The same PCA axes and individuals read on the plain map and on the entropy-rescaled maps of Taleb et al.: which statements about the nearest group, the side and the position between two groups change, and which do not.
Mvagnus Albertus · Popualtion Geneticist
Published · 15 min read
A plain PCA map and an entropy-rescaled map are built from the same axes and the same individuals; they differ only in how a position is turned into a distance. This article asks what that changes for modern English groups. Three modern English groups and modern Orkney are read against 27 comparison groups on the conventional map C and on the entropy-rescaled maps A and B (Taleb et al.1), over three reference panels (two projections on WE8, one on EU5), 16 origin shifts of the rescaled maps and a bootstrap of individuals. Maps A and B agree almost completely (rank correlation 0.9992 to 1.00; no side statement differs). Map C agrees less with them (0.95 to 0.97 on WE8, 0.89 on EU5) and differs from them in one of 24 side statements on each WE8 projection and in four on EU5. For Cornwall the nearest class is Iron Age Britain on map C and a modern English group on maps A and B, in all three panels. Kent's side in the contrast of the Late-Middle Iron Age with the continental Saxon group of Anderten is resolved by the bootstrap on maps A and B and not on map C. The comparison shows where the choice of map matters; it does not show which map is closer to the true relationships. Positions on a map are not ancestry shares.
Taleb et al.1 replace each coordinate of a PCA map by its information value before distances are measured, so that a step near the centre of the map counts little and a step near the edge counts a lot. On the plain map the same step counts the same everywhere. Both maps read the same individuals on the same axes. This article asks what the choice of map changes in a concrete series: for three modern English groups, with modern Orkney as a comparison, which statements about the nearest group, about the side of a pair of groups and about the position between two groups change when the plain map is replaced by a rescaled one, and which do not.
The data are the PCA-distance series of the Nexju platform on the custom dataset English_DG_studynachbau_1240K_v4 (base aadr-v66-ext-3) that the article on modern English groups describes in full. 31 groups with 1,252 individuals are projected. Four are targets: Modern English, Kent (27); Modern English, Cornwall (27); Modern English, Human Origins (12); and Modern Orkney (GBR) (25). The 27 comparison groups are 19 groups of England from the Neolithic to the early modern period, Modern Orcadians, Modern Scottish and Modern French, and five continental Germanic groups. No reference population of the axes appears as a group.
Panel WE8 has eight present-day reference populations (Basque, Spanish, North Italian, Sardinian, Finnish, Russian, Adygei and Druze; 234 individuals; 992,097 SNPs) and is projected with shrinkmode (the base panel) and with plain projection. Panel EU5 has five (Basque, Spanish, North Italian, Finnish and Russian; 141 individuals; 970,217 SNPs; shrinkmode). Each PCA has six components. Three panels times three maps give nine settings. Table 1 names the maps.
Table 1 | The three maps of the series
Map | Distance is measured on | Origin shifts |
|---|---|---|
C | the conventional PCA coordinates, each axis as ρ·√λ | none |
A | the entropy-rescaled coordinates of Taleb et al.: for each individual sgn(ρ)·(−½ ln(1−ρ²))·√λ, group positions are the means of the transformed individuals | 8 |
B | the second entropy-rescaled map, in bits | 8 |
The 16 shifts move the origin of map A or map B by ±0.5 and ±1 standard deviations of the base panel's ρ on each of the two axes. Map C is not shifted.
Before the run the following statements were declared: four targets, nine classes of groups for the nearest group, six contrasts of two groups, three pairs for betweenness and six cluster pairs. Each statement is counted over the nine settings and over the 16 shifts. Q = d(target, P1) / d(target, P2) is the distance of a target to the first group of a contrast over its distance to the second; below 1 the target is on the side of the first group. Intervals are 95 % bootstrap intervals (2,000 resamples of individuals within groups) on the base panel. The comparison below sets the same statements side by side on map C and on maps A and B.
Fig. 1 shows the 31 group centroids on the plain map C and the rescaled map A for the base panel (WE8, shrinkmode). The statements below are not a visual reading of this figure. Each is a count over the nine settings and the 16 origin shifts, or a bootstrap on the base panel.
