multidimensional d2 phase dispersion statistic · multidimensional d2 phase dispersion statistic...
TRANSCRIPT
Multidimensional D2 phase dispersionstatisticNigul OlspertAstroMHD Research Group Meeting
Monday 22nd August, 2016
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
2
Harmonics and periodsSpectral analysisThinking in terms of harmonics
I Power spectral density→ periodogramI Unevenly sampled data: Lomb-Scargle periodogram
PDM methodsThinking in terms of periods
I By default support uneven samplingI Generalizable to multiple dimensions
SS(P) =n∑
i=1
(yi(P)− yi(P))2 (1)
Lafler & Kinman: yi(P) = yi−1(P), Renson: weighted versionStellingwerf: yi(P) = y i(P), where y i(P) is a bin mean.
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
3
Examples of phase diagramCorrect period
Wrong period
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
4
From periods to cyclesD2 methodThinking in terms of cycles
D2(P, tcoh)=
∑i,j>i
g(ti , tj ,P, tcoh)||f (ti)− f (tj)||2
2σ2∑
i,j>ig(ti , tj ,P, tcoh)
, (2)
where g = g1(ti , tj ,P)g2(ti , tj , tcoh)
I Essentially a generalized PDM methodI In addition to selection function g1 in phase, introduces a
selection function g2 in time lagWe use:g1 = 2 cos(2πν(tj − ti))) + 1g2 = exp(− ln 2((tj − ti)/tcoh)
2)
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
5
Significance estimation
I Null hypothesis: Gaussian white noiseI Stellingwerf showed that the spectral line has then F
distribution.I In many cases analytical form is hard or impossible to
derive→ use permutation testI Is white noise correct null hypothesis?I Also in red noise (or Brownian noise) patterns of periodicity
can occur.I Red noise is a special case of AR(1) process
Xt = c + ϕXt−1 + εt with ϕ > 0I How to use red noise as null hypothesis?I What if we could resample from the same process instead?
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
6
Example of red noise spectrum
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
7
Test casesRotating particle Oscillations in a box
(a)
(b)
(c)
0.95
1
1.05
1.1
ν
0.975
0.98
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
D2
0.95
1
1.05
1.1ν
0.984
0.986
0.988
0.99
0.992
0.994
0.996
0.998
1
1.002
D2
5 10 15 20 25 30lcoh
0.95
1
1.05
1.1
ν
0.997
0.998
0.999
1
1.001
1.002
1.003
1.004
D2
(a)
(b)
(c)
0.95
1
1.05
1.1
ν
0.997
0.998
0.999
1
1.001
1.002
1.003
1.004
1.005
1.006
1.007
D2
0.95
1
1.05
1.1
ν
0.996
0.998
1
1.002
1.004
1.006
1.008
1.01
D2
5 10 15 20 25 30lcoh
0.95
1
1.05
1.1
ν0.999
1
1.001
1.002
1.003
1.004
1.005
D2
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
8
Patterns
Periodic signal(a)
−0.6−0.4−0.2
00.20.40.6
0 5 10 15 20
(b)
−0.2−0.1
00.10.20.3
0 5 10 15 20
5 10 15 20 25 30lcoh
0.8
0.9
1
1.1
1.2
1.3
1.4
ν
0.20.30.40.50.60.70.80.911.11.2
D2
V
a
l
u
e
Time
5 10 15 20 25 30lcoh
0.8
0.9
1
1.1
1.2
1.3
1.4
ν
0.980.9850.990.99511.0051.011.0151.021.0251.03
D2
V
a
l
u
e
Time
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
9
Patterns
Cyclic signal(a)
−0.6−0.4−0.2
00.20.40.6
0 5 10 15 20
(b)
−0.3−0.2−0.1
00.10.2
0 5 10 15 20
5 10 15 20 25 30lcoh
0.8
0.9
1
1.1
1.2
1.3
1.4
ν
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
D2
V
a
l
u
e
Time
5 10 15 20 25 30lcoh
0.8
0.9
1
1.1
1.2
1.3
1.4
ν
0.985
0.99
0.995
1
1.005
1.01
1.015
1.02
D2
V
a
l
u
e
Time
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
10
Patterns
Temporary signal(a)
−0.4−0.2
00.20.4
0 5 10 15 20
(b)
−0.4−0.2
00.20.4
0 5 10 15 20
5 10 15 20 25 30lcoh
0.95
1
1.05
1.1
ν
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
1.05
D2
V
a
l
u
e
Time
5 10 15 20 25 30lcoh
0.95
1
1.05
1.1
ν
0.75
0.8
0.85
0.9
0.95
1
D2
V
a
l
u
e
Time
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
11
Cycles in PENCIL-Millennium data
Mean cycle length estimates
Cycle noBr Bθ Bφ
N S N S N SI 0.47 0.47 0.48 0.48 0.46 0.46II 5.12 4.98 5.13 4.98 5.17 5.02III 49.2 43.0 46.2 40.2 50.8 46.06IV 108.4 105.1 108.0 106.0 107 .5 104.1
Notes: the numbers in italic represent cycles appearing only inthe bottom of the convection zone, otherwise the cycle exists inthe full hemisphere
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
12
Cycles in PENCIL-Millennium data5 year cycle
6.7
5.0
4.0
3.3
P [yr]
Br (north) Br (south) Br (full)
6.7
5.0
4.0
3.3
P [yr]
Bθ (north) Bθ (south) Bθ (full)
5 10 15 20 25 30lcoh
6.7
5.0
4.0
3.3
P [yr]
Bϕ (north)
5 10 15 20 25 30lcoh
Bϕ (south)
5 10 15 20 25 30lcoh
Bϕ (full)
0.560.640.720.800.880.961.041.12
θ2
0.480.560.640.720.800.880.961.04
θ2
0.600.650.700.750.800.850.900.951.00
θ2
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
13
Cycles in PENCIL-Millennium data100 year cycle
142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4
P [yr]
Br (north) Br (south)
142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4
P [yr]
Bθ (north) Bθ (south)
1 2 3 4 5lcoh
142.9 125.0 111.1 100.0 90.9 83.3 76.9 71.4
P [yr]
Bϕ (north)
1 2 3 4 5lcoh
Bϕ (south)
1.0001.0021.0041.0061.0081.0101.0121.014
θ2
0.9850.9900.9951.0001.0051.0101.0151.020
θ2
0.9600.9680.9760.9840.9921.0001.008
θ2
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
14
Cycles in PENCIL-Millennium data
0.5 year cycle
1 2 3 4 5 6 7 8 9 10lcoh
0.67
0.57
0.5
0.44
0.4
0.36
0.33
0.31
P[y
r]
0.80.820.840.860.880.90.920.940.960.9811.02
D2
Multidimensional D2 phase dispersion statisticMonday 22nd August, 2016
15
Cycles in PENCIL-Millennium data50 year cycle
66.7
50.0
40.0
33.3
P [yr]
Br (north) Br (south)
66.7
50.0
40.0
33.3
P [yr]
Bθ (north) Bθ (south)
1 2 3 4 5 6 7 8 9 10lcoh
66.7
50.0
40.0
33.3
P [yr]
Bϕ (north)
1 2 3 4 5 6 7 8 9 10lcoh
Bϕ (south)
1.0851.0871.0901.0931.0951.0971.1001.1031.105
θ2
1.0681.0721.0761.0801.0841.0881.0921.0961.100
θ2
1.0651.0701.0751.0801.0851.0901.0951.100
θ2