Statistics, Data Mining, and Machine Learning in Astronomy Visual Figure Index This is a visual listing of the figures within the book The first number below each thumbnail gives the figure number wit[.]
Visual Figure Index This is a visual listing of the figures within the book The first number below each thumbnail gives the figure number within the text; the second (in parentheses) gives the page number on which the figure can be found Stars Plate = 1615, MJD = 53166, Fiber = 513 14 15 16 17 18 19 20 21 22 200 −1 2 1 2.0 15 150 log10 [g/(cm/s2 )] 1.5 0.6 0.4 r−i 1.0 0.8 14 250 rpetrosian −1 1.0 300 Flux r Galaxies 14 15 16 17 18 19 20 21 22 16 0.2 2.5 3.0 3.5 0.0 r−i 100 4.0 17 −0.2 0 4.5 50 −1 −1 g−r −1 −1 g−r 3000 4000 5000 6000 7000 8000 9000 −0.4 18 1.0 10000 1.5 2.0 2.5 1.2 (21) 4.0 4.5 Example of phased light curve 5.0 8000 7500 7000 6500 6000 Teff (K) redshift 1.3 (23) 101 2.5 3.5 u−r ˚ λ(A) 1.1 (19) 3.0 1.4 (24) 5500 5000 4500 1.5 (26) 0.30 2.5 2.5 0.25 2.0 2.0 0.20 1.5 1.5 14.5 15.5 10−1 0.2 1.0 0.4 0.6 phase 0.8 1.0 r−i 0.5 0.0 0.5 1.0 1.5 2.0 −1 g−r Inner 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.4 1.11 (34) 0.00 −0.2 −0.1 0.0 0.1 0.2 0.3 0.4 a∗ Scaling of Search Algorithms −0.5 2.6 2.8 3.0 Semimajor Axis (AU) 3.2 3.4 −0.5 3.6 2.0 2.2 2.4 2.6 2.8 3.0 3.2 a(AU) 0.0 0.5 1.0 g−r 1.5 2.0 60◦ 0.0 0.5 Relative sort time 1.5 2.0 2.5 HEALPix Pixels (Mollweide) Aitoff projection 60◦ 30◦ −120◦ −60◦ 0◦ 60◦ 120◦ −120◦ −60◦ 0◦ 0◦ 60◦ 120◦ −30◦ −60◦ −60◦ Lambert projection Raw WMAP data Mollweide projection 60◦ 30◦ −120◦ −60◦ 0◦ 0◦ 60◦ −120◦ −60◦ 120◦ 0◦ 60◦ 120◦ −30◦ −60◦ −150◦−120◦ −90◦ −60◦ −30◦ 0◦ 30◦ 60◦ 90◦ 120◦ 150◦ 1.13 (35) 1.14 (36) ∆T (mK) kd-tree Example Quad-tree Example 1.15 (38) Ball-tree Example level level level level level level level level level level level level list sort NumPy sort O[N log N] 101 O[N] 100 10−1 10−2 106 Length of Array 1.0 y y p(A ∩ B) p(B) 0.5 0.0 x p(A ∪ B) = p(A) + p(B) − p(A ∩ B) 3.1 (70) ×10−3 13.5 12.0 10.5 9.0 7.5 6.0 4.5 3.0 1.5 0.0 Conditional Probability p(x|1.5) Joint Probability p(x, y) p(y) 1.5 2.3 (58) 2.4 (59) T 3.2 (72) 1.0 1.4 px (x) = Uniform(x) − f P f P D 0.0 0.5 1.0 x 1.5 1.0 0.8 0.6 0.6 f N − f N 3.3 (76) 1.6 1.6 1.2 1.2 0.8 mag = −2.5 log10 (flux) 0.8 0.4 0.4 2.0 20% flux error y = exp(x) py (y) = px (ln y)/y 1.2 0.4 0.2 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 x 2.5 (61) 0.8 p(x|1.0) 2.0 p(A) 107 2.2 (52) p(x) 2.1 (45) 105 p(mag) 10−3 108 p(flux) 107 Length of Array p(x|0.5) 106 py (y) 10−3 1.0 g−r 1.10 (33) Hammer projection 30◦ 0◦ −0.5 2.5 1.9 (32) −30◦ Scaling of Sort Algorithms 102 efficient search (O[log N]) Relative search time 0.0 −0.5 -1 linear search (O[N]) 10−4 0.15 0.05 −0.6 100 10−2 0.0 Mercator projection 1.12 (34) 10−1 2.4 75◦ 60◦ 45◦ 30◦ 15◦ 0◦ −15◦ −30◦ −45◦ −60◦ −75◦ 0.10 −0.4 −0.8 2.2 Outer 0.20 0.0 −0.2 101 102 103 −2.5 −1.5 −0.5 0.5 −2.5 −1.5 −0.5 0.5 mean [Fe/H] in pixel mean [Fe/H] in pixel number in pixel Mid 0.25 0.2 8000 7000 6000 5000 8000 7000 6000 5000 8000 7000 6000 5000 Teff Teff Teff 2.0 101 1.8 (31) sin(i) 1.5 2.0 2.5 3.0 3.5 4.0 4.5 101 100 Period (days) 1.7 (29) i − z log(g) 1.6 (28) 1.5 2.0 2.5 3.0 3.5 4.0 4.5 10−1 1.0 0.5 0.5 0.10 0.00 0.0 g−r 100 1.0 0.05 −0.5 −0.5 0.15 px (x) r−i 0.0 r−i 15.0 r−i magnitude 1.5 100 Sine of Inclination Angle Period (days) 2.0 0.00 0.25 0.50 0.75 1.00 x 3.4 (77) 0.0 0.0 1.2 1.6 y 2.0 2.4 2.8 0.0 0.5 1.0 flux 1.5 3.5 (78) 2.0 0.0 −1.0 −0.5 0.0 mag 0.5 1.0 • 528 Visual Figure Index Skew Σ and Kurtosis K 0.7 Binomial Distribution Gaussian Distribution Uniform Distribution mod Gauss, Σ = −0.36 log normal, Σ = 11.2 0.6 µ = 0, W = µ = 0, W = µ = 0, W = 1.0 0.40 b = 0.2, n = 20 b = 0.6, n = 20 b = 0.6, n = 40 µ = 0, σ = 0.5 µ = 0, σ = 1.0 µ = 0, σ = 2.0 0.8 0.7 µ=1 µ=5 µ = 15 0.35 0.20 p(x) 0.4 0.30 0.3 0.6 