How far the maps agree. Table 2 gives, for each panel, the rank correlation between two maps and the number of statements whose answer differs between them. Maps A and B agree almost completely on every panel, and no side or betweenness statement differs between them. Map C agrees less with the rescaled maps: the rank correlation is 0.95 to 0.97 on WE8 and 0.89 on EU5. Of the 24 side statements, one differs between map C and map A (and between C and B) on each WE8 projection and four on EU5. Of the 12 betweenness statements, three differ on WE8 with shrinkmode, two with plain projection and one on EU5.
Table 2 | Agreement between the maps within each panel
Panel | C and A | C and B | A and B |
|---|---|---|---|
WE8, shrinkmode | 0.952 (1/3) | 0.948 (1/3) | 0.9995 (0/0) |
WE8, plain | 0.967 (1/2) | 0.966 (1/2) | 0.9992 (0/0) |
EU5, shrinkmode | 0.886 (4/1) | 0.886 (4/1) | 1.000 (0/0) |
Each cell gives the Spearman rank correlation between the two maps and, in brackets, the number of side statements (of 24) and of betweenness statements (of 12) whose answer differs between them.
Panel and origin. Fig. 2 adds how each map responds to the choice of panel and to the origin. Between WE8 and EU5 no side statement changes on map C and three change on each rescaled map; betweenness statements change in four to six on map C and in six to seven on maps A and B. The rank correlation between panels is somewhat lower on the rescaled maps (WE8 shrinkmode against EU5: 0.883 on C, 0.873 on A, 0.878 on B; WE8 plain against EU5: 0.840, 0.798, 0.812). Within WE8, shrinkmode against plain projection changes no side statement on any map (rank correlation 0.991 on C, 0.972 on A, 0.974 on B). A shift of the origin, which applies to the rescaled maps only, changes at most one of the 24 side statements and at most three of the 12 betweenness statements, but up to eight nearest-group statements; the largest change comes from a shift of +1 along axis 2 (eight on both maps). The series does not identify the reason for the stronger response to the panel. One property of EU5 is noted here without a test: its eigenvalue spectrum is flat (PC2 1.40 %, PC3 1.39 %).
| Origin shift | Sides | Betweenness | Nearest |
|---|---|---|---|
| A PC1 -1 | 0 | 0 | 0 |
| A PC1 -0.5 | 0 | 0 | 0 |
| A PC1 +0.5 | 0 | 0 | 0 |
| A PC1 +1 | 0 | 1 | 2 |
| A PC2 -1 | 0 | 3 | 4 |
| A PC2 -0.5 | 0 | 3 | 1 |
| A PC2 +0.5 | 1 | 1 | 5 |
| A PC2 +1 | 1 | 1 | 8 |
| B PC1 -1 | 0 | 0 | 0 |
| B PC1 -0.5 | 0 | 0 | 0 |
| B PC1 +0.5 | 0 | 0 | 1 |
| B PC1 +1 | 1 | 2 | 3 |
| B PC2 -1 | 0 | 3 | 5 |
| B PC2 -0.5 | 0 | 3 | 2 |
| B PC2 +0.5 | 1 | 1 | 5 |
| B PC2 +1 | 1 | 1 | 8 |
| Map | Panels | Spearman | Sides flipped | Betweenness flipped |
|---|---|---|---|---|
| C | WE8-shrinkmode – WE8-plain | 0.991 | 0 | 2 |
| C | WE8-shrinkmode – EU5-shrinkmode | 0.883 | 0 | 4 |
| C | WE8-plain – EU5-shrinkmode | 0.840 | 0 | 6 |
| A | WE8-shrinkmode – WE8-plain | 0.972 | 0 | 1 |
| A | WE8-shrinkmode – EU5-shrinkmode | 0.873 | 3 | 6 |
| A | WE8-plain – EU5-shrinkmode | 0.798 | 3 | 7 |
| B | WE8-shrinkmode – WE8-plain | 0.974 | 0 | 1 |
| B | WE8-shrinkmode – EU5-shrinkmode | 0.878 | 3 | 6 |
| B | WE8-plain – EU5-shrinkmode | 0.812 | 3 | 7 |
Nearest groups. Fig. 3 reads each target by the class of its nearest group in every setting. Modern English, Human Origins is nearest to a modern English group in all nine settings, on the plain and on the rescaled maps. For the other three targets the map matters (Table 3). Cornwall is nearest to Iron Age Britain on map C in all three panels and to a modern English group on maps A and B in all three. Kent is nearest to Roman Britain in seven settings; the two exceptions are maps A and B of EU5, where Medieval England is nearest. Orkney is nearest to Modern Scotland and France in seven settings; on maps A and B of EU5 it is Bronze Age Britain. The series states both answers for Cornwall and does not decide between them.
