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[.]
Trang 1Visual 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.
14
17
20
Galaxies
g− r
−1
0
1
2
14
17
20
Stars
g− r
−1
0
1
2
1.1 (19)
λ( ˚A)
50 100 150 200 250 300
Plate = 1615, MJD = 53166, Fiber = 513
1.2 (21)
u− r
14
15
16
17
18
1.3 (23)
redshift
−0.4
0.0
0.2
0.4
0.6
0.8
1.0
1.4 (24)
4500 5000 5500 6000 6500 7000 7500 8000
Teff(K)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
1.5 (26)
g− r
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
1.6 (28)
0.00.20.40.60.81.0
phase 14
15
Example of phased light curve
g− r
−1
0 1
10−1
100 101
Period (days)
1.7 (29)
Semimajor Axis (AU)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
1.8 (31)
g− r
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
1.9 (32)
g− r
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
1.10 (33)
5000
7000
T eff
1 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
number in pixel
5000
7000
T eff
1 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
−2.5 −1.5 −0.5 0.5
mean [Fe /H]inpixel
5000
7000
T eff
1 .5
2 .0
2 .5
3 .0
3 .5
4 .0
4 .5
−2.5 −1.5 −0.5 0.5
mean [Fe /H]inpixel
1.11 (34)
−0.2−0.1 0.0 0.1 0.2 0.3 0.4
a∗
−0.8
−0.4
0.0 0.2 0.4
2.0 2.2 2.4 2.6 2.8 3.0 3.2 a(AU) 0.00 0.05 0.10 0.15 0.20 0.25
1.12 (34)
−75◦
−60◦
−45◦
−30◦
−15◦
0◦
15◦
30◦
45◦
60◦
75◦
Mercator projection
1.13 (35)
−120◦−60◦ 60◦120◦
−60 ◦
−30◦
30◦
60◦Hammer projection
−120◦−60◦ 60◦120◦
−60 ◦
−30◦
30◦
60◦Aitoff projection
−120 ◦ −60 ◦
60◦120◦
−60 ◦
−30◦
30◦
60◦
Mollweide projection
−120 ◦ −60 ◦
60◦120◦
Lambert projection
1.14 (36)
HEALPix Pixels (Mollweide)
Raw WMAP data
1.15 (38)
Length of Array
10−4
10−3
10−2
10−1
100
101
Scaling of Search Algorithms
linear search (O[N])
efficient search (O[log N])
2.1 (45)
Length of Array
10−3 10−2 10−1
100 101 102
Scaling of Sort Algorithms list sort NumPy sort
O[N log N]
O[N]
2.2 (52)
Quad-tree Example
2.3 (58)
kd-tree Example
2.4 (59)
Ball-tree Example
2.5 (61)
p(A ∪ B) = p(A) + p(B) − p(A ∩ B)
3.1 (70)
x
×10−3
Joint Probability
0.0 0.5 1.0 1.5 2.0
x
p(y)
0.0 0.5 1.0 1.5 2.0
0.0 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5
)p(x
Conditional Probability
0.0 0.5 1.0 1.5 2.0
x
3.2 (72)
0
1 D
3.3 (76)
0 .00 0.25 0.50 0.75 1.00 x
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2
1 .4
p x x) = Uniform(x)
1 .2 1.6 2.0 2.4 2.8 y
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
y = exp(x)
p y y) = p x(ln y)/y
3.4 (77)
flux
0 .0
0 .4
0 .8
1 .2
1 .6
20% flux error
−1.0 −0.5 0.0 0 .5 1 .0
mag
0 .0
0 .4
0 .8
1 .2
1 .6
mag = −2.5log 10(flux)
3.5 (78)
Trang 20.1
0.2
0.3
0.4
0.5
0.6
0.7
Skew Σ and KurtosisK
Gaussian, Σ = 0
mod Gauss, Σ = −0.36
log normal, Σ = 11.2
x
0.0
0.1
0.2
0.3
0.4
0.5
Laplace, K = +3
