generalization of the cross-wavelet functionwsoon/myownpapers-d/... · time series. colwell et al....
TRANSCRIPT
New Astronomy 56 (2017) 86–93
Contents lists available at ScienceDirect
New Astronomy
journal homepage: www.elsevier.com/locate/newast
Generalization of the cross-wavelet function
V.M. Velasco Herrera
a , W. Soon
b , ∗, G. Velasco Herrera
c , R. Traversi d , K. Horiuchi e
a Instituto de Geofísica, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacán, 04510, México D.F., México b Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA c Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Universidad Nacional Autónoma de México, Ciudad Universitaria, Coyoacán, 04510, México D.F.,
México d Department of Chemistry “Ugo Schif”, University of Florence, Sesto F. no, 50019 Florence, Italy e Graduate School of Science and Technology, Hirosaki University, Japan
h i g h l i g h t s
• A new time-frequency method of analysis, based on the generalization of Einstein’s cross-functions, is introduced. • The method is relevant for n > 2 time series. • We test our algorithm for the study of solar activity variations.
a r t i c l e i n f o
Article history:
Received 28 December 2016
Revised 17 April 2017
Accepted 24 April 2017
Available online 25 April 2017
a b s t r a c t
We introduce the method of multiple cross-wavelet algorithm, hereafter also as Einstein’s cross func-
tions, for the time-frequency study of solar activity records or any astronomical and geophysical time
series in general. The main purpose of this algorithm is to allow the simultaneous examination of the
time-frequency information contents in n > 2 time series available. Previous cross-wavelet algorithm only
permit the study of two time series at a time and was not extended to the generalized n > 2 problems
until now. Furthermore, our new work lifted the restriction from the original formulation that are valid
only for stationary processes. We applied our new algorithm to several of the solar activity proxies avail-
able in order to demonstrate the broad and powerful utility of this technique. We have used solar activity
proxy records that are obtained under different geophysical archives and time periods which are, in turn,
suitable for studying both the statistical and physical properties for solar variations valid on timescales of
multi-century, millennium to several millennia. We focus on documenting the methodology in this paper
rather than any elaborate interpretation of the results.
Published by Elsevier B.V.
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1. Introduction
“Let us suppose that the quantity y ( t ) (for example, the number
of sunspots) [ sic. ] is determined empirically as a function of time
for a very large interval “T ”. How can we characterize the statistical
properties of y ( t ) ?”
In 1914, one Einstein (1914) posted this question in a most
insightful and original note that appeared in Archives de Sciences
Physiques et Naturalles ; a publication of the Swiss Physical Society,
written in the form:
M (�) = 〈 x (t) y (t + �) 〉 (1)
∗ Corresponding author.
E-mail addresses: [email protected] (V.M. Velasco Herrera),
[email protected] (W. Soon).
R
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http://dx.doi.org/10.1016/j.newast.2017.04.012
1384-1076/Published by Elsevier B.V.
here the operations 〈◦〉 indicate smoothing in time and � de-
otes the choice of time interval/delay ( Einstein, 1914 ).
The physicist, Yaglom (1987) , noted that Einstein (1914) had in-
ependently proposed the so-called Wiener–Khintchine theorem
or relating power spectrum and correlation function
(M (�) ).
aglom (1987) stated that “neither Wiener nor Khintchine’s proof
s as physically lucid as the one proposed by Einstein in 1914.” This
s mainly because Enstein’s original derivation and deduction man-
ged to avoid any excessive reliance on probabilistic concepts. It is
ather sure that even Wiener himself would approve of Yaglom’s
igh praise about Einstein’s physical intuition because one can find
iener in 1930, while commenting on harmonic analysis by Lord
ayleigh and Arthur Schuster, remarked that “one is astonished by
he skill with which the authors use clumsy and unsuitable tools
o obtain the right results, and one is led to admire the unfailing
euristic insight of the true physicist” (see p. 204 of Masani, 1986 ;
r p. 127 of Wiener, 1930 ). Also relevant for any students of solar
V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93 87
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ctivity is the fact that A. Einstein at the turn of the 20th cen-
ury could not find steady work and has to earn “extra money by
erforming calculations related to sunspot research, working un-
er [Professor Alfred Wolfer (1854–1931)], who in those years was
irector of the Swiss Astronomical Observatory.” (p. 8 of Yaglom,
987 ).
The cross function ( R xy ( τ )) of two stationary functions x ( t ) and
( t ), is a technique to find the interrelation (or interdependence)
etween two physical phenomena and for measuring the similarity
nd difference of one signal to another with a time delay τ . Such
cross function was originally proposed by Wiener (1930) as:
xy (τ ) =
∫ t 2
t 1
x (t) y ∗(t − τ ) dt (2)
here ( ∗) denotes complex conjugation.
