Applica'on of 'me series and spectral methods in Solar & Astrophysics Ding Yuan Email: Ding.Yuan@wis.kuleuven.be Centre for mathema'cal Plasma Astrophysics Department of Mathema'cs Katholieke Universiteit Leuven (KU Leuven) Celes'jnenlaan 200B, bus 2400 B-­‐3001 Leuven, Belgium Dr. D. Yuan, KU Leuven Content •
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Fundamentals of sta's'cs Pre-­‐processing methods Fourier transform Windowed Fourier transform Wavelet Periodogram Date-­‐compensated Discrete Fourier transform Filtering method: 'me and spectral domain Significance tests and noise analysis Dr. D. Yuan, KU Leuven Fundamental of Sta's'cs y = {y1, y2 ,..., yN }
y j = x j + s j , where x is a signal and s is a random noise
E[y] = E[x] = E[s] = 0; Var[x] = σ 2x ; Var[s] = σ 2s
Var[y] = E[y 2 ] − (E[y])2
= E[x 2 ] + E[2xs] + E[s 2 ] − (E[x])2
= Var[x] + E[2xs] + E[s 2 ]
= σ 2x + σ 2s (error propagation)
E[(s / σ s )2 ] = E[ χ 12] =1,
χ 12 is a χ 2 -distribution with 1-degree of freedom
E[2xs] ≈ 0 by assuming
s are independent.
Dr. D. Yx
uan, and
KU Leuven Time series: sunspot cycle @Courtesy of NASA Dr. D. Yuan, KU Leuven Time series: sunspot waves ysical Journal, 792:41 (7pp), 2014 September 1
Time series (20 points detrending)
(a)
UF1
(b)
0.3
0.2
0.1
0.0
-0.1
-0.2
10
Frequency (mHz)
Relative Intensity
0.4
Yua
0.0 0.2 0.4 0.6 0.8 1.0
Norm. Counts (d)
(c)
8
6
4
2
0
20
40
60
Time (min)
80
100
0.2 0.4 0.6 0.8 1.0
Norm. Power
Baseline-ratio intensity variation of a pixel taken at U1. The red curve is a single harmonic fit to the time series. Other panels are (b) the histo
) the Morlet wavelet spectrum, and (d) the periodogram.
on of this figure is available in the online journal.)
Yuan et al 2014, ApJ 792 Dr. D. Yuan, KU Leuven Time series: flare pulsa'ons Anfinogentov et al.
d with
amped
2002).
n et al.
xcited
of the
2
140
∆U Magnitude
120
100
intensity
2007).
dment
emonypical
with a
recipiis pronamic
ectron
roduce
mission
80
60
6
4
2
0
40
0
2
4
6
hours from the flare peak
20
0
0.0
0.5
1.0
1.5
hours from the flare peak
2.0
Figure 1. The U-band light curve of the YZ CMi megaflare observed on 2009
Anfinogentov et al. 2013 ApJ 773 January 16 (inset). An expanded version of the light curve (black dots) during the
decay phase appears in the main
solid line shows the least-squares
Dr. figure.
D. Yuan, KThe
U Leuven approximation of the long-term component of the flare profile.
Time series: MHD waves T. J. Wang et al.: Propagating low-frequency slow magnetoacoustic w
Doppler shift
Relative intensity
!!
Wang et al. 21 009 &A 03 Lprofiles
25 of
Fig. 3. Wavelet analysis
for oscillation
at y = A
150
. a)5Time
Doppler shift and relative intensity. Here positive values for the Doppler
Dr. D. Yuan, KU Leuven shift represent the blueshifted emission.
b) The wavelet power spectrum
D
Fig. 4. Wavelet analys
are same as in Fig. 3.
