Use Geary's C normality variance in spatial_autocorr (fixes #1183)

[gaoflow]

Description

squidpy.gr.spatial_autocorr shares a single _analytic_pval implementation between mode="moran" and mode="geary", swapping only E[score]. The variance it applies is the Moran's I normality variance (Cliff & Ord 1981); Geary's C has a different normality variance, so the analytic p-value (pval_norm / var_norm) for mode="geary" was miscalibrated.

Fixes #1183.

Fix

In _analytic_pval, branch on the statistic and use the Geary's C normality variance (matching pysal/esda's Geary):

Var[C] = ((2·S1 + S2)(n − 1) − 4·S0²) / (2·(n + 1)·S0²)

Moran's I is left exactly as before.

Verification

Reproduced with row-standardised 6-NN weights (the same setup as the issue):

	squidpy var_norm (Geary)	esda Geary.VC_norm	ratio
before	1.4583e-03	1.7014e-03	0.857
after	1.7014e-03	1.7014e-03	1.000

Computing the Geary closed-form from squidpy's own row-standardised weight moments reproduces the post-fix var_norm exactly (relative diff 0.0), and it clearly differs from the old Moran value — confirming the previous code applied the wrong formula. The reporter's 100k-trial null study shows the analytic p-value going from badly non-uniform (KS p ≈ 1e-108) to well-calibrated.

Tests

Added test_spatial_autocorr_var_norm_formula (parametrised over moran/geary) asserting the emitted var_norm matches the closed-form variance of the chosen statistic. It passes with the fix and fails on the Geary case without it. All existing spatial_autocorr tests continue to pass; Moran's I output is unchanged.

[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 76.55%. Comparing base (55572c4) to head (b466c26).

Additional details and impacted files

@@           Coverage Diff           @@
##             main    #1197   +/-   ##
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  Coverage   76.55%   76.55%           
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  Files          63       63           
  Lines        9067     9069    +2     
  Branches     1521     1522    +1     
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+ Hits         6941     6943    +2     
  Misses       1541     1541           
  Partials      585      585           

Files with missing lines	Coverage Δ
src/squidpy/gr/_ppatterns.py	80.60% <100.00%> (+0.16%)	⬆️

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[selmanozleyen]

Thanks a lot! I will merge this as soon as I get approval of someone else as well