Methodology, data and limitations

Data sources

Data snapshot: FBref tables fetched 14 September 2026, snapshot commit ef7dbd4 · code: src/, config/ · status: exploratory, not pre-registered · all limitations

• FBref (Sports Reference), read with soccerdata 1.9.1: player season tables (standard, playing time) for the 9 headline leagues, the Czech First League, the peer domestic leagues and the German second tier; the country page "Players from Czechia" for pool discovery; the nationality column for peer counts. Season tables fetched 14 September 2026; the Big-5 history back to 1995/96 is in the snapshot committed 22 September 2026 (Sports Reference, 2026; Robberechts, 2026)
• Wikipedia: national-team squad tables (2024–25 Nations League, 2026 FIFA World Cup, 2026 World Cup qualification, UEFA Euro 2024, UEFA European Under-21 Championship 2025), accessed 14 September 2026; text under CC BY-SA 4.0 (Wikipedia contributors, 2026)
• Wikidata / Wikimedia Commons: player portraits (P18) matched on name, citizenship and date of birth; Wikidata is CC0, each Commons file carries its own licence (credits in the footer) (Wikidata contributors, 2026)
• UEFA association coefficients, men’s five-season ranking (2022/23–2026/27, as published on the fetch date), primary source uefa.com, read from Wikipedia’s transcription of UEFA’s table on 14 September 2026 because ClubElo was unreachable; ranks 1–9 of that ranking are exactly the nine headline leagues (England, Italy, Spain, Germany, France, Portugal, Belgium, the Netherlands, Türkiye) (UEFA, 2026)
• Eurostat: population on 1 January 2024, table demo_pjan, accessed 14 September 2026; reuse under the European Commission’s reuse policy with attribution (Eurostat, 2024)

Data availability

The processed tables behind every page are committed in the public repository under data/snapshot/cze/ (snapshot commit ef7dbd4) and listed with their SHA-256 prefixes under Download the tables. Raw FBref pages are not redistributed; FBref’s own terms of use apply to its data. Squad lists from Wikipedia are CC BY-SA 4.0 and are credited as such; Wikidata is CC0; Eurostat data are reused with attribution; the UEFA coefficients are public rankings quoted with their source. League or club headshots are not used: a portrait appears only where a Wikimedia Commons file exists, credited in the footer with a link to its file page. The atlas’s own text, figures and data are CC BY 4.0 and its code MIT.

From raw tables to a feature vector

One 2025/26 player-season, traced from its raw FBref row to the five-number vector the rest of this chapter builds on: Vladimír Coufal, the Czech player with the most 2025/26 minutes among those who cleared the inclusion floor.

Vladimír Coufal: raw row and feature row

Raw FBref row, 2025/26

Column	Value
league	GER-Bundesliga
season	2025-2026
team	Hoffenheim
player	Vladimír Coufal
nation	CZE
pos	DF
born	1992
age	32
mp	34
min	3012
gls	1
ast	8
pk	0
crdy	4
crdr	0

Feature row after the pipeline

Feature	Raw	Shrunk	Quality-adjusted	Z-score
npg_p90	0.030	0.034	0.027	0.21
ast_p90	0.239	0.193	0.152	4.31
min_share	0.984	0.984	0.984	1.67
age	32.000	32.000	32.000	1.45
cards_p90	0.119	0.135	0.135	−0.90

Six wrangling checks turn the raw rows above into the pool used everywhere else in this report, recomputed on every run:

• Women's entries filtered: 92 entries.
• Namesakes in the pool: 2 players.
• Pool players without season tables: 125 players.
• Split-season rows collapsed: 441 rows.
• Unmatched call-up names: 18 names.
• Missing birth years: 0 rows.
• Unjoined goalkeeper rows: 12 rows.
• Czech First League rows without a nationality: 232 rows.

The full log, with the recorded pipeline incidents, is in the data-quality log below.

Every position group shares the same five-number vector (spec §5): a raw column becomes a per-90 rate, shrunk toward its league-season median (Efron and Morris, 1975), then multiplied by the league quality projection.

• npg_p90 — non-penalty goals per 90 minutes: penalties are shot quality, not open-play production, and inflate a taker's raw goal count for reasons that have nothing to do with how the player creates chances
• ast_p90 — assists per 90 minutes: the direct creative complement to non-penalty goals, on the same basic table for every league in the corpus
• min_share — minutes played ÷ (club matches × 90): the coach's own read of the player, sturdier than counting appearances (see below)
• age — age at the season's start (start year − birth year): median production still shifts by age band in this corpus (see below), so it stays as its own axis rather than being folded into anything else
• cards_p90 — yellow + 2 × red cards per 90 minutes: a discipline signal folded into one rate because red cards alone are too sparse a signal on their own (see below)

Five candidates considered for the 2025/26 vector and what the data said about each, corpus-wide (every fetched nationality, the same population the rest of this chapter describes):

Candidate	Statistic	Value	Decision
gls_p90	Correlation with npg_p90	0.973	Replaced by npg_p90
mp	Correlation with minutes played	0.844	Replaced by minutes share
crdr_p90	Share of player-seasons with zero red cards	0.854	Folded into cards
age	Spread of median goals + assists per 90 across age bands (max − min band)	0.028	Kept
born	Share of player-seasons missing a birth year	0.003	Kept — required for the player key

Despite the high correlation, goals and non-penalty goals diverge for the players who take penalties: Nabil Touaizi (POR-Primeira Liga, 100 %); Yohan Croizet (HUN-NB I, 100 %); Luca Zuffi (SUI-Super League, 100 %). Appearances (mp) correlate strongly with minutes but not perfectly — a substitute cameo counts the same as 90 minutes started, which is why minutes share, not appearances, measures playing time here. Median goals + assists per 90 by age band runs from 0.147 (30+) – 0.175 (23-25).

The picture behind league adjustment: the same raw rate reads very differently depending on the league it was earned in.

What shrinkage does to a low-minute player: at the inclusion floor the league median still carries most of the weight; Antonín Růsek's rate moves the most of any Czech-eligible player this season.