A raw 'me series 1.  Check for faulty data: missing points (zero or interpola'on), outliers 2.  Data metrics: uniform or uneven 'me series, mean, variance, histogram. 3.  Detrending: moving average, polynomial (exponen'al, linear) fit, running difference. Dr. D. Yuan, KU Leuven A27 (2010)
Outliers (spikes) Type-­‐I outlier: removal by thresholding Chorley 2010 A&A 513 Type-­‐II: jump of mean value Removal by thresholding in deriva've Dr. D. Yuan, KU Leuven o. 2, 2009
ULTRA-LONG-PERIOD
OSCILLATIONS IN
EUV FILAMENTS
Detrending: polynomial fit (a) 160
(DN/s)
140
120
100
80
Period (hour)
(c)
50
100
150
50
100
150
-0.0
-0.1
-0.1
0
10
95%
95%
I/Io -1
60
0
(b) 0.1
0.0
100
0
50
100
Time (hour)
Dr. D. 150
Yuan, KU Leuven EUV filament and the other i
are related, we compare cha
amplitudes of intensity varia
dataet sets.
is A
presented
Foullon al. This
2009 pJ 700 fo
for Sequence 2, starting 2003
in Figure 5 for Sequence 1, sta
by F04. In both figures, the
where I is (a) the 12 minut
intensity time series in the R
CELIAS/SEM 304 Å flux (te
first-order CELIAS/SEM 260
dominant periodicities detecte
a 50 hr boxcar running average
running average in Sequence
10–20 hr oscillations (Time =
panel (a) despite the 50 hr box
in panel (b).
Nevertheless, there are inte
time series appear similar, wit
for a 4-day interval indicated
respective power spectra are
(c) and (d) below. As illustrat
Detrending: exponen'al fit Anfinogentov et al.
∆U Magnitude
120
100
80
60
4
20
2
10
0
40
The Astrophysical Journal, 773:156 (5pp), 2013 August 20
6
intensity
140
0
2
4
6
hours from the flare peak
0
−10
20
−20
0
0.5
1.0
1.5
hours from the flare peak
2.0
ure 1. The U-band light curve of the YZ CMi megaflare observed on 2009
ary 16 (inset). An expanded version of the light curve (black dots) during the
y phase appears in the main figure. The solid line shows the least-squares
oximation of the long-term component of the flare profile.
60
30
60
90
minutes from the flare peak
120
Lomb−Scargle periodogram
50
power
0
0.0
Guessed function fitting
40
30
20
10
band emission was almost 6 mag brighter than the quiescent
0
e. As this was one of the longest and most-energetic flares
10
20
30
40
50
60
70
80
period,
min
r observed in white-light on an isolated low-mass star, it
Figure 2. Upper panel: the short-term intensity variations (gray dots) extracted from the U-band light curve. The solid line shows the best
Anfinogentov et of a32 l. 2013 Atime
pJ of 746 73 labeled as a “megaflare.” The observations
were
acquired
sinusoidal
oscillation,
with a period
minutes
and a damping
minutes. Lower panel: the Lomb–Scargle periodogram
horizontal
dashed
line
shows
the
confidence
level
of
99.9%.
h the New Mexico State University 1 m Telescope at the
D. Ywas
uan, KU Leuven ache Point Observatory, New Mexico, USA.
TheDr. star
I = 6.88. Subtracting the best-fit power-law function from the
0
Autocorrelat
Uniform or non-­‐uniform data •  Uniform data (or uniformly interpolated data): FFT, Windowed FFT, wavelet, etc. •  Non-­‐uniform data: Periodogram, DCDFT, etc. •  Short 'me series (a few oscilla'on cycles): Nonlinear fit (mpfit.pro, hdps://www.physics.wisc.edu/
~craigm/idl/figng.html), Op'miza'on methods (IDL rou'ne:powell.pro), Bayesian inference (Marsh ApJ 2008, 681; Irregui, ApJ, 2011 740). Dr. D. Yuan, KU Leuven is used alternatively in the following text). The
Discrete Fourier Transform (DFT) Yk =
N
−1
!
j
−i2πk N
yj e
j=0
, k = 0, · · · , N − 1
#
•  DFT requires O(N"2) operations;
2 operations, while FFT com
ation
requires
O
N
•  FFT is an algorithm that compute accurate DFT with
operations.
(N O(NlogN)
log N ) operations.
