Extracting the multi-timescale activity patterns of
online financial markets
Teruyoshi Kobayashi
Anna Sapienza
Emilio Ferrara
February 2018
Discussion Paper No.1809

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANExtracting the multi-timescale activity patterns of online financial
markets
Teruyoshi Kobayashi,1 Anna Sapienza,2 Emilio Ferrara2,∗
1

2

Department of Economics, Kobe University, Kobe, Japan
University of Southern California, Information Sciences Institute, Los Angeles (CA), USA∗
Corresponding author: emiliofe@usc.edu
February 21, 2018

Abstract
Online financial markets can be represented as complex systems where trading dynamics can be captured and characterized at different resolutions and time scales. In this work, we develop a methodology
based on non-negative tensor factorization (NTF) aimed at extracting and revealing the multi-timescaletrading dynamics governing online financial systems. We demonstrate the advantage of our strategy firstusing synthetic data, and then on real-world data capturing all interbank transactions (over a million)
occurred in an Italian online financial market (e-MID) between 2001 and 2015. Our results demonstrate
how NTF can uncover hidden activity patterns that characterize groups of banks exhibiting different
trading strategies (normal vs. early vs. flash trading, etc.). We further illustrate how our methodology
can reveal “crisis modalities” in trading triggered by endogenous and exogenous system shocks: as an
example, we reveal and characterize trading anomalies in the midst of the 2008 financial crisis.

Introduction
Social activity of individuals follows certain rhythms at different time scales. Many of the individual
activities, such as sending e-mails and making phone calls, are likely to be done in particular timeintervals within a day (i.e., diurnal or circadian cycles), and the total daily activity could heavily dependon the day of the week (i.e., weekly cycles) [1–6]. In general, these social activity rhythms emerging atdifferent time scales may be correlated with each other; for instance, it has been shown that face-to-facecontacts between classmates in a primary school follow a common diurnal cycle driven by the daily classschedule [7, 8], yet at the same time they would also share activity rhythms at longer scales such as
weekly and monthly, reflecting the annual school schedule.
As social communications between humans form temporal social networks, financial transactions
between banks also shape time-varying networks [9–13]. In the interbank market, for instance, overnightbilateral lending and borrowing between banks organize temporal networks whose structure changes ona daily basis, because the overnight financial contracts last for only one day [11, 12]. Thus, financial
markets, similarly to social networks, can be interpreted as complex systems where each agent’s activity
should be captured and characterized at appropriate time scales [11, 12]. In fact, many commonalities
have recently been found between social communication patterns of humans and financial interaction
patters of banks at particular temporal resolutions [12, 14, 15].

1

In recent years, non-negative tensor factorization (NTF) has been frequently used to extract temporal
activity patterns in various social contexts, such as face-to-face contacts [8, 16], Twitter posts [17] andplayers’ matches in online games [18]. Among these studies, Gauvin et al. [8] showed that NTF is highly
effective in detecting diurnal rhythms of students’ activity in a primary school [7, 19]. Characterizing
diurnal rhythms lead the authors to uncover the multi-timescale community structure formed by classmates: students’ temporal activity cycles, rather than the aggregated history of contacts, allows to reveal
meaningful patterns that explain the complexity of children contact networks and communities.
Given the similarity between social and financial temporal dynamics, justified by underlying humanfactors, and the fact that multiple activity cycles are present at different frequencies in human social
activities, our question is whether similar rhythms exist also in financial systems. In this work, we
uncover hidden multi-timescale patterns of banks’ activities in the Italian online interbank market, eMID [20]. In previous studies, it has been recognized that there exist activity patterns in banks’ financialtransactions at a particular time scale, such as inter-day [12] and intra-day scales [21–23]. However, it
is still unknown whether these patterns coexisting at different time scales are dependent on each other,in which case banks exhibiting a given interday activity pattern are likely to follow a particular intraday
pattern.
We employ NTF as a tool to detect multi-timescale patterns, in which banks’ activities are captured
by a tensor having three dimensions: the list of banks, time of the day, and date. We will show that
the PARAFAC decomposition of the 3-way tensor [24, 25] indicates that banks’ trading activities can
be classified into several patterns over the data period of 2001–2015. In particular, the NTF allows us
to identify an anomalous pattern around the Lehman Brothers’ collapse in September 2008. Our multitimescale NTF approach reveals not only how banks facing the financial crisis changed their diurnal
trading rhythms, but also on what dates such anomalies emerged. Our framework to extract multitimescale patterns advances the understanding of when and how banks react to systemic shocks, and in
the future it could contribute to controlling the spread of financial systemic risk [9, 10, 26–28].