League multipliers (quality projection)

npG/90_quality = npG/90_shrunk × m_league

League	Multiplier
ENG-Premier League	1.000
ITA-Serie A	0.856
ESP-La Liga	0.807
GER-Bundesliga	0.788
FRA-Ligue 1	0.666
POR-Primeira Liga	0.630
BEL-Pro League	0.573
NED-Eredivisie	0.510
TUR-Süper Lig	0.484
GER-2. Bundesliga	0.473
POL-Ekstraklasa	0.438
CZE-First League	0.434
NOR-Eliteserien	0.374
DEN-Superliga	0.371
SUI-Super League	0.293
AUT-Bundesliga	0.268
HUN-NB I	0.265
CRO-HNL	0.249
SVK-Super Liga	0.217

Method: uefa_coefficient. ClubElo unavailable at run time (HTTP error); fell back to Wikipedia's UEFA men's association coefficient, current 5-year ranking (2022–23–2026–27 seasons). tier-1 multiplier = coef[country] / max(coef); tier-2 multiplier = 0.6 * tier-1 multiplier of the same country (stated assumption, not derived from data). Strongest tier-1 country = 1.00. The sensitivity analysis below shows the ranking's robustness to ±20 % on any one multiplier.

League strength: two estimates

A season in one league is not automatically worth the same as a season in another: goals and assists come easier in some competitions than others. This section works out an exchange rate between leagues by watching the same players before and after they change league, the way a manager judges a new signing by how his output changes at the new club.

Full method, figures and diagnostics

How much is a Czech First League season worth in Premier League terms? The UEFA multiplier above answers that from countries' continental results; this model answers the same question from the players who actually changed leagues.

Movers — 2125 players observed in at least two leagues across 8108 qualifying player-seasons — anchor the model, since only a player's own before/after change of league separates their level from the league's scoring environment. A hierarchical Poisson model of non-penalty goals plus assists per 90 (Gelman et al., 2013) fits a league effect and a player effect together, sampled with NUTS (Hoffman and Gelman, 2014) in PyMC (Abril-Pla et al., 2023) and diagnosed with ArviZ (Kumar et al., 2019). Partial pooling keeps the league effects regularised while leaving player effects close to unpooled, the same within-subject logic behind plus-minus and RAPM ratings elsewhere in team sports (Kharrat, McHale and Peña, 2020; Hvattum, 2019).

In these terms, a Czech First League season converts to 0.67 of a Premier League one (90 % HDI 0.59–0.75).

League	m_L (median)	90 % HDI	Transitions	UEFA
ENG-Premier League	1.000	1.000–1.000	588	1.000
ITA-Serie A	0.935	0.890–0.981	569	0.856
ESP-La Liga	0.908	0.859–0.953	435	0.807
FRA-Ligue 1	0.810	0.772–0.848	672	0.666
GER-Bundesliga	0.777	0.741–0.814	588	0.788
HUN-NB I	0.743	0.625–0.864	35	0.265
POR-Primeira Liga	0.721	0.680–0.762	350	0.630
BEL-Pro League	0.671	0.637–0.710	474	0.573
CZE-First League	0.665	0.585–0.749	64	0.434
TUR-Süper Lig	0.660	0.625–0.696	472	0.484
POL-Ekstraklasa	0.659	0.599–0.729	131	0.438
AUT-Bundesliga	0.657	0.576–0.729	80	0.268
DEN-Superliga	0.635	0.576–0.702	117	0.371
GER-2. Bundesliga	0.600	0.555–0.645	230	0.473
NED-Eredivisie	0.597	0.567–0.632	387	0.510
CRO-HNL	0.597	0.525–0.671	64	0.249
SUI-Super League	0.597	0.538–0.649	128	0.293
NOR-Eliteserien	0.574	0.507–0.639	64	0.374

Refit on seasons before 2025/26, the model predicts each mover's first 2025/26 row after a league change — 863 such moves — against two baselines: the same rate as before, and that rate scaled by the ratio of UEFA multipliers. This is the same population CIES Football Observatory's expatriate-player reports track (Poli, Ravenel and Besson, 2024).

Method	Log predictive density	MAE (rate)
Same rate as before	−2.628	0.137
Rate × UEFA ratio	−2.734	0.151
Transfer-graph model	−2.174	0.127

Against 18 leagues in common, the model's medians and the UEFA multipliers correlate at Spearman's rho = 0.74.

• HUN-NB I: model rank 6 vs UEFA rank 17 (m_L 0.74 vs multiplier 0.27).
• NED-Eredivisie: model rank 15 vs UEFA rank 8 (m_L 0.60 vs multiplier 0.51).
• NOR-Eliteserien: model rank 18 vs UEFA rank 13 (m_L 0.57 vs multiplier 0.37).

R-hat ≤ 1.003, minimum bulk ESS 1224, 0 divergent transitions across 8108 player-seasons from 2125 movers; fit in 228 s.

Posterior predictive check

Observed vs. replicated non-penalty goals plus assists per player-season (Vehtari, Gelman and Gabry, 2017): 15 % vs 14 % share of zeros, mean 4.45 vs 4.45, 90th percentile 11.00 vs 10.85.

What the movers do not show: they are not a random sample — players tend to move up when they are good and down when they are older — so the within-player contrast describes the league difference for the kind of player who moves, and the age term absorbs only the part of that selection that is age. The interval is the model's uncertainty, not the selection's.

The report's rankings keep the UEFA-coefficient multiplier throughout; the model above is shown alongside it as a check on that multiplier's own assumption — that continental results track player-level strength — not as a replacement for it.

Five methods, one task, five seasons

Here we ask a simple question of five different methods: given a player's numbers this season, how well can each one guess his numbers next season. Each method is tested only on seasons it has not already seen, the way a scout's judgement only really counts for what he predicts before a season starts, not after it has finished.

Full method, figures and diagnostics

What does a player's season tell us about the next one, given where he plays, how old he is and — for exports — at what age he moved? In model terms: predict a player's league-adjusted production next season (non-penalty goals plus assists per 90, quality-adjusted) from this season's numbers, for every player-season pair with at least 450 minutes in both seasons, every nationality in the corpus.