Our research also deal with “Method for the determination of
he statistical values of observations concerning quantities sub-
ect to irregular fluctuations” as originally answered by Einstein
1914) . Here we present a generalization of M (�) given by Ein-
tein using wavelet functional basis for multiple systems and to
pell out a new practical method of time series analysis that is
uitable for the application in studies of time-frequency informa-
ion contents in many solar activity proxy records, extending well
eyond the monthly sunspot records available from 1750-present
see e.g., original studies by Schuster, 1906; Izenman, 1985; Frick
t al., 1997; Lomb, 2013 ). Those indirect solar activity proxies we
tudied in this paper were derived from
14 C in tree-rings and
10 Be
nd nitrate (NO
−3
) concentrations in ice cores for the last 10,0 0 0
ears or more (see e.g., Soon et al., 2014; Horiuchi et al., 2016;
raversi et al., 2016 ). Our previous paper focused on both the sta-
istical and physical relations between the solar activity indices
nd proxy climatic records ( Soon et al., 2014 ) and in this paper
e wish to focus more on the formal derivation and discussion of
he total multiple cross-wavelet algorithm.
In general, we consider both the homogeneous and heteroge-
ous kinds of multiple system. Examples of a homogenous type of
multiple system can be found within various sensors or detec-
ors that can record information from a single parameter, like the
ravitational waves by the Laser Interferometer Gravitational-Wave
bservatory (LIGO) detectors, seismic waves by a network of seis-
ometers or the High Altitude Water Cherenkov Experiment for
tudying the most energetic and origin of cosmic rays. An exam-
le of a heterogenous multiple system is one that involves mea-
urements of several parameters simultaneously like those by land-
ased weather stations or the so-called ARGO floats across the
orld oceans. This is why there are clear needs for developing a
eliable and generic algorithm in order to better decipher physi-
ally relevant information from the multiple systems.
Several previous, perhaps through independent derivations be-
ause those researchers are mostly from different areas of sci-
nces and communities, effort s have constructed and defined the
ross-wavelet functions for two time series ( Hudgins et al., 1993;
esme-Ribes et al., 1995; Hudgins and Huang, 1996; Torrence and
ompo, 1998 ). Both Torrence and Compo (1998) and a later work
y Grinsted et al. (2004) offer two very widely used toolboxes for
alculating cross wavelet transform and wavelet coherence. Before
e describe our multiple algorithm utilizing wavelets as the an-
lyzing function, we should also briefly acknowledge other recent
rogress and innovative application of wavelet transforms for time
eries analysis. Frick et al. (1998) arrived at a most practical al-
orithm for a wavelet analysis of data series with data gaps fre-
uently found in astrophysical and geophysical observational and
easurement programs. Ng and Chan (2012) recently introduced
he method of calculating partial wavelet coherence in order to
etter deduce inter-relationship between two general geophysical
ime series. Colwell et al. (2014) explored the possibility of loop
oles in the so-called Feynman–Hellmann theorem on signal de-
ection and identification from true cross correlations of variables
n very large data sets, for example in the field of biological sci-
nce. Important practical problems in crossing information when
ata have different formats, for example a) historical data, b) digi-
al data, and c) analog data must also be carefully considered.
. Method: Einstein’s cross function
In order to generalize the cross function originally proposed
y Einstein, we used the wavelet transform and will invoke the
adamard product ( �) for u x v matrices. For example if A = (αi j )
nd B = (βi j ) are two u x v matrices then the Hadamard product of
and B is a u x v matrix C = A � B = B � A . Each element of matrix
= (αi j ) , is calculated from element-by-element multiplication of
atrices A and B , that is to say c i j = αi j βi j (see the Appendix for
nother formulation).
For an analysis of x ( t ) and y ( t ) time series ( n = 2 ), the cross-
avelet was used which measures the common power among
hese time series accounting for the degree of synchronization in
hase, frequency and/or amplitude. The generalized multiple cross-
avelet transform algorithm to study inter-relations in multiple ( n
2) time series is described next.
Applying wavelet transform to Einstein’s cross function ( Eq. (1) )
nd � = 0 , we obtained the cross wavelet analysis introduced by
udgins et al. (1993) and defined for two time series x ( t ) and y ( t ),
ith “l ” elements each time series:
x (t) = [ x (t 1 ) , x (t 2 ) , x (t 3 ) , . . . , x (t l )]
(t) = [ y (t 1 ) , y (t 2 ) , y (t 3 ) , . . . , y (t l )]
nd wavelet transforms W x and W y , respectively, in the following
ay:
( M xy ) = < W xy ∗ (t, s ) > [ t,s ] = < W x (t, s ) � W
∗y (t, s ) > [ t,s ]
here 〈◦〉 [ t, s ] indicates for the wavelet spectrum smoothing in
oth time ( t ) and scale ( s ) ( Torrence and Compo, 1998 ).