Data
We use the time-stamped data on bilateral interbank transactions occurred between January 2001 andDecember 2015 in the Italian online interbank market (e-MID). e-MID is an online platform for financial
institutions provided by the Italian company e-MID SIM S.p.A. based in Milan, Italy (henceforth, wecall financial institutions as “banks” for brevity). Banks in need of lending or borrowing funds may finda counterpart by posting an order on the online platform, which makes e-MID a marketplace in which
lenders and borrowers are matched. Each transaction record in this data represents when a loan contract
agreement has been reached between two banks and how much funds have been lent (in million Euros).
In addition, the data contains information about the type of each loan agreement, i.e., a lender-proposedtransaction or a borrower-proposed transaction. We define a lender-proposed (resp., borrower-proposed)transaction to be a transaction proposed by a bank that lends to (resp., borrows from) its counterpart.
As in the other interbank markets, the vast majority of transactions (> 86%) in the e-MID market
are overnight lending and borrowing (i.e., a loan contract lasts just for one day), while there are othertypes of transactions having longer maturity lengths such as two weeks, three month, etc. To eliminate
the influence of differences in maturity length, we focus on overnight transactions of unsecured Euro
deposits labeled as “ON” (i.e., overnight) or “ONL” (i.e., overnight large, namely overnight transactions
greater than 100 million Euros). Table 1 summarizes the basic statistics of this dataset.
There are 3,839 business days over the data period between Jan 2, 2001 and Dec 31, 2015, on eachof which transactions are made between 8:00 and 18:00. There are 289 banks that conducted at least
one transaction over this period: 194 of them are Italian banks while the rest are foreign banks. The
total number of transactions (ON and ONL) that we use in the analysis amounts to 1,148,699. It

2

(b) 800
600

100# trades

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01

# active banks(a) 150

Figure 1: Daily activities in the e-MID market. 20-day moving average is shown. (a) Number of active banks having
at least one transaction on a day. (b) Number of trades.
Table 1: Summary statistics of the Italian interbank market, e-MID.

Data period
No. days
Trading time
Maturity length
No. banks (Italian banks)
No. total transactions
Average no. daily participants
Average no. daily transactions

Jan 2, 2001 – Dec 31, 2015
3,839
8:00 – 18:00
overnight (ON and ONL)
289 (194)
1,148,699
95.10
299.22should be noted that the 289 banks are not always active over the data period (cf., Fig. 1a). In fact,
the daily average of the number of participating banks is around 95, and the average number of dailytransactions is approximately 300 (cf., Fig. 1b). The dataset is commercially available from e-MID SIM
S.p.A (http://www.e-mid.it/).

MethodAs noted in the previous section, banks are not always active in the interbank market. At the daily scale,some banks have transactions only on a certain fraction of business days while having no transactions
on the other days [12, 21]. If we look at intraday time scales, on the other hand, the frequency of a bankhaving at least one transaction is not uniformly distributed over the time intervals [21, 22].
Since the maturity of loan contracts is overnight, it is important to identify what are the days in whichbanks are more likely to be active. Conversely, since the participation of banks in the trading activity
can change over the course of a day, it is also crucial to understand intraday activity changes. Due to themulti-scale nature of bank trading activity, directly studying the number of transactions of each bankover time would not allow us to detect inter- and intra-day dynamics. Thus, we need to disentangle
each bank’s activity across the two dimensions (i.e., time scales): interday and intraday. To this aim,
we represent our data in a three-dimensional array (i.e., tensor), whose entries represent the amount oftrades each bank performs at a given time of a given day. Given this multi-dimensional representation,
our main goal is to extract some meaningful correlated multi-timescale activity patterns related to groupsof banks sharing a similar number of transactions at the same time (respectively, intraday and interday).
This can be achieved by means of non-negative tensor factorization (NTF), as described in the following.

