Medieval Germany and England in entropy-rescaled PCA distances

Do medieval individuals from Germany and England lie nearer to modern Germans and English, or to particular ancient groups, when they are placed on entropy-rescaled PCA maps? Fifty-three groups (1,371 individuals: 10 recent-pole groups, 19 medieval target groups, 4 Viking-age groups, 2 context groups and 18 ancient anchors) are projected on four PCA scaffolds built from modern reference populations. Each scaffold is drawn in three maps, a conventional one (C) and two entropy-rescaled ones (A, natural logarithm; B, bits) after Taleb et al. (2025). The question is which ordinal statements (nearest group, side in a contrast, betweenness) survive a change of panel, of map and of origin. Design, endpoints and survival rule were written down before the first PCA run.

Of 57 nearest-neighbour statements 4 are stable, of 152 side statements 56, and of 171 betweenness statements 4. Coarse statements survive, fine-grained ones do not, and the panel matters more than the map. Only distance methods are used, on the Human Origins SNP set, no all-SNPs mode; no qpAdm is run. Positions on a map are not ancestry.

The question was whether medieval German and English individuals lie nearer to modern Germans and English or to particular ancient groups (Roman Britain, the Bronze and Iron Age of Britain and Central Europe, Iron Age Denmark, Roman-period Poland) when they are placed on the same entropy-rescaled PCA maps as in the earlier Erfurt and Southeast Balkans series, and which ordinal statements (nearest neighbours, side in a contrast, betweenness) survive a change of panel, of map and of origin.

A statement is called stable only if it holds in at least 10 of the 12 panel-by-map settings (the declared 12 of 15, scaled to the four scaffolds that passed the batch check), in at least 13 of 16 origin shifts of the base panel, and, for sides and betweenness, in the bootstrap on the base panel in all three maps. Panel-stable means that it holds in at least 10 of 12 settings only. Of 57 nearest-neighbour statements 4 are stable, of 152 side statements 56 (and 37 more are panel-stable only), and of 171 betweenness statements 4 (and 8 more are panel-stable only). A further 84 betweenness statements say stably that a group does not lie between two anchors (Table 1). Fine-grained statements do not survive; coarse ones do.

No modern Germans in the data. The reference data contain no modern Germans, Dutch, Danes, Swedes, Austrians, Irish or Welsh, and the one recent German sample, a Bonn sample, carries contamination and exclusion flags and is removed. The recent poles are therefore the English (Human Origins English and 1000 Genomes GBR, pooled, 50 individuals), Scottish, Orcadian, Norwegian, Icelandic, Czech and French groups and two groups of historical Flemish individuals (AD 1600 to 1750 and AD 1700 to 1850, ancient DNA from Sint-Truiden and Mechelen, not living people). Statements about modern Germans are not possible in this series and none is made. The Czech sample (Prague) is the only Central European pole; on the base panel it lies close to the English (distance 0.0027 in the conventional map, 14 percent of the mean distance between groups).

Nearest groups. No nearest ancient anchor is stable for any of the 19 targets. Four of the 57 statements are stable: the nearest recent pole of the Slavic-period group of Saxony-Anhalt is the Norwegian sample (12 of 12 settings, 16 of 16 shifts), the nearest recent pole of late medieval Cambridgeshire is the Flemish 1700 to 1850 group (10 of 12 settings, 16 of 16 shifts), the nearest medieval group of medieval Cambridgeshire is late medieval Cambridgeshire (10 of 12 settings, 16 of 16 shifts) and the nearest recent pole of Viking-age England is the Flemish 1700 to 1850 group (11 of 12 settings, 16 of 16 shifts). The class of the nearest group (recent pole or ancient anchor) is more informative. Among the nine recent poles and the 18 ancient anchors the single nearest group is an ancient anchor, in at least 10 of 12 settings and at least 13 of 16 shifts, for 7 targets (early medieval Saxon England, other early medieval England, Viking-age England, Schleswig, the Frisian terp group, early medieval Belgium and high medieval Belgium), and a recent pole in all settings and shifts for Obermöllern and the Slavic-period group of Saxony-Anhalt; for the other ten targets it depends on panel or origin. Which anchor is nearest is unstable: Roman-period Poland (the Wielbark group) is the most frequent nearest anchor of 8 targets (6 to 9 of 12 settings) but never reaches the threshold.