| Target | Panel · map | Origin shifts | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Most often | WE8-shrinkmode C | WE8-shrinkmode A | WE8-shrinkmode B | WE8-plain C | WE8-plain A | WE8-plain B | EU5-shrinkmode C | EU5-shrinkmode A | EU5-shrinkmode B | A PC1 -1 | A PC1 -0.5 | A PC1 +0.5 | A PC1 +1 | A PC2 -1 | A PC2 -0.5 | A PC2 +0.5 | A PC2 +1 | B PC1 -1 | B PC1 -0.5 | B PC1 +0.5 | B PC1 +1 | B PC2 -1 | B PC2 -0.5 | B PC2 +0.5 | B PC2 +1 | |
| Modern English, Kent (n=27) | Roman Britain 7/9 · 16/16 | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Medieval England | Medieval England | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain | Roman Britain |
| Modern English, Cornwall (n=27) | Modern English 6/9 · 10/16 | Iron Age Britain | Modern English | Modern English | Iron Age Britain | Modern English | Modern English | Iron Age Britain | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Iron Age Britain | Iron Age Britain | Modern English | Roman Britain | Modern English | Modern English | Modern English | Modern English | Iron Age Britain | Iron Age Britain | Modern English | Roman Britain |
| Modern English, Human Origins (n=12) | Modern English 9/9 · 16/16 | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English | Modern English |
| Modern Orkney (GBR, n=25) | Modern Scotland and France 7/9 · 12/16 | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Bronze Age Britain | Bronze Age Britain | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Roman Britain | Modern Scotland and France | Modern Scotland and France | Bronze Age Britain | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Modern Scotland and France | Roman Britain | Modern Scotland and France | Modern Scotland and France | Bronze Age Britain |
Table 3 | Class of the nearest group in each setting
Setting | Kent | Cornwall | Human Origins | Orkney |
|---|---|---|---|---|
WE8 shrinkmode, map C | Roman Britain | Iron Age Britain | Modern English | Modern Scotland and France |
WE8 shrinkmode, map A | Roman Britain | Modern English | Modern English | Modern Scotland and France |
WE8 shrinkmode, map B | Roman Britain | Modern English | Modern English | Modern Scotland and France |
WE8 plain, map C | Roman Britain | Iron Age Britain | Modern English | Modern Scotland and France |
WE8 plain, map A | Roman Britain | Modern English | Modern English | Modern Scotland and France |
WE8 plain, map B | Roman Britain | Modern English | Modern English | Modern Scotland and France |
EU5 shrinkmode, map C | Roman Britain | Iron Age Britain | Modern English | Modern Scotland and France |
EU5 shrinkmode, map A | Medieval England | Modern English | Modern English | Bronze Age Britain |
EU5 shrinkmode, map B | Medieval England | Modern English | Modern English | Bronze Age Britain |
Targets: Modern English, Kent; Modern English, Cornwall; Modern English, Human Origins; Modern Orkney (GBR). The nine classes are Modern English, Modern Scotland and France, Neolithic Britain, Bronze Age Britain, Iron Age Britain, Roman Britain, Early medieval England, Medieval England and Continental Germanic. Read from the nearest-group view of the series (Fig. 3).