Evaluated by rolling-origin cross-validation (Hyndman and Athanasopoulos, 2021): for each of 5 target seasons in turn, every model trains only on pairs whose target season came earlier and is tested on that season's pairs — the same discipline a forecaster uses when the future is genuinely unknown at fit time, and what makes the per-season table below a real "performance over time" read rather than one blended number. Two baselines — persistence (next season = this season) and shrinkage to the league mean — sit alongside three fitted models sharing the same eight features: a hierarchical Bayesian regression (target ~ Normal(μ, σ), partial pooling on league and player, NUTS, 2 chains × 400 draws per origin (Gelman et al., 2013)), a gradient-boosted regressor and a small multilayer perceptron (scikit-learn (Pedregosa et al., 2011), hidden layers 64/32 — a small MLP, not an embedding model: PyTorch is not part of this pipeline). The Bayesian model's training rows are subsampled to at most 2500 per origin to keep five origins' worth of fits inside the runtime budget; the other four models train on the full split. The target is adjusted with the multiplier of the league the player is in next season; the features only know this season's league, so a move is a change the models cannot see coming — a limitation shared by all five.

Model	2021/22	2022/23	2023/24	2024/25	2025/26	Pooled
Persistence	0.0768 (0.054)	0.0782 (0.056)	0.0789 (0.056)	0.0787 (0.057)	0.0684 (0.048)	0.0751 (0.053)
Shrinkage to league mean	0.1343 (0.104)	0.1293 (0.100)	0.1311 (0.098)	0.1271 (0.098)	0.1099 (0.089)	0.1225 (0.095)
Hierarchical Bayesian	—	0.0715 (0.053)	0.0760 (0.054)	0.0722 (0.053)	0.0681 (0.052)	0.0714 (0.053)
Gradient boosting	—	0.0723 (0.051)	0.0744 (0.051)	0.0698 (0.051)	0.0677 (0.051)	0.0706 (0.051)
Small MLP	—	0.0794 (0.059)	0.0792 (0.057)	0.0735 (0.054)	0.0714 (0.054)	0.0753 (0.056)

Each cell: RMSE (MAE), in league-adjusted npG+A per 90. The first origin's training set is empty by construction — the corpus's earliest feature season leaves no earlier target season to train on — so only the two baselines are reported there.

The Bayesian model's 90 % predictive interval covered the observed value 91 % of the time, pooled across the 4 origins it was fit for (2022/23: 90 %, 2023/24: 90 %, 2024/25: 91 %, 2025/26: 93 %).

By pooled RMSE, Gradient boosting wins (0.071 vs 0.075 for persistence, 6 % lower).

The clearest season-to-season move in the winner's own RMSE is between 2023/24 and 2024/25 (0.005) — the kind of drift this rolling-origin table exists to surface.

Dating the break and one forecast

This section finds the two seasons when the count of home-nation players in Europe's five biggest leagues changed level for good — the rise and the fall — and puts a probability on each being the true turning point rather than an ordinary dip. It also makes one forecast for next season, as a demonstration of the method, not a prediction about any player.

Full method, figures and diagnostics

The series is modelled as a local level in state space (Durbin and Koopman, 2012): the log of the season count follows a Gaussian random walk (σ ~ HalfNormal(0.2)), plus two ordered step changes δ₁, δ₂ in the level at unknown seasons τ₁ < τ₂ — on the series since 1995/96 one step is misspecified wherever the count both rises and falls, and a single step lands on whichever change buys more likelihood. NUTS only samples continuous parameters, so the τ pair is not sampled directly — every ordered pair of candidate seasons at least 3 seasons from either end and from each other is marginalised out of the model with a single log-sum-exp potential, and each break's own posterior is recovered afterwards from the continuous draws, the standard move for a marginalised discrete parameter (Gelman et al., 2013); the changepoint idea itself is due to (Adams and MacKay, 2007), applied here to a batch, two-break setting. The report's "break" is the step that lowers the level; the other is the rise. Each step is dated to the season where its own marginal posterior peaks, and a step within ±5 % of no change is not dated. A rolling-origin backtest (Hyndman and Athanasopoulos, 2021) refits the same local level without the step at each of 15 origins, forecasting one season ahead and scoring against the naive "same as last season" baseline — the honest forecaster's read, since no real origin knows in advance which side of a break it sits on. The one forecast below, from the same change-point-free model fitted on the full series, is a demonstration of the method on a count of players, not a statement about any player.

For Czechia, the model dates the break to 2014/15 (58 % posterior probability), a ×0.59 (0.37–1.02, 90 % HDI) change in the level; the random walk's own innovation scale is σ = 0.059.

The rise before it is dated to 2000/01 (56 % posterior), a ×1.67 (0.74–2.40) change in the level. The most probable seasons for each:

• rise: 2000/01: 56 % · 1999/00: 10 % · 2002/03: 7 %
• fall: 2014/15: 58 % · 2013/14: 7 % · 2012/13: 7 %

The same model, fit separately for the two contrast countries:

• Denmark: not dated; the fit does not meet the convergence rule.
• Croatia: 2003/04 (30 % posterior), ×0.75 (0.51–1.16). Rise: 2014/15 (27 %), ×1.31.

One-step-ahead rolling-origin backtest of the plain local level (no change point) against the naive "same as last season", 15 origins from 2010/11 to 2024/25:

Origin	Forecast season	Actual	Model median (90 % interval)	Naive
2010/11	2011/12	20	20 (11–31)	19
2011/12	2012/13	16	20 (11–31)	20
2012/13	2013/14	19	18 (10–29)	16
2013/14	2014/15	10	18 (10–29)	19
2014/15	2015/16	6	15 (7–25)	10
2015/16	2016/17	11	11 (5–19)	6
2016/17	2017/18	12	11 (5–19)	11
2017/18	2018/19	12	11 (5–20)	12
2018/19	2019/20	15	12 (5–20)	12
2019/20	2020/21	11	13 (6–22)	15
2020/21	2021/22	11	12 (5–21)	11
2021/22	2022/23	9	12 (5–20)	11
2022/23	2023/24	12	10 (5–18)	9
2023/24	2024/25	12	11 (5–19)	12
2024/25	2025/26	10	11 (5–19)	12

Pooled across 15 origins: MAE 2.47 for the model against 2.73 for the naive baseline, 93 % of the 90 % intervals covered the observed value.