Gaussian, K = 0
Cosine, K = −0.59
Uniform, K = −1.2
3.6 (80)
x
0.0
0.2
0.4
0.6
0.8
1.0
Uniform Distribution
µ = 0, W = 1
3.7 (86)
x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Gaussian Distribution
µ = 0, σ = 0.5
3.8 (87)
x
0.00
0.05
0.10
0.15
0.20
0.25
Binomial Distribution
b = 0.2, n = 20
3.9 (90)
x
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
Poisson Distribution
µ = 1
µ = 15
3.10 (92)
x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Cauchy Distribution
µ = 0, γ = 0.5
3.11 (93)
−6
−2
0 4
mean median robust mean (mixture) robust mean (sigma-clip)
Sample Size
−60
−20
0 20 60
3.12 (94)
x
0.0
0.2
0.4
0.6
0.8
1.0
Laplace Distribution
µ = 0, ∆ = 0.5
3.13 (96)
Q
0.0
0.1
0.2
0.3
0.4
0.5
χ2Distribution
k = 1
k = 7
3.14 (97)
x
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Student’st Distribution
t(k = ∞)
t(k = 2.0)
t(k = 1.0)
t(k = 0.5)
3.15 (99)
x
0.0
0.2
0.4
0.6
0.8
1.0
|d1
,d2
Fisher’s Distribution
d= 1, d2= 1
d= 5, d2= 2
d= 2, d2= 5
d= 10, d2= 50
3.16 (101)
x
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Beta Distribution
α = 0.5, β = 0.5
α = 0.5, β = 1.5
3.17 (102)
x
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Gamma Distribution
k = 1.0, θ = 2.0
k = 5.0, θ = 0.5
3.18 (103)
x
0.0
0.1
0.2
0.3
0.4
0.5
Weibull Distribution
k = 0.5, λ = 1
k = 2.0, λ = 2
3.19 (104)
0.4
0.8
1.2
1.6
2.0
N = 2
0.5
1.0
1.5
2.0
2.5
N = 3
1 3
N = 10
3.20 (106)
−0.15
0.00
0.05
0.10
0.15
¯
µ = mean(x)
σ = 1 √ 12 · W
N
−0.03
0.00
0.01
0.02
0.03
¯
µ = 12[max(x) + min(x)]
σ = 1 √ 12 · 2W N
3.21 (107)
x
−4
−2
0 2 4
σ1= 2
σ2= 1
α = π/4
σx= 1.58
σy= 1.58 σxy= 1.50
3.22 (110)
x
6 8 10 12 14
5% outliers
Input Fit Robust Fit
x
15% outliers
Input Fit Robust Fit
3.23 (113)
0 10 20
Pearson-r
No Outliers
0 8 10 18
Spearman-r
0.32 0.34 0.36 0.38 0τ 40 0.42 0.44 0.46 0.48
0 10 25
Kendall-τ
3.24 (119)
x
0 50 100 200 300
Input data distribution
x
0.0
0.2
0.4
0.6
0.8
1.0
Cumulative Distribution
0.00.20.40.60.81.0 p(< x)
−3
−1
0 2
Inverse Cuml Distribution
x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
KS test: Cloned Distribution
3.25 (121)
9
10
11
correct errors
ˆ
µ = 9.99 χ2dof= 0.96 (−0.2 σ)
overestimated errors
ˆ
µ = 9.99 χ2dof= 0.24 (−3.8 σ)
observations
9
10
11
underestimated errors
ˆ
dof= 3.84 (14 σ)
observations
incorrect model
ˆ
µ = 10.16 χ2
dof= 2.85 (9.1 σ)
4.1 (133)
−6−4−2 0 2 4 6 x
0 .00
0 .05
0 .10
0 .15
0 .20
0 .25
0 .30
0 .35
Best-fit Mixture
1 2 3 4 5 6 7 8 9 10
n components 3800 3850 3900 3950 4000 4050
AIC
−6−4−2 0 2 4 6 x
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
4.2 (139)
σ
0 5 10 15 20
σ (std.dev.)
σG (quartile)
4.3 (142)
σ ∗
0 100 200 300 400 500
σ ∗ (std .dev.)
σ ∗
G (quartile)
0 .5 0.6 0.7 0.8 0.9 1.0 1.1 σ
0 5 10 15 20
σ (std.dev.)