Cosine, K = −0.59 Uniform, K = −1.2 0.6 0.25 0.15 0.5 0.4 p(x|µ) Laplace, K = +3 Gaussian, K = 0.5 p(x|b, n) 0.0 p(x|µ, W ) 0.1 p(x|µ, σ) 0.8 0.2 0.20 0.10 0.15 0.3 0.4 0.4 p(x) Poisson Distribution 0.25 Gaussian, Σ = 0.5 0.3 0.10 0.2 0.05 0.2 0.2 0.05 0.1 0.1 0.0 −2 x 3.6 (80) −0.5 0.5 1.0 0.0 1.5 3.7 (86) Cauchy Distribution −4 −2 x Value 15 20 x 25 30 0.00 35 χ2 Distribution 0.5 10 15 x 20 25 30 3.10 (92) Student’s t Distribution 0.45 t(k = ∞) k=1 k=2 k=5 k=7 0.4 0.40 t(k = 2.0) 0.35 t(k = 0.5) t(k = 1.0) −2 −4 mean median 0.30 robust mean (mixture) robust mean (sigma-clip) 0.6 60 0.3 0.3 p(x|µ, ∆) −6 40 Value 10 µ = 0, ∆ = 0.5 µ = 0, ∆ = 1.0 µ = 0, ∆ = 2.0 0.8 0.4 0.1 Laplace Distribution 1.0 0.2 3.9 (90) 0.5 0.00 3.8 (87) µ = 0, γ = 0.5 µ = 0, γ = 1.0 µ = 0, γ = 2.0 0.6 p(x|µ, γ) 0.0 x 0.25 p(x|k) −4 −1.0 p(Q|k) 0.0 −1.5 0.4 0.2 0.2 0.1 0.20 0.15 20 −20 0.10 −40 0.05 −60 −4 −2 x 20 40 60 80 0.0 −6 100 0.0 −2 x Sample Size 3.11 (93) 3.12 (94) Fisher’s Distribution 6 0.00 10 2.5 d1 = 10, d2 = 50 3.14 (97) 0.35 x 2.0 N =2 1.6 k = 0.5, λ = k = 1.0, λ = k = 2.0, λ = k = 2.0, λ = 0.5 0.30 −2 3.15 (99) Weibull Distribution k = 1.0, θ = 2.0 k = 2.0, θ = 1.0 k = 3.0, θ = 1.0 k = 5.0, θ = 0.5 0.40 2.0 −4 Q 0.45 α = 0.5, β = 0.5 α = 1.5, β = 1.5 α = 3.0, β = 3.0 α = 0.5, β = 1.5 d1 = 5, d2 = d1 = 2, d2 = Gamma Distribution 3.0 d1 = 1, d2 = 0.8 3.13 (96) Beta Distribution 1.0 1.2 0.8 0.4 2.5 0.4 N =3 2.0 0.25 0.20 1.5 p(x) 1.5 p(x|k, λ) p(x|k, θ) 0.6 p(x|α, β) p(x|d1 , d2 ) −4 p(x) 0.0 0.3 1.0 0.5 0.4 1.0 0.15 0.2 N = 10 0.10 0.5 p(x) 0.2 0.1 0.05 0.2 0.4 0.6 x 0.8 1.0 3.17 (102) µ¯ = mean(x) 0.15 0.10 σ1 = σx = 1.58 σ2 = σy = 1.58 x 10 3.18 (103) 15% outliers 10 0.01 −0.02 0.05 χ2 dof = 3.84 (14 σ) χ2 dof = 2.85 (9.1 σ) µ ˆ = 10.16 observations observations 4.1 (133) 10 x 12 14 10 x 12 3850 0.6 0.4 0.2 10 0.85 0.90 0.95 1.00 σ 1.05 1.10 Ji p(x) ax ( ) n = 20 x 10 x 4.8 (167) 0.14 true distribution observed distribution 0.12 0.6 observed sample 0.2 random sample 0.06 0.04 100 0.00 5.1 (188) 12 k 16 20 0.3 0.2 0.1 0.02 10−1 0.4 xobs 5.2 (192) 0.0 −4 −3 −2 −1 xobs − xtrue bias/scatter (mag) p(xobs − xtrue ) reg ion p(xobs ) 0.08 sa mp led 101 0.45 0.50 rs 0.55 0.60 0.65 0.70 15 0.0 −2 −1 x −3 −2 −1 x Cloned Distribution 0.6 0.5 0.4 KS test: D = 0.00 p = 1.00 0.3 −1 0.2 10 −2 −3 0.32 0.34 0.36 0.38 0.40 τ 0.42 0.44 0.46 0.48 0.1 0.0 0.2 0.4 0.6 p(< x) 0.8 500 σ∗ (std dev.) 400 σG∗ (quartile) σ (std dev.) 20 300 200 0.0 −3 1.0 −2 −1 x 3.25 (121) xc = 120 hB (x) 0.035 σG (quartile) 0.030 15 0.025 10 100 0 0.020 0.6 1.004 1.008 0.005 0.5 0.6 0.7 0.8 0.9 1.0 1.1 σ 0.1 0.0 1.4 5580 points 0.8 1.2 1.0 0.6 0.8 0.4 0.6 p(x) 0.2 p(y) 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 x x, y 0.4 0.2 mobs 23 24 5.3 (195) 22.5 20.0 17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.0 100 120 160 180 200 u − r > 2.22 u − r < 2.22 24 22 −21.0 20 18 16 14 −21.5 12 −22.0 0.08 0.09 0.10 z 0.11 10 0.12 0.08 0.09 0.10 z 0.11 0.12 100 u − r < 2.22 N = 45010 −20.0 10−1 −20.5 −21.0 10−2 10−3 −21.5 −22.0 0.0 L(µ, σ) for x ¯ = 1, V = 4, n = 10 −0.5 −1.0 10−4 0.08 0.09 0.10 z 0.11 u − r > 2.22 u − r < 2.22 10−5 −20 0.12 −21 −22 M −23 100 100 10−1 10−1 10−2 10−2 −1.5 10−3 0.4 3.5 −2.0 3.0 −2.5 0.3 10−4 −3 −2 −1 −3.0 2.5 0.2 −3.5 2.0 0.0 140 x u − r > 2.22 N = 114152 −20.5 4.0 −4.0 0.1 22 80 4.10 (171) 1.5 21 hS (x) 60 −20.0 1.0 1.6 4.5 0.5 0.000 4.5 (145) 5.0 p=2 p=4 (x > xc classified as sources) 0.010 1.000 4.9 (170) 0.7 scatter bias 0.5 0.10 k x) x σ absolute magnitude bias = 0.000 = 0.001 = 0.1 = 0.01 p(x) −2 4.7 (153) b∗ = 0.5 b∗ = 0.1 ym Jk 0.00 −4 n = 10 (x 0.05 102 ax (xk ,yk ) 0.15 100 4.6 (149) 103 (xi ,yi ) ym xmax Anderson-Darling: A2 = 194.50 Kolmogorov-Smirnov: D = 0.28 Shapiro-Wilk: W = 0.94 Z1 = 32.2 Z2 = 2.5 0.10 10−2 0.40 4.4 (143) y xmax −1 y −2 0.20 10−4 0.35 1.8 