.. _sample-10p75: BGS VLIM log M\ :sub:`*` ≥ 10.75, z < 0.31 — detailed systematic analysis =============================================================================== Intermediate-mass BGS VLIM sample (2,802,710 galaxies, z < 0.31). The LRT rejects the additive null at NSIDE 64 (:math:`\lambda_{\rm LR} = 89.1`). .. contents:: On this page :local: :depth: 1 .. seealso:: :doc:`results_ls10` — summary tables and figures for all nine samples. ---- Sample statistics ----------------- .. csv-table:: :header: "Parameter", "NSIDE 32", "NSIDE 64", "NSIDE 128", "NSIDE 256" :widths: 36, 16, 16, 16, 16 "Stellar-mass threshold", "log M* ≥ 10.75", "log M* ≥ 10.75", "log M* ≥ 10.75", "log M* ≥ 10.75" "Redshift limit", "z < 0.31", "z < 0.31", "z < 0.31", "z < 0.31" "N\ :sub:`gal`", "2,802,710", "2,802,710", "2,802,710", "2,802,710" "N\ :sub:`pix` (good footprint)", "5,618", "21,662", "84,824", "332,812" "N\ :sub:`templates`", "11", "11", "11", "11" "MCMC walkers", "210", "210", "210", "210" "MCMC steps after burn-in", "1500", "1500", "1500", "1500" ---- Goodness-of-fit: :math:`\hat{\sigma}` by method and resolution ------------------------------------------------------------------ The noise parameter :math:`\hat{\sigma}` measures residual scatter after systematic subtraction — lower is better. Results are shown for NSIDE 32, 64, 128, and 256. ISD-3 is unavailable at NSIDE 128 and 256 (no partial files generated at those resolutions). .. csv-table:: :header: "Method", ":math:`\hat{\sigma}` (N32)", ":math:`\hat{\sigma}` (N64)", ":math:`\hat{\sigma}` (N128)", ":math:`\hat{\sigma}` (N256)", "Notes" :widths: 14, 12, 12, 12, 12, 38 "OLS", "0.3057", "0.2831", "0.4190", "0.6422", "closed-form least-squares" "ElasticNet", "0.3069", "0.2831", "0.4194", "0.6423", "L1+L2 regularised; 3-fold CV" "ISD-1", "0.3057", "0.2831", "0.4190", "0.6422", "iterative self-calibration; poly order 1" "ISD-3 †", "0.6315", "0.4658", "0.9265", "0.6894", "† degree-3 polynomial; unavailable at NSIDE 128/256" "MCMC-add", "0.3061", "0.2831", "0.4191", "0.6422", "MCMC additive; acc N32=0.388 N64=0.392" "MCMC-comb", "0.3982", "0.3009", "0.4300", "0.6305", "MCMC combined; acc N32=0.295 N64=0.281" † ISD-3 uses a degree-3 polynomial expansion and is ill-conditioned with correlated templates. Likelihood Ratio Test (additive vs combined model) -------------------------------------------------- .. csv-table:: :header: "Resolution", ":math:`\lambda_{\rm LR}`", "dof", "p-value", "Reject H₀" :widths: 14, 14, 8, 22, 12 "NSIDE 32", "1169.2", "11", "< 10\ :sup:`-244`", "**Yes**" "NSIDE 64", "89.1", "11", "< 10\ :sup:`-14`", "**Yes**" "NSIDE 128", "287.4", "11", "< 10\ :sup:`-55`", "**Yes**" "NSIDE 256", "580.6", "11", "< 10\ :sup:`-117`", "**Yes**" MCMC acceptance fractions: NSIDE 32: add 0.388, comb 0.295 NSIDE 64: add 0.392, comb 0.281 NSIDE 128: add 0.388, comb 0.293 NSIDE 256: add 0.388, comb 0.297. Healthy range: 0.15–0.50. ---- Template amplitude ranking — additive model (MCMC-add, NSIDE 64) ------------------------------------------------------------------ All 11 templates sorted by absolute MCMC-add additive amplitude :math:`|\hat{a}_i|`. OLS shown for comparison. .. csv-table:: :header: "Rank", "Template", "NSIDE", ":math:`\hat{a}_i` (MCMC-add)", ":math:`\hat{a}_i` (OLS)", "Physical meaning" :widths: 5, 28, 6, 14, 14, 50 :stub-columns: 1 "1", "**GAIA_nstar_faint**", "64", "+0.3222", "+0.3268", "GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)" "2", "**GAIA_nstar_medium**", "64", "-0.2216", "-0.2251", "GAIA medium stellar density (crowding and deblending near bright stars)" "3", "**GAIA_phot_rp_mean_flux**", "64", "-0.0415", "-0.0418", "GAIA mean stellar flux in RP band (red scattered light)" "4", "**GAIA_phot_g_mean_flux**", "64", "+0.0290", "+0.0292", "GAIA mean stellar flux in G band (scattered-light / sky-background variations)" "5", "**GAIA_phot_bp_mean_flux**", "64", "-0.0149", "-0.0146", "GAIA mean stellar flux in BP band (blue scattered light)" "6", "LS10_EBV_NSIDE_0064", "?", "-0.0122", "-0.0122", "LS10_EBV_NSIDE_0064" "7", "LS10_GALDEPTH_R_NSIDE_0064", "?", "+0.0076", "+0.0075", "LS10_GALDEPTH_R_NSIDE_0064" "8", "LS10_PSFSIZE_R_NSIDE_0064", "?", "-0.0047", "-0.0048", "LS10_PSFSIZE_R_NSIDE_0064" "9", "LS10_GALDEPTH_Z_NSIDE_0064", "?", "+0.0045", "+0.0045", "LS10_GALDEPTH_Z_NSIDE_0064" "10", "LS10_NOBS_R_NSIDE_0064", "?", "-0.0023", "-0.0022", "LS10_NOBS_R_NSIDE_0064" "11", "LS10_GALDEPTH_G_NSIDE_0064", "?", "+0.0017", "+0.0016", "LS10_GALDEPTH_G_NSIDE_0064" ---- Template amplitude ranking — multiplicative model (MCMC-comb, NSIDE 64) -------------------------------------------------------------------------- All 11 templates sorted by absolute MCMC-comb multiplicative amplitude :math:`|\hat{b}_i|`. .. csv-table:: :header: "Rank", "Template", "NSIDE", ":math:`\hat{b}_i` (MCMC-comb)", "Physical meaning" :widths: 5, 30, 6, 16, 50 :stub-columns: 1 "1", "**GAIA_nstar_faint**", "64", "+0.6170", "GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)" "2", "**GAIA_nstar_medium**", "64", "-0.3227", "GAIA medium stellar density (crowding and deblending near bright stars)" "3", "**GAIA_phot_rp_mean_flux**", "64", "-0.0498", "GAIA mean stellar flux in RP band (red scattered light)" "4", "**GAIA_phot_g_mean_flux**", "64", "+0.0250", "GAIA mean stellar flux in G band (scattered-light / sky-background variations)" "5", "**LS10_NOBS_R_NSIDE_0064**", "?", "+0.0166", "LS10_NOBS_R_NSIDE_0064" "6", "LS10_PSFSIZE_R_NSIDE_0064", "?", "+0.0129", "LS10_PSFSIZE_R_NSIDE_0064" "7", "LS10_EBV_NSIDE_0064", "?", "+0.0117", "LS10_EBV_NSIDE_0064" "8", "LS10_GALDEPTH_R_NSIDE_0064", "?", "+0.0109", "LS10_GALDEPTH_R_NSIDE_0064" "9", "GAIA_phot_bp_mean_flux", "64", "-0.0108", "GAIA mean stellar flux in BP band (blue scattered light)" "10", "LS10_GALDEPTH_Z_NSIDE_0064", "?", "-0.0076", "LS10_GALDEPTH_Z_NSIDE_0064" "11", "LS10_GALDEPTH_G_NSIDE_0064", "?", "-0.0050", "LS10_GALDEPTH_G_NSIDE_0064" ---- Per-galaxy weight statistics (NSIDE 64) ---------------------------------------- From the ``*_NSIDE0064_WEIGHTS.fits`` file. Mean ≈ 1 and small std indicate a well-behaved weight distribution. .. csv-table:: :header: "Method", "N", "mean", "std", "p1", "p5", "p50", "p95", "p99" :widths: 14, 12, 8, 8, 8, 8, 8, 8, 8 "OLS", "2,802,710", "0.9976", "0.0325", "0.9371", "0.9566", "0.9932", "1.0548", "1.0866" "ElasticNet", "2,802,710", "0.9970", "0.0311", "0.9376", "0.9573", "0.9931", "1.0509", "1.0817" "ISD-1", "2,802,710", "0.9975", "0.0326", "0.9368", "0.9565", "0.9932", "1.0548", "1.0869" "ISD-3 †", "2,802,710", "1.4256", "6.3201", "0.6910", "0.8561", "0.9801", "1.2456", "1.7699" "MCMC-add", "2,802,710", "0.9975", "0.0324", "0.9368", "0.9567", "0.9932", "1.0544", "1.0861" "MCMC-comb", "2,802,710", "1.0658", "0.0391", "0.9604", "1.0028", "1.0661", "1.1213", "1.1747" ---- Systematic weight maps ---------------------- .. raw:: html