The one forecast, for next season (2026/27), from the same change-point-free model fitted on the full series:

Country	Season	Median	90 % interval
Czechia	2026/27	11	5–19
Denmark	2026/27	35	23–51
Croatia	2026/27	24	15–35

A demonstration of the method on a count of players, not a statement about any player.

Change-point fit, 4 chains × 2000 draws across 31 seasons: max R-hat 1.004, minimum bulk ESS 2321, minimum tail ESS 1830, 0 divergent transitions. Every fit in this section is checked against a convergence rule fixed on 1 October 2026, before the refit (design/edition-refit-protocol.md): max R-hat ≤ 1.01, bulk and tail ESS ≥ 400, no divergent transitions, with a stricter step size only for a fit that fails. The peers' change-point fits and the forecast fits: 4 of 5 pass. Backtest origins whose fit fails, kept in the scores: 0 of 15.

Cross-country youth-minutes panel

This section asks whether the pattern from the funnel above — countries that give young players more minutes at home tend to have a deeper pool in the strongest leagues — holds up across every country in the comparison set, not only the two shown earlier. Two seasons and a handful of countries is a small sample, so the range around the estimate is wide.

Full method, figures and diagnostics

For each of the 8 peer countries, the 2024/25 → 2025/26 average U21 share of domestic-league minutes (x) against the 2024/25 → 2025/26 average top-9 players per million (y), one row per country. A Bayesian simple regression, y ~ Normal(α + β·x, σ), weakly informative priors (4 chains × 1000 draws (Abril-Pla et al., 2023)), answers the between-country question of the youth-minutes page: does a country with a higher average U21 share also have a deeper pool, on average (Gelman et al., 2013). Cross-checked against a plain pooled least-squares slope on the full two-season panel (numpy polyfit, ignoring country structure entirely, 1000 percentile-bootstrap resamples; statsmodels is not a dependency here), which should agree in sign. Two seasons per country (previous and metrics; the current season is partial and left out); a peer whose top flight FBref does not track is absent from the panel, which is why n can be below the peer count.

β = +10.11 per 10 percentage points of U21 share (90 % HDI −1.69–23.26), R² = 0.27; OLS on the same 8 country means lands close by, at +10.65 (−3.85–20.33), and the pooled-panel OLS slope agrees in sign at +7.76 (1.12–13.99). n = 8; the interval is wide because the panel is small.

R-hat ≤ 1.001, minimum bulk ESS 2126, 0 divergent transitions.

A separate check: the same regression, but with a country random intercept fit on the full two-season panel instead of country means. Within-country changes across the two seasons carry no signal — β_within = −0.16 per 10 percentage points (90 % HDI −2.53–2.17), n = 16 country-seasons (R-hat ≤ 1.004, minimum bulk ESS 851, 0 divergent transitions; posterior-median country-intercept scale σ_country = 5.19). With only two seasons per country the intercept absorbs almost all of the between-country pattern the fit above isolates, leaving this estimate driven by season-to-season noise alone; reported here as a robustness check, not a second headline.

Descriptive only: a country's average U21 share and its average per-capita count are shown together because they are the numbers on hand, not because one is claimed to produce the other.

Panel rows (n = 16)

Country	Season	U21 share	Per million
CRO	2024/25	12.8 %	15.54
DEN	2024/25	13.2 %	13.42
NOR	2024/25	10.5 %	9.01
SUI	2024/25	9.2 %	5.36
AUT	2024/25	6.4 %	5.02
CZE	2024/25	11.1 %	2.20
HUN	2024/25	12.2 %	1.36
POL	2024/25	12.5 %	1.26
DEN	2025/26	15.3 %	12.58
CRO	2025/26	14.0 %	12.18
NOR	2025/26	11.5 %	9.37
SUI	2025/26	7.7 %	6.03
AUT	2025/26	9.2 %	4.80
CZE	2025/26	6.3 %	2.39
HUN	2025/26	14.1 %	1.57
POL	2025/26	9.3 %	1.23

What the gap is made of

The gap between the home nation and a comparison country, in players per million, can be split into pieces that line up with three things this report already measures: how much young players play at home, how strong the domestic league is, and how old players are when they move abroad. This does not say which of the three causes the gap, only how much of it moves together with each one — the way a manager might note that a poor season lines up with an injury crisis without claiming the injuries alone explain it.

Full method, figures and diagnostics

Ridge regression (α = 1.00, standardised channels, converted back to original units) of top-9 players per million on three channels — U21 share of domestic minutes, domestic league strength (M2's m_L where the country's top flight is in that model's fitted set, else the UEFA-coefficient multiplier (every country in this run used the model estimate)) and median export age (recent entrants, or the all-time median for a country with none recently) — over the 8 peer countries with data on all three channels, metrics season. For one contrast country at a time, the fitted model's own prediction splits the gap exactly into one term per channel — the linear-model form of the Blinder-Oaxaca wage decomposition (Oaxaca, 1973; Blinder, 1973), applied here to a per-capita player count. Three linear channels means this split is Shapley-equivalent: the Shapley value of an additive term in a linear model equals its own coefficient's contribution, so no permutation machinery is implemented. The residual is whatever the three channels do not carry.

Contrast	Channel	Contribution	Share of gap	90 % interval
Norway Gap (players per million): +6.98	U21 minutes	+4.22	60 %	+1.22 – +5.91
	League strength	+5.12	73 %	+1.90 – +6.77
	Export age	0.00	0 %	0.00 – 0.00
	Residual	−2.37	—	—
Denmark Gap (players per million): +10.19	U21 minutes	+7.35	72 %	+2.17 – +10.27
	League strength	+1.67	16 %	+0.65 – +2.26
	Export age	0.00	0 %	0.00 – 0.00
	Residual	+1.17	—	—

Bootstrap 90 % intervals on each channel's contribution, 1000 resamples of the panel's rows (the model refit on each resample; the home/contrast countries' own values held fixed); the residual itself is not bootstrapped — it is the two countries' own observed counts minus the fitted gap.