For the generalization of the Einstein’s cross-function
Einstein, 1914 ), instead of considering the functions x ( t )
nd y ( t ) in Eq. (1) , we consider two matrices X and X
T
ith “n ”-time series (x 1 (t) , x 2 (t) , x 3 (t) , · · · , x n (t) ; x i (t) = x i (t 1 ) , x i (t 2 ) , x i (t 3 ) , . . . , x i (t l )]) ; 1 ≤ i ≤ n ) in each matrix and
ts elements are the time-dependent variables:
=
⎛
⎜ ⎜ ⎜ ⎝
x 1 (t)
x 2 (t)
. . .
x n (t)
⎞
⎟ ⎟ ⎟ ⎠
T = ( x 1 (t) x 2 (t ) · · · x n (t ) )
here superscript T indicates transpose of the matrix.
However, we write the X and X
T matrices using X and X
T , with
he purpose of using the properties of the Hadamard product:
=
n −columns ︷ ︸︸ ︷ (X X · · · X
)=
⎛
⎜ ⎜ ⎜ ⎝
x 1 (t) x 1 (t) · · · x 1 (t)
x 2 (t) x 2 (t) · · · x 2 (t)
. . . . . .
. . . . . .
x n (t) x n (t) · · · x n (t)
⎞
⎟ ⎟ ⎟ ⎠
︸ ︷︷ ︸ n −columns
88 V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93
w
<
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�
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�
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W
X
T =
⎛
⎜ ⎜ ⎜ ⎝
X
T
X
T
. . .
X
T
⎞
⎟ ⎟ ⎟ ⎠
=
⎛
⎜ ⎜ ⎜ ⎝
x 1 (t) x 2 (t) · · · x n (t)
x 1 (t) x 2 (t) · · · x n (t)
. . . . . .
. . . . . .
x 1 (t) x 2 (t) · · · x n (t)
⎞
⎟ ⎟ ⎟ ⎠
⎫ ⎪ ⎪ ⎪ ⎬
⎪ ⎪ ⎪ ⎭
n − rows
X
∗T =
⎛
⎜ ⎜ ⎜ ⎝
x ∗1 (t) x ∗2 (t) · · · x ∗n (t)
x ∗1 (t) x ∗2 (t) · · · x ∗n (t)
. . . . . .
. . . . . .
x ∗1 (t) x ∗2 (t) · · · x ∗n (t)
⎞
⎟ ⎟ ⎟ ⎠
Then the Einstein’s cross function ( M ) using X and X
∗T can be
written as follows:
M =
⟨X � X
∗T ⟩=
⟨ ⎛
⎜ ⎜ ⎜ ⎝
x 1 (t) x 1 (t) · · · x 1 (t)
x 2 (t) x 2 (t) · · · x 2 (t)
. . . . . .
. . . . . .
x n (t) x n (t) · · · x n (t)
⎞
⎟ ⎟ ⎟ ⎠
⟩
�
⟨ ⎛
⎜ ⎜ ⎜ ⎝
x ∗1 (t) x ∗2 (t) · · · x ∗n (t)
x ∗1 (t) x ∗2 (t) · · · x ∗n (t)
. . . . . .
. . . . . .
x ∗1 (t) x ∗2 (t) · · · x ∗n (t)
⎞
⎟ ⎟ ⎟ ⎠
⟩
=
⟨ ⎛
⎜ ⎜ ⎜ ⎝
x 1 (t) x ∗1 (t) x 1 (t) x ∗2 (t) · · · x 1 (t) x ∗n (t)
x 2 (t) x ∗1 (t) x 2 (t) x ∗2 (t) · · · x 2 (t) x ∗n (t)
. . . . . .
. . . . . .
x n (t) x ∗1 (t) x n (t) x ∗2 (t) · · · x n (t) x ∗n (t)
⎞
⎟ ⎟ ⎟ ⎠
⟩
=
⟨ ⎛
⎜ ⎜ ⎜ ⎝
c 11 c 12 · · · c 1 n
c 21 c 22 · · · c 2 n
. . . . . .
. . . . . .
c n 1 c n 2 · · · c nn
⎞
⎟ ⎟ ⎟ ⎠
⟩
=
⎛
⎜ ⎜ ⎜ ⎝
< c 11 > < c 12 > · · · < c 1 n >
< c 21 > < c 22 > · · · < c 2 n >
. . . . . .
. . . . . .
