BGS VLIM log M_* ≥ 11.0, z < 0.35 — detailed systematic analysis

High-mass BGS VLIM sample (1,619,838 galaxies, z < 0.35). Shot noise dominates per-pixel statistics at both resolutions. The LRT rejects the additive null at NSIDE 64 (\(\lambda_{\rm LR} = 75.1\)).

Sample statistics 

Parameter	NSIDE 32	NSIDE 64	NSIDE 128	NSIDE 256
Stellar-mass threshold	log M* ≥ 11.0	log M* ≥ 11.0	log M* ≥ 11.0	log M* ≥ 11.0
Redshift limit	z < 0.35	z < 0.35	z < 0.35	z < 0.35
N_gal	1,619,838	1,619,838	1,619,838	1,619,838
N_pix (good footprint)	5,614	21,646	84,719	332,484
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	0.2971	0.2974	0.4580	0.7557	closed-form least-squares
ElasticNet	0.2985	0.2974	0.4580	0.7558	L1+L2 regularised; 3-fold CV
ISD-1	0.2971	0.2974	0.4580	0.7558	iterative self-calibration; poly order 1
ISD-3 †	0.7040	0.4975	0.4899	0.7786	† degree-3 polynomial; unavailable at NSIDE 128/256
MCMC-add	0.2974	0.2975	0.4580	0.7558	MCMC additive; acc N32=0.389 N64=0.390
MCMC-comb	0.3927	0.3052	0.4716	0.7256	MCMC combined; acc N32=0.292 N64=0.289

† 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	997.4	11	< 10^-207	Yes
NSIDE 64	75.1	11	< 10^-11	Yes
NSIDE 128	206.1	11	< 10^-38	Yes
NSIDE 256	682.0	11	< 10^-139	Yes

MCMC acceptance fractions: NSIDE 32: add 0.389, comb 0.292 NSIDE 64: add 0.390, comb 0.289 NSIDE 128: add 0.389, comb 0.291 NSIDE 256: add 0.388, comb 0.288. 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	+0.3608	+0.3616	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.2525	-0.2531	GAIA medium stellar density (crowding and deblending near bright stars)
3	GAIA_phot_rp_mean_flux	64	-0.0427	-0.0428	GAIA mean stellar flux in RP band (red scattered light)
4	GAIA_phot_g_mean_flux	64	+0.0316	+0.0314	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
5	GAIA_phot_bp_mean_flux	64	-0.0174	-0.0170	GAIA mean stellar flux in BP band (blue scattered light)
6	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0067	+0.0068	LS10_GALDEPTH_G_NSIDE_0064
7	LS10_PSFSIZE_R_NSIDE_0064	?	-0.0067	-0.0067	LS10_PSFSIZE_R_NSIDE_0064
8	LS10_EBV_NSIDE_0064	?	-0.0060	-0.0060	LS10_EBV_NSIDE_0064
9	LS10_GALDEPTH_Z_NSIDE_0064	?	+0.0055	+0.0055	LS10_GALDEPTH_Z_NSIDE_0064
10	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0046	+0.0046	LS10_GALDEPTH_R_NSIDE_0064
11	LS10_NOBS_R_NSIDE_0064	?	+0.0011	+0.0012	LS10_NOBS_R_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	+0.3700	GAIA faint stellar density (photometric mis-classification of faint stars as galaxies)
2	GAIA_nstar_medium	64	-0.2100	GAIA medium stellar density (crowding and deblending near bright stars)
3	GAIA_phot_rp_mean_flux	64	-0.0359	GAIA mean stellar flux in RP band (red scattered light)
4	GAIA_phot_g_mean_flux	64	+0.0332	GAIA mean stellar flux in G band (scattered-light / sky-background variations)
5	LS10_EBV_NSIDE_0064	?	+0.0297	LS10_EBV_NSIDE_0064
6	GAIA_phot_bp_mean_flux	64	-0.0251	GAIA mean stellar flux in BP band (blue scattered light)
7	LS10_GALDEPTH_R_NSIDE_0064	?	+0.0139	LS10_GALDEPTH_R_NSIDE_0064
8	LS10_NOBS_R_NSIDE_0064	?	+0.0101	LS10_NOBS_R_NSIDE_0064
9	LS10_GALDEPTH_G_NSIDE_0064	?	+0.0049	LS10_GALDEPTH_G_NSIDE_0064
10	LS10_PSFSIZE_R_NSIDE_0064	?	+0.0044	LS10_PSFSIZE_R_NSIDE_0064
11	LS10_GALDEPTH_Z_NSIDE_0064	?	-0.0026	LS10_GALDEPTH_Z_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	1,619,838	0.9968	0.0337	0.9268	0.9536	0.9926	1.0563	1.0898
ElasticNet	1,619,838	0.9963	0.0323	0.9268	0.9543	0.9927	1.0524	1.0847
ISD-1	1,619,838	0.9968	0.0339	0.9257	0.9533	0.9925	1.0567	1.0904
ISD-3 †	1,619,838	1.8086	8.4772	0.5103	0.7973	1.0347	1.3246	4.4536
MCMC-add	1,619,838	0.9968	0.0337	0.9267	0.9535	0.9926	1.0563	1.0897
MCMC-comb	1,619,838	1.0292	0.0352	0.9239	0.9707	1.0333	1.0763	1.1131

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) (\(\delta w/w \approx -0.1\%\) at 30 arcmin).

Without correction: suitable even without correction.
After correction: suitable — correction provides marginal improvement.
Large-angle warning (\(\theta > 2°\)): max correction 11.5% at 181 arcmin — any analysis using angular scales > 1° must apply WEIGHT_COMB.

LRT (NSIDE 64): \(\lambda_{\rm LR} = 75.1\) (dof = 11), p = 1.3e-11 → Reject H₀ — multiplicative contamination is statistically detected.

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

BGS VLIM log M* ≥ 11.0, z < 0.35 — detailed systematic analysis

BGS VLIM log M_* ≥ 11.0, z < 0.35 — detailed systematic analysis