BGS VLIM log M_* ≥ 11.5, z < 0.35 — detailed systematic analysis

High-mass BGS VLIM sample (120,882 galaxies, z < 0.35). Shot noise dominates per-pixel statistics at both resolutions. The LRT rejects the additive null at NSIDE 64 (\(\lambda_{\rm LR} = 151.5\)).

Sample statistics 

Parameter	NSIDE 32	NSIDE 64	NSIDE 128	NSIDE 256
Stellar-mass threshold	log M* ≥ 11.5	log M* ≥ 11.5	log M* ≥ 11.5	log M* ≥ 11.5
Redshift limit	z < 0.35	z < 0.35	z < 0.35	z < 0.35
N_gal	120,882	120,882	120,882	120,882
N_pix (good footprint)	5,571	21,344	83,244	180,102
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	0.4451	0.6438	1.3802	2.1098	closed-form least-squares
ElasticNet	0.4453	0.6440	1.3803	2.1098	L1+L2 regularised; 3-fold CV
ISD-1	0.4451	0.6438	1.3802	2.1098	iterative self-calibration; poly order 1
ISD-3 †	2.1206	1.0507	1.4249	2.1189	† degree-3 polynomial; unavailable at NSIDE 128/256
MCMC-add	0.4455	0.6441	1.3803	2.1098	MCMC additive; acc N32=0.387 N64=0.388
MCMC-comb	0.5862	0.7104	1.4304	2.0211	MCMC combined; acc N32=0.281 N64=0.287

† 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	352.1	11	< 10^-69	Yes
NSIDE 64	151.5	11	< 10^-27	Yes
NSIDE 128	196.7	11	< 10^-36	Yes
NSIDE 256	575.1	11	< 10^-116	Yes

MCMC acceptance fractions: NSIDE 32: add 0.387, comb 0.281 NSIDE 64: add 0.388, comb 0.287 NSIDE 128: add 0.388, comb 0.297 NSIDE 256: add 0.387, comb 0.283. 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_medium	64	-0.1024	-0.1062	GAIA medium stellar density (crowding and deblending near bright stars)
2	GAIA_nstar_faint	64	+0.0901	+0.0952	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
3	GAIA_phot_rp_mean_flux	64	-0.0527	-0.0531	GAIA mean stellar flux in RP band (red scattered light)
4	GAIA_phot_g_mean_flux	64	+0.0501	+0.0507	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
5	GAIA_phot_bp_mean_flux	64	-0.0288	-0.0292	GAIA mean stellar flux in BP band (blue scattered light)
6	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0193	+0.0192	LS10_GALDEPTH_G_NSIDE_0064
7	LS10_PSFSIZE_R_NSIDE_0064	?	-0.0098	-0.0098	LS10_PSFSIZE_R_NSIDE_0064
8	LS10_GALDEPTH_Z_NSIDE_0064	?	+0.0083	+0.0084	LS10_GALDEPTH_Z_NSIDE_0064
9	LS10_NOBS_R_NSIDE_0064	?	+0.0073	+0.0074	LS10_NOBS_R_NSIDE_0064
10	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0065	+0.0065	LS10_GALDEPTH_R_NSIDE_0064
11	LS10_EBV_NSIDE_0064	?	+0.0031	+0.0032	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	+0.6752	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.3323	GAIA medium stellar density (crowding and deblending near bright stars)
3	LS10_EBV_NSIDE_0064	?	+0.0415	LS10_EBV_NSIDE_0064
4	GAIA_phot_bp_mean_flux	64	-0.0356	GAIA mean stellar flux in BP band (blue scattered light)
5	GAIA_phot_g_mean_flux	64	+0.0188	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
6	GAIA_phot_rp_mean_flux	64	+0.0143	GAIA mean stellar flux in RP band (red scattered light)
7	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0134	LS10_GALDEPTH_R_NSIDE_0064
8	LS10_NOBS_R_NSIDE_0064	?	+0.0119	LS10_NOBS_R_NSIDE_0064
9	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0077	LS10_GALDEPTH_G_NSIDE_0064
10	LS10_GALDEPTH_Z_NSIDE_0064	?	+0.0047	LS10_GALDEPTH_Z_NSIDE_0064
11	LS10_PSFSIZE_R_NSIDE_0064	?	-0.0027	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	120,882	0.9677	0.0399	0.8457	0.9089	0.9695	1.0270	1.0636
ElasticNet	120,882	0.9717	0.0339	0.8605	0.9216	0.9759	1.0171	1.0473
ISD-1	120,882	0.9678	0.0397	0.8441	0.9085	0.9697	1.0266	1.0620
ISD-3 †	120,882	3.3404	14.7097	0.2588	0.5345	0.9954	2.0296	100.0000
MCMC-add	120,882	0.9677	0.0397	0.8457	0.9091	0.9696	1.0265	1.0630
MCMC-comb	120,882	1.1084	0.0572	0.9525	1.0000	1.1186	1.1849	1.2112

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 sub-percent contamination (below noise floor) (\(\delta w/w \approx +0.7\%\) at 30 arcmin).

Without correction: suitable even without correction.
After correction: suitable — correction provides marginal improvement.
Large-angle warning (\(\theta > 2°\)): max correction 9.7% at 97 arcmin — any analysis using angular scales > 1° must apply WEIGHT_COMB.

LRT (NSIDE 64): \(\lambda_{\rm LR} = 151.5\) (dof = 11), p = 7.3e-27 → Reject H₀ — multiplicative contamination is statistically detected.

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

BGS VLIM log M* ≥ 11.5, z < 0.35 — detailed systematic analysis

BGS VLIM log M_* ≥ 11.5, z < 0.35 — detailed systematic analysis