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![Instrumental Gain Modelling G(ta) = g(ta) (1 + ˆϵ ) Smooth gain 1/f noise (stochastic fractional gain) Flicker Noise Pˆϵ(f ) = { 0, |f | < fc, (f0/|f |)α , |f | ≥ fc. g(ta) = 4∑ n=0 an Pn(x), x = 2(ta − tmin) tmax − tmin − 1. 0 200 400 600 800 1000 Time [s] 0.0003 0.0002 0.0001 0.0000 0.0001 0.0002 Fractional Gain Error 1/f vs Time 2. Data Model 2.1. Smooth Gain 4/20](slides/04.svg)
![Stochastic Gain Model G(ta) = g(ta) (1 + ˆϵ ) Smooth gain 1/f noise (stochastic fractional gain) Flicker Noise (Complete Model) Gaussianity and stationarity . Pˆϵ(f ) = { 0, |f | < fc, (f0/|f |)α , |f | ≥ fc. g(ta) = 4∑ n=0 an Pn(x), x = 2(ta − tmin) tmax − tmin − 1. 0 200 400 600 800 1000 Time [s] 0.0003 0.0002 0.0001 0.0000 0.0001 0.0002 Fractional Gain Error 1/f vs Time 2. Data Model 2.2. Stochastic Gain 5/20](slides/05.svg)
![Stochastic Gain Model But how to implement the 1 /f noise modelling? • Case 2: Diagonal in CFT space (Ncorr)aa′ = ξ(ta′ − ta) ξ(τ ) = Θα 0 πτ Re [ Γ(µ, iΘc) e−i π 2 µ ] µ = 1 − α, Θ0 = τ f0, Θc = τ fc /uni00000010/uni00000018/uni00000013/uni00000013/uni00000013/uni00000013/uni00000018/uni00000013/uni00000013/uni00000013 /uni0000002f/uni00000044/uni0000004a/uni00000003/uni0000000b/uni00000056/uni0000000c /uni00000010/uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000048/uni00000010/uni0000001a /uni00000026/uni00000044/uni00000056/uni00000048/uni00000003/uni00000014/uni0000001d/uni00000003/uni00000037/uni0000004c/uni00000050/uni00000048/uni00000010/uni00000057/uni0000004c/uni00000050/uni00000048/uni00000003/uni00000026/uni00000052/uni00000055/uni00000055/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051 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/uni00000026/uni00000044/uni00000056/uni00000048/uni00000003/uni00000015/uni0000001d/uni00000003/uni00000033/uni00000036/uni00000027/uni00000003/uni0000000b/uni00000051/uni00000052/uni00000055/uni00000050/uni00000044/uni0000004f/uni0000004c/uni00000056/uni00000048/uni00000047/uni0000000c /uni00000013/uni00000015/uni00000013/uni00000013/uni00000013/uni00000017/uni00000013/uni00000013/uni00000013 /uni00000037/uni0000004c/uni00000050/uni00000048/uni00000003/uni0000000b/uni00000056/uni0000000c /uni00000013 /uni00000014/uni00000013/uni00000013/uni00000013 /uni00000015/uni00000013/uni00000013/uni00000013 /uni00000016/uni00000013/uni00000013/uni00000013 /uni00000017/uni00000013/uni00000013/uni00000013 /uni00000018/uni00000013/uni00000013/uni00000013 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 2. Data Model 2.2. Stochastic Gain 7/20](slides/07.svg)