< c n 1 > < c n 2 > · · · < c nn >
⎞
⎟ ⎟ ⎟ ⎠
where < c i j > = < x i x ∗j > = < x i (t) x ∗
j (t) > is an Einstein’s cross func-
tion ( Eq. (1) ). By applying the wavelet transform ( W ) to X and X
∗T ,
we obtained the multiple cross-wavelet spectrum ( �):
� =
⟨W [ X ] � W [ X
∗T ] ⟩[ t,s ]
=
⟨ ⎛
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
W [ x 1 (t)] W [ x 1 (t)] · · · W [ x 1 (t)]
W [ x 2 (t)] W [ x 2 (t)] · · · W [ x 2 (t)]
. . . . . .
. . . . . .
W [ x n (t)] W [ x n (t)] · · · W [ x n (t)]
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⟩
[ t,s ]
�
⟨ ⎛
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
W [ x ∗1 (t)] W [ x ∗2 (t)] · · · W [ x ∗n (t)]
W [ x ∗1 (t)] W [ x ∗2 (t)] · · · W [ x ∗n (t)]
. . . . . .
. . . . . .
W [ x ∗1 (t)] W [ x ∗2 (t)] · · · W [ x ∗n (t)]
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⟩
[ t,s ]
=
⟨ ⎛
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
W 11 W 12 · · · W 1 n
W 21 W 22 · · · W 2 n
. . . . . .
. . . . . .
W n 1 W n 2 · · · W nn
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
⟩
[ t,s ]
=
⎛
⎜ ⎜ ⎜ ⎜ ⎜ ⎝
< W 11 > [ t,s ] < W 12 > [ t,s ] · · · < W 1 n > [ t,s ]
< W 21 > [ t,s ] < W 22 > [ t,s ] · · · < W 2 n > [ t,s ]
. . . . . .
. . . . . .
< W n 1 > [ t,s ] < W n 2 > [ t,s ] · · · < W nn > [ t,s ]
⎞
⎟ ⎟ ⎟ ⎟ ⎟ ⎠
here each element of � is a cross wavelet:
W i j > [ t,s ] = < W x i x ∗j (t, s ) > [ t,s ] = < W x i (t, s ) � W
∗x j (t, s ) > [ t,s ]
Using the properties of Hadamard product, � can be written
s:
= �total � �partial
ith
total =
⟨ ⎛
⎜ ⎜ ⎜ ⎝
W 11 1 · · · 1
1 W 22 · · · 1
. . . . . .
. . . . . .
1 1 · · · W nn
⎞
⎟ ⎟ ⎟ ⎠
⟩
[ t,s ]
=
⎛
⎜ ⎜ ⎜ ⎝
< W 11 > [ t,s ] 1 · · · 1
1 < W 22 > [ t,s ] · · · 1
. . . . . .
. . . . . .
1 1 · · · < W nn > [ t,s ]
⎞
⎟ ⎟ ⎟ ⎠
partial =
⟨ ⎛
⎜ ⎜ ⎜ ⎝
1 W 12 · · · W 1 n
W 21 1 · · · W 2 n
. . . . . .
. . . . . .
W n 1 W n 2 · · · 1
⎞
⎟ ⎟ ⎟ ⎠
⟩
[ t,s ]
=
⎛
⎜ ⎜ ⎜ ⎝
1 < W 12 > [ t,s ] · · · < W 1 n > [ t,s ]
< W 21 > [ t,s ] 1 · · · < W 2 n > [ t,s ]
. . . . . .
. . . . . .
< W n 1 > [ t,s ] < W n 2 > [ t,s ] · · · 1
⎞
⎟ ⎟ ⎟ ⎠
here �total is the total cross-wavelet spectrum and �partial is the
artial cross-wavelet spectrum. We note that �partial were created
n order to eliminate the influence of one variable or more on a
et of other variables. The implications of these partial cross func-
ions and algorithms will be discussed elsewhere because these al-
orithms were not fully invoked in this paper.
The squared multiple total cross-wavelet ( ��) spectrum is de-
ned as the product ( Track ) of the diagonal elements in �total and
s given by the formula:
� = ��(t, s ) = Track
(�total
)=
i = n ∏
i =1
�total ii
= < W 11 � W 22 � . . . � W nn > [ t,s ] (3)
The multiple cross-wavelet spectrum can be written as the am-
litude/magnitude ( W(t, s ) ) and multiple cross-wavelet phase dif-
erence ( φ( t, s )) as:
(t, s ) =
(Re {(��(t, s )) 1 / 2
}2 + Im
{( ��(t, s )) 1 / 2
}2 )1 / 2
(4)
V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93 89
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(t, s ) = tan
−1
(Im
{ 〈 (��(t,s )) 1 / 2 〉 [ t,s ]
} Re
{ 〈 (��(t,s )) 1 / 2 〉 [ t,s ]
} )
(5)
For the sake of historical completeness in our formulation of
ultiple cross wavelet calculations, we also listed here an ear-
ier published version of our algorithm. Soon et al. (2014) have
roposed the following equation to calculate the multiple cross-
avelet ( W ):
(t, s ) =
n ∏
j=1 j � = i
< W x i � W
∗x j
> [ t,s ]
= < W x i �
(W x 1 � W x 2 � W x 3 · · · � W x j . . .