σG (quartile)
4.4 (143)
x
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
0.040
hB(x)
hS(x)
xc = 120
(
x > xc classified as sources)
4.5 (145)
p = 1 − HB(i)
10−3
10−2
10−1
100
=
=
=
0001
4.6 (149)
0.0
0.1
0.2
0.3
0.4
0.5
Anderson-Darling:A2 = 0.29
Kolmogorov-Smirnov:Shapiro-Wilk:D = 0.0076 W = 1 Z1 = 0.2
x
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Anderson-Darling:A2 = 194.50
Kolmogorov-Smirnov:Shapiro-Wilk:W = 0.94 Z1 = 32.2 D = 0.28 Z2 = 2.5
4.7 (153)
x
y
max(x)
( x i ,y i
J i
x
ymax
( x)
xmax (
x k ,y k
J k
4.8 (167)
0 .0 0.2 0.4 0.6 0.8 1.0
x, y
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2
1 .4
1 .6
1 .8
p(x)
0 .0 0.2 0.4 0.6 0.8 1.0 x
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
5580 points
0 .0
2 .5
5 .0
7 .5
10 .0
12 .5
15 .0
17 .5
20 .0
22 .5
4.9 (170)
0.080.090.100.110.12 z
−22.0
−21.0
−20.0
u − r > 2.22
0.080.090.100.110.12 z
10 16 22
u − r > 2.22
−23
−22
−21
−20 M 10−5 10−3 10−1
100
u − r > 2.22
0.080.090.100.110.12 z
−22.0
−21.0
−20.0
u − r < 2.22
N = 45010
4.10 (171)
k
10−1
100
101
102
103
n = 10
b∗= 0.5
b∗= 0.1
k
n = 20
5.1 (188)
xobs 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14
sampled region
true distribution observed distribution
−4 −3 −2 −1 0 1 2 3 4
xobs− xtrue 0.0 0.1 0.2 0.3 0.4 0.5 0.6
xobs
observed sample random sample
5.2 (192)
mobs 0.0 0.1 0.2
scatter bias
σπ/π
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
p = 2
5.3 (195)
µ
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
L(µ, σ) for ¯x = 1, V = 4, n = 10
−5.0
−4.0
−3.0
−2.0
−1.0
0.0
5.4 (199)
10−4 10−2
100
µ
0.0
0.2
0.4
0.6
0.8
1.0
10−4 10−2
100
0.0
0.2
0.4
0.6
0.8
1.0
5.5 (201)
Trang 3Visual Figure Index • 529
x
0.00
0.05
0.10
0.15
0.20
0.25
σ fit
σG fit
5.6 (203)
µ
1 2 3 4 5
L(µ, σ) for ¯x = 1, σtrue = 1, n = 10
−5.0
−4.0
−3.0
−2.0
−1.0
0.0
5.7 (205)
−3 −2 −1 0 1 2 3 4 5 µ
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
σ
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
marginalized approximate
5.8 (206)
0 .0 0.2 0.4 0.6 0.8 1.0 b
0 .0
0 .5
1 .0
1 .5
2 .0
2 .5
3 .0
0 .0 0.2 0.4 0.6 0.8 1.0 b
10−6
10 −5
10 −4
10−3
10 −2
10−1
10 0
5.9 (208)
µ
1 2 3 4 5
L(µ, γ) for ¯x = 0, γ = 2, n = 10
−5.0
−4.0
−3.0
−2.0
−1.0
0.0
5.10 (209)
µ
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 γ
0.0
0.2
0.4
0.6
0.8
1.0
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 γ
0.0
0.2
0.4
0.6
0.8
1.0
5.11 (210)
L(µ, W ) uniform, n = 100
−6.4
−4.8
−3.2
−1.6
0.0
µ
p(W )
9.8
10.0
10.2
10.4
10.6
5.12 (212)
0.6 0.8 1.0 1.2 σ1.4 1.6 1.8 2.0
0.0
0.2
0.4
0.6
0.8
1.0
L(σ, A) (Gauss + bkgd, n = 200)
−5.0
−3.5
−2.0
−0.5
0.0
x
0.0
0.1
0.2
0.3
5.13 (214)
x
0 .00
0 .02
0 .04
0 .06
0 .08
0 .10
0 .12
0 .14
0 .16
0 .18
−0.01 0 .00 0 .01 0 .02 a
0 50 100 150 200 250 300
500 pts
20 pts
5.14 (216)
−5
0 10 20
y i
50 points
5 bins
a∗
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Poisson Likelihood Gaussian Likelihood
x
−1
0 2 4
y i
50 points
40 bins
0.02
0.04
0.06
0.08
0.10
0.12
0.14
Poisson Likelihood Gaussian Likelihood