0.25 10−6 p = − HB (i) 0.2 50 Inverse Cuml Distribution σ∗ Anderson-Darling: A2 = 0.29 Kolmogorov-Smirnov: D = 0.0076 Shapiro-Wilk: W = Z1 = 0.2 Z2 = 1.0 −3 0.4 100 0.70 −3 1.15 0.3 0.0 −4 0.6 0.015 0.1 10−8 0.65 0.040 4.3 (142) 0.30 10−10 0.60 Kendall-τ σ (std.dev.) 0.2 10−3 10−12 0.55 3.24 (119) 0.4 10−2 rp p(x) 3900 1.0 150 Spearman-r σG (quartile) 3950 0.5 10−1 0.50 20 14 15 4.2 (139) 100 0.8 0.00 3800 0.0 −6−4−2 6 10 −6−4−2 x x n components 0.45 counts per pixel 0.15 0.10 10 0.40 20 1.0 AIC BIC 4000 p(σ|x, I) 0.20 incorrect model 4050 class χ2 dof = 0.24 (−3.8 σ) µ ˆ = 9.99 underestimated errors 11 µ ˆ = 9.99 3.23 (113) p(class|x) p(x) 0.25 χ2 dof = 0.96 (−0.2 σ) class Best-fit Mixture 0.30 µ ˆ = 9.99 x class 0.35 overestimated errors 10 0.35 M −2 3.22 (110) 11 18 16 14 12 10 −4 information criterion 3.21 (107) correct errors Input Fit Robust Fit −4 104 1.0000 0.8 p(µ|x, I) 2W N 0.8000 Cumulative Distribution 300 log(L) · 0.6000 x 200 p(σ|x, I ) √1 12 103 N N (σ ∗ ) σ= 102 normalized distribution −0.03 0.4000 Input data distribution Pearson-r 25 N(τ) Input Fit Robust Fit −0.01 0.2000 250 −2 0.00 0.0000 3.20 (106) N(x) 12 0.02 µ ¯ x W √ N N(rs ) · y √1 12 y σ= µ¯ = 12 [max(x) + min(x)] 0.03 Luminosity 10 y µ ¯ −0.15 Luminosity x No Outliers 1% Outliers 15 −0.10 normalized C(p) −0.05 O21 20 14 5% outliers α = π/4 σxy = 1.50 0.05 0.00 0 3.19 (104) p(< x) 4.0 p(x)dx 3.5 ρ(z)/[z/0.08]2 3.0 Φ(M) 2.5 p(σ|x, I) 2.0 x 10−3 µ 1.0 0.8 0.8 0.6 0.2 σπ /π 0.3 1.0 −3 0.4 2 σ 5 0.6 0.4 −4.5 0.2 0.1 10−4 1.0 P (< σ|x, I) 1.5 0.0 M 1.0 3.16 (101) 0.00 P (< µ|x, I) 0.5 0.0 0.0 N(rp ) 0.0 0.0 −2 −1 µ 5.4 (199) −5.0 0.0 −3 −2 −1 0.2 µ 5.5 (201) 0.0 σ • Visual Figure Index marginalized approximate −1.5 p(b|x, I ) p(σ) 0.4 −3.0 0.6 p(µ) −2.5 σ 0.4 100 2.5 10−1 2.0 10−2 1.5 1.0 −4.0 x −3 5.6 (203) −2 −1 µ 1.0 0.8 0.8 p(γ|x, I) −2.4 −3.2 10.2 −4.0 0.0 −1 µ 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 γ P (< γ|x, I) 0.2 0.0 −1.0 0.12 −3.5 0.8 1.0 1.2 σ 1.4 1.6 1.8 −5.0 2.0 5.0 µ 5.2 x p(g1 |µ0 ) (bad point) 1.2 p(g1 ) g1 0.015 5.16 (219) µ −4 −2 µ 5.17 (222) 0.4 0.0 x 0.2 0.4 0.6 g1 0.8 1.0 20 20 0.00 0.02 a∗ 0.04 non-Gaussian distribution Scott’s Rule: 38 bins Scott’s Rule: 24 bins Freed.-Diac.: 49 bins Freed.-Diac.: 97 bins −4 100 −2 x Knuth’s Rule: 99 bins −4 −2 x 5.20 (227) 1.10 Single Gaussian fit 1.05 1.2 σ p(x) 10 0.95 0.90 σ1 0.85 0.4 0.80 0.3 0.75 0.3 0.4 0.5 µ 0.6 0.7 0.8 1.25 0.3 1.00 0.5 x 0.4 σ2 15 Generating Distribution Knuth Histogram Bayesian Blocks 80 1.15 0.6 p(x) 10 5.15 (218) 40 60 Sample Size N 1.6 true distribution best fit normal γ Knuth’s Rule: 38 bins 0.2 5000 points 0 µ1 = 0; σ1 = µ2 = 1; σ2 = ratio = 0.1 0.08 0.02 −1 Gaussian distribution 0.8 0.3 0.02 20 Input pdf and sampled data 0.7 0.01 a −20 0.8 0.0 0.00 5.19 (224) 0.4 0 −0.01 10 40 −60 0.2 0.04 −40 5.18 (223) Generating Distribution Knuth Histogram Bayesian Blocks 104 p(g1 |µ0 ) (good point) 500 points 0.3 0.02 a∗ Poisson Likelihood Gaussian Likelihood 108 60 0.2 0.020 0.00 1012 µ2 0.010 a 10 0.10 1016 100 80 p(g1 ) (good point) 0.005 0.14 1020 0.8 0.4 0.000 x 1024 1.0 0.6 −2 0.12 1028 p(g1 ) (bad point) −4 50 points 40 bins 0.06 5.14 (216) 1.4 0.0 10 0.2 0.04 100 0.04 1.6 0.4 0.08 0.06 −5 150 50 1.8 0.6 Poisson Likelihood Gaussian Likelihood 0.12 0.10 0.04 5.13 (214) 0.8 50 0.14 0.0 150 100 50 points bins 0.02 0.08 0.06 5.4 −5.0 10 200 0.10 0.3 1.0 continuous discrete, 1000 bins discrete, bins −4.5 0.6 µ 15 −4.0 0.0 −2 20 500 pts 100 pts 20 pts 0.02 4.8 5.12 (212) 200 −4 300 0.00 5.11 (210) 1.0 0.1 4.6 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 γ 0.8 5.10 (209) p(x) 0.6 p(x) 0.4 b 250 OCG for N points 0.2 −3.0 0.2 −4.0 −4.5 0.0 0.14 Sample Value µ 1.0 0.16 −2.0 −2.5 0.0 −1 0.8 0.18 −0.5 0.2 −2 0.6 10−6 5.9 (208) 0.2 0.4 0.0 −3 0.4 −3.0 b 0.4 p(µ) P (< µ|x, I) −1.5 p(W ) 0.6 