Multidimensional D2 phase dispersion statistic · Multidimensional D2 phase dispersion statistic Monday 22nd August, 2016 2 Harmonics and periods Spectral analysis Thinking in terms

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.


3

Examples of phase diagramCorrect period

Wrong period


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)


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?


6

Example of red noise spectrum


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


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


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


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


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


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


13


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


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


15


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

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Multidimensional D2 phase dispersion statistic · Multidimensional D2 phase dispersion statistic Monday 22nd August, 2016 2 Harmonics and periods Spectral analysis Thinking in terms