5.15 (218)
a
0
50
100
150
200
continuous
discrete, 1000 bins
discrete, 2 bins
5.16 (219)
µ
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
µ
5.17 (222)
g1
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
p(g1) (bad point) p(g1|µ0) (bad point)
p(g1) (good point) p(g1|µ0) (good point)
5.18 (223)
100
104
108
1012
1016
1020
1024
1028
Sample SizeN
−60
−20
0 20 60
5.19 (224)
Scott’s Rule:
38 bins Gaussian distribution
Scott’s Rule:
24 bins non-Gaussian distribution
Freed.-Diac.:
49 bins Freed.-Diac.:
97 bins
x
Knuth’s Rule:
38 bins
x
Knuth’s Rule:
99 bins
5.20 (227)
0.0
0.1
0.2
0.3
0.4
Knuth Histogram
Bayesian Blocks
x
0.0
0.1
0.2
0.3
0.4
Knuth Histogram
Bayesian Blocks
5.21 (229)
0 1 2 4
5.22 (232)
x
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
µ1= 0;σ1= 0.3
µ2= 1;σ2= 1.0
ratio = 1.5
Input pdf and sampled data true distribution best fit normal
5.23 (235)
0.30.40.5 µ0.60.70.8
0.75
0.80
0.85
0.90
0.95
1.00
1.05
1.10
1.15
Single Gaussian fit
0.8
1.2
1.6
0.2
0.3
0.4
0.75
1.00
1.25
−0.2 −0.1 0.0 µ1 0.1
0.6
1.2
1.8
0.8 µ21.21.60.20.30.40.75 σ21.001.25
5.24 (236)
µ
0 1 2 3 4 5
5.25 (239)
5.8
6.0
6.2
6.4
6.6
0.10.20.30.40.50.6
A
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
5.8 6.0 6.2 µ 6.4 6.6
x
0.0
0.1
0.2
0.3
0.4
0.5
yobs
5.26 (240)
x
0.0
0.2
0.4
0.6
0.8
1.0
x
0.0
0.2
0.4
0.6
0.8
1.0
x
0.0
0.2
0.4
0.6
0.8
1.0
x
6.1 (252)
u
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Gaussian Exponential Top-hat
6.2 (253)
−350
−250
−300 −200 −100 0 100
y (Mpc)
−350
−250
top-hat ( h = 10)
−300 −200 −100 0 100
y (Mpc)
exponential ( h = 5)
6.3 (255)
−350
−250
−300 −200 −100 0 100
y (Mpc)
−350
−250
k-neighbors (k = 5)
−300 −200 −100 0 100
y (Mpc) k-neighbors (k = 40)
6.4 (259)
0.0
0.1
0.2
0.3
0.4
Nearest Neighbors (k=10)
Kernel Density (h=0.1)
Bayesian Blocks
x
0.0
0.1
0.2
0.3
0.4
Nearest Neighbors (k=100)
Kernel Density (h=0.1)
Bayesian Blocks
6.5 (260)
−0.9−0.6−0.3 0.0
[Fe /H]
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
Input
0 2 4 6 8 10 12 14
N components
−34500
−33500
−32500
−0.9−0.6−0.3 0.0
[Fe /H]
0 .0
0 .1
0 .2
0 .3
0 .4
0 .5
Converged
6.6 (261)
−350
−250
y (Mpc)
−350
−250
6.7 (262)
0.0
0.1
0.2
0.3
0.4
Mixture Model (3 components) Kernel Density (h = 0.1)
Bayesian Blocks
x
0.0
0.1
0.2
0.3
0.4
Mixture Model (10 components)
h = 0.1)
Bayesian Blocks
6.8 (263)
0 20 60 100
N = 100 points
0 20 60 100
N = 1000 points
N = 10000 points
n clusters
16.0
16.5
17.0
17.5
18.0
18.5
N=100 N=1000 N=10000
6.9 (265)
−2 0 2 4 6 8
x
−2
0
2
4
6
8
Input Distribution
−2 0 2 4 6 8
x
Density Model
−2 0 2 4 6 8
x
Cloned Distribution
6.10 (266)
−5
0 10
x
−5
0 10
Extreme Deconvolution resampling
x
Extreme Deconvolution cluster locations
6.11 (268)
−0.5
0.0
0.5
1.0
1.5
g − r
−0.5
0.0
0.5
1.0
1.5
Extreme Deconvolution resampling
g − r
Extreme Deconvolution cluster locations
0 10 30 50
w = −0.227g + 0.792r
−0.567i + 0.05
single epoch
σG = 0.016
standard stars
σG = 0.010
XD resampled
σG = 0.008
6.12 (269)
[Fe/H]