Hence FFT is much fast
•  FFTPACK (Fortran), FFT (IDL), numpy.fft(Python)
sufficiently
small
intervals, therefore
•  IDL and Fortran
FFTfrequency
comparison (www.ssec.wisc.edu/
~paulv/fft/fft_comparison.html)
ith negligible errors.
ram
Dr. D. Yuan, KU Leuven #1: How to use FFT (IDL) Ø d: the vector of *me; Ø xx: the vector of variables Ø n=n_elements(xx); number of samples Ø d=d-­‐d[0] ; *me invariance Ø xx=xx-­‐mean(xx) ; remove mean value Ø dt=d[1:n-­‐1]-­‐d[0:n-­‐2] ; calculate cadence Ø dt_min=min(dt,max=dt_max) Ø '=0.5*(dt_min+dt_max); ensure uniform data Dr. D. Yuan, KU Leuven #2: How to use FFT (IDL) Ø temp = r(xx) ; calculate FFT Ø if n mod 2 eq 0 then begin ; even number Ø freq = findgen(n/2+1)/(n*') Ø pow_r = abs(temp[0:n/2])^2 Ø phase_r=atan(temp[0:n/2],/phase) Ø endif else begin ; odd number Ø freq=findgen((n+1)/2)/(n*') Ø pow_r=abs(temp[0:(n-­‐1)/2])^2 Ø phase_r=atan(temp[0:(n-­‐1)/2],/phase) Ø endelse Dr. D. Yuan, KU Leuven #3: How to use FFT (IDL) Ø norm=variance(xx)/float(N) Ø pow_r=pow_r/norm ; Normaliza'on Ø power=pow_r Ø fs=0.01 & Num=1000. Ø d=findgen(Num)*fs Ø xx=2*sin(2*!pi*15*d)-­‐cos(2*!pi*16*d)
+sin(2*!pi*30*d) ; A 'me series of freq=[15, 16, 30] Dr. D. Yuan, KU Leuven #4: How to use FFT (IDL) Without normaliza'on Normalized Dr. D. Yuan, KU Leuven Windowing •  Problems in FFT: aliasing (discre'za'on), spectral leakage (finite 'me span) •  Windowing -­‐> a) Select a desired range; b) Apply weights to the data; c) Reduce the noise Without normaliza'on Normalized by repe''ve measurements. •  Harris, 1978 “On the use of windows for harmonic analysis with the Discrete Fourier Transform” Dr. D. Yuan, KU Leuven Windowing Without normaliza'on Normalized Harris, 1978 The selected sequence has to contain sufficient informa'on. Dr. D. Yuan, KU Leuven function over the signal in the time domain. The window function is only non-zero
Windowed FFT (Short-­‐'me FFT) obtained. Mathematically, windowed
Fourier transform
Sliding DFT is expressed as,
over a certain range in time and is padded with zero over most of time. Usually
the non-zero part is moving (sliding) over time, therefore the dynamical spectra are
Yk,m =
N
−1
"
j=0
j
yj wj−m e−i2πk N , k = 0, · · · , N − 1
= {Yk " Wk }(m),
(1.89)
(1.90)
1. WFFT dynamic ('me-­‐dependent) spectrum. where
wj and rWeveals function
and its Fourier transform,
respectively,
k are theawindow
the convolution
operation.
Thereware
a number of
and
(") iin
(1.90) denotes
2. It s aEq.
rbitrary to choose a proper window idth. normaliza'on Normalized commonlyWithout used window
functions, i.e. a rectangle
window, cosine bell window,
3. The spectral resolu'on depends on the window width. 4. Different windows produce slightly different spectra. 28
Dr. D. Yuan, KU Leuven Windowed FFT window width=1/8 'me span Ø xx=2*sin(2*!pi*15*d)-­‐cos(2*!pi*16*d)+sin(2*!
pi*30*d) ; A 'me series of freq=[15, 16, 30] Without normaliza'on Normalized Dr. D. Yuan, KU Leuven Windowed FFT window width=1/4 'me span Ø xx=2*sin(2*!pi*15*d)-­‐cos(2*!pi*16*d)+sin(2*!
pi*30*d) ; A 'me series of freq=[15, 16, 30] Without normaliza'on Normalized Dr. D. Yuan, KU Leuven Windowed FFT window width=1/2 'me span Ø xx=2*sin(2*!pi*15*d)-­‐cos(2*!pi*16*d)+sin(2*!
pi*30*d) ; A 'me series of freq=[15, 16, 30] Without normaliza'on Normalized Dr. D. Yuan, KU Leuven , it is related to the Fourier period. Ψ is the normalised
& Compo 1998,
for details).