A decomposition of a correlation this panel happens to show, not a causal accounting; with n = 8 and three correlated national-level channels, the shares are indicative, not precise. A channel's share of the gap is only shown when the gap itself is at least 3 players per million — below that, a small denominator can send a share past 100 % in either direction; the contribution itself, in players per million, is always reported.

Age at export

This section asks whether players who reach one of the top-ranked leagues younger produce more there in their first two seasons, holding the strength of the league they came from fixed. Clubs are also more willing to take an early chance on a player they already rate highly, so the curve cannot separate that selection from any effect of leaving young.

Full method, figures and diagnostics

For every peer-nationality player (home nation included) whose first top-9-league season lies inside the fetched window and isn't censored (the same censoring rule as "Where do Czech players go when they leave?", reused here) — 115 players, ages 17–31 — age at that season is modelled against league-adjusted production over the player's first one or two top-9 seasons. f(age) is a natural cubic spline with knots at 19, 21, 23 and 25 (a plain quadratic below n = 150; the quadratic branch was used here), alongside origin-league strength (the transfer-graph model, § League strength), position and a partially pooled country effect (Gelman et al., 2013), sampled with NUTS (Hoffman and Gelman, 2014), 4 chains × 1000 draws; every design column and the outcome were standardised before fitting, the youth-minutes panel's own lesson about raw-scale priors on differently-scaled covariates.

Age	Expected G+A/90 (median)	90 % HDI
19	0.14	0.12 – 0.16
21	0.14	0.13 – 0.16
23	0.14	0.12 – 0.16
25	0.14	0.12 – 0.16
27	0.14	0.12 – 0.16

β, per one-unit increase in origin-league strength (m_L): −0.03 (−0.11–0.05).

Czech exports' own country effect: 0.00 (−0.02–0.02); Czech exports arrive at a median age of 23.

The same model, fit again without origin-league strength: the country-effect scale (σ_n) is 0.135 with the league term in the model and 0.139 without it — what moves between the two is what the league term is absorbing.

Leave-one-nation-out: excluding Czechia's own 8 exports (n = 107 remaining) and refitting, the 21-vs-24 difference is 0.00 (−0.01–0.01), against 0.00 in the full fit. The refit meets the convergence rule (max R-hat 1.003, minimum bulk ESS 1201, minimum tail ESS 1473, 0 divergent transitions).

Posterior predictive check

Observed vs. replicated y (mean league-adjusted G+A/90 over the first two top-9 seasons): mean 0.14 vs 0.14, sd 0.10 vs 0.10, 10th percentile 0.02 vs 0.01, 90th percentile 0.27 vs 0.28.

R-hat ≤ 1.002, minimum bulk ESS 1190, 0 divergent transitions across 115 players; fit in 6.1 s.

What this model does not separate: the corpus is not a random sample of players who could have left later. Players who leave earlier tend to be the ones judged ready earliest — a selection effect the age curve mixes with any genuine development effect of arriving young, and this report does not try to tell the two apart.

Bayesian shrinkage

Per-90 rates of players with few minutes are shrunk towards the median of their league and season (players with at least 900 minutes) using the empirical Bayes formula (Efron and Morris, 1975), with K = 10 phantom matches expressed as 900 minutes:

shrunk_rate = (events + K × league_median) / (minutes / 90 + K),   K = 10

At 450 minutes (the inclusion floor) the league median carries 67 % of the weight; at 900 minutes the player's own rate and the median weigh the same. Minutes share and age are not shrunk.

Goalkeepers are counted the same way as outfield players throughout this report, with one difference: because a goalkeeper's save numbers swing around a lot from game to game, the model needs a bigger sample of shots faced before it trusts a keeper's own numbers over the league average.

Goalkeeper rates (chapter II's counter-example) use the same formula. GA/90 and saves/90 are shrunk toward their league-season median with the same K = 900 minutes, and GA/90 is then quality-adjusted by the league multiplier — a goal conceded in a stronger league counts less. Save percentage is shrunk the same way but against shots on target faced, not minutes: a single season's shot count sits far below K = 900, so save_pct_shrunk compresses hard toward the league median for almost every goalkeeper — a large gap in the raw, unshrunk save percentage is the more informative read there.

PCA loadings

One five-feature vector per position group (npg_p90, ast_p90, min_share, age, cards_p90), standardised, reduced to two components per projection. Style uses the shrunk rates; quality multiplies the rates by the league multiplier first.

Loadings table

Group	Projection	PC	% variance	npG/90	A/90	min share	age	cards/90
FW	style	PC1	25.2 %	0.533	0.215	0.651	0.430	−0.246
FW	style	PC2	23.6 %	−0.206	−0.271	−0.120	0.823	0.439
FW	quality	PC1	28.9 %	0.636	0.380	0.482	0.217	−0.415
FW	quality	PC2	24.9 %	−0.108	−0.114	0.133	0.910	0.361
MF	style	PC1	29.5 %	0.597	0.601	0.383	0.073	−0.361
MF	style	PC2	23.2 %	−0.245	−0.117	0.484	0.828	0.082
MF	quality	PC1	30.0 %	0.590	0.608	0.382	0.133	−0.344
MF	quality	PC2	24.2 %	−0.244	−0.199	0.490	0.809	0.087
DF	style	PC1	25.0 %	0.546	0.429	0.505	0.183	−0.480
DF	style	PC2	21.1 %	−0.343	−0.378	0.278	0.804	−0.128
DF	quality	PC1	25.8 %	0.432	0.356	0.578	0.327	−0.496
DF	quality	PC2	21.7 %	−0.462	−0.375	0.104	0.797	−0.023

Sensitivity analysis (±20 % multipliers)

For each scenario the quality-adjusted ranking of Czech-eligible players within each position group was recomputed and compared with the baseline; "top-10" is the union of the 3 groups' own top tens (30 players at baseline). Of the 41 scenarios, 37 change nobody in that set; the largest churn is 1 (CZE-First League multiplier -20%, mean rank shift 0.88 in the top twenty)*.