� · · · � W x n −1 � W x n
)∗> [ t,s ]
The multiple cross-wavelet phase difference is then calculated
s follows:
(t, s ) = tan
−1 (
Im { 〈 W (t,s ) 〉 [ t,s ] } Re { 〈 W (t,s )) 〉 [ t,s ] }
)�� is the wave function of a physical system in time-frequency
pace and | ��( t, s )| 2 is the spectral power of the physical system.
n this work, we introduce the global multiple cross wavelet spec-
rum and the global multiple cross wavelet phase difference re-
pectively, as:
�global (s ) =
∑
t
��(t, s ) (6)
global (s ) =
∑
t
φ(t, s ) (7)
Finally, the Einstein’s multiple cross-functions ( M ) are calcu-
ated in the following manner:
= W
−1 [ ��] (8)
here W
−1 is the inverse wavelet transform.
Applying wavelet transform to Eq. (8) , we obtain the squared
ultiple total cross wavelet:
� = W [ M ] (9)
In summary, the generalization of Einstein’s cross function is
reated as an optimization issue but in the time-frequency space.
ere the purpose of the optimization and transformation is to find
ocal symmetries for each of the periodicities that are in common
etween the “n ”-time series (signals). We wish to remind that our
eneralization of the Einstein’s cross function ( Eq. (1) ) was ob-
ained under the condition that � = 0 , i.e., with no time delay.
owever, it is also possible to re-write Eq. (3) , taking into account
ach of the delay times of the “n ”-time series (signals).
�(�) = Track
(�total (�)
)=
i = n ∏
i =1
�total ii (�)
= < W 11 (�1 ) � . . . � W nn (�n ) > [ t,s ]
here W ii (�i ) = W ii (t, s ) � W
∗ii (t − �i , s ) . The average time delay
�s ) between “n ”-time series (signals) for each wavelet scale ( s ),
an be obtained as follows:
s =
⟨φ(t, s ) T s
2 π
⟩here T s is the periodicity and φt ( s ) is the phase difference.
Finally, for a broad application to most multiple systems, we
reate two algorithms: a) An algorithm is required which in at
east in one channel the signal is registered, then Trace is used
sum) or b) in which an algorithm is required to know if all chan-
els the same signal is registered, so Track (multiplication) is used.
Thus, we have:
+ = Trace (�total
)=
i = n ∑
i =1
�total ii = < W 11 + W 22 + · · · + W nn > [ t,s ]
(10)
� = Track
(�total
)=
i = n ∏
i =1
�total ii = < W 11 � W 22 � . . . � W nn > [ t,s ]
(11)
. Choice of data series
For this paper, we have chosen to illustrate our data analyses
nd testing of our algorithm using both artificially generated time
eries and real-world data records. For the real-world time series,
e have chosen to focus on various known and indirect proxies
f solar magnetic variations that are obtained from Earth paleocli-
atic archives and this fact has been discussed and reviewed in
etails in our previous papers ( Traversi et al., 2012; Soon et al.,
014; Horiuchi et al., 2016 ).
Our core solar activity proxy data are based on
14 C in tree rings,
oth
10 Be and nitrate contents in ice core records covering the
olocene period of past 10,0 0 0 years. In addition for this paper, we
dded the latest high-resolution records of cosmogenic 10 Be from
arine sediment cores (500-yr resolution) and an ice core (100-yr
esolution) recently published by Horiuchi et al. (2016) covering an
nterval of 200 to 170 thousand years ago across the Iceland Basin
eomagnetic excursion (ca. 190 thousand years ago).
The sedimentary 10 Be records were obtained from the western
quatorial Pacific (West Caroline Basin) by analyzing the authigenic0 Be/ 9 Be ratio in sediments. The ice-core 10 Be record was obtained
rom the eastern inland Antarctica (Dome Fuji station: 39 o 42 ′ E,
7 o 19 ′ S) by analyzing the concentration and flux of 10 Be in ice
amples. During a large minimum of geomagnetic intensity, such
s the Iceland Basin excursion, solar modulation in
10 Be production
ay be relatively more enhanced than during other periods. In-
eed, several multi-centennial to bi-millennial periodicities of pos-
ible solar origin were detected in this interval, the detailed nature
f which was discussed in Horiuchi et al. (2016) .