3

Non-negative tensor factorization (NTF) at different time scales
Let us consider a three-dimensional tensor X ∈ RI×J×K , where I = N is the number of banks, J = T is
the number of time intervals of bank activity in a day (here between 8:00 and 18:00), and K = D is thenumber of days. The entry xijk ∈ Z+ of the tensor X denotes the total amounts of trades (in million
Euros) conducted by the i-th bank during j-th time interval of the k-th day. To extract the intra- andinter-day trading patterns characterizing banks with similar number of transactions over time, we rely on
the NTF, which decomposes the tensor X into the sum of several rank-one tensors, namely components.
Here, we implement the NTF by applying the well-established method called PARAFAC (parallelfactor analysis) or CANDECOMP (canonical decomposition) with non-negativity constraints [24, 25, 29].
The decomposition can be written as
X ≈

R
X

ar ◦ br ◦ cr ,(1)

air bjr ckr ,

(2)

r=1

xijk ≈

RX
r=1

where R ∈ N denotes the number of components (i.e., the rank of the tensor), and the operator ◦represents outer product. Eq. (1) is the so-called canonical polyadic decomposition (CP) of a tensor X ,T
D
where ar ∈ RN
+ , br ∈ R+ and cr ∈ R+ represent r-th component factors that respectively encode themembership of a bank to the component, the intervals in a day and the days in which the component is
active (i.e. br and cr are the intra- and inter-day activity patterns of the groups of banks in ar ).
×R
Let A ∈ RN
, B ∈ RT+×R and C ∈ RD×R
denote the factor matrices, whose r-th columns are++
the vectors ar , br and cr , respectively. The factor matrices A, B and C are calculated by solving thefollowing minimization problem with non-negativity constraints:minkX − JA, B, CKk2F ,A≥0,B≥0,C≥0(3)

where JA, B, CK represents the Kruscal form of the tensor decomposition (i.e., the RHS of Eq. (1)), andk·kF denotes the Frobenius norm.
This minimization problem can be rewritten to perform the minimization with respect to one factor
matrix at a time. Let X(n) denote a matrix created by the mode-n matricization (or flattening) of tensor
X , where each row of X(n) consists of a vector corresponding to a given index of mode-n. That is,X(1) , X(2) and X(3) are N × T D, T × N D and D × N T matrices, respectively. They can be expressedin terms of factor matrices as follows:
X(1) = A(C

B)> ,

(4)

X(2) = B(C

>

A) ,

(5)

X(3) = C(B

>

(6)

A) ,

where denotes the Khatri–Rao product, which is the “matching columnwise” Kronecker product definedas A B = [a1 ⊗ b1 , . . . , aR ⊗ bR ]. The minimization problem for PARAFAC decomposition Eq. (3)can be now reformulated as


> 2
> 2
> 2
minkX(1) − A(C B) kF , minkX(2) − B(C A) kF , minkX(3) − C(B A) kF , . (7)
A≥0

B≥0

C≥0

The minimizers of A, B and C of this problem are computed using the non-negative alternate leastsquares method (ANLS) combined with the Block Principal Pivoting method (BPP) developed by [30].
Our implementation is based on the Tensor Toolbox [31, 32] and the MATLAB codes available from [33].