0.2 p(a) −6.4 0.8 0.4 σ 0.6 −5.6 9.8 0.6 0.8 −4.8 10.0 1.0 0.8 L(σ, A) (Gauss + bkgd, n = 200) −0.8 −1.6 0.2 −2 1.0 1.0 0.0 0.4 0.0 0.0 5.8 (206) 10.4 0.2 −3 0.0 −3 −2 −1 µ −5.0 0.6 W p(µ|x, I) L(µ, W ) uniform, n = 100 10.6 0.4 5.7 (205) 1.0 0.6 log(L) −2 p(x) −4 A −6 log L(µ, W ) 0.00 −2.5 −3.5 10−5 0.0 0.0 −2.0 10−4 0.5 −4.5 −1.0 −1.5 10−3 p(a) 0.05 0.2 −0.5 −3.5 0.2 0.0 L(µ, γ) for x ¯ = 0, γ = 2, n = 10 σ p(x) 0.10 0.6 −2.0 log(L) 0.15 3.0 log(L) 0.8 b∗ 1.0 0.8 b∗ −1.0 γ 1.0 −0.5 yi 0.0 L(µ, σ) for x ¯ = 1, σtrue = 1, n = 10 yi σ fit σG fit 0.20 P (< b|x, I ) 0.25 529 1.00 0.2 0.1 0.1 1.8 ratio p(x) 0.75 0.2 1.2 0.6 −4 5.21 (229) µ 5.22 (232) 0.5 −2 −1 x 0.6 Gaussian Exponential Top-hat p(x) 0.6 yobs 0.3 0.5 0.4 0.2 0.2 0.0 0.1 −2 −1 0 x 10 −2 −1 0.4 x 0.6 p(x) 0.8 0.8 0.2 0.0 0.5 σ 1.0 0.4 0.8 0.3 p(x) 0.0 5.8 6.0 6.2 µ 6.4 5.26 (240) −2 −1 500 points 10 15 20 0.4 Generating Distribution Nearest Neighbors (k=100) Kernel Density (h=0.1) Bayesian Blocks 0.5 Input −31500 0.4 −32000 0.3 −32500 0.2 −33000 0.1 −0.9 −0.6 −0.3 0.0 [Fe/H] 0.0 10 x 15 20 6.5 (260) Input Distribution Density Model −34500 15 1.00 σ2 −2 −1 0.4 Gaussian (h = 5) exponential (h = 5) −200 input KDE: Gaussian (h = 5) k-neighbors (k = 5) k-neighbors (k = 40) −250 −300 −200 100 −300 −200 −100 y (Mpc) −250 −300 −300 −200 −100 y (Mpc) 100 100 −300 −200 −100 y (Mpc) 500 points 18.5 N = 100 points 60 0.1 −250 0.3 −0.9 −0.6 −0.3 0.0 [Fe/H] 16.0 −300 15 −200 −100 100 200 100 Standard Stars Single Epoch Extreme Deconvolution resampling Extreme Deconvolution cluster locations n clusters 60 20 10 x 15 20 20 40 x 60 80 100 20 40 x 60 80 6.9 (265) 0.5 0.5 0.4 0.4 0.3 0.3 0.5 y 0.0 −0.5 1.5 1.0 r−i 0.5 y 15 −0.5 −0.5 Extreme Deconvolution resampling Extreme Deconvolution cluster locations 0 6.10 (266) 1.0 1.5 −0.5 0.0 0.5 g − r 1.0 w = −0.227g + 0.792r −0.567i + 0.05 1.5 0.2 0.2 N(w) y 40 −2 −2 −2 x x x 0.5 g − r single epoch σG = 0.016 standard stars σG = 0.010 XD resampled σG = 0.008 50 10 −2 0.0 [α/Fe] 0.0 −5 30 0.1 0.1 20 10 −5 x 12 6.11 (268) x 12 0.0 −0.06 −0.04 −0.02 0.0 −0.9 0.00 w 6.12 (269) 0.02 0.04 0.06 N = 10000 points 40 6.8 (263) 1.0 N = 1000 points 80 0.0 6.7 (262) 20 Generating Distribution Mixture Model (10 components) Kernel Density (h = 0.1) Bayesian Blocks 0.2 y (Mpc) r−i 10 0.1 −350 1.5 5000 points −300 Noisy Distribution 17.0 20 0.0 0.0 17.5 16.5 40 0.4 N=100 N=1000 N=10000 18.0 80 0.2 0.1 10 12 14 N components True Distribution 100 Generating Distribution Mixture Model (3 components) Kernel Density (h = 0.1) Bayesian Blocks 0.3 −200 100 6.4 (259) −300 −350 −350 −250 0.2 −350 top-hat (h = 10) 6.3 (255) 0.3 µ 5.25 (239) input −300 −200 −100 y (Mpc) −3 1.25 0.4 Converged 10 0.75 −300 −200 0.5 AIC BIC 6.6 (261) Cloned Distribution u 6.2 (253) −34000 0.0 0.1 −2 x −33500 0.2 0.4 [α/Fe] p(x) [α/Fe] p(x) 0.0 5000 points −31000 Generating Distribution Nearest Neighbors (k=10) Kernel Density (h=0.1) Bayesian Blocks 0.1 −1 6.1 (252) 0.2 0.3 −2 x 0.4 0.3 −4 0.0 6.6 p(x) 0.6 p(x) 0.5 x (Mpc) 0.4 A x (Mpc) 0.3 [α/Fe] 0.2 0.3 σ1 −250 0.2 0.1 0.2 −300 0.4 0.1 1.6 −350 0.1 0.6 0.2 1.2 µ2 −250 −200 0.2 0.6 0.8 −350 0.3 0.4 0.7 0.1 −200 1.0 5.8 K(u) 0.0 x 6.0 x (Mpc) 6.2 −0.2 −0.1 0.0 µ1 5.24 (236) 0.8 0.4 6.4 5.23 (235) 1.0 6.6 µ −2 BIC/N 20 x (Mpc) 15 y 10 x y x (Mpc) x (Mpc) 0.0 0.0 −0.6 −0.3 [Fe/H] 6.13 (272) 0.0 −0.9 −0.6 −0.3 [Fe/H] 6.14 (274) 0.0 100 • 530 Visual Figure Index 101 −300 u − r < 2.22 N = 16883 y r12 −350 r13 −200 x (Mpc) y 101 u − r > 2.22 N = 38017 r12 −250 −250 r23 r12 −300 r14 −350 100 100 10−1 10−1 x w(θ) ˆ x (Mpc) −200 x r24 r31 r34 10−2 10−2 (a) point U1 = X2 V1T Σ1 U2 V2T Σ2 component component component component component component component component component component 101 3000 4000 5000 6000 7000 ˚ wavelength (A) 3000 4000 5000 6000 7000 ˚ wavelength (A) 3000 