As concerning the nitrate series used here, we chose nitrate
ecord from TALDICE ice core (East Antarctica, 159 o 11 ′ E, 72 o 49 ′ , 2315 m a.s.l.) spanning the last 11,400 years before present (BP).
he TALDICE (TALos Dome Ice CorE) ice core was processed during
everal sessions from 2006 to 2008 at the Alfred Wegener Institute
n Bremerhaven (Germany) and analyzed in different European lab-
ratories.
Different resolutions were chosen for discrete sample collection.
ere we use the “bag mean” data, obtained at the University of
lorence, by melting a whole 1-m long strip (bag mean) and col-
ecting it in a single vial. The ice samples covering the Holocene
645 − 11400 years BP, 73 − 660 m depth) were dated using the
ALDICE official timescale (TALDICE-1, Buiron et al., 2011 ). The bag
eans correspond to about 12 years at present time and about 30
ears at the beginning of Holocene. In the considered temporal pe-
iod, the dating uncertainty of the TALDICE core fluctuates between
bout 200 and 450 years showing values higher than 300 year in
ost of the Holocene.
90 V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93
Fig. 1. Multiple cross wavelet analysis for the 3 idealized time series: F 1 (t) = 2 cos ( 2 πt/ 11 ) + cos ( 2 πt/ 100 ) + cos ( 2 πt/ 120 ) ; F 2 (t) = 3 cos ( 2 πt/ 11 ) + cos ( 2 πt/ 200 ) +
cos ( 2 πt/ 120 ) + cos ( 2 πt/ 60 ) and F 3 (t) = 3 cos ( 2 πt/ 11 ) + cos ( 2 πt/ 30 ) + cos ( 2 πt/ 120 ) + cos ( 2 πt/ 5 ) (black, blue and red curves, respectively, in the top-most panel). In
the center panel for the multiple Einstein’s cross wavelet result, the orientation of the arrows shows relative phasing of the time series at each timescale; arrows at 0 °(pointing to the right, → ) indicate that all time series are perfectly positively correlated (in phase) and arrows at 180 ° (pointing to the left, ← ) indicate that they are
perfectly negatively correlated (180 ° out of phase), both of these two perfect cases implying a linear relationship between the considered phenomena; non-horizontal arrows
( ↗ , ↘ , ↙ , ↖ , ↑ , ↓ ) indicate an out of phase situation and a more complex non-linear relationship. Two common periodicities are successfully detected at 11 and 120 time
units. Unique but uncommon periods in individual time series did not showed up in the cross wavelet spectrum and thus proving the successful working of the algorithm.
In this figure, top-most panel shows the original time series, left-most panel and right-most panel show the global multiple cross wavelet spectrum Eq. (6) ) and global
multiple cross wavelet phase difference ( Eq. (7) ) for each timescale while the bottom panel shows the instantaneous evolution of the multiple Einstein’s cross function (blue
curve, Eq. (8) ) and multiple cross wavelet phase difference (black curve, Eq. (5) ) for the selected timescale or oscillation at time unit = 11. Red dashed line in the left-most
panel represents the significance level in referenced to the power of red noise level at the 95% confidence interval. (For interpretation of the references to color in this figure
legend, the reader is referred to the web version of this article.)
a
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4. Results and discussion
We start by studying the three idealized time series shown in
Fig. 1 .
F 1 (t) = 2 cos ( 2 πt/ 11 ) + cos ( 2 πt/ 100 ) + cos ( 2 πt/ 120 ) ;
F 2 (t) = 3 cos ( 2 πt/ 11 ) + cos ( 2 πt/ 200 ) + cos ( 2 πt/ 120 ) +
cos ( 2 πt/ 60 ) ;
F 3 (t) = 3 cos ( 2 πt/ 11 ) + cos ( 2 πt/ 30 ) + cos ( 2 πt/ 120 ) +
cos ( 2 πt/ 5 ) .
In examining the inter-relationship among these three time se-
ries, the matrix �total can be written explicitly as follows:
�total =
⟨ ( W 11 1 1
1 W 22 1
1 1 W 33
) ⟩ [ t,s ]
=
⎛
⎝
< W 11 > [ t,s ] 1 1
1 < W 22 > [ t,s ] 1
1 1 < W 33 > [ t,s ]
⎞
⎠
The squared multiple total cross-wavelet ( ��) spectrum for
three time series ( F 1 ( t ), F 2 ( t ), F 3 ( t )) is given by the formula:
�� = Track
(�total
)=
i =3 ∏
i =1
�total ii = < W 11 � W 22 � W 33 > [ t,s ]
(12)
M = W
−1 [ ��] = W
−1 [ < W 11 � W 22 � W 33 > [ t,s ] ] (13)
The result in Fig. 1 shows the correctness of our formulation
nd derivation of the generalized Einstein’s cross function and its
ultiple cross spectrum for example summarized in Eqs. (8) and
9) , respectively, adopting three idealized time series ( F 1 ( t ), F 2 ( t ),
3 ( t )). From Fig. 1 , one can find that we can essentially recover the
wo common periodicities, at 11 and 120 time units, seeded in the
hree artificial records with the straightforward application of our
ultiple cross-wavelet algorithm. This fact adds confidence in ap-
lying our new algorithm to study realistic and more complex so-
ar and geophysical phenomena as we recently discussed in Soon
t al. (2014) .