4

Rank size
To determine the number of components R (i.e., rank size) used in the NTF model, we rely on the Core
Consistency Diagnostic [24]. Given the tensor element xijk , we can rewrite Eq. (2) as
xijk =

R X
R X
R
X

λnmp ain bjm ckp ,(8)

n=1 m=1 p=1

where λnmp denotes the (n, m, p) element of the superdiagonal binary tensor L (i.e., λnmp = δnm δmp δnp ).
Now let gijk denote the (i, j, k) element of the core tensor G, which is obtained by fitting the data to the
Tucker3 model [24]. In the Tucker3 model, the minimization problem reads
minkX − AG(C ⊗ B)> k2F ,G(9)where X is a N × T D matrix converted from tensor X , and G is also a matricized version (Rn × Rm Rp )of the core tensor G. With the Tucker3 decomposition, the tensor is written as
xijk =

Rp
Rn X
Rm X
X
n

m

gnmp ain bjm ckp .

(10)

p

From the lemma of Bro and Kiers [24], the core tensor G will be identical to the superdiagonal tensor L if
the Tucker3 model is perfectly fitted and the factors have full column rank. If G is significantly different
from the superdiagonal tensor L, by contrast, it means that there are non-negligible interactions between
factors and the PARAFAC model is not appropriate. Thus, we could assess whether the data should be
fitted to the PARAFAC model or the Tucker3 model by measuring the distance between G and L.
We employ the Core Consistency (CC) value proposed by Bro and Kier [24] as a measure of the
distance between G and L:!
PR PR PR2
n=1
m=1
p=1 (gnmp − λnmp )CC = 100 × 1 −,(11)R
where we imposed the constraint Rn = Rm = Rp = R in implementing the Tucker3 decomposition. CC
takes 100 if the PARAFAC model perfectly fits the data and less than 100 (possibly a negative value)
if the model does not fit perfectly. Note that a rise in R will reduce fitting errors while increasing thepossibility of over-fitting. In general, as R increases, there arises more interactions between components,
which makes the core tensor G far from superdiagonal, resulting in a low value of CC. Therefore, it isreasonable to stop increasing R before the symptom of over-fitting emerges. Our determination of R isgiven asR = max R0 ∈ {R0 : CC(R0 ) > Lcc , R0 ∈ N, },(12)

where CC(·) denotes the core consistency as a function of the number of components, and Lcc is a threshold parameter. In line with [16], we set Lcc = 85, and to minimize randomness created by a PARAFAC
decomposition, we implement PARAFAC decomposition 20 times for a given rank size and use theirmean as the value for CC(R0 ). Our implementation uses the MATLAB code available form [34] which isbased on the alternating nonnegativity-constrained least squares with block principal pivoting [35].

Results
Synthetic temporal financial markets
To validate the accuracy of our multi-timescale decomposition, we first implement NTF on a synthetic
financial market whose properties are fully known ex ante. Suppose that there are three groups of banks,5

(a) 1

participation probability

0.8

fitness(b)group 1group 2group 3

0.6
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days

Figure 2: Predefined intra- and interday trading patterns for the synthetic financial market. (a) Predefined intraday
activity patterns, represented by the fluctuation of fitness. (b) Predefined interday activity patterns, given by theparticipation probability.

each of which having particular intra- and inter-day trading patterns. First, we introduce intraday
trading patterns by considering the “fitness” of banks. In the fitness model, the probability that banks
i and j trade at time t is given by
pij,t = ai,t aj,t ,(13)

where ai,t ∈ [0, 1] denotes the fitness of bank i within the time interval [t − ∆, t] [12, 36]. ∆ is thelength of a time interval (in a day) expressed in minutes. Differences in the intraday trading patterns
across groups are then represented by different values of fitness. Let us assume that the fitness evolves as
illustrated in Fig. 2a. Banks belonging to group 1 are most active in the early morning (blue solid), andbanks in group 2 exhibit the highest trading activity around noon (red solid), and banks in group 3 aremost active at the end of the day (black dotted). More specifically, the fitness value of a bank belonging
to group s at time interval t is given as
asi,t = f (t; µs , σ), t = 1, . . . , T,