Sides and intervals. Fig. 4 and Table 4 give Q on each map for every target and contrast of the series. On the base panel the point estimate of Q is on the same side of 1 on all three maps for 23 of the 24 side statements. The exception is Orkney, Late-Middle Iron Age against Saxon England, where Q is 1.08 on map C and 0.71 on maps A and B, and where the bootstrap interval includes 1 on all three maps. The bootstrap interval excludes 1 on all three maps for 16 of the 24 statements and on none for seven: Saxon England against Anderten for all four targets, French against Orcadians for Human Origins, and the two contrasts of Orkney that set the Late-Middle Iron Age against Saxon England and against Anderten. For one statement the maps differ: Kent, Late-Middle Iron Age against Anderten, where the interval includes 1 on map C (Q 0.30, 0.10 to 1.04) and excludes it on map A (0.26, 0.065 to 0.91) and on map B (0.24, 0.062 to 0.87).
The size of Q changes more than its side. For Orkney it is higher on the rescaled maps for Beaker against Late-Middle Iron Age (2.97 on C, 5.18 on A, 5.23 on B) and for Neolithic against Iron Age (16.0, 22.7, 22.9), and lower for French against Orcadians (12.5, 7.74, 7.92). For the English groups the direction depends on the contrast: for the Neolithic against the Iron Age Cornwall has 63.8 on C and 46.4 and 50.7 on A and B, Kent 38.8 on C and 49.4 and 53.1. The largest ratio between two maps is 1.8 (Orkney, Beaker against Late-Middle Iron Age).
Table 4 | Q for every target and contrast on the three maps, base panel WE8
Target | Contrast | Q on C | Q on A | Q on B |
|---|---|---|---|---|
Kent | Iron Age vs Saxon England | 0.235 | 0.186 | 0.177 |
Kent | Iron Age vs Anderten | 0.301† | 0.258 | 0.240 |
Kent | Saxon England vs Anderten | 1.28† | 1.38† | 1.36† |
Kent | Neolithic vs Iron Age | 38.8 | 49.4 | 53.1 |
Kent | Beaker vs Iron Age | 9.29 | 13.6 | 14.6 |
Kent | French vs Orcadian | 3.20 | 2.34 | 2.24 |
Cornwall | Iron Age vs Saxon England | 0.118 | 0.173 | 0.161 |
Cornwall | Iron Age vs Anderten | 0.143 | 0.227 | 0.208 |
Cornwall | Saxon England vs Anderten | 1.21† | 1.31† | 1.30† |
Cornwall | Neolithic vs Iron Age | 63.8 | 46.4 | 50.7 |
Cornwall | Beaker vs Iron Age | 17.0 | 13.7 | 14.9 |
Cornwall | French vs Orcadian | 2.03 | 1.58 | 1.49 |
Human Origins | Iron Age vs Saxon England | 0.178 | 0.234 | 0.219 |
Human Origins | Iron Age vs Anderten | 0.214 | 0.305 | 0.282 |
Human Origins | Saxon England vs Anderten | 1.20† | 1.31† | 1.29† |
Human Origins | Neolithic vs Iron Age | 37.9 | 31.9 | 34.7 |
Human Origins | Beaker vs Iron Age | 10.7 | 9.69 | 10.5 |
Human Origins | French vs Orcadian | 1.59† | 1.26† | 1.19† |
Orkney | Iron Age vs Saxon England | 1.08† | 0.712† | 0.712† |
Orkney | Iron Age vs Anderten | 1.44† | 1.01† | 1.01† |
Orkney | Saxon England vs Anderten | 1.34† | 1.42† | 1.42† |
Orkney | Neolithic vs Iron Age | 16.0 | 22.7 | 22.9 |
Orkney | Beaker vs Iron Age | 2.97 | 5.18 | 5.23 |
Orkney | French vs Orcadian | 12.5 | 7.74 | 7.92 |
† The 95 % bootstrap interval of Q includes 1 on that map. Q = d(target, P1) / d(target, P2) for the groups in the order named; below 1 the target is on the side of the first. Iron Age: England Late-Middle Iron Age (74); Saxon England: England Saxon (84); Anderten: Germany Medieval Saxon, Anderten (6); Neolithic: England Neolithic, Megalithic (60); Beaker: England Beaker (28); French: Modern French (31); Orcadian: Modern Orcadians (17). Targets: Kent, Modern English, Kent (27); Cornwall, Modern English, Cornwall (27); Human Origins, Modern English, Human Origins (12); Orkney, Modern Orkney (GBR) (25).
