and therefore how much a flat map discards.
A note on attribution¶
Gower observes that comparable results had been reported by Torgerson (1958), but that the earlier work does not appear to recognize the connection with principal components analysis. The construction is accordingly often called Torgerson scaling or Torgerson-Gower scaling. Hifuku cites Gower for the derivation of the necessary and sufficient condition, which is the part the implementation depends on.
In-document navigation¶
| Section | Content |
|---|---|
| Section 1 | Introduction: association matrices and numerical classification |
| Section 2 | Principal component analysis and the implied interpoint distance |
| Section 3 | Coordinates from distances, equations 2 to 8, the p.s.d. condition, and the five-step procedure |
| Section 4 | Duality of Q and R techniques |
| Section 5 | Factor analysis and distances between factor scores |
| Section 6 | Proximity analysis |
| Section 7 | Conclusion |