(14)

where f (t; µs , σ) is the p.d.f. of normal distribution with mean µs and standard deviation σ. We set(µ1 , µ2 , µ3 ) = (0, T /2, T ) and σ = T /4.
Second, the interday trading patterns are captured by variations in a bank’s participation probability,
which is the probability that a bank participates in the market on a given day, apart from whether or
not the bank is able to find a trading partner ex post. For day d = 1, . . . , D, the participation probabilityof bank i, denoted by qi,d , is given as (Fig. 2b)

0.5
if i ∈ group 1, 2 (d − 1)
if i ∈ group 2 and d ≤ D/2,
qi,d = D
(15)
−2


(d − D) if i ∈ group 2 and d > D/2,

D


1
(d − 1)
if i ∈ group 3.
D
For simplicity, we assume that trade volumes are identical for all trades so that the (i, j, k) element of
the 3-dimensional tensor is equal to the total number of trades that bank i has during time interval j on
day k. We set N = 120, T = 20 (i.e., resolution ∆ = 30 for hours between 8:00 and 18:00 in each day)and D = 1000.
Using the above synthetic tensor, we can now implement our multi-timescale tensor factorization.
The factorization of tensor X allows us to measure the activity of each individual element: a bank, an
intra-day time interval, and a day. The activity of intra-day time interval j of component r is given by

6Component 1

intraday activity(a) 0.6

Component 2
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0

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8:00 10:00 12:00 14:00 16:00 18:00

time

(b)
interday activity

Component 3

0.6

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days
Figure 3: PARAFAC decomposition for the synthetic financial markets. (a) Intra-day activity for each component.
Activity in time interval j of component r is given by bjr . (b) Inter-day activity for each component. Activity of day
k of component r is given by ckr .
100Core-Consistency

50
0
-50
-100
-150
-200
-250

1

2

3

4

Figure 4: Core-Consistency for the synthetic markets. Dashed line denotes Lcc = 85.

bjr , and analogously the activity of day k of component r is represented by ckr . Fig. 3 illustrates that
PARAFAC decomposition well extracts the true multi-timescale patterns. The core-consistency value
strongly suggests R = 3 since CC(4) takes a large negative value.

Empirical data
Now we analyze empirical data. Given the accuracy of our method in the synthetic model, we expect
the NTF method to extract latent multi-timescale patterns in the real-world financial market, e-MID.
Here, we implement the N × T × D tensor factorization described in Method, where we set intradaytime resolution ∆ = {3, 5, 10, 15, 30, 45, 50}, and D is set at the total number of business days between
January 2001 and December 2015 (i.e., D = 3,839). The number of active banks N , which traded atleast once during the period, is 289. The Core-Consistency will reveal the number of components R, if
any, which is equivalent to the the variety of existent activity patterns.

Core-Consistency
The CC value for a given rank size is illustrated in Fig. 5a for ∆ = 15. The CC value becomes lower
than 85 for R ≥ 4, suggesting that R = 3 should be selected. In general, imposing different intraday7Core-Consistency

(a)

(b)

= 15
100

6

95

5

90

4

85

3

80

2

75

1

70

1

2

3

4

0

5

0 5

15

30

45

60

Figure 5: Determination of rank size. (a) Core-Consistency values for ∆ = 15. Error bar denotes the 95% confidence
interval of the mean Core-Consistency value, calculated by 20 runs of PARAFAC decomposition. (b) Selected valueof R for a given time resolution ∆ ∈ {3, 5, 10, 15, 30, 45, 60}.

resolutions may lead us to choose different values of R (Fig. 5b). There is a tendency that a higher
resolution requires a larger number of components for the PARAFAC decomposition to be appropriate.
In the following, we employ ∆ = 15 and thereby R = 3 as the benchmark case, the rationale of which
will be discussed in the next subsection.