Mollweide weight maps — all six methods

weight maps NSIDE 32 weight maps
NSIDE 32 weight maps
weight maps NSIDE 64 weight maps
NSIDE 64 weight maps
weight maps NSIDE 128 weight maps
NSIDE 128 weight maps
weight maps NSIDE 256 weight maps
NSIDE 256 weight maps
Systematic weight distributions ------------------------------- Narrow peaks near 1 indicate stable weight estimates. ElasticNet weights may be exactly 1 when cross-validation selects zero amplitudes. .. raw:: html

Per-galaxy weight distributions — all six methods

weight distributions NSIDE 32 weight distributions
NSIDE 32 weight distributions
weight distributions NSIDE 64 weight distributions
NSIDE 64 weight distributions
weight distributions NSIDE 128 weight distributions
NSIDE 128 weight distributions
weight distributions NSIDE 256 weight distributions
NSIDE 256 weight distributions
Angular clustering w(θ) before and after correction ----------------------------------------------------- Each panel shows the observed angular two-point correlation function (solid black) and the corrected :math:`w(\theta)` for all six methods. A well-corrected sample shows suppressed excess clustering at all scales. Each panel corresponds to one map resolution. .. raw:: html

w(θ): observed vs corrected — all six methods

wtheta NSIDE 32
NSIDE 32
wtheta NSIDE 64
NSIDE 64
wtheta NSIDE 128
NSIDE 128
wtheta NSIDE 256
NSIDE 256
---- Cosmological analysis verdict ----------------------------- Sub-degree scales: unknown (no empirical two-point measurement available). * Without correction: uncertain. * After correction: **suitable** after applying ``WEIGHT_COMB``. LRT (NSIDE 64): :math:`\lambda_{\rm LR} = 89.1` (dof = 11), p = 2.5e-14 → **Reject H₀** — multiplicative contamination is statistically detected. **Recommendation**: use ``WEIGHT_COMB`` (``WEIGHT_SYS``) for all science-grade analyses.