Scenario table

Scenario	Description	Top-10 overlap	Top-10 churn	Mean Δ rank (top 20)
baseline	current multipliers from config/league_quality.yaml	30 / 30	0	0.00
ENG-Premier League_minus20	ENG-Premier League multiplier -20%	30 / 30	0	0.10
ENG-Premier League_plus20	ENG-Premier League multiplier +20%	30 / 30	0	0.07
ITA-Serie A_minus20	ITA-Serie A multiplier -20%	30 / 30	0	0.00
ITA-Serie A_plus20	ITA-Serie A multiplier +20%	30 / 30	0	0.00
ESP-La Liga_minus20	ESP-La Liga multiplier -20%	30 / 30	0	0.00
ESP-La Liga_plus20	ESP-La Liga multiplier +20%	30 / 30	0	0.00
GER-Bundesliga_minus20	GER-Bundesliga multiplier -20%	30 / 30	0	0.13
GER-Bundesliga_plus20	GER-Bundesliga multiplier +20%	30 / 30	0	0.20
FRA-Ligue 1_minus20	FRA-Ligue 1 multiplier -20%	30 / 30	0	0.00
FRA-Ligue 1_plus20	FRA-Ligue 1 multiplier +20%	30 / 30	0	0.03
NED-Eredivisie_minus20	NED-Eredivisie multiplier -20%	30 / 30	0	0.20
NED-Eredivisie_plus20	NED-Eredivisie multiplier +20%	29 / 30	1	0.23
POR-Primeira Liga_minus20	POR-Primeira Liga multiplier -20%	30 / 30	0	0.00
POR-Primeira Liga_plus20	POR-Primeira Liga multiplier +20%	30 / 30	0	0.00
BEL-Pro League_minus20	BEL-Pro League multiplier -20%	30 / 30	0	0.00
BEL-Pro League_plus20	BEL-Pro League multiplier +20%	30 / 30	0	0.02
TUR-Süper Lig_minus20	TUR-Süper Lig multiplier -20%	30 / 30	0	0.13
TUR-Süper Lig_plus20	TUR-Süper Lig multiplier +20%	30 / 30	0	0.05
CZE-First League_minus20	CZE-First League multiplier -20%	29 / 30	1	0.88
CZE-First League_plus20	CZE-First League multiplier +20%	29 / 30	1	0.75
SVK-Super Liga_minus20	SVK-Super Liga multiplier -20%	30 / 30	0	0.00
SVK-Super Liga_plus20	SVK-Super Liga multiplier +20%	30 / 30	0	0.00
AUT-Bundesliga_minus20	AUT-Bundesliga multiplier -20%	30 / 30	0	0.00
AUT-Bundesliga_plus20	AUT-Bundesliga multiplier +20%	30 / 30	0	0.00
HUN-NB I_minus20	HUN-NB I multiplier -20%	30 / 30	0	0.00
HUN-NB I_plus20	HUN-NB I multiplier +20%	30 / 30	0	0.00
POL-Ekstraklasa_minus20	POL-Ekstraklasa multiplier -20%	29 / 30	1	0.37
POL-Ekstraklasa_plus20	POL-Ekstraklasa multiplier +20%	30 / 30	0	0.12
CRO-HNL_minus20	CRO-HNL multiplier -20%	30 / 30	0	0.00
CRO-HNL_plus20	CRO-HNL multiplier +20%	30 / 30	0	0.00
DEN-Superliga_minus20	DEN-Superliga multiplier -20%	30 / 30	0	0.00
DEN-Superliga_plus20	DEN-Superliga multiplier +20%	30 / 30	0	0.00
SUI-Super League_minus20	SUI-Super League multiplier -20%	30 / 30	0	0.00
SUI-Super League_plus20	SUI-Super League multiplier +20%	30 / 30	0	0.00
NOR-Eliteserien_minus20	NOR-Eliteserien multiplier -20%	30 / 30	0	0.00
NOR-Eliteserien_plus20	NOR-Eliteserien multiplier +20%	30 / 30	0	0.00
GER-2. Bundesliga_minus20	GER-2. Bundesliga multiplier -20%	30 / 30	0	0.00
GER-2. Bundesliga_plus20	GER-2. Bundesliga multiplier +20%	30 / 30	0	0.00
all_minus20	every league multiplier -20%	30 / 30	0	0.00
all_plus20	every league multiplier +20%	30 / 30	0	0.00

* Churn = baseline top-10 members that leave the set under the scenario; mean Δ rank = mean absolute rank change over the baseline top-20 union. Scenarios: baseline, every league ±20 % on its own, and all leagues ±20 % at once.

The table above is fixed at the config multipliers; the panel below is the same Czech-eligible top ten made interactive — drag any league's slider (0.5×–1.5× of its default) and the ranking recomputes in the browser.

Data-quality log

8 recomputed checks · 6 recorded incidents

Every wrangling decision that changed a count, with the count. The first block is recomputed on every run; the second is the incident record (dates and counts as recorded at the time).

Check	Count	What it counts
Women's entries filtered	92 entries	FBref country-page entries dropped for a surname ending in -ová (see Limitations).
Namesakes in the pool	2 players	Active pool players sharing a normalised name (e.g. father and son), disambiguated by club.
Pool players without season tables	125 players	Czech professionals on FBref's country page who play in a league without season tables and carry no metrics.
Split-season rows collapsed	441 rows	Player-season-group rows merged into one after a mid-season transfer (minutes summed, rates minutes-weighted).
Unmatched call-up names	18 names	National-team squad-table names that match no Czech-eligible row in the feature tables.
Missing birth years	0 rows	Season-table rows of nation CZE with no birth year, which cannot form a player_key.
Unjoined goalkeeper rows	12 rows	Keeper-page rows with no matching GK row in the season tables on (league, season, team, player_key); dropped from every goalkeeper exhibit.
Czech First League rows without a nationality	232 rows	Season-table rows in the Czech First League where FBref records no nationality — the highest rate of any league in this pipeline. Every “own nationals” share reads that column as its numerator while the denominator keeps the league’s full minutes, so those shares are floors, not point estimates.

