BGS VLIM log M_* ≥ 11.25, z < 0.35 — detailed systematic analysis

High-mass BGS VLIM sample (541,855 galaxies, z < 0.35). Shot noise dominates per-pixel statistics at both resolutions. The LRT rejects the additive null at NSIDE 64 (\(\lambda_{\rm LR} = 123.4\)).

Sample statistics 

Parameter	NSIDE 32	NSIDE 64	NSIDE 128	NSIDE 256
Stellar-mass threshold	log M* ≥ 11.25	log M* ≥ 11.25	log M* ≥ 11.25	log M* ≥ 11.25
Redshift limit	z < 0.35	z < 0.35	z < 0.35	z < 0.35
N_gal	541,855	541,855	541,855	541,855
N_pix (good footprint)	5,609	21,555	84,131	325,324
N_templates	11	11	11	11
MCMC walkers	210	210	210	210
MCMC steps after burn-in	1500	1500	1500	1500

Goodness-of-fit: \(\hat{\sigma}\) by method and resolution 

The noise parameter \(\hat{\sigma}\) 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).

Method	\(\hat{\sigma}\) (N32)	\(\hat{\sigma}\) (N64)	\(\hat{\sigma}\) (N128)	\(\hat{\sigma}\) (N256)	Notes
OLS	0.3308	0.3842	0.6405	1.2602	closed-form least-squares
ElasticNet	0.3319	0.3846	0.6406	1.2603	L1+L2 regularised; 3-fold CV
ISD-1	0.3308	0.3842	0.6405	1.2602	iterative self-calibration; poly order 1
ISD-3 †	0.7031	0.6196	0.6454	1.3131	† degree-3 polynomial; unavailable at NSIDE 128/256
MCMC-add	0.3313	0.3843	0.6406	1.2602	MCMC additive; acc N32=0.386 N64=0.389
MCMC-comb	0.4062	0.3930	0.6329	1.2117	MCMC combined; acc N32=0.293 N64=0.288

† ISD-3 uses a degree-3 polynomial expansion and is ill-conditioned with correlated templates.

Likelihood Ratio Test (additive vs combined model)

Resolution	\(\lambda_{\rm LR}\)	dof	p-value	Reject H₀
NSIDE 32	613.5	11	< 10^-124	Yes
NSIDE 64	123.4	11	< 10^-21	Yes
NSIDE 128	140.8	11	< 10^-24	Yes
NSIDE 256	597.3	11	< 10^-121	Yes

MCMC acceptance fractions: NSIDE 32: add 0.386, comb 0.293 NSIDE 64: add 0.389, comb 0.288 NSIDE 128: add 0.387, comb 0.287 NSIDE 256: add 0.390, comb 0.301. 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 \(|\hat{a}_i|\). OLS shown for comparison.

Rank	Template	NSIDE	\(\hat{a}_i\) (MCMC-add)	\(\hat{a}_i\) (OLS)	Physical meaning
1	GAIA_nstar_faint	64	+0.2755	+0.2785	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.1944	-0.1970	GAIA medium stellar density (crowding and deblending near bright stars)
3	GAIA_phot_g_mean_flux	64	+0.0530	+0.0522	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
4	GAIA_phot_rp_mean_flux	64	-0.0511	-0.0504	GAIA mean stellar flux in RP band (red scattered light)
5	GAIA_phot_bp_mean_flux	64	-0.0334	-0.0331	GAIA mean stellar flux in BP band (blue scattered light)
6	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0132	+0.0131	LS10_GALDEPTH_G_NSIDE_0064
7	LS10_PSFSIZE_R_NSIDE_0064	?	-0.0079	-0.0079	LS10_PSFSIZE_R_NSIDE_0064
8	LS10_GALDEPTH_Z_NSIDE_0064	?	+0.0054	+0.0054	LS10_GALDEPTH_Z_NSIDE_0064
9	LS10_NOBS_R_NSIDE_0064	?	+0.0044	+0.0045	LS10_NOBS_R_NSIDE_0064
10	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0034	+0.0034	LS10_GALDEPTH_R_NSIDE_0064
11	LS10_EBV_NSIDE_0064	?	-0.0029	-0.0028	LS10_EBV_NSIDE_0064

Template amplitude ranking — multiplicative model (MCMC-comb, NSIDE 64)

All 11 templates sorted by absolute MCMC-comb multiplicative amplitude \(|\hat{b}_i|\).

Rank	Template	NSIDE	\(\hat{b}_i\) (MCMC-comb)	Physical meaning
1	GAIA_nstar_faint	64	+0.3944	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.2422	GAIA medium stellar density (crowding and deblending near bright stars)
3	LS10_EBV_NSIDE_0064	?	+0.0513	LS10_EBV_NSIDE_0064
4	GAIA_phot_bp_mean_flux	64	-0.0343	GAIA mean stellar flux in BP band (blue scattered light)
5	GAIA_phot_g_mean_flux	64	+0.0288	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
6	GAIA_phot_rp_mean_flux	64	-0.0168	GAIA mean stellar flux in RP band (red scattered light)
7	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0137	LS10_GALDEPTH_G_NSIDE_0064
8	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0098	LS10_GALDEPTH_R_NSIDE_0064
9	LS10_GALDEPTH_Z_NSIDE_0064	?	-0.0063	LS10_GALDEPTH_Z_NSIDE_0064
10	LS10_NOBS_R_NSIDE_0064	?	+0.0054	LS10_NOBS_R_NSIDE_0064
11	LS10_PSFSIZE_R_NSIDE_0064	?	-0.0016	LS10_PSFSIZE_R_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.

Method	N	mean	std	p1	p5	p50	p95	p99
OLS	541,855	0.9914	0.0352	0.8993	0.9443	0.9892	1.0495	1.0841
ElasticNet	541,855	0.9906	0.0252	0.9133	0.9564	0.9927	1.0248	1.0468
ISD-1	541,855	0.9914	0.0355	0.8980	0.9439	0.9893	1.0500	1.0844
ISD-3 †	541,855	1.8450	9.0095	0.7238	0.8593	0.9633	1.2506	5.9101
MCMC-add	541,855	0.9914	0.0352	0.8990	0.9445	0.9892	1.0493	1.0843
MCMC-comb	541,855	1.0292	0.0470	0.9159	0.9462	1.0363	1.0932	1.1294

Systematic weight maps 

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.

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 \(w(\theta)\) for all six methods. A well-corrected sample shows suppressed excess clustering at all scales. Each panel corresponds to one map resolution.

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

Cosmological analysis verdict 

Sub-degree scales (\(\theta < 30'\)): regime is moderately contaminated (\(\delta w/w \approx +2.2\%\) at 30 arcmin).

Without correction: borderline without correction.
After correction: suitable after applying WEIGHT_COMB.
Large-angle warning (\(\theta > 2°\)): max correction 17.4% at 181 arcmin — any analysis using angular scales > 1° must apply WEIGHT_COMB.

LRT (NSIDE 64): \(\lambda_{\rm LR} = 123.4\) (dof = 11), p = 3.7e-21 → Reject H₀ — multiplicative contamination is statistically detected.

Recommendation: use WEIGHT_COMB (WEIGHT_SYS) for all science-grade analyses.

BGS VLIM log M* ≥ 11.25, z < 0.35 — detailed systematic analysis

BGS VLIM log M_* ≥ 11.25, z < 0.35 — detailed systematic analysis