It must have zero mean
Wavelet Transform me and frequency spaces to be admissible as a wavelet
( j − h)δ t Wa v e l e t i s d e f i n e d a s t h e
Wh (s) = ∑ y j Ψ [
] convolution of a time series with a
s
is one of j=0
the most commonly
usedmother
mother
function
to
scaled
function
Ψ.
N −1
*
N −1
Torrence
& modulated
Compo, A practical
It consists
of a* planeiωwave
function
by
a
h
δ
t
k
1992):
= ∑ Yk Ψ̂ (sω k )e
guide to wavelet, BAMS 1998.
k=0
Ψ0 (η) = π
−1/4 iω0 η −η2 /2
e
e
,
Ψ0 is the original mother function
(1.92)
mensional frequency that satisfies the admissibility con-
er using larger ω0 was found to improve the spectral
ood 2000).
Dr. D. Yuan, KU Leuven wavelet analysis are taken from Farge (1992), Weng
and Lau (1994), and Meyers et al. (1993). Each section is illustrated with examples using the Niño3 SST.
smoothing) or a Gaussian window (Kaiser 1994).
As discussed by Kaiser (1994), the WFT represents
an inaccurate and inefficient method of time–frequency localization, as it imposes a scale or “response
interval” T into the analysis. The inaccuracy arises
from the aliasing of high- and low-frequency components that do not fall within the frequency range of the
window. The inefficiency comes from the T/(2δt) frequencies,
which
must be fanalyzed
at each
Morlet mother unc'on is time step,
regardless of the window size or the dominant frequenfrequently used in awindow
nalyzing cies
present. In addition,
several
lengths must
usually
be analyzed
to determine the most approprioscillatory signals. ate choice. For analyses where a predetermined scal may not be appropriate because of a wide range
ing
ofTorrence dominant frequencies,
a method
of time–frequency
&
C
ompo 1
998, BAMS localization that is scale independent, such as waverou'nes: letIDL analysis,
should be employed.
Mother func'on ψ (t / s)
^
ψ
(s ω)
a. Morlet
0.3
6
4
0.0
-0.3
-4
2
-2
0
2
4
b. Paul (m=4)
0.3
0
-2
-1
0
6
4
0.0
-0.3
-4
-2
0
2
4
c. DOG (m=2)
0.3
-1
0
-1
0
6
4
0.0
-0.3
-4
2
-2
0
2
4
0
-2
2
hdp://paos.colorado.edu/
2
0
-2
1
b. Wavelet transform
research/wavelets/ The wavelet transform can be used to analyze time
series that contain nonstationary power at many dif1
2
Farge, 1992, Annu. Rev. Fluid ferent frequencies (Daubechies 1990). Assume that
one
has a time series, xn, with equal time spacing δt
Mech. and n = 0 … N − 1. Also assume that one has a waveMoortel 2002 A&A 81, 311 letDe function,
ψ0(η), that
depends
on a 3
nondimensional
“time”
parameter
To be
“admissible”
wavelet,
Sych 2008 ηS.ol. Phys. 248, as3a95 this function must have zero mean and be localized in
time and frequency space (Farge 1992). An exDr. D. Yuan, Kboth
U Leuven 1
2 ample is the Morlet wavelet, consisting of a plane
Examples: umbral waves Journal, 792:41 (7pp), 2014 September 1
Time series (20 points detrending)
(a)
UF1
(b)
0.3
0.2
0.1
0.0
-0.1
-0.2
10
Frequency (mHz)
Relative Intensity
0.4
Yuan et al.