0.0
0.1
0.2
0.3
0.4
0.5
6.13 (272)
[Fe/H]
0.0
0.1
0.2
0.3
0.4
0.5
6.14 (274)
Trang 4−250
−350
−250
y (Mpc)
−350
−250
6.15 (276)
r12
r12
r23
r31
r12
r13
r24
r14
r23r34
6.16 (278)
θ (deg)
10−2 10−1 100
10 1
u − r > 2.22
N = 38017
θ (deg)
10−2 10−1 100
10 1
u − r < 2.22
N = 16883
6.17 (280)
3000 4000 5000 6000 7000 wavelength ( ˚A)
3000 4000 5000 6000 7000 wavelength ( ˚A)
3000 4000 5000 6000 7000 wavelength ( ˚A)
7.1 (291)
x
y
x y
7.2 (293)
1
=
=
7.3 (295)
mean PCA components
component 1
component 2
component 3
3000 4000 5000 6000 7000 wavelength ( ˚A) component 4
mean ICA components
component 1
component 2
component 3
3000 4000 5000 6000 7000 wavelength ( ˚A) component 4
component 1 NMF components
component 2
component 3
component 4
3000 4000 5000 6000 7000 wavelength ( ˚A) component 5
7.4 (298)
10−3 10−1
100 102
Eigenvalue Number
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
7.5 (299)
0 10
mean
0 10
mean + 4 components (σ2tot = 0.85)
0 10
mean + 8 components (σ2tot = 0.93)
wavelength ( ˚A) 0
10
mean + 20 components (σ2tot = 0.94)
7.6 (300)
λ ( ˚ A)
True spectrum (nev=10)
λ ( ˚ A)
λ ( ˚ A)
7.7 (304) PCA projection
7.8 (307)
absorption galaxy galaxy emission galaxy narrow-line QSO broad-line QSO
−1.0
0.0
0.5
1.0
c2
c1
−0.8
0.0
0.4
0.8
c3
c2
absorption galaxy galaxy emission galaxy narrow-line QSO broad-line QSO
−0.04
0.00
0.02
−0.010 −0.0050.000 c10.0050.010
−0.04
0.00
0.02
0.04
−0.04 −0.02 c20.00 0.02
7.9 (310)
x
−2.0
−0.5
0.0
0.5
1.0
1.5
2.0
x1 x2 x3 x4
True fit fit to{x1, x2, x3}
−0.5
0.0
0.5
1.0
0.5 1θ1 0 1.5 2.0
−0.5
0.0
0.5
1.0
x4
8.1 (323)
38 42 46
χ2dof = 1.57
Straight-line Regression
χ2dof = 1.02
4th degree Polynomial Regression
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 z
38 42 46
χ2dof = 1.09
Gaussian Basis Function Regression
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 z χ2dof = 1.11
Gaussian Kernel Regression
8.2 (328)
θ1
θ2
θnormal equation
θridge
θ2
θnormal equation
θlasso
r
8.3 (333)
36
42
48
Linear Regression
0.0 0.5 1.0 1.5
z
−15
−5
0
10
×1012
Linear Regression
Ridge Regression
0.0 0.5 1.0 1.5
z
−2
0
2
4
Ridge Regression
Lasso Regression
0.0 0.5 1.0 1.5
z
−0.5
0.0
0.5
1.0
1.5
2.0
Lasso Regression
8.4 (335)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
z
36 38 40 42 44 46
100 observations
0.1 0.2 0.3 0.4 0.5 0.6 0.7 ΩM 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1
8.5 (341)
x
100 200 300 400 500 600
1.8 2.0 2.2 2.4 2.6 slope
−60
−20
0 20 60 100
8.6 (344)
t
0 10 20 30 40 50
c = 1
c = 2
c = 5
c = ∞
8.7 (346)
100 200 300 400 500 600 700
squared loss:
y = 1.08x + 213.3
Huber loss:
y = 1.96x + 70.0
8.8 (347)
100
300
500
700
intercept
0.6
0.8
1.0
1.2
1.4
1.6
no outlier correction
(dotted fit)
intercept
2.0
2.2
2.4
2.6
2.8
mixture model
(dashed fit)
intercept
2.0
2.2
2.4
2.6
2.8
outlier rejection
(solid fit)
8.9 (348)
−3
−1
0 2
−3
−1
0 2
−2.0
−0.5
0.0
0.5
1.0
1.5
2.0
2.5
−1.5
0.0
0.5
1.0
1.5
2.0
8.10 (350)
36 38 40 42 44 46 48
8.11 (352)
x
0.0
0.5
1.0
1.5
2.0
d = 1