Fig. 2 gives the first illustrative application of our multiple cross
avelet algorithm. In this analysis, we utilize 4 time series that
epresent the indirect proxies of solar activity variations covering
early the full Holocene time interval of past 10,0 0 0 years. The pri-
ary purpose of our analysis is to show not only the practical util-
ty of our new algorithm but also to demonstrate the power of the
echnique in gaining more physical insights for understanding how
he Sun’s magnetism over a broad range of physical timescales. The
lobal spectrum (left most panel in Fig. 2 ) give the evidence of the
rominence of the 20 0 0-yr, 940-yr, 340-yr, 240-yr and 120-yr os-
illations common in all four solar activity proxy records.
The bi-millennial, millennial, bi-centennial and centennial scale
ignals are previously well-known quasi-regular periods of solar
ctivity named by solar physics community as the Hallstatt, Eddy,
uess-de Vries and Gleissberg-Yoshimura cycles respectively, while
he 340-yr variation is a new feature based on the current analy-
V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93 91
Fig. 2. Multiple cross wavelet analysis of four solar activity proxy time series (center panel): Nitrate concentration ( Traversi et al., 2012 ; green series in top panel), Solar
Modulation Parameter, �, derived from
10 Be of the Greenland Ice Core Project ( Vonmoos et al., 2006 ; black series in top panel) and from
14 C production rate ( Roth and Joos,
2013 ; blue series in top panel) and the composite Solar Modulation Parameter, �, based on 10 Be and 14 C records ( Steinhilber et al., 2012 ; red series in top panel). Please
consult Soon et al. (2014) for additional details of the data records; this re-analysis revises and improves from the original Fig. 3 of that paper. The meaning of each panel
is exactly the same as first described for Fig. 1 . The bottom panel specifically shows the instantaneous evolution of the multiple Einstein’s cross function amplitude (blue
curve, Eq. (8) ) and multiple cross wavelet phase difference (black curve, Eq. (5) ) for the selected timescale or oscillation at 20 0 0 years; the result suggests that the four solar
activity proxies shared largest common power during the early to mid-Holocene interval roughly from 90 0 0 to 30 0 0 year before present and that their phase relationship at
this bimillennial scale is roughly linear. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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is. As noted in the introduction, Soon et al. (2014) has discussed
he details of the physical processes involved in the interpretations
f some of the noted oscillations while combining the solar activ-
ty and climatic records of the Holocene across the world. This is
hy physical mechanism and interpretation will not be a focus of
he current paper. But one can be re-assured of the role and im-
ortance of the multiple algorithm is simply because if one just
roduce the cross-wavelet using only two time series (say, using
nly times series from
10 Be and
14 C solar activity proxies), one
ould not have found the coherent signals from 340-yr and 10 0 0-
r timescales. In this sense, it is physically relevant to point out
hat the 340-yr signal has also been noted in another newly re-
onstructed
10 Be record by Adolphi et al. (2014) that covers the
lacial period from 22,500 to 10,0 0 0 years ago (see their Figure S4
s well as our own unpublished wavelet analysis of this new
10 Be
ecord).
The global phase spectrum (right most panel in Fig. 2 ) offers an
verall views on the relative phasing of various time-scales of vari-
bility. Over most of the timescales from 60 to 800 yrs, we find
hat most signals are relatively coherent with near-in-phase rela-
ionships among 4 records. For the prominent 90 0-10 0 0-yr Eddy
cale, one find that the phase relationship among these 4 solar
roxies are more complex and yielding a negative inter-phase rela-
ion. Finally, the bottom panel illustrates the time variation for the
i-millennial Hallstatt scale, showing the clear dominance of the
i-millennial scale variation during the early to middle Holocene
ith the modulating signal almost disappearing by 40 0 0 yr BP.
he time-frequency summary plot in the center or middle panel of
ig. 2 also shows that the bicentennial and centennial scales are in-
i
tead more prominent post mid-Holocene and the modulating sig-
als persist almost until the present day.