0.0 0.2 0.4 0.6 0.8 1.0
Norm. Counts (d)
(c)
8
6
4
2
0
20
40
60
Time (min)
80
100
0.2 0.4 0.6 0.8 1.0
Norm. Power
-ratio intensity variation of a pixel taken at U1. The red curve is a single harmonic fit to the time series. Other panels are (b) the histogram of
orlet wavelet spectrum, and (d) the periodogram.
s figure is available in the online journal.)
Yuan et al 2014, ApJ 792 Dr. D. Yuan, KU Leuven 3-­‐min slow wave & instrumental effect Period (second)
100
50
50
0
0
-50
-50
1000
1000 10 20 30 40 50 60
Global wavelet
200
400
Time (min)
600
200
200
300
300
400
500
1000
Period (second)
(a) 27-Oct-11 04:30:01
100
400
500
0
100
200
400
Time (min)
600
500
1000.0 0.2 0.4 0.6 0.8 1.0
Periodogram
200
WFFT: fixed window width for all periods 200
400
Time (min)
300
400
500
Dr. D. Yuan, 600
0.0KU 0.2Leuven 0.4 0.6
50
!
0
-50
Wavelet: wider window for 400
longer periods 200
300
Histogram
Y (arcsec)
Values
Time series (50) points detrending)
-100
-250 -200 -150 -100 -50
X (arcsec)
Figure 2.4: a) AIA 171 Å image of active
2011 at 04:30:01
is shown
with the flux
Yuan, 2UT
013, PhD thesis, labels a University bright loop in
backgound, u
of dark
Warwick detaled in Sec. 2.2. A cut that was taken t
with aA black
bar. to b)mThe
trade-­‐off ake brunning
etwenn differe
at 04:30:01
UT.
R2temporal is the first
half of R1
spectral and resolu5on covers about 10 cycles of the propagating
0.8 1.0
subtracted time-distance plot D1 . The firs
The reliability and efficiency of periodogram was studied in Scargl
ere we follow that discussion. The periodogram Py (ω) for an angular f
onent ω is defined as
Periodogram $%
&
%
2
[ j yj sin ω(tj − τ )]2
1 [ j yj cos ω(tj − τ )]
%
Py (ω) =
+ %
,
2
2
2
ω(tj −tτj .) The parameter
j − τ ) as
j cossequence
j sin τ ω(t
physical observable
at time
is defined

 

#
#



tan(2ωτ ) =
sin 2ωtj /
cos 2ωtj  .
j
(1.78)
j
Periodogram is equivalent to least-­‐square fit method that Pis y (ω)
τ ensures
the time-invariant
properties
of the power
spectrum.
in extrac'ng component in uat
nevenly ted effec've at any frequency
ω. Itperiodic is normally
calculated
a set of M fre-
spaced data. Scargle, 1982 ApJ, Horne & Baliunas 1986, ApJ e frequencies may coincide with the natural frequencies used in the
IDL rou'ne: is. hdp://www.arm.ac.uk/~csj/idl/PRIMITIVE/scargle.pro Dr. D. Yuan, KU Leuven -compensated Fourier transform
that the Py (ωn ) are independent random
26variables (Scargle 1982). In
Example: CCD temperature-­‐induced EUV image intensity varia'on A&A 533, A116 (2011)
(a)
Yuan al. 2011 Fig. 1. a) The field of view
overet AR8253
taken A
at &A 1998
includes the fan-like structure and is re-sized to the left
of interest (128 × 128 pixels) showing the fan-like stru
17 GHz radio emission over AR8253 at 1998-07-01 01
oscillations in sunspots, the possibility of the acou
Dr. D. Yuan, Kage
U Leuven into the corona has recently been demonstrate
Date-­‐compensated DFT •  Orthogonal Basis: Orthonormal basis: h0 = a0 H 0
H 0 (t j ) = 1
!N −1
h
=
a
H
−
a
h
<
h
,
H
>
1
1
1
1
0
0
1
The angle bracket denotes the inner product < y1 , y2 >= 0
y
H1 (t coefficients
t
j ) = cosaω
,
a way that
0 a1j and a2 are
h2 determined
= a2 H 2 −inasuch
2 h0 < h0 , H 2 >
H 2 (t j ) = sin ω t j
< h,H >
< −a
h0 , h
2h
0 1>=< 1h1 , h12 >=< h2 , h2 > .