8.12 (353)
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
0.0 0.5 1.0 1.5 2.0
d = 2
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
d = 3
0.0 0.5 1.0 1.5 2.0 2.5 3.0
x
d = 19
8.13 (354)
polynomial degree
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
cross-validation
training
polynomial degree
0
20
60
100
cross-validation
training
8.14 (356)
Number of training points
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
training
Number of training points
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
training
8.15 (358)
x
0.0
0.1
0.2
0.3
0.4
0.5
g1(x)
g2(x)
9.1 (370)
x
−1
0 1 2 3 4 5
9.2 (373)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u − g
−0.1
0.0 0.1 0.2 0.3
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
9.3 (374)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u g
−0.1
0.0
0.1
0.2
0.3
0.0
0.2
0.4
0.6
0.8
1.0
N colors
0.0
0.2
0.4
0.6
0.8
1.0
9.4 (376)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u g
−0.1
0.0 0.1 0.2 0.3
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
9.5 (376)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u g
−0.1
0.0 0.1 0.2 0.3
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
N=1
9.6 (378)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u − g
−0.1
0.0 0.1 0.2 0.3
k = 10
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
k=1 k=10
9.7 (380)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u − g
−0.1
0.0 0.1 0.2 0.3
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
9.8 (382)
Trang 5Visual Figure Index • 531
x
−1
0
1
2
3
4
9.9 (383)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u − g
−0.1
0.0 0.1 0.2 0.3
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
9.10 (385)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u − g
−0.1
0.0 0.1 0.2 0.3
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
9.11 (386)
69509/ 346
split ong − r
2841/ 333
split onu − g
66668/ 13
split ong − r
1666/ 23
split ong − r
1175/ 310
split onr − i
1645/ 11
split onu − g
65023/ 2
split onr − i
392/ 16
split oni − z
1274/ 7
split onu − g
756/ 41
split onr − i
419/ 269
split oni − z
1616/ 3
split onu − g
29/ 8
split onr − i
6649/ 2
split onu − g
58374/ 0
non- variable
126 / 1 split on i − z
266 / 15 split on g − r
1001 / 2 split on i − z
273 / 5 split on g − r
379 / 0
non-variable
377 / 41 split on u − g
123 / 18 split on g − r
296 / 251 split on i − z
1296 / 0
non-variable
320 / 3 split on i − z
21 / 1 split on g − r
8 / 7 split on r − i
5200 / 0
non-variable
1449 / 2 split on u − g
Numbers are count of non-variable / RR Lyrae
in each node
Training Set Size:
69855 objects Cross-Validation, with
137 RR Lyraes (positive)
23149 non-variables (negative) false positives: 53 (43.4%) false negatives: 68 (0.3%)
9.12 (387)
0.7 0.8 0.9 1.0 1.1 1.2 1.3
u − g
−0.1
0.0 0.1 0.2 0.3
depth = 12
0.0 0.2 0.4 0.6 0.8 1.0
N colors 0.0 0.2 0.4 0.6 0.8 1.0
depth=7 depth=12
9.13 (388)
depth of tree
0.01
0.02
0.03
0.04
cross-validation
training set
ztrue
0.0