Intra-day and inter-day activity
Fig. 6a shows intra- and inter-day activities for ∆ = 15, in which case there are three different patterns.
The intra-day activity of Component 1 is characterized by its bimodal pattern.1 It has two peaks around
10:00 and 17:00. On the other hand, both Components 2 and 3 have a single distinct peak in the earlymorning, after which their activities are very low. However, the distribution of intraday activity ofComponent 3 is more skewed than that of Component 2, which makes Components 1 and 3 two polarcases and Component 2 in between them. We would call the trading patterns of Components 1, 2 and 3
as normal trading, early trading, and flash trading, respectively.
The activity patterns for different temporal resolutions are also presented in Fig. S1. For the case of afiner intra-day resolution (e.g., ∆ = 5, Fig. S1a), there are more than three components, and the activityof an additional component (i.e., Component 4) turns out to be very similar to the flash-trading pattern
we saw in Fig. 6a. On the other hand, when the temporal resolution is low (e.g., ∆ = 45, Fig. S1c), there
are only two components and the flash-trading pattern is no longer detected. Given these observations,
it is natural to set ∆ = 15 as a benchmark case, where apparently independent intra-day patterns canbe captured. It should be noted that the following results are not sensitive to the temporal resolution
level as long as R = 3 is to be selected (e.g., ∆ = 30, Fig. S1b).
Fig. 6b illustrates the inter-day activity of each component. For Components 1 and 2, the activity
decreased radically during the global financial crisis of 2007–2009, which was initiated by a significant
decline of the US house prices (the peak date is indicated by blue dotted). In contrast, the activity ofComponent 3 spiked around the collapse of Lehman Brothers in September 2008. Fig. 6c shows how theshare of a given component evolved over time.
These observations suggests that Component 3 captures the “crisis mode” of bank trading while
Components 1 and 2 represent more normal trading patterns. The crisis-mode interpretation is reinforcedby the intra-day trading pattern of Component 3, in which banks trade only in the very early morning.
In fact, we will show that, although we did not use information about the directionality of trades, this1 Since the assignment of component index is originally arbitrary in the NTF implementation, we re-assign index in theascending order of cumulative intra-day activity between 8:00 and 10:00.

8

resol=15
Component 1

Component 2
1

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Activity
Peak of US house price
Lehman failure

20

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(c)

10:00

time

(b)
activity

Component 3

1

20

activity

(a)

years

Figure 6: Intra- and inter-day activities. Peak of the US house price (July 2006) and Lehman collapse (September2008) are indicated by blue dotted and red dashed, respectively. The data for the US house price is taken from FRED
(S&P/Case-Shiller U.S. National Home Price Index) [37] (a) Intra-day activity. (b) Inter-day activity. (c) Share of acomponent in the daily activity. In panels (b) and (c), 20-day moving average is shown.

anomaly comes from the fact that banks in need of liquidity tried to obtain loans, rather than to extend
credit, in the midst of the financial crisis.

Banks’ affiliation to components
Each element of vector ar represents the extent to which a bank’s trading pattern is captured by Component r. Thus, we can extract a subgroup of banks whose trading patterns are characterized by Componentr at least to some extent. Here, the set of banks belonging to component r is given by Ir = {i0 } forwhich ai0 r is within the 90th percentile of {air }N
i=1 (Fig. 7a). It should be noted that a bank may belong
to multiple components if its trading pattern has multiple features (Fig. 7b).
The bank activity patterns emerging at the daily scale, captured by the row average of ar c>r , exhibit
explicit differences across components (Fig. 7c). While all types of banks have not been active since 2009,
the normal-trading feature of banks belonging to Component 1 represents their behavior until 2008, andthe trading pattern of banks belonging to Component 2 are dominated by the early-trading pattern up
to 2006. On the other hand, the banks belonging to Component 3 exhibit daily patterns similar to theother types of banks prior to 2008, but suddenly their activity level spiked around the collapse of Lehman
Brothers.

9

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Banks belonging to Component 3

20

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20

Component 1
Component 2
Component 3

0.008

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03

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20

membership level

0.01

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(c)

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(a)

years
Figure 7: Affiliation of banks to components. The set of banks belonging to component r is given by Ir = {i0 } forwhich ai0 r is within the 90th percentile of {air }N
i=1 . (a) Number of banks that belong to a component. (b) Jaccard
index for the overlap of banks belonging to multiple components. (c) Average membership level on a given day.
Membership level for a bank is given by a row of ar c>r .