N
∑
< y1, Fourier
y2 >= transform
y1 (t j )y
2 (t j ) is com
The date-compensated discrete
(DCDFT)
DCDFT: Every frequency 1
component shares a frac'on of the P (ω) =F (ω)F ∗ (ω)
mean value. √
DFT: Only the zero frequency F (ω) = < y, h1 + ih2 > /a0 2,
component contains the mean value. √
Dr. D. Yuan, KU Leuven Ferraz-­‐Mello theAJ imaginary un
where (∗) denotes complex conjugate,
i = −1 1is981 Examples: DFDFT vs FFT Ø xx=2*sin(2*!pi*15*d)-­‐cos(2*!pi*16*d)+sin(2*!
pi*30*d) ; A 'me series of freq=[15, 16, 30] Dr. D. Yuan, KU Leuven Advantages of DCDFT and Lamb-­‐Scargle periodogram •  Applicable to unevenly spaced data; •  Compute the power of any frequency or a selected range instantly ; •  Periodogram is associated with a significance test; •  DCDFT es'mate the amplitude (power) beder than other methods. •  DCDFT could es'mate the phase, amplitude and residue, therefore could extract any frequency component without resort to spectral domain (Harmonic filter). Dr. D. Yuan, KU Leuven Frequency filter •  Apply a window func'on in frequency domain •  Remove unwanted signal or noise y1 (t) = y(t)∗ w(t)
Y1 (ω ) = Y (ω )⋅W (ω )
Y (ω ) = FFT[y(t)]
W (ω ) = FFT[w(t)]
IDL usage: y1=FFT(W*FFT(y),/inverse); Dr. D. Yuan, KU Leuven Mul'-­‐mode QPP detected with NoRH A. R. Inglis and V. M. Nakariakov: A multi-periodic oscillatory event in a solar flare
Inglis & Nakariakov A&A 2009 Dr. D. Yuan, KU Leuven 263
Time-­‐domain Filter: Harmonic filter y1 (t) = y(t) − a − b cos ω t − csin ω t
a,b, and c are calculated with DCDFT Ferraz-­‐Mello AJ 1981 Yuan et al A&A 2011 Dr. D. Yuan, KU Leuven Removing TRACE orbital periods D. Yuan et al.: Leakage of long-period oscillations from the chromosphereto the corona
(a)
(b)
(c)
(d)
Dr. D. Yuan, KU Leuven (e)
(f)
Frequency vs 'me domain filter •  Frequency filter: No clean removal of a single spectral component due to aliasing, •  Easy to implement and capable of wide band filtering. •  Time domain filter: clean removal •  Good at removing one spectral component. Dr. D. Yuan, KU Leuven Noise es'mate in FFT PkN = NYk2 / 2σ 2 is the normalized Fourier power
Yk is the FFT of y j
σ 2 is the total variance of y j
α is the lag-1 auto-correlation coefficient of y j
The red noise spectrum is
1−α 2
Pk =
1 + α 2 − 2α co(2π k / N )
Pk = 1 (normalized mean variance) for white noise α = 0.
Torrence & Compo 1998 Dr. D. Yuan, KU Leuven χ 22
(17)
⇒” indicates “is disg distribution for the
s
χ2
(18)
3
a. α=0.00
14
95%
12
Mean
1
100
10
Period (δt)
1
8
6
2
0
b. α=0.70
64
95%
32
16
8
4
2
Period (years)
1
0.5
FIG. 6. Fourier power spectrum from Fig. 3, smoothed with a
five-point running average (thin solid line). The thick solid line is
the global wavelet spectrum for the Niño3 SST. The lower dashed
line is the mean red-noise spectrum, while the upper dashed line
is the 95% confidence level for the global wavelet spectrum,
assuming α = 0.72.