4000 5000 6000 7000 ˚ wavelength (A) 20 10−2 15 0.90 0.70 102 103 3000 0.0 c2 0.5 7000 8000 5800 5900 θ2 emission galaxy −1.0 −0.5 0.0 x 0.5 1.0 38 1.5 χ2 dof = 1.57 0.0 Gaussian Kernel Regression µ x2 42 40 θ2 1.5 0.0 0.5 1.0 z 1.5 1.5 1.6 600 f(x) slope −2 2.6 200 240 intercept 2.4 2.4 −3 280 −3 10 80 40 intercept 80 120 8.9 (348) 8 polynomial degree 10 12 100 cross-validation training 0.35 rms error BIC 40 y 1.5 d=2 1.0 1.0 x x 10 0.5 1.0 z 1.5 2.0 8.11 (352) 0.5 cross-validation training d=2 0.4 0.0 0.5 1.0 1.5 x 2.0 2.5 3.0 d = 19 8.12 (353) 8.13 (354) 20 30 40 50 60 70 Number of training points 80 90 100 cross-validation training d=3 1.0 0.3 0.2 0.0 −0.1 0.25 0.20 g2 (x) 0.1 0.15 40 50 60 70 Number of training points −0.1 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u g 9.4 (376) 0.2 0.2 g − r contamination 0.0 −1 1.0 0.8 0.6 0.0 0.4 0.2 −0.1 0.0 N colors 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u g 9.5 (376) −1 −1 9.1 (370) 0.3 0.4 0.2 0.2 1.0 0.8 0.6 0.0 0.4 0.2 −0.1 0.0 N colors 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u g 9.6 (378) 9.2 (373) 0.3 0.6 0.4 0.2 0.2 N=1 N=3 1.0 0.8 0.6 0.0 0.4 −0.1 0.2 0.0 N colors k = 10 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u−g 9.7 (380) 0.2 1.0 0.8 0.6 0.4 0.2 N colors N colors 9.3 (374) 1.0 0.8 0.3 0.6 0.4 0.2 0.2 k=1 k=10 0.0 0.1 0.4 0.0 1.0 0.8 0.0 0.1 x 1.0 0.8 0.6 0.0 0.1 x completeness 0.4 completeness 0.3 0.6 0.0 0.1 0.0 −2 100 1.0 0.8 contamination completeness 0.2 90 8.15 (358) 1.0 0.3 80 1.0 0.0 0.4 0.2 −0.1 0.0 N colors 0.8 0.6 0.4 0.2 0.0 0.1 0.8 0.6 completeness 30 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u−g 9.8 (382) contamination 8.14 (356) 20 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u−g g −r 0.00 10 14 completeness 12 contamination 10 g −r polynomial degree 0.8 0.0 0.1 g1 (x) 0.2 contamination 350 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x x x 3.5 0.3 g − r 300 0.0 0.0 −0.5 0.05 250 d=3 0.10 20 200 x 0.5 36 0.0 0.30 60 g − r 2.0 1.5 0.40 80 150 38 0.00 10 14 100 0.0 y rms error 0.10 50 8.8 (347) −1.0 −1.5 10 0.15 0.05 0.5 0.25 0.2 100 40 0.20 0.1 Φ(t) 42 0.30 0.3 10 −0.5 0.35 0.5 0.4 2.0 10 0.40 cross-validation training 0.6 t d=1 8.10 (350) 0.8 0.7 squared loss: y = 1.08x + 213.3 Huber loss: y = 1.96x + 70.0 −5 8.7 (346) 0.5 0.0 −2.0 200 −10 1.5 −1.0 −40 120 2.6 46 1.0 −1.5 40 intercept 2.2 2.4 slope 2.0 0.5 −0.5 2.0 −2 2.0 f(x) 2.0 1.8 48 1.5 slope slope 2.2 2.0 250 44 1.0 2.2 200 −1 2.5 outlier rejection (solid fit) 150 x 300 c=1 µ 160 2.8 100 0.8 0.6 mixture model (dashed fit) c=2 10 −60 400 c=3 −20 50 500 c=5 20 8.6 (344) 300 100 150 200 250 300 350 x 20 p(x) y 0.1 0.2 0.3 0.4 0.5 0.6 0.7 ΩM −1 c=∞ 30 100 1.0 2.6 600 40 −40 0.4 1.2 rms error 300 200 no outlier correction (dotted fit) 1.4 200 intercept y ΩΛ 0.7 500 400 700 40 60 8.5 (341) 700 8.3 (333) 50 400 0.8 36 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 z 8.4 (335) −40 8.2 (328) 80 0.5 1.0 z 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 z 100 0.6 0.5 θ1 r χ2 dof = 1.11 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 z 2.0 600 38 0.0 1.5 y −0.5 1.0 θ1 y 0.0 −2 0.5 decision boundary −1 50 x4 2.0 500 40 0.5 −5 1.5 θ1 1.1 1.0 −15 2.8 1.0 1.5 −10 0.0 0.5 100 observations µ µ θ 100 0.02 0.9 1.0 z χ2 dof = 1.09 x3 0.00 c2 8.1 (323) 42 0.5 −0.02 1.0 0.0 −0.04 0.010 g −r c3 38 0.000 0.005 c1 θ1 r 0.0 −0.5 44 Lasso Regression θnormal equation θlasso 44 x1 1.0 0.5 46 2.0 6300 θridge χ2 dof = 1.02 Gaussian Basis Function Regression 46 −0.5 absorption galaxy 7.9 (310) Ridge Regression 6200 galaxy −0.04 −0.010 −0.005 6100 θ2 θnormal equation 40 0.00 −0.02 6000 θ2 4th degree Polynomial Regression 0.02 Lasso Regression 6300 48 0.04 Ridge Regression 6200 ˚ λ (A) 1.0 1.0 narrow-line QSO −0.04 52 50 Linear Regression 48 46 44 42 40 38 36 12 15 ×10 