Fig. 3 serves as another independent test of our multiple al-
orithm in detecting common scale and oscillation in three real-
orld records of cosmogenic 10 Be as both proxies of solar-cosmic
ay activity as well as geomagnetic variations. In the top part of
ig. 3 , we show the cross wavelet spectrum result from the three
ime series records from Horiuchi et al. (2016) while the bottom of
art of Fig. 3 shows the direct and simply wavelet spectrum from
he 3-series stacked record. The close agreement of the two results
s very encouraging in the sense that it shows how well our multi-
le cross wavelet algorithm can confidently recover the result from
he post-processed and 3 time series-stacked
10 Be record.
The physical interpretation and information from our cross
avelet results will be of wide interest and possible importance
or the solar and cosmic ray and Earth science communities. It is
lear that the result provides a very clear detection of the 1500-
800-yr-like signals that have been considered as a solar activity
illennial and bi-millennial scale modulation of the 10 Be records
nd has been studied intensively e.g., by Soon et al. (2014) and
oriuchi et al. (2016) recently.
In conclusion, our goal of documenting and testing our new
eneralization of cross-function adopting the original work by Ein-
tein, here termed as Einstein’s cross function, using the wavelet
asis function has been accomplished. We welcome a broad and
ide-ranging application of our newly derived and tested multiple
ross wavelet algorithm to study any solar, astronomical and geo-
hysical records towards the goal of not only being able to confirm
r deny any physical signals but also to make full use of the phase
nformation to better ascertain and clarify any physical processes
92 V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93
Fig. 3. (a) Top Panel: Multiple cross wavelet analysis of (1) and (2) two 10 Be/ 9 Be (black and blue series in top panel) ratio records from the ocean sediment cores of the West
Caroline Basin (Western equatorial Pacific) and (3) 10 Be flux (red series in top panel) from Dome Fuji ice core (East Antarctica) over the geological and paleoclimatic epoch
of Iceland Basin excursion of 200–170 kyr ago reported and studied by Horiuchi et al. (2016) . The meaning of the five individual panels is exactly the same as described
for Fig. 1 . The bottom-most panel in (a) specifically shows the instantaneous evolution of the cross-function amplitude (blue curve) and phase (black curve) for the selected
timescale or oscillation at 60 0 0 years. (b) Bottom Panel: The direct wavelet analysis of the 3-records stacked 10 Be time series of 500-yr resolution for a comparison with
the multiple cross wavelet result shown in the top panel (a). The fact that multiple cross wavelet result closely represented the direct wavelet analysis of the stacked series,
although with added information on the inter-relationship among the three individual time series shown in the top panel, provided further confidence in the multiple cross
wavelet algorithm introduced, for the first time, in this paper. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of
this article.)
[
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more data-rich world.
and mechanisms in all the real-world phenomena. The first appli-
cation of our algorithm has been performed and reported in Soon
et al. (2014) and we look forward to further collaborative works. It
is perhaps relevant to be reminded by Yaglom (1987 , p. 10) com-
ment that around 1914, “no applications could have existed for
Einstein’s] concept” about the link between power spectral inten-
ity and cross-correlation functions, the time is indeed now for us
o apply Einstein’s cross wavelet algorithm to gain deeper physi-
al insights on solar and geophysical phenomena in our relatively
V.M. Velasco Herrera et al. / New Astronomy 56 (2017) 86–93 93
A
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WY
cknowledgments
We thank two anonymous referees for the critical comments
hat have led to the simplification and improvement to our
anuscript. We are open to share and/or collaborate with every-
ne interested in applying the generalized algorithm presented in
his paper.
V.M.V.H.’s works are supported by CONACYT-167750 grant. KH’s
ork was partially supported by the Grant-in-Aid for Scientific Re-
earch (A) (No. 25247082) from the Japan Society for the Promo-
ion of Science.
W.S.’s works were partially supported by two past and
ne current SAO grants (proposals ID: 0 0 0 0 0 0 0 0 0 0 0 01061-V101,
0 0 0 0 0 0 0 0 0 0 01062-V101, and 0 0 0 0 0 0 0 0 0 0 03010-V101, respec-
ively). W.S. wishes to dedicate his part of the work to the memory
f three dear friends: Professor Robert M. Carter, Professor William
. Gray and Mr. Allan Ariffin Tan.
ppendix
A u x v matrix C = A � B = B � A can be written in tensor form
s follows:
i j = { A u v • B pq } i j = δu v i δpq
j A up B v q (14)
here • is also the Hadamard product, but in tensorial form we
ill call it symmetric product and δ jk i
is the Kronecker delta with
hree indices and defined as follows
jk i
=
{1 if i = j = k
0 if i � = j � = k
eferences
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