Characterization of components
Aside from the intra- and inter-day trading patters, it would be of interest to see whether we may
characterize each component by inherent attributes of the belonging banks. In the e-MID data set, a
trading bank is classified into either of the following four types: aggressor lender, quoter lender, aggressor
borrower and quoter borrower. In the e-MID online platform, a bank posts a request for loans or a
proposal for lending, and a loan contract is made if another bank accepts the posted request. A bankis called “aggressor lender” when the bank accepts a request for loan posted on the e-MID platform,
and the bank posted the request and borrowed fund is called “quoter borrower”. “quoter lender” and“aggressor borrower” can also be understood analogously. In addition, we can also ask to what extent
the nationality of banks can explain the differences in trading patterns.
In fact, the majority of transactions in the e-MID market were done between aggressor lenders andquoter borrowers, which is a common property to all the component types (Fig. 8a). However, morethan half of the banks belonging to Component 3 are quoter borrowers and the share of aggressor lenders
is significantly lower than that. As for the nationality, the share of Italian banks among all the banks
belonging to Component 1 is significantly low while the corresponding fraction is significantly large forthe banks in Component 2 (Fig. 8b) at 90% statistical significance, respectively. This suggests that
foreign banks (i.e., banks from countries outside Italy) are likely to exhibit the normal-trading pattern,
and Italian banks are more likely to employ the early-trading pattern than foreign banks do.
These observations give us important implications about the multi-timescale trading patterns in thefinancial market. First, flash trading, which represents the intra-day trading pattern of Component 3,
can be attributed mostly to quoter borrowers. Recall that flash trading was observed mostly in the
period around the Lehman collapse in September 2008, but we do not know ex ante whether such a
trading pattern was driven either by demand or supply. The fact that Component 3 is largely attributed

10

(a)

(b)

Figure 8: Role of bank attributes in the characterization of components. (a) Average frequency of a bank being an
aggressor lender, a quoter borrower, an aggressor borrower and a quoter lender. Error bar denotes 95% confidence
interval. (b) Share of Italian banks in each group. “Random” denotes the average share of Italian banks under random
chance, and error bar denotes the 90% interval calculated from a binomial distribution with parameter (Nr , p), where
Nr is the number of banks belonging to component r and p ≡ #Italian banks/N .

to quoter borrowers suggests that the flash-trading pattern during the financial crisis was conducted bybanks that attempted to obtain liquidity as early as possible by posting quotes for loans. This may be
regarded as evidence that some banks in fact faced a serious liquidity shortage at the time of the Lehman
collapse [38, 39]. Our multi-timescale NTF approach reveals not only how banks reacted to the fear ofliquidity shortage (i.e., intra-day pattern), but also specific dates on which the fear was most evident(i.e., inter-day pattern).
Second, differences in the nationality of banks can lead to the variety of trading patterns. It turns outthat foreign banks (Italian banks) are more likely to employ normal trading (early trading) than early
trading (normal trading). On the other hand, the source of the crisis modality cannot be explained by
the nationality of banks (Fig. 8b), suggesting that banks had similar chance of being in a crisis mode
regardless of their nationalities.

Discussion
In this work, we presented an analytic framework based on non-negative tensor factorization (NTF)
to extract temporal activity patterns of financial systems. Despite previous studies on online financial
markets recognized the existence of trading activity patterns at specific time scales (e.g., inter-day [12],or intra-day [21–23]), we demonstrated—to the best of our knowledge for the first time—that activity
patterns coexist at different time scales and depend upon each other. Our methodology allowed us
to uncover the hidden multi-timescale patterns of trading activities in an online interbank market (eMID [20]).
Within our framework, banks’ activities were represented by means of a tensor of three dimensions,
namely the list of banks, time of the day, and date. Leveraging the power of non-negative tensor
factorization (based on the PARAFAC decomposition of the 3-way tensor [24, 25]), we uncovered themulti-timescale nature of financial trading dynamics in e-MID, suggesting that banks’ trading patterns
could be classified into subgroups exhibiting significantly different multi-timescale patterns. Our framework also allowed us to attribute roles to banking institutions (aggressor lender/borrower, or quoter
lender/borrower) based on their trading patterns, yielding interpretable analytic insights.
By modeling the multi-timescale dynamics occurred over the period covered by our data (2001–2015),
the proposed NTF-based framework allowed us to identify trading anomalies in the midst of the financial