15
10
Mean
5
0
1000
10
4
0
1000
20
Variance (σ2)
2
100
10
Period (δt)
1
midpoint of n1 and n2, and na = n2 − n1 + 1 is the number of points averaged over. By repeating (21) at each
time step, one creates a wavelet plot smoothed by a
certain window.
The extreme case of (21) is when the average is over
all the local wavelet spectra, which gives the global
wavelet spectrum
Torrence & Compo 1998 FIG. 5. (a) Monte Carlo results for local wavelet spectra of white
Dr. D. Yuan, KU Leuven noise (α = 0.0). The lower thin line is the theoretical mean whitenoise spectrum, while the black dots are the mean at each scale
sure of the backgrou
in the local wavelet
al. 1998).
By smoothing the
can increase the deg
increase the signific
To determine the DO
dependent points. F
the power at each fre
ers, and the average
each with two DOF,
of freedom (Spiege
wavelet spectrum, on
χ22 distributed, yet F
points are no longer
both time and scale
time appears to len
wavelet function bro
one expects ν ∝ na a
to consider is to def
such that ν = 2naδ t/
show that this τ is too
even though one is
correlated, some add
The Monte Carlo
shows the mean and
4
3
N −1
)
an background speche distribution for the
4
Variance (σ2)
s shown that the local
ean Fourier spectrum.
nts are normally disients (the bandpassed
ould also be normally
e wavelet power specstributed. The upper
the 95% Fourier rede 95% level from the
ious section. Thus, at
ming a red-noise prote that for a wavelet
function, such as the
there is only one dend the distribution is
FFT and Wavelet noise level Variance (σ2)
DOFs, denoted by χ22
termine the 95% con%), one multiplies the
95th percentile value
e 95% Fourier confiST is the upper dashed
few frequencies now
95%
space-time maps constructed from this loop portion in the 171 Å (top) and 193 Å (bottom) channels. T
the final peak positions derived from the linear fit. The slopes of these lines give the propagation speeds.
mark the location of the row used in the wavelet analysis for periodicity estimation.
Long period oscilla'ng in ac've region loops Krishna Prasad et al A&A 2012 Dr. D. Yuan, KU Leuven False alarm probability in periodogram PN (ω k ) = Py (ω k ) / σ y2 follows exponential distribution.
let Z = max{PN (ω k )}, the probability that Z is above a certain power level
Pr{Z > z} = 1 − [1 − e − z ]M , M is the number of independent frequencies
At a small probality p0 , a random noise generate a power at level z0 ,
z0 = − ln[1 − (1 − p0 )1/M ]
Above level z0 , the power is signficant at a confidence level of 1- p0
p0 =0.01,0.03,0.05, e.g.
Horne & Baliunas 1986, ApJ Dr. D. Yuan, KU Leuven Interac've significance test for mul' peaks Yuan et al 2011 A&A z
at
a
confidence
level
of
1p
=
0.95
0
0
riodograms of 171 Å data shown after iteratively subtracting the highest peak in the spectrum w
Dr. D. Yuan, KU Leuven Fisher’s randomiza'on test Randomly permute two data,
y j = {y0 , y1, y2 ,...yN −1 } ⇒ P
yj
m
yrj = {yr 0 , yr1, yr 2 ,...yrN −1 } ⇒ P
yrj
m
if the time series is better organized
in favor of a dominant peak at frequencey m then,
P >P
yrj
m
yj
m
Repeat M times, and in R cases such scenario occur,
then peak at m is false at a probalbity of p0 = R / M
Linnell Nemec & Nemec 1985 AJ 90, Yuan et al 2011 A&A Starlink-­‐PERIOD package: hdp://starlink.eao.hawaii.edu/starlink Dr. D. Yuan, KU Leuven Summary •
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Fundamentals of sta's'cs Pre-­‐processing methods Fourier transform Windowed Fourier transform Wavelet Periodogram Date-­‐compensate Discrete Fourier transform Filtering method: 'me and spectral domain Significance tests and noise analysis Dr. D. Yuan, KU Leuven