Where the choice of map matters. In this series it matters for a minority of statements. Maps A and B give the same answers. Map C gives a different answer in one of 24 side statements on each WE8 projection and in four on EU5, in one to three of the 12 betweenness statements, and in the nearest class of three of the four targets in at least some settings. The clearest case is Cornwall, whose nearest class is Iron Age Britain on map C and a modern English group on maps A and B in all three panels. One side statement is resolved by the bootstrap on the rescaled maps and not on map C: Kent, Late-Middle Iron Age against Anderten.
What does not change. For most statements the three maps agree. Sixteen of the 24 side statements are resolved by the bootstrap on all three maps, Modern English, Human Origins has a modern English group as its nearest class in all nine settings, and the rescaled maps do not respond less to the choice of panel than the plain map in this series: between WE8 and EU5 three side statements change on maps A and B and none on map C.
What the comparison does not show. The series has no ground truth. A different answer on map C and on map A is not an error of either, and the comparison does not test which map is closer to the true relationships between the groups. The flat centre and steep edge of the rescaling are not tested here against known relationships; such a test would need groups whose relationships are known in advance.
The result is a position on a map, not an ancestry share, and no qpAdm model is used. It is one series on one dataset with four targets and 31 groups; the 72 declared statements are not independent and the counts cannot be carried over to other regions. None of the reference populations of WE8 or EU5 is British or Germanic; panel EU5 has a flat eigenvalue spectrum (PC2 1.40 %, PC3 1.39 %), and the first two components of WE8 carry 1.31 % and 1.07 % of the variance. Several groups are small (Modern Scottish 3, the Anglian group 6, Imperial Roman Britannia 6, England Roman 7, the Saxon group of Anderten 6); the bootstrap of Q takes this into account, the nearest-group and betweenness counts do not. The thresholds of 8 of 9 settings and 13 of 16 shifts were fixed in advance and are conventions. Origin shifts are applied to the rescaled maps only; map C is not shifted. The group Modern English, Human Origins (12) holds two SGDP re-sequences of 1000 Genomes genomes.
Data and methods. Custom Nexju dataset English_DG_studynachbau_1240K_v4 (base aadr-v66-ext-3); PCA-distance run 0d6f72b9 with the base panel WE8-shrinkmode, the settings WE8-plain and EU5-shrinkmode, and the maps C, A and B. Panel WE8: Basque, Spanish, North Italian, Sardinian, Finnish, Russian, Adygei, Druze; 992,097 SNPs, 234 reference individuals. Panel EU5: Basque, Spanish, North Italian, Finnish, Russian; 970,217 SNPs, 141 reference individuals. Six components each. 16 origin shifts of the rescaled maps (±0.5 and ±1 standard deviations of the base panel's ρ on each of two axes of maps A and B). Bootstrap of individuals within groups, 2,000 resamples, 95 %. A statement is stable when it holds in at least 8 of 9 settings, at least 13 of 16 shifts and in the bootstrap, panel-stable when it has the same answer in all settings and shifts but the bootstrap interval of Q includes 1 on at least one map, and depends otherwise. Targets, classes, contrasts, betweenness pairs and cluster pairs were declared before the run. The same run underlies the article on modern English groups. No qpAdm model is used in this article, and no position in it is an ancestry share.
Taleb, N. N., Zalloua, P., Elbassioni, K., Hatzikirou, H., Henschel, A. & Platt, D. E. Informational rescaling of PCA maps with application to genetic distance. Comput Struct Biotechnol J 27, 48–56 (2025).
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