Sides. Seven groups, Viking-age England, medieval Cambridgeshire, late medieval Cambridgeshire, Alt-Inden, early medieval southern Germany, early medieval Belgium and high medieval Belgium, are stably nearer Iron Age Britain than the pooled English (K1) and also nearer Roman Britain (K2). Five groups, Deersheim, Obermöllern, the Saxon sites of Lower Saxony, the Slavic-period group of Saxony-Anhalt and the Netherlands Saxon group, are nearer the pooled English than Iron Age Britain (K1), and the same groups without the Netherlands Saxon group are nearer the pooled English than Roman Britain (K2). Against the Flemish 1700 to 1850 group (K5), the East Anglian and Kent Saxon group, Deersheim, Obermöllern, the Saxon sites of Lower Saxony, the Slavic-period group of Saxony-Anhalt and the Netherlands Saxon group are nearer the pooled English, and medieval Cambridgeshire, late medieval Cambridgeshire, early medieval southern Germany, early medieval Belgium and high medieval Belgium are nearer the Flemish group; for Viking-age England and Alt-Inden neither side of K5 is stable. These contrasts largely sort the groups along PC1 (Spearman correlation of log Q with PC1 across the 19 targets on the base panel, conventional map: −0.90 for K1, −0.86 for K2, −0.87 for K5), so they describe a position on one axis, not a separate relationship to each pole.

Betweenness. Four statements are stable: early medieval Saxon England between early Iron Age Denmark and Iron Age Britain (B3), early medieval Saxon England between early Iron Age Denmark and the Flemish 1700 to 1850 group (B7), early medieval Saxon England between Hallstatt and early Iron Age Denmark (B8) and the East Anglian and Kent Saxon group between early Iron Age Denmark and the pooled English (B4). All four have early Iron Age Denmark at one end of the segment, and the criterion is a tolerance of 10 percent in a two-dimensional map: it says nothing about ancestry or admixture. Eight more statements hold in at least 10 of 12 settings and 13 of 16 shifts but not in the bootstrap (Table 9).

Clusters. At k = 10 the only declared pair that clusters in at least 10 of 12 settings is the East Anglian and Kent Saxon group with the medieval Saxon sites of the Netherlands (12 of 12). The pairs most discussed do not cluster: the East Anglian and Kent Saxon group with the pooled English in 4 of 12 settings, early medieval Saxon England in 1, medieval Cambridgeshire in 0, and the Flemish 1700 to 1850 group with the pooled English in 0.

Dependence on panel, map and origin. Adding East Asian or African reference populations changes the first axes from a West Eurasian to an outgroup-driven structure (first eigenvalue 4.7 in MG0, 38.2 in MG1, 37.0 in MG2, 47.7 in MG3). The rank correlation of the 1,326 centroid distances between MG0 and MG3 is 0.58 (conventional map), 0.24 and 0.24 (rescaled maps), and up to 79 of the 152 side statements change side between two scaffolds. On the base panel, shifting the origin by one standard deviation changes 4 to 15 side statements, and the choice among the three maps changes at most 9. The shrinkage scaffold MG4 failed the declared batch-invariance check by a small margin and is excluded, as the design requires; as a post hoc control it changes no direction of any side or betweenness statement (Table 13).

Figure 1 sums up the status of all 380 declared statements: 57 nearest-group statements, 152 side statements and 171 betweenness statements.

Method as in the earlier Erfurt and Southeast Balkans series, after Taleb et al. (2025, Comput Struct Biotechnol J 27:48 to 56). Principal components are computed on a reference panel of modern populations with the smartpca normalisation; all other individuals (ancient, medieval and the recent poles) are projected by least squares and do not influence the axes. For each individual the platform returns the coordinate ρ on component j (unit-norm eigenvector scale; for projected individuals a projection, not a correlation) and the eigenvalue λj. Three maps are drawn from the same coordinates:

A group is placed at the centroid of its transformed individual coordinates and distances are Euclidean in the PC1 by PC2 plane. The transformation is measured from the origin, so the dependence on origin and panel is part of the result. No qpAdm is used.