• 2026-09-14 A stale FBref season index made soccerdata fetch the season-less URL, which FBref serves as the season in progress: nine leagues' 2025/26 tables were 2026/27 after four rounds. Fixed by checking the page's own heading against the requested season. (9 leagues, recorded)
• 2026-09-14 FBref intermittently served a squads-only page (no season table yet) for a season still in progress. Fixed by a single refetch, in the same page-heading guard.
• 2026-09-13 ClubElo's API answered 502 for the whole run; league multipliers fell back to UEFA association coefficients.
• 2026-09-13 The Wikidata portrait query matched on name, citizenship and birth date only; seven portraits belonged to namesakes in other sports until an occupation filter was added. (7 portraits, recorded)
• 2026-09-13 Slovakia's top flight is not on FBref at all; this report's Slovak exhibits rest entirely on players abroad.
• 2026-09-14 The first out-of-sample comparison of the league-strength model applied the UEFA-multiplier baseline in the wrong direction and started both baselines from a raw previous-season rate (a zero-goal season predicted zero); found before publication. Corrected: multiplier ratio m_prev / m_new, baselines from the shrunk rate. The model's margin over the baselines shrank from about 3 nats to 0.5 and is the figure reported.

Limitations of this analysis

Validation & robustness

Each model in this pipeline carries its own validation next to where it is described; this section collects one headline diagnostic from each as it lands. So far:

• Season-to-season model comparison (M1): rolling-origin evaluation over 5 seasons, pooled RMSE favours Gradient boosting (0.071 vs 0.075 for persistence), the Bayesian model's 90 % interval covered 91 % of observed values.
• League strength (M2): R-hat ≤ 1.003, 0 divergent transitions; out-of-sample log predictive density Same rate as before −2.628; Rate × UEFA ratio −2.734; Transfer-graph model −2.174; Spearman rho = 0.74 against the UEFA multipliers.
• The break and forecast (M4): the break dates to 2014/15 (58 % posterior) for Czechia, a ×0.59 level change; the plain local level beats the naive baseline on MAE (2.47 vs 2.73) with 93 % of the 90 % intervals covering the observed value over 15 rolling-origin backtests.
• The youth-minutes panel (M3): across 8 countries (country means, n = 8), the between-country slope is +10.11 per 10 percentage points of U21 share (−1.69–23.26), R² = 0.27; a plain pooled OLS slope agrees in sign at +7.76 (1.12–13.99) — the interval is wide because the panel is small. A within-country check (country-random-intercept fit on the full two-season panel) finds no signal: β_within = −0.16.
• The gap decomposition (M5): ridge fit (α = 1.00) on n = 8 peer countries; for Norway, the residual is −2.37 of a 6.98 gap — a decomposition of a correlation, not a causal accounting.
• Age at export (M6): R-hat ≤ 1.002, 0 divergent transitions across 115 players; β on origin-league strength is −0.03; leaving out 8 Czech exports and refitting shifts the 21-vs-24 difference by +0.0010.

Reproducibility

The pipeline is public: github.com/sandovabarbora/czefootball-player-pool-atlas, code under the MIT licence. From a clean clone, uv sync && make restore-snapshot && make figures && make pages rebuilds this edition from the committed data snapshot (commit ef7dbd4; FBref tables fetched 14 September 2026); make figures refits the models with their seeds (about 15 minutes an edition), which reproduces the model outputs in the snapshot, and redraws the figures. The findings on the front page come from the cross-edition file: run make restore-snapshot && make figures for every edition (NATION=den, eng, esp, ger, nor and the default), then make nations, then make pages for each edition. make test runs the automated checks; make all refetches everything and reruns the whole pipeline. Seed 42 for every stochastic step: KMeans, every PyMC fit (random_seed), the bootstrap on the minutes share and the gap decomposition’s bootstrap (42 plus the contrast’s index). Fetchers cache raw pages; the render step never touches the network. Each question page names the module that computes its data in its metadata line.

The processed tables also load into BigQuery unchanged: infra/bigquery/ has generated schemas, a bq load script and a key/unit contract for the model-output JSONs the tables don't cover. Run infra/bigquery/load.sh -n <dataset> <nation> to see the load commands without running them.

Internal automated checks

320 automated tests (pytest) check the pipeline and the site build on every change. No external review of the atlas has taken place.

Related methods and what was taken from them

The methods below shaped this report's design. Each entry states what the method is, what this report took from it, and what was left out and why.

One pairing here has no single citation behind it: the transfer-graph league-strength model (§ League strength) and the age-at-export curve (§ Age at export) are fit independently and joined only through origin-league strength as a covariate — a within-player league-identification model feeding a spline-in-age production curve is, to the author's knowledge, this report's own combination, not drawn whole from any one method below.