Linear Regression 10 6100 7.7 (304) Straight-line Regression x1 −2.0 −1.5 c2 6000 42 0.5 7.8 (307) 6000 44 broad-line QSO 0.00 5000 46 x2 x3 µ y c2 c3 −1.0 −0.02 6300 x4 −1.5 −0.5 4000 48 True fit fit to {x1 , x2 , x3 } 0.5 0.4 −1.0 6200 0.0 0.0 0.02 5900 ˚ wavelength (A) 1.0 −0.4 IsoMap projection 6100 7.6 (300) −0.5 0.5 10 101 6000 ˚ λ (A) mean + 20 components = 0.94) (σtot 15 0.75 2.0 absorption galaxy 0.0 c1 5800 20 0.80 5900 ˚ λ (A) mean + components (σtot = 0.93) 10 0.85 1.5 galaxy −0.5 5800 20 0.95 emission galaxy 0.8 LLE projection True spectrum reconstruction (nev=10) mean + components (σtot = 0.85) 10 10−3 1.00 narrow-line QSO −0.5 −1.0 −0.8 −1.0 mean 15 broad-line QSO 0.5 7.2 (293) 10 20 10−1 Eigenvalue Number 1.0 3000 4000 5000 6000 7000 ˚ wavelength (A) 15 100 7.5 (299) 0.0 3000 4000 5000 6000 7000 ˚ wavelength (A) 7.1 (291) 101 0.65 100 7.4 (298) PCA projection 100 θ (deg) 102 NMF components component component 3000 4000 5000 6000 7000 ˚ wavelength (A) 7.3 (295) 10−1 6.17 (280) ICA components mean component 10−2 101 10−2 100 θ (deg) 6.16 (278) PCA components mean = X1 10−1 (c) point completeness 6.15 (276) (b) point contamination 200 flux 100 flux y (Mpc) flux −100 Normalized Eigenvalues −200 Cumulative Eigenvalues −300 normalized flux −350 normalized flux −300 normalized flux r23 −250 flux x (Mpc) −200 1.0 0.8 0.6 0.4 0.2 0.0 • Visual Figure Index 0.2 −0.1 0.0 N colors 0.04 0.4 0.4 N colors 9.11 (386) 1.0 N = 500 rms = 0.018 cross-validation training set 0.03 0.3 0.2 0.02 zfit rms error 0.02 zfit rms error zfit 0.2 0.02 0.1 0.1 0.2 0.1 0.6 GNB LDA QDA LR KNN DT GMMB 0.4 0.2 0.2 ztrue 0.3 0.4 9.14 (390) 10 15 depth of tree 20 0.0 0.1 0.2 ztrue 0.3 0.4 9.15 (392) 2.0 Time Domain: Multiplication window W (x) D amplitude H(f) h(t) H(f) Undersampled data: ∆t > tc H(f) Sampled signal: pointwise multiplication 0.0 0.2 ×10−18 0.8 −100 k 50 1.0 time (s) 1.5 2.0 w(t; t0 , f0 , Q) 10−42 10−44 102 h(t) real part imag part 0.0 0.4 0.6 efficiency g −r 0.8 1.0 w(t) Wavelet PSD completeness 0.6 0.00 0.03 0.06 0.09 0.12 0.15 false positive rate 0.3 Data PSD 0.2 0.1 0.0 0.2 0.4 0.6 0.8 −0.1 1.0 Window PSD −20 −10 t 0.8 10 20 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.1 0.2 0.3 0.4 f 0.5 0.6 0.7 0.8 10.5 (416) 1.5 Input Signal Filtered Signal Wiener Savitzky-Golay 0.5 0.0 −0.5 20 40 60 80 20 40 60 λ f0 = 10 Q = 0.5 80 λ Input PSD 3000 0.5 Wavelet PSD f0 f0 0.7 9.18 (396) 0.0 Filtered PSD −0.5 1/4 2000 1000 −1.0 4200 4400 4600 4800 5000 λ 10 10 20 30 40 50 60 70 80 90 λ 10.11 (424) 0.8 0.6 0.4 0.2 0.0 500 1000 f 1500 2000 10.12 (425) 14 P = 8.76 hr mag 12 ID = 14752041 ID = 1009459 P = 14.78 hr P = 3.31 hr 10 15 t 20 25 30 1.0 20 0.4 0.2 P = 13.93 hr 14.5 15.6 mag 10 PLS (ω) 0.6 ID = 10025796 17 P = 2.58 hr 18 19 20 ω 21 16.2 10.16 (439) 15.5 ID = 11375941 0.0 0.2 0.4 0.6 phase 0.8 1.0 0.4 8.59 ID = 18525697 0.0 0.2 0.4 0.6 phase 0.8 8.60 1.0 10.17 (440) 0.6 0.8 −1.0 −1.5 0.5 0.5 0.8 12 11 10 20 40 60 80 100 time (days) 40 0.8 30 0.6 20 0.4 10 0.2 −10 0.0 0.1 0.2 0.3 0.4 period (days) 0.5 0.6 0.7 0.8 0.9 0 ω 14.8 15.2 ω0 = 8.61 P0 = 17.52 hours 0.0 0.2 0.4 0.6 phase 8.61 0.4 0.6 −0.5 −1.0 10 15 −1.5 20 0.5 ω0 = 8.61 0.0 zoomed view 26775 26750 26725 26700 26675 15000 10000 5000 0.8 1.0 0 −1.0 10 N frequencies 15 1.1 −1.5 20 10.19 (443) 0.9 −0.5 20 12 15 10 10 0.8 0.7 −0.5 0.0 0.5 1.0 g−i 1.5 2.0 0.2 0.8 A 1.0 1.2 1.4 10.20 (445) −1.5 0.6 f 20000 −0.5 −1.0 0.4 0.5 0.6 1.0 0.2 13 25000 a log(P ) log(P ) u−g −0.5 −1.0 0.0 10.15 (437) zoomed view 1.0 0.0 −0.5 1.0 Data PSD 0.8 (10x errors) 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f 22850 22825 22800 22775 22750 1.0 1.5 0.8 ω0 = 17.22 15000 5000 2.0 0.0 80 1.0 Window PSD 0.8 0.6 0.4 0.2 0.