0.1
0.2
0.3
0.4
depth = 13
rms = 0.020
9.14 (390)
depth of tree 0.01 0.02 0.03 0.04
cross-validation training set
ztrue 0.0 0.1 0.2 0.3 0.4
zfit
depth = 20 rms = 0.017
9.15 (392)
number of boosts 0.01 0.02 0.03
Tree depth: 3
cross-validation training set
ztrue 0.0 0.1 0.2 0.3 0.4
N = 500 rms = 0.018
9.16 (395)
0 .000 0.008 0.016 0.024 0.032 0.040
false positive rate
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
GNB QDA LR KNN DT GMMB
0 .0 0.2 0.4 0.6 0.8 1.0
efficiency
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
9.17 (396)
−0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
u − g
−0.2
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
0.00 0.03 0.06 0.09 0.12 0.15 false positive rate 0.6 0.7 0.8 0.9 1.0
GNB
QDA LR KNN DT GMMB
9.18 (396)
1 mode
3 modes
phase
8 modes
10.1 (407)
0.0
0.5
1.0
1.5
2.0
data windowW (x)
Convolution:
[
D ∗ W ](x)
0.20.40.6 0.8 x
0.0
0.5
1.0
1.5
[D ∗ W ](x)= F−1{F[D] · F[W ]}
F(D)
F(W )
Pointwise product:
F(D) · F(W )
k
10.2 (411)
Signal and Sampling Window Sampling Rate ∆t
Time Domain: Multiplication
FT of Signal and Sampling Window
∆f = 1/∆t
Frequency Domain: Convolution
t
Sampled signal: pointwise multiplication
f
Convolution of signal FT and window FT Well-sampled data: ∆t < tc
Signal and Sampling Window Sampling Rate ∆t
Time Domain: Multiplication
FT of Signal and Sampling Window
∆f = 1/∆t
Frequency Domain: Convolution
t
Sampled signal: pointwise multiplication
f
Convolution of signal FT and window FT Undersampled data: ∆t > tc
10.3 (413)
−1.5
0 .0
0 .5
1 .0
1 .5
Data
0 .0 0.2 0.4 0.6 0.8 1.0
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
Data PSD
t
−0.2 0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
1 .2
1 .4
Window
0 .0 0.2 0.4 0.6 0.8 1.0 f
0 .0
0 .2
0 .4
0 .6
0 .8
1 .0
Window PSD
10.4 (414)
t
−0.1
0.0
0.1
0.2
0.3
f
0.0
0.2
0.4
0.6
0.8
10.5 (416)
time (s)
−1.0
0.0
0.5
1.0
×10−18
frequency (Hz)
10−46
10−40
Top-hat window
frequency (Hz)
10−46
10−40
Hanning (cosine) window
10.6 (417)
−2
0
Input Signal:
Localized Gaussian noise
−1.0
0.0
0.5
1.0
Example Wavelet
t0 = 0, f0 = 1.5, Q = 1.0
w(t; t0, f0, Q) = e−[f0(t−t0)/Q] 2 e2πif0(t−t0)
real part imag part
t
1 4 7
Wavelet PSD
10.7 (419)
−1.0
0.0
0.5
1.0
1.5
2.0
Input Signal:
Localized spike plus noise
−1.0
0.0
0.5
1.0
Example Wavelet
t0 = 0, f0 = 1/8, Q = 0.3
w(t; t0, f0, Q) = e−[f0(t−t0)/Q] 2 e2πif0(t−t0)
real part imag part
t
1/8 1/4 1/2
Wavelet PSD
10.8 (420)
−1.0
0.0
0.5
1.0
f0 = 5
t
−1.0
0.0
0.5
1.0
f0 = 10
Q = 1.0
t
f0 = 10
Q = 0.5
10.9 (421)
λ
−0.5
0.0
0.5
1.0
1.5
Input Signal
λ
Filtered Signal Wiener Savitzky-Golay
f
0 1000 3000
Input PSD
f
Filtered PSD
10.10 (423)
λ
0 .0
0 .1
0 .2
0 .3
0 .4
Effective Wiener
Filter Kernel
0 10 20 30 40 50 60 70 80 90
λ
−0.5
0 .0
0 .5
1 .0
1 .5
Kernel smoothing
result
10.11 (424)
λ ( ˚A)
30 60 90 100
f
0.0
0.2
0.4
0.6
0.8
1.0
10.12 (425)
λ ( ˚A)
30 60 90 100
SDSS white dwarf 52199-659-381
f
10−8
10−7
10−6
10−5
10−4
10−3
10−2
10101010−1012
10.13 (426)
t
−1.5
0.0 0.5 1.0 1.5