Table 2 lists the scaffolds. MG0 is the declared base panel of 27 populations (Abkhasian, Albanian, Algerian, Assyrian, Azeri, Balkar, Bulgarian, Chechen, Croatian, Cypriot, Egyptian, Estonian, Finnish, Georgian, Hungarian, Jordanian, Kumyk, Lebanese, Lezgin, Maltese, Moroccan, Ossetian, Romanian, Saudi, Northern Spanish, Syrian, Tunisian), the panel of the Erfurt series without the five populations that serve here as projected poles (Czech, English, Icelandic, Norwegian, Scottish); MG1 adds Han and Japanese; MG2 adds Yoruba and Mandenka; MG3 adds all four; MG4 is MG0 with shrinkage (shrinkmode). Ten components were requested and the first two are used; no outlier removal and no LD pruning. On MG0 the 50 European groups of the series (all but the two Erfurt context groups) occupy only 8 percent of the PC1 span and 7 percent of the PC2 span of the reference populations, which is why small coordinate differences decide the nearest neighbours.

The platform allows at most 64 populations per run, so every scaffold was run in two batches (A and B) with identical reference populations and three bridge groups in both (English, Iron Age Britain, Brücken). The invariance check (identical eigenvalues, reference means and bridge centroids; bridge centroids within 10⁻³) passed exactly for MG0 to MG3. MG4 failed it by a small margin (largest bridge difference 1.23e-03 against the limit 10⁻³, reference means up to 5.7e-03); as declared, it is reported as failed and not used, and the survival thresholds were scaled from 12 of 15 to 10 of 12 settings (addendum 1, written before any endpoint was computed).

All results rest on the Human Origins SNP set, no all-SNPs mode. The projected set has 53 groups with 1,371 individuals (Table 3): ten groups of recent poles (nine poles, since English and GBR are pooled); six groups of medieval England; nine of medieval Germany (Saxony-Anhalt migration period: Brücken, Deersheim, Rathewitz, Obermöllern; Saxon sites of Lower Saxony; Schleswig; Alt-Inden in the Rhineland; early medieval southern Germany; and a Slavic-period group of Saxony-Anhalt as contrast); four of the medieval Low Countries; four Viking-age groups (Denmark, Norway, Sweden, Iceland); two context groups of medieval Erfurt Jewish individuals from the Erfurt series; and 18 ancient anchors (seven in Britain from Bell Beaker to Roman, seven in Central Europe from Corded Ware to La Tène, Iron Age Denmark in an early and a late group, Roman-period Poland and the Roman Netherlands).

Inclusion: at least 100,000 SNPs on the Human Origins set, none of the flags for contamination, sex mismatch, coverage, damage or analysis exclusion, no outlier-suffixed or duplicate label, one library per individual, and the lower-coverage member of each listed related pair removed (135 individuals removed from 230 listed pairs among the eligible individuals); groups with more eligible individuals than the cap (40, or 60 for Middle and Late Iron Age Britain and early Iron Age Denmark) were reduced by a seeded draw (seed 20261022). The platform attached a warning to 155 of the 1,371 individuals: an assessment warning to 61 (based on contamination estimates from ANGSD or hapConX, which are below about 0.015 and therefore below the exclusion rule; acknowledged, individuals kept), a relatedness note to 102 (relatives outside the selection may remain) and a duplicate-library note to 1; some individuals carry more than one. The 97 modern Human Origins individuals carry the warning that they have no 1240K SNPs (they are array data); the PCA uses the SNP set of the dataset, so this is a suitability warning, not an exclusion. The Cambridgeshire medieval group carries an AADR placeholder date of 700 BP.

All endpoints are reported whatever the result.

Bootstrap: 2,000 resamples of individuals within each group on MG0 (seed 20261022), the same draws for the three maps. Sensitivity: map A with the cosine scale on MG0.

Figure 2 maps the sampling sites of the 537 individuals of the 19 medieval target groups, which are drawn in four regions for display only: England (231 individuals), Saxony-Anhalt (137), the North Sea coast (65) and West and south (104). Figure 3 gives their dates, and Figure 4 places them on the base panel MG0, one panel per region, with the centroid of each target group and outlines round the modern English (pooled), Iron Age Britain, early Iron Age Denmark and the Flemish 1700 to 1850 groups. The regions are a device for drawing, not a result: the statements of this article are made on the 52 group centroids. Positions on a PCA plane are not ancestry.