• Empirical-Bayes shrinkage of a sparse per-unit rate toward a group mean (Efron and Morris, 1975) · taken: per-90 rates are shrunk toward the league-season median with a K = 10 phantom-match prior before the quality projection (§ Bayesian shrinkage) · left out: the fully hierarchical variance-component estimate the original method also supports, since one shared K fits this corpus's minutes floor well enough for a descriptive report.
• Multilevel (partial-pooling) regression and posterior predictive checking as a model-diagnosis routine (Gelman et al., 2013) · taken: the league-strength and youth-panel models pool leagues and countries partially rather than fitting each alone or merging them into one, and the league-strength posterior predictive check (§ League strength) compares simulated to observed production · left out: model comparison by WAIC/LOO, since every model here is instead scored on seasons it never trained on (rolling-origin backtests), a stronger check for this report's purpose.
• The No-U-Turn Sampler, the gradient-based MCMC method PyMC uses by default (Hoffman and Gelman, 2014) · taken: every Bayesian model in this report — league strength, model comparison, the change-point series, the youth panel — is fit with it and diagnosed on R-hat and divergences · left out: variational inference as a faster approximate alternative, since none of the four models is slow enough to need it.
• Plus-minus and regularised adjusted plus-minus ratings, which isolate a player's contribution from teammates' and opponents' by regression (Kharrat, McHale and Peña, 2020; Hvattum, 2019) · taken: the league-strength model's within-player logic — only a mover's own before/after change of league separates their level from the league's scoring environment — follows the same identification idea, applied to leagues rather than teammates · left out: an actual RAPM fit over lineup data, which this corpus's season-level tables (no lineups, no possession data) cannot support.
• Bayesian online change-point detection, a sequential method for locating a shift in a data-generating process (Adams and MacKay, 2007) · taken: unknown break seasons with a marginalised discrete location parameter, here two ordered change points (a rise and a fall) on a short batch series rather than sequentially · left out: the online, sequential setting and more than two change points, since one country’s Big-5 count per season is short and fixed.
• The local-level model — a random walk plus noise for a slowly drifting series — in the state-space tradition (Durbin and Koopman, 2012) · taken: the change-point model (§ Dating the break) is exactly this local level in log space, with two added step changes at the breaks · left out: a local linear trend or seasonal component, since the series is annual and too short for a trend term to be identifiable.
• Rolling-origin (time-series) cross-validation: refit on data up to each origin, score only on what came after it (Hyndman and Athanasopoulos, 2021) · taken: both the model-comparison exercise (§ Three models, one task) and the change-point backtest (§ Dating the break) are scored this way, never on a random split that could leak future seasons into training · left out: expanding-vs-sliding-window variants beyond the single expanding-window scheme, since the corpus's five metrics seasons leave little room to compare schemes.
• The Oaxaca–Blinder decomposition, splitting a gap between two groups' means into an explained and an unexplained part via a linear model (Oaxaca, 1973; Blinder, 1973) · taken: the gap-decomposition exhibit (§ What the gap is made of) splits the per-capita gap into the three measured channels plus a residual the same way · left out: the detailed, coefficient-level decomposition of the explained share, since three channels are few enough to read directly off the coefficients themselves.
• The silhouette coefficient, a per-point measure of how well a clustering separates its groups (Rousseeuw, 1987) · taken: used to sanity-check the PCA cluster counts per position group and projection before they were fixed (§ Cluster archetypes) · left out: a silhouette sweep reported in the text, since the chosen cluster counts are stable across position groups and projections.
• Gradient-boosted trees and a small multilayer perceptron, two non-Bayesian machine-learning baselines standard in this kind of comparison (Pedregosa et al., 2011) · taken: both sit alongside the persistence, shrinkage and Bayesian models in § Three models, one task, on the same eight features and the same rolling-origin split · left out: hyperparameter search beyond scikit-learn's defaults (plus the MLP's 64/32 hidden layers), since the comparison's point is model family, not a tuned leaderboard.
• CIES Football Observatory’s periodic counts of footballers playing outside their home association (Poli, Ravenel and Besson, 2024) · taken: the population of players who changed league, which the out-of-sample league-strength check tracks, and the framing of the atlas’s contribution on the front page · left out: CIES’s own counts, since the atlas computes its per-head and pathway measures from its own fetched tables.

Next steps not attempted

• Player-season embeddings and graph methods over the transfer network — clubs and moves as a graph, players as nodes with learned representations — are natural next tools (graph neural networks, specifically) for a pool this size, but were not attempted here.
• Event- and tracking-derived features (pressing intensity, progressive carries, expected threat) would sharpen the style axis beyond the five box-score numbers used here, but no tracking data source was available for this corpus.

Glossary

Terms used in the sections above, each in one plain sentence.

Per 90

A rate scaled to a full match: a player with three goals in five matches, each played the full ninety minutes, has a rate of 0.6 goals per 90 — it lets a player who came on as a substitute be compared fairly with one who started every match.

Minutes share

How much of a club's available playing time a player actually got, out of every minute the club's matches could have offered that season; a player who played every minute of every match has a minutes share of 100 percent.

Shrinkage

A way of not trusting a small sample too much: a player with only a handful of matches has his numbers pulled part of the way toward the league's typical number, the way a manager waits for more than one good game before trusting that a young player's form is real.

League multiplier / league strength

A number saying how much a goal, an assist or a minute is worth in one league compared with another; a striker's goal in a weaker league counts for less once it is adjusted by that league's own multiplier — the same idea as judging a transfer by the level the player is coming from.

Quality-adjusted

A player's raw numbers after they have been multiplied by the league multiplier above, so a rate earned in a strong league and one earned in a weaker league can be compared on the same footing.

Interval (90 percent)

A range around a number that shows how sure the model is, not a single guess; a ninety percent interval means the model thinks the true value falls inside that range about nine times out of ten.

Posterior

What the model believes about a number after it has seen the data, expressed as a range of plausible values rather than one single figure; the probability attached to a break season in this report is a posterior probability.

Out-of-sample

Checking a method only on matches, players or seasons it was not shown while it was being built — the way a manager judges a scouting report by what actually happens once the player signs, not by how well the report described what had already happened.

Cluster

A group of players whose season numbers look similar to each other and different from other groups, found automatically from the data rather than assigned by hand; a cluster is a style label such as "high-volume scorers", not a formal position.

Change point

The point in a series of seasons where the level genuinely shifts to a new one and stays there, rather than just one unusually high or low season on its own; this report finds one for the count of home-nation players in Europe's biggest leagues.

90 % HDI and bootstrap interval

A 90 % highest-density interval is the narrowest range holding 90 % of a Bayesian model’s posterior; a 90 % bootstrap interval is the 5th to 95th percentile of an estimate recomputed on resampled data.

Top-9 leagues

The nine leagues ranked 1–9 in UEFA’s men’s association coefficient ranking on the fetch date (England, Italy, Spain, Germany, France, Portugal, Belgium, the Netherlands, Türkiye); the Big-5 are the first five.

Export age (two measures)

Age at the first season with real playing time in a foreign league (the pathway measures and the brief), and age at the first season in a top-9 league (the destination tables and the gap decomposition, recent entrants only); the two are different numbers for the same players.

Players counted

Per head: players with the country’s FBref nationality who appeared in a top-9 league in the season, all positions. Destinations: Czech-eligible outfield players with at least 450 minutes in a covered league, one row per position group. Season changes: the same players, one row per player. Pool list: every Czech-eligible player with a season row. Country page: every professional FBref lists for the country, including those in leagues without tables.

Sideways move

A player abroad in a league whose UEFA-coefficient multiplier is no higher than his home league’s.

Stepping-stone league

A covered league below the top nine that is no peer’s home league; in this configuration only the German second tier.

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