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f 10000 0.5 0.0 1.0 0.5 60 0.6 25000 14.8 2.5 1.5 40 10.14 (431) 10.18 (442) 0.5 2.0 2000 14.4 0.0 −10 22 20 0.4 f 1.0 Data PSD 0.8 0.6 0.4 0.2 0.0 100 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 f t 30000 terms term 15.0 0.0 17 0.2 10.10 (423) 0.0 0.2 15.9 1500 15.2 ω0 = 17.22 P0 = 8.76 hours 0.0 0.2 0.4 17.18 17.19 17.20 17.21 17.22 17.23 0.6 0.0 0.2 20000 0.4 0.8 ID = 10022663 14.8 0.0 t Data 30000 14.4 ∆BIC 30 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 5000 14.4 0.0 16 50 40 1000 f terms term 0.6 14.0 mag standard generalized 500 0.2 15 0.8 0.8 15.2 13.6 4800 1.0 P = 2.95 hr 14.8 15.0 4600 10.13 (426) 14.8 11 10 102 101 100 10−1 10−2 10−3 10−4 10−5 10−6 10−7 10−8 14.6 14.4 13 1.0 4400 −0.2 0.2 r(t) −5 4200 0.0 t ˚ λ (A) PLS (ω) −10 −0.5 −0.2 40 PLS (f ) 0.0 30 SDSS white dwarf 52199-659-381 4000 P SD(f ) scaled P SD(f ) 0.0 20 10.9 (421) 110 100 90 80 70 60 50 40 30 ˚ λ (A) 1.0 0.1 10 10.8 (420) 110 100 90 80 70 60 50 40 30 4000 0.5 t mag 0.2 −10 mag flux 0.3 −20 hobs flux Kernel smoothing result 1.0 −30 flux 1.5 1/8 −40 10.7 (419) Effective Wiener Filter Kernel 0.4 flux 10.6 (417) log(P ) frequency (Hz) log(P ) t A −1 PLS (f ) −2 PLS (f ) −3 y(t) −4 ∆BIC 103 ∆BIC 102 ∆BIC 10−46 power 10−44 y(t) 0.2 f0 = Q = 0.5 f0 = 10 Q = 1.0 1.0 Hanning (cosine) window 10−42 GNB LDA QDA LR KNN DT GMMB 0.8 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 u−g −1.0 0.9 1.0 0.0 w(t; t0 , f0 , Q) = e−[f0 (t−t0 )/Q] e2πif0 (t−t0 ) N colors −0.2 f f0 = Q = 1.0 1/2 10−40 skew real part imag part 0.5 103 10−38 40 0.2 0.0 0.3 0.0 −0.5 −1.0 w(t; t0 , f0 , Q) = e−[f0 (t−t0 )/Q] e2πif0 (t−t0 ) 0.4 −0.5 Example Wavelet t0 = 0, f0 = 1/8, Q = 0.3 1.0 0.5 −0.5 frequency (Hz) 10−36 20 0.4 0.2 1.0 0.8 0.6 0.4 0.2 0.0 60 80 100 0.0 1.0 Window 0.8 0.6 0.4 0.2 0.0 60 80 100 0.0 40 0.4 1.0 0.6 0.5 0.0 Example Wavelet t0 = 0, f0 = 1.5, Q = 1.0 1.0 1.0 0.5 −0.5 −1.0 10−46 20 0.6 0.5 10.4 (414) −1.0 1.0 10−40 0.6 t Input Signal: Localized spike plus noise 1.5 2.0 Top-hat window 10−38 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 −0.2 0.8 0.0 0.8 0.7 Data f 10.3 (413) w(t; t0 , f0 , Q) 0.5 10−36 H(f) t −2 0.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 100 −1 −1.0 K (λ) −50 h(t) h(t) 0.6 Input Signal: Localized Gaussian noise 0.5 P SD(f) x 10.2 (411) 1.0 P SD(f) 0.4 Convolution of signal FT and window FT h(t) [D ∗ W ](x)= F −1 {F [D] · F [W ]} 0.0 FT of Signal and Sampling Window ∆f = 1/∆t h(t) DW Signal and Sampling Window Sampling Rate ∆t 0.5 −0.5 Frequency Domain: Convolution 1.0 modes 10.1 (407) f Time Domain: Multiplication 1.5 2.0 Convolution of signal FT and window FT h(t) Pointwise product: F (D) · F (W ) 1.0 9.17 (396) t Convolution: [D ∗ W ](x) 1.5 0.4 FT of Signal and Sampling Window ∆f = 1/∆t Sampled signal: pointwise multiplication 0.0 1.0 phase 0.3 Frequency Domain: Convolution Signal and Sampling Window Sampling Rate ∆t 0.5 0.5 0.2 ztrue 1.0 F (W ) 0.0 0.1 Well-sampled data: ∆t < tc F (D) data D(x) 1.5 modes 0.0 9.16 (395) mode PLS (ω) 100 200 300 400 500 number of boosts 1.2 0.8 flux 0.1 1.4 0.9 P SD(f ) 0.0 1.0 P (f ) 20 126 / split on i − z P (f ) 10 15 depth of tree depth=7 depth=12 1.0 9.13 (388) 0.0 0.2 0.000 0.008 0.016 0.024 0.032 0.040 0.0 false positive rate y(t) 0.0 y(t) 0.01 Tree depth: 0.0 0.01 w(t) 0.0 0.01 0.2 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u−g 0.8 0.3 0.4 depth = 12 1001 / split on i − z 266 / 15 split on g − r 392 / 16 split on i − z 9.12 (387) 0.4 depth = 20 rms = 0.017 cross-validation training set −0.1 273 / split on g − r 1274 / split on u − g 1666 / 23 split on g − r Cross-Validation, with 137 RR Lyraes (positive) 23149 non-variables (negative) false positives: 53 (43.4%) false negatives: 68 (0.3%) 0.03 0.03 379 / non-variable 2841 / 333 split on u − g Training Set Size: 69855 objects 0.0 377 / 41 split on u − g 756 / 41 split on r − i 0.2 0.6 