Entropy-Rescaled PCA, Step by Step

Entropy-rescaled PCA changes how positions on a principal-component map are turned into distances between groups. Each individual’s coordinate on an axis is replaced by its information value, so that differences near the centre of the map count very little and differences near the edge count a lot. Group positions are then averaged from the rescaled individuals, and the straight distance between groups is measured. This note explains the method in four parts: the rescaling curve, the four steps, a reading of a plain versus a rescaled map of a Southeast Balkans series, and a worked example with two individuals. The result is a position on a map, not an ancestry share, and it depends on the reference populations that build the axes.

A plain PCA map treats a step of the same size on an axis alike, whether it happens near the centre or near the edge. A small shift among central groups therefore weighs as much in a distance as the same shift among outer groups.

Entropy-rescaled PCA rescales each coordinate before anything is measured. The method is that of Taleb et al.¹, who call it informational rescaling of PCA maps. A coordinate between −1 and 1 is converted into the information it carries about its axis. That value stays flat near the centre and rises steeply toward the edge. The sections below follow the conversion from the general method to a concrete series of Southeast Balkans groups.

The core of the method is one function. An individual at position ρ on an axis gets the rescaled coordinate

Near the centre this value is almost flat: a coordinate of 0.10 becomes 0.005 and 0.30 becomes 0.047. Near the edge it climbs steeply: 0.85 becomes 0.64, and the curve keeps rising as ρ approaches 1. The sign is kept, so the curve is mirrored through the origin. Table 1 sets the rescaled coordinate against the plain PCA coordinate, which is simply ρ itself.

The rescaling is the third of four steps (Table 2). The other three decide where individuals are placed and how groups are compared. A shorter distance therefore means that two groups are closer on the same two axes.

On the rescaled map the centre is pulled together and the outer groups are pushed apart. The plain PCA map and the rescaled map of the series show the same groups, and group positions are averages of individuals.

Doclea and Bulgarian stay far out on the left. Turkish, Cypriot, Lazio Roman and West Anatolia Byzantine are stretched toward the lower right. Çinamak BA/IA, Illyria S IA and the modern Albanians collapse into one tight group near the origin, with Albania medieval beside them.

The axis scales differ between the two maps. Only the arrangement of the groups can be compared across them, not the numbers on the axes.

A point is converted one axis at a time, and each converted value keeps its sign. Two individuals, P and Q, are read off both axes and converted with the same curve (Table 3).

The point keeps its direction and moves only along PC1 and PC2, so it never jumps to the other side of the origin. The straight distance between the converted points is 1.03: the differences are 0.98 on PC1 and 0.32 on PC2, and √(0.98² + 0.32²) ≈ 1.03. The scale of the converted points differs from that of the original ones.

A rescaled map shows positions, not ancestry shares. Four points decide how it can be read.

Data and methods. The rescaling follows Taleb et al.¹: a coordinate ρ between −1 and 1 on one axis is replaced by sign(ρ) · (−½ ln(1 − ρ²)), with the natural logarithm. The values in Tables 1–3 are computed from this formula and rounded; Tables 2 and 3 are schematic and not from a run. In the Southeast Balkans series the axes are built from the reference populations before any group is placed, and group positions are averages of individuals. No qpAdm model is used in this note, and no position in it is an ancestry share.

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