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: vmv@geofisica.unam.mx (V.M. Velasco Herrera),

wsoon@cfa.harvard.edu (W. Soon).

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o

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

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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

φ

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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

m

(

F

t

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p

l

e

w

r

n

m

i

t

t

g

p

c

s

a

S

t

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.)

s

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p

p

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y

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c

g

a

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b

b

w

T

F

s

n

g

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r

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t

p

t

i

p

t

w

f

c

1

m

a

H

g

s

b

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c

p

o

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.)

[

s

t

c

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

t

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w

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0

t

o

M

A

a

C

w

w

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δ

R

A

B

C

EF

FG

H

HH

IL

MN

N

R

SS

ST

T

T

V

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

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matsii)