Data
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
f
0.0 0.2 0.4 0.6 0.8 1.0
PLS
Window PSD 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
f
0.0 0.2 0.4 0.6 0.8 1.0
Data PSD
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
f
0.0 0.2 0.4 0.6 0.8 1.0
PLS
Data PSD (10x errors)
10.14 (431)
time (days) 7 9 10 12
period (days)
0.0
0.2
0.4
0.6
0.8
−10
0 10 30
10.15 (437)
t
8
10
12
14
ω
0.0
0.2
0.4
0.6
0.8
1.0
PLS
standard
generalized
−10
0
10
30
50
10.16 (439)
14.4
14.8
15.2
ID = 14752041
P = 8.76 hr
14.6
14.8
15.0
ID = 1009459
P = 2.95 hr
13.6
14.0
14.4
14.8
ID = 10022663
P = 14.78 hr 15
17ID = 10025796
P = 3.31 hr
0.00.20.40.60.81.0
phase
15.6
15.9
16.2
ID = 11375941
P = 2.58 hr
0.00.20.40.60.81.0
phase
14.5
15.0
15.5ID = 18525697
P = 13.93 hr
10.17 (440)
17.18 17.19 17.20 17.21 17.22 17.23 0.0 0.2 0.4 0.6 0.8 1.0
6 terms
1 term
0.0 0.2 0.4 0.6 0.8 1.0
14.4 14.8 15.2
ω0= 17.22
P0= 8.76 hours
ω
0.0 0.2 0.4 0.6 0.8 1.0
6 terms
1 term
0.0 0.2 0.4 0.6 0.8 1.0 phase
14.4 14.8 15.2
ω0= 8.61
P0= 17.52 hours
10.18 (442)
0 5000 10000 20000 30000
ω0= 17.22
N frequencies 0
5000 10000 20000 30000
ω0= 8.61
22750 22825 zoomed view
26675 26750 zoomed view
10.19 (443)
−1.5
−0.5
0.0
0.5
−0.50.00.51.01.52.0
g − i
−1.5
−0.5
0.0
0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4 A
10.20 (445)
−0.50.00.51.01.52.0
g − i
−1.5
0.0
0.5
1.0
1.5
2.0
−0.50.00.51.01.52.0
g − i
1.0
1.5
2.0
2.5
−0.50.00.51.01.52.0
g − i
0.5
1.0
1.5
2.0
2.5
−0.50.00.51.01.52.0
g − i
−0.2
0.0
0.2
0.4
0.6
0.8
1.0
10.21 (446)
−1.5
−0.5
0.0
0.5
−0.50.00.51.01.52.0
g − i
−1.5
−0.5
0.0
0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4 A
10.22 (448)
−1.5
−0.5
0.0
0.5
−0.50.00.51.01.52.0
g − i
−1.5
−0.5
0.0
0.5
0.2 0.4 0.6 0.8 1.0 1.2 1.4 A
10.23 (449)
0.6
0.7
0.8
0.9
1.0
1.1
0.0
0.5
1.0
1.5
3.8
3.9
4.0
4.1
4.2
0.6 0.7 0.8 0.9 1.0 1.1
φ
0 10
t
10.24 (452)
2 6 10
46 50 54
b0
0.00
0.05
0.10
0.15
0.20
0.25
T
t
8 10 15
10.25 (454)
Trang 60 20 60 80 100
t
2
10
20
4.0
4.5
5.0
5.5
6.0
0.07
0.08
0.09
0.10
0.11
b0
0.0098
0.0099
0.0100
0.0101
0.0102
0.0103
0.0104
4.0 4.5 5.0 5.5 6.0
ω
10.26 (455)
0.76
0.78
0.80
0.82
0.088
0.096
0.104
0.112
29.8530.0030.15 T
0.01980
0.01995
0.02010
0.760.780.800.82
ω
t
−1.5
0.0
0.5
1.0
1.5
2.0
2.5
10.27 (456)
−2
0 2
Input Signal: chirp
t
0.1
0.2
0.3
0.4
0.5
0.6
Wavelet PSD
10.28 (457)
t
−1.5
−0.5
0.0
0.5
1.0
1.5
P (f) ∝ f−1
f
10 10
−1
0
101
t
P (f) ∝ f−2
f
10.29 (459)
t (days)
19.2
19.4
19.6
19.8
20.0
20.2
20.4
20.6
20.8
t (days)
−1.0
0.0
0.5
1.0
Scargle True Edelson-Krolik
10.30 (462)
np.arange(3) + 5
np.ones((3, 3)) + np.arange(3)
0
2
0
2
0
2
np.arange(3).reshape((3, 1)) + np.arange(3)
A.1 (493)
x
−1.0
−0.5
0.0
0.5
1.0
Simple Sinusoid Plot
A.2 (496)
x
−1.0
−0.5
0.0
0.5
1.0
Simple Sinusoid Plot
A.3 (496)
x 0 10 20 30 40 50 60
A.4 (497)
−1.0
−0.5
0.0
0.5
1.0
A.5 (502)
Wavelength (Angstroms)
0.0
0.1
0.2
0.3
0.4
0.5
SDSS Filters and Reference Spectrum
C.1 (516)
t
−1.0
0.0
0.5
1.0
Re[h]
Im[h]
f
−1.5
0.0
0.5
1.0
1.5
E.1 (524)