0.0 123 / 18 split on g − r 1175 / 310 split on r − i 0.0 0.4 depth = 13 rms = 0.020 0.3 0.6 0.1 296 / 251 split on i − z 419 / 269 split on i − z 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u−g 9.10 (385) cross-validation training set 0.8 0.8 contamination 0.4 1296 / non-variable 69509 / 346 split on g − r true positive rate x 0.04 rms error 0.0 0.7 0.8 0.9 1.0 1.1 1.2 1.3 u−g 9.9 (383) 0.6 320 / split on i − z 1616 / split on u − g g − r 1645 / 11 split on u − g 1.0 0.2 21 / split on g − r h(t) 0.8 8/7 split on r − i 29 / split on r − i 0.2 0.3 5200 / non-variable 66668 / 13 split on g − r 0.4 0.0 0.1 1449 / split on u − g 6649 / split on u − g 0.6 P SD(f ) 1.0 1.0 58374 / non-variable 65023 / split on r − i 0.8 completeness −0.1 0.2 0.2 contamination −1 −1 0.4 contamination g −r y 0.1 0.0 −2 0.3 0.6 0.0 −3 0.8 g −r 0.2 Numbers are count of non-variable / RR Lyrae in each node 1.0 completeness completeness 1.0 0.3 true positive rate 531 10 17 16 15 14 13 12 11 10 20 40 60 80 100 t 54 t 1.5 −1.5 0.5 1.0 g−i 1.5 −0.5 2.0 0.0 0.5 1.0 g−i 1.5 2.0 52 1.0 T 0.0 φ −0.5 50 0.5 2.5 48 1.0 0.0 0.0 0.0 46 0.25 0.6 0.4 log(P ) −0.5 4.2 0.20 −0.5 4.1 0.15 0.2 0.5 −1.0 ω 1.0 −1.0 0.0 0.5 1.0 g−i 1.5 2.0 −0.2 4.0 0.10 3.9 0.0 −0.5 α 1.5 log(P ) 0.8 J −K i−K 2.0 0.05 −0.5 10.21 (446) 0.0 0.5 1.0 g−i 1.5 2.0 −1.5 −0.5 0.0 0.5 1.0 g−i 1.5 2.0 0.2 10.22 (448) 0.4 0.6 0.8 A 1.0 1.2 1.4 −1.5 3.8 −0.5 0.0 0.5 1.0 g−i 1.5 2.0 0.2 10.23 (449) 0.4 0.6 0.8 A 1.0 1.2 1.4 0.00 10 11 12 13 r0 0.6 0.7 0.8 0.9 a 1.0 10.24 (452) 1.1 0.0 0.5 1.0 φ 1.5 9.5 10.0 10.5 b0 11.0 10 A 10.25 (454) 12 46 48 50 T 52 54 • Visual Figure Index 0.80 0.78 20 40 60 80 0.76 100 2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 20.4 20 40 60 80 −1.5 0.104 ω Wavelet PSD P SD(f ) f0 0.02010 0.3 β β 0.0102 0.01995 11.0 4.0 4.5 5.0 A 5.5 6.0 0.07 0.08 0.09 0.10 0.11 ω 10.26 (455) 29.85 30.00 T 30.15 0.76 0.80 A 0.82 0.088 0.096 0.104 0.112 ω 10.27 (456) np.arange(3) + 0.78 20 40 5 19.4 19.2 10 60 80 + 2 = 3 np.arange(3).reshape((3, 1)) + np.arange(3) 0 1 2 2 2 1.0 0.5 0.5 0.0 0.0 −0.5 = 2 3 A.1 (493) 101 10−1 100 −1.0 101 f Re[h] Im[h] h(t) H(f ) normalized flux / filter transmission 0.0 −1.0 0.2 0.1 t 1.5 RMS Error = 1.4e-13 Re[H] 1.0 Im[H] 0.5 0.0 −0.5 −1.0 C.1 (516) 500 −1.5 z 9000 10000 11000 0.5 30 0.0 −0.5 −1.0 A.3 (496) 1.0 0.3 i 400 20 −0.5 r 200 t (days) 40 0.5 1.0 1.5 E.1 (524) 2.0 f −4 −3 −2 −1 x 0.5 6000 7000 8000 Wavelength (Angstroms) 100 10 SDSS Filters and Reference Spectrum 5000 300 Scargle True Edelson-Krolik 10.30 (462) −1.0 A.2 (496) 0.4 g 1000 60 −0.5 0.5 4000 800 50 x 0.0 3000 600 0.0 −0.5 100 10.29 (459) Simple Sinusoid Plot 1.0 −1.0 + 400 t (days) 10−3 f N(x) 1 200 0.5 10−6 10−1 100 10.28 (457) y 1 y 1 1.0 np.ones((3, 3)) + np.arange(3) 20.0 19.8 t 1.0 = 10−2 t Simple Sinusoid Plot + 10 10−5 0.1 0.01980 0.0098 10.5 b0 10−4 0.2 0.0100 10.0 10−1 0.4 0.0104 9.5 100 0.088 0.0099 P (f ) ∝ f −2 101 0.5 0.0101 t 0.096 0.08 20.2 19.6 P (f ) ∝ f −1 0.6 0.09 0.0103 0.0 −0.5 −1.0 −2 0.11 ω 0.5 −1 100 t 0.112 0.07 20.6 1.0 t 0.10 20.8 1.5 Input Signal: chirp observed flux 4.0 0.82 ACF(t) 20 18 16 14 12 10 counts 4.5 h(t) A 5.5 5.0 hobs hobs 6.0 A 532 2.5 3.0 3.5 A.4 (497) x 4 A.5 (502) 10 12 14 16 ... Signal and Sampling Window Sampling Rate ∆t 0.5 −0.5 Frequency Domain: Convolution 1.0 modes 10.1 (407) f Time Domain: Multiplication 1.5 2.0 Convolution of signal FT and window FT h(t) Pointwise... 1.5 0.4 FT of Signal and Sampling Window ∆f = 1/∆t Sampled signal: pointwise multiplication 0.0 1.0 phase 0.3 Frequency Domain: Convolution Signal and Sampling Window Sampling Rate ∆t 0.5 0.5... 0.5 cross-validation training d=2 0.4 0.0 0.5 1.0 1.5 x 2.0 2.5 3.0 d = 19 8.12 (353) 8.13 (354) 20 30 40 50 60 70 Number of training points 80 90 100 cross-validation training d=3 1.0 0.3 0.2 0.0