![System Temperature Model Sky + Receiver + Ground + Atmospheric + ... T sys j,p (ta) = ∑ X BX p (ta) ∗ T sky,X + T el p (ta) + T nd j,p (ta) + T rec j,p Linear Model ⃗Tsys = Uce ⃗pce + Ures ⃗pres 0 1000 2000 3000 4000 5000 6000 Time [s] 1.00 0.75 0.50 0.25 0.00 0.25 0.50 0.75 1.00 Linear Response of System T emperature Parameters Sky #167 Sky #33 Sky #15 Sky #316 Other modes Other modes Other modes Other modes Other modes 2. Data Model 2.3. System T emperature 8/20](slides/08.svg)
![Joint Bayesian Analysis Framework • The data model can be rewritten as d(ta) ≃ g(ta)Tsys(ta) [1 + ˆw + ˆϵ] . • the likelihood function of the data model is given by L(n|g, Tsys, N) = ( 2π)− N 2 |N|− 1 2 exp {[ − 1 2 nT N−1n ]} , where N = Nw + Ncorr. • Gibbs sampling steps: g(i+1) ← Ppost(g|d, T (i) sys, N(i)) N(i+1) ← Ppost(N|d, T (i) sys, g(i+1)) T (i+1) sys ← Ppost(Tsys|d, g(i+1), N(i+1)) 3. Bayesian Workflow 3.0. 9/20](slides/09.svg)
![Tractable Implementation for Gain and Tsys samplers Iterative GLS sampler • Conditional on noise, the data model can be generally written as d = (Up + µ) ◦ (1 + n). • Rearrangement - effective additive form: d ′ = Up + ϵ, ϵ ∼ N (0, Σ), where d ′ = d − µ and the noise covariance structure is given by: Σ = Diag(Up+µ) N Diag(Up+µ). Estimating Σ with iterative GLS: 1. Initialize p(0) using OLS: p(0) = (U⊤U)−1U⊤d ′. 2. At iteration k , compute Σ(k ) = diag(Up(k ) + µ) N diag(Up(k ) + µ). 3. Update p by solving: [ U⊤ ( Σ(k ) ) −1 U ] p(k +1) = U⊤ ( Σ(k ) ) −1 d ′. 4. Repeat until convergence: ∥p(k +1) − p(k )∥ < tol . 3. Bayesian Workflow 3.1. Iterative GLS 10/20](slides/10.svg)
![Tractable Implementation for Gain and Tsys samplers Gaussian Constrained Realisation (GCR) equations Per TOD set sampling: ( C−1 + U⊤Σ−1U ) psample = U⊤ ( Σ−1d ′ + Σ− 1 2 ω ) + C−1 ¯p + C− 1 2 η, Multiple TOD sets joint fitting: C−1 + ∑ j Uj ⊤Σ−1 j Uj psample = ∑ j Uj ⊤ [ Σ−1 j d ′ j + Σ − 1 2 j ωj ] + C−1 ¯p + C− 1 2 η, 3. Bayesian Workflow 3.1. Iterative GLS 11/20](slides/11.svg)
![Tractable Implementation for Sampling 1/f Parameters COMAT: An O(n2) implementation Ncorr = ξ0 ξ1 ξ2 · · · ξn−1 ξ1 ξ0 ξ1 · · · ξn−2 ξ2 ξ1 ξ0 · · · ξn−3 ... ... ... . . . ... ξn−1 ξn−2 ξn−3 · · · ξ0 • Recursive Determinant det A B C D = det(A)det(D−CA−1B). • Levinson-Durbin Recursion • Perturbative Scenario (Λ + P)−1 ≃ Λ−1 − m−1∑ k =0 Λ−1P [ −PΛ−1]k Λ−1. ln[det(Λ + P)] ≈ m−1∑ k =0 (−1)k k + 1 Tr [( PΛ−1) k +1] + ln[det(Λ)] 3. Bayesian Workflow 3.2. 1/f Sampler 12/20](slides/12.svg)
![An illustrating Example Histogram of the residuals across pixels 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Tresidual = Tsample sky Ttrue sky [K] 0 5 10 15 20 25Probability Density 16th 84th pct = [-0.13, 0.17] K Figure: 1×TOD; 1 CalSrc 1.5 1.0 0.5 0.0 0.5 1.0 1.5 Tresidual = Tsample sky Ttrue sky [K] 0 10 20 30 40Probability Density 16th 84th pct = [-0.10, 0.03] K Figure: 2×TOD; 1 CalSrc 4. Demonstration 4.3. Results 17/20](slides/17.svg)