11

crisis. For example, around the time of Lehman Brothers’ collapse in September 2008, our approach
showed how banks changed their trading rhythms, and on what dates such anomalies emerged.
Understanding trading patterns in financial systems is of fundamental importance for many reasons:first, a rigorous characterization of such systems’ trading dynamics could help central banks to effectively
intervene in the interbank market. In turn, this would contribute to reducing systemic risk in the financial
system as a whole. Financial linkages created by bilateral transactions between banks can lead to a
global network of interconnected risk, in which a failure of one bank can immediately spread over theentire system [9, 26, 27]. Extracting multi-timescale patterns of financial markets allows for a better
understanding of when and how banks react to systemic shocks.
Over the last couple of years, many proposals for measuring and controlling systemic risk have beenpresented, yet most of them focus on the static nature of financial networks [40–42]. However, as we
showed in this study of the Italian e-MID interbank market, various patterns and anomalies systematicallyemerge at different time scales, which should be taken into account in the design of financial regulations.
This work lays out the foundations for other researchers to study how patterns and anomalies emerge infinancial systems at different temporal resolutions: in the future, such a framework could contribute tocontrolling the spread of financial systemic risk [9, 10, 26–28].
In this spirit, we will investigate whether our method could be used both in predictive and prescriptive
ways to help determine what could happen under certain risk conditions, as well as to provide possible
intervention strategies prior to shocks or cascading collapses in financial systems.

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Acknowledgements
TK acknowledges financial support from the Japan Society for the Promotion of Science Grants no.
15H05729 and 16K03551. EF is grateful to DARPA for support (grant #D16AP00115). This project
does not necessarily reflect the position/policy of the Government; no official endorsement should be
inferred. Approved for public release; unlimited distribution.

Author contributions
TK and EF conceived the research and wrote the manuscript. TK performed the analysis. TK, AS and
EF discussed the results and reviewed the manuscript.

Competing interests
The authors declare no competing interests.

14years1in the daily activity. (a) ∆ = 5, (b) ∆ = 30, and (c) ∆ = 45. See the caption of Fig. 6.
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Supplementary Information“Extracting the multi-timescale activity patterns of online
financial markets”
Teruyoshi Kobayashi, Anna Sapienza and Emilio Ferrara

Intra- and inter-day activity of banks

The r-th rows of factor matrices B and C (i.e., br and cr ) respectively represent the intraand inter-day activities of component r. Fig. S1 illustrates intra- and inter-day activities for agiven temporal resolution ∆ ∈ {5, 30, 45}. The selected rank size, based on the Core-Consistency
measure, is R = 4 for ∆ = 5, R = 3 for ∆ = 30, and R = 2 for ∆ = 45. We note that the choiceof ∆ does not affect the activity distributions for a fixed value of R.

resol = 5=5Component 4

16:00
18:00

Figure S1: Intra- and inter-day activities. Upper: Intra-day activity. Middle: Inter-day activity. Lower: Share of a component

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sha1_base64="KLDaCTe85460c7HkQK4FyA/EUao=">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</latexit><latexit<latexit

12:00

20

1

20
0

activity(c)01

activity

03

20

activity
10:00

20

activity (share)

05

01

20

20

0
8:00

20

03

01

20

activity (share)

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sha1_base64="VvrHidPxVM1hpFvo9yCoZdTeZZ4=">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</latexit><latexit

sha1_base64="VvrHidPxVM1hpFvo9yCoZdTeZZ4=">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</latexit><latexit<latexit(b)20

resol = 30

= 30
Component 3

0.2

time

16:00
18:00

years

resol = 45