Granger Causality for Predictability in Dynamic Mode Decomposition G. Revati Syed Shadab K. Sonam S. R. Wagh and N. M. Singh Abstract The dynamic mode decomposition DMD tech-

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Granger Causality for Predictability in Dynamic Mode Decomposition

G. Revati, Syed Shadab, K. Sonam, S. R. Wagh, and N. M. Singh

Abstract— The dynamic mode decomposition (DMD) tech-

nique extracts the dominant modes characterizing the innate

dynamical behavior of the system within the measurement data.

For appropriate identiﬁcation of dominant modes from the

measurement data, the DMD algorithm necessitates ensuring

the quality of the input measurement data sequences. On that

account, for validating the usability of the dataset for the DMD

algorithm, the paper proposed two conditions: Persistence of

excitation (PE) and the Granger Causality Test (GCT). The

virtual data sequences are designed with the hankel matrix rep-

resentation such that the dimensions of the subspace spanning

the essential system modes are increased with the addition of

new state variables. The PE condition provides the lower bound

for the trajectory length, and the GCT provides the order

of the model. Satisfying the PE condition enables estimating

an approximate linear model, but the predictability with the

identiﬁed model is only assured with the temporal causation

among data searched with GCT. The proposed methodology

is validated with the application for coherency identiﬁcation

(CI) in a multi-machine power system (MMPS), an essential

phenomenon in transient stability analysis. The signiﬁcance of

PE condition and GCT is demonstrated through various case

studies implemented on 22 bus six generator system.

Index Terms— Coherency Identiﬁcation, Dynamic Mode De-

composition (DMD), Granger causality, Hankel, Persistence of

Excitation (PE).

I. INTRODUCTION

With the growing emphasis on data-driven modeling, un-

derstanding the interactions and connections among the time

series drawn from observational data is a ﬁeld of interest.

Causality is the intersection of philosophy and sciences

[1], deriving the generalizations and theories from speciﬁc

observations by analyzing the cause and effects among the

observational data. A primary approach for understanding

the information ﬂow amongst the time series is to determine

the cross-correlation [2] among the two time series and

to discover the existence of a peak in the correlation at

some non-zero lag. The causal inferences drawn from the

correlation are misleading since the correlation reveals only

whether the two variables are statistically linked. The causal

relationship amongst two variables can be direct, or indirect

due to confounding effect [3] i.e., besides the variables under

study there is some additional unnoticed variable correlated

with the considered variables. Furthermore, the correlation

being a symmetric measure fails to provide any information

about the causality direction.

G. Revati, Syed Shadab, S. R. Wagh, and N. M. Singh are with Control

and Decision Research Centre (CDRC), EED, Veermata Jijabai Tech-

nological Institute, Mumbai 400019, India cdrc@ee.vjti.ac.in.

K. Sonam is with the Computer Science and Engineering Department,

University of South Carolina, USA

As per the principle of time asymmetry of causation, clas-

sical physics employs the precedence of causes over effects.

Accounting for the direction of causation, a dynamical model

identiﬁcation is another approach. The concept of dynamical

model identiﬁcation is fundamentally developed on the fact

that law drives the system and enables the evolution of the

same state in a similar manner [4] i.e., similar effects are

produced by the same causes mentioned as per physical

determination. The laws deﬁning the system dynamics are

identiﬁed from the regression of observational data achieved

through the evaluation of correlation. For detecting and

quantifying the temporal causality amidst the time series,

a powerful statistical test known as Granger Causality (GC)

was ﬁrst proposed in [5]. The widespread applications of the

GC in neuroscience [6], economy [7], and climate modelling

[8] are mentioned in the literature. The fundamental notion

behind GC is the enhancement in the prediction of one

variable with the introduction of past information of another

variable along with the past information of the considered

variable itself.

Conventionally the control applications extensively opted

for the system identiﬁcation methods ﬁtting the data to the

model parameterized priori [9]. The growing complexity and

huge amount of available system data challenged the standard

strategies for learning the dynamical system. Alternatively,

the paradigm shift occurred towards the identiﬁcation of a

dynamical system from the raw measurement data of the

system. The data-driven modeling approaches are generally

dependent on searching for the accurate combination of the

known trajectory in order to achieve a reliable prediction

which is usually an ill-conditioned problem [10]. For deal-

ing with such problems, the Moore-Penrose pseudoinverse

[11] solving the least norm problems is preferred due to

computational simplicity. One such data-driven subspace

prediction strategy is the Dynamic Mode Decomposition

(DMD) [12] which decomposes the high dimensional data

into spatiotemporal coherent modes.

DMD is a dimensionality reduction technique [13] pio-

neered in the ﬂuid dynamics community by Peter Schmid for

identifying the linear approximation from the data compris-

ing the dominant modes describing the dynamical behavior

of the system [14]. The quality identiﬁcation of the dominant

modes capturing the dynamics depends on the quality of the

measurement data exploited for the strategy. For capturing

the modes of the system, the cardinality of the measurement

sequences utilized in the DMD should be greater than or

equal to the underlying system modes. Hence the dimensions

of the subspace spanning the essential dynamical modes

are increased with the Hankel matrix [15] introducing the

arXiv:2210.12737v1 [eess.SY] 23 Oct 2022

new state variables. The temporal evolution of the system

lies in the column space and the row space deﬁnes the

spatial structure of the system modes hence for the estima-

tion of an accurate approximate linear model a sufﬁcient

number of rows and columns of the input data matrices are

necessary. Hence for identifying the accurate linear model

with predictability, the paper has proposed two conditions:

Persistence of excitation (PE) and Granger Causality Test

(GCT). The PE is a necessary condition and provides the

lower bound on the trajectory length and the GCT which

being a sufﬁcient condition is informative to detect the

causation among the measurement sequences and to ﬁnd the

appropriate order of the model ensuring the predictability of

the identiﬁed linear model.

The suitability of the proposed approach is veriﬁed with

the application of coherency identiﬁcation in a multimachine

power system (MMPS). Coherency is a property of gen-

erators to swing together the coherency identiﬁcation is a

necessary phenomenon for the transient stability analysis in

MMPS. The relevance of PE condition for capturing the

dominant dynamical modes of the system and the signiﬁ-

cance of GCT to establish the vital role of causation to ensure

predictability is demonstrated with the various experimental

case studies implemented on the 22 bus 6 generator power

system.

The remaining paper is structured as follows: The concepts

including Dynamic mode decomposition, the persistence of

excitation, vector autoregression, and Granger causality are

discussed brieﬂy as preliminaries in Section II. The proposed

methodology explaining the details of the PE condition and

the Granger Causality Test along with the DMD algorithm is

presented in Section III. The results for the comprehensive

case study along with the test and error analysis are illus-

trated in Section IV. The paper is concluded with the future

work in Section V.

II. PRELIMINARIES

A. Dynamic Mode Decomposition (DMD)

DMD is designed to extract spatially coherent modes,

oscillate at a ﬁxed frequency, and decay or growth at a ﬁxed

rate [16]. DMD is a data-driven technique that emerged from

the ﬂuid dynamics community [17] identifying a dynamical

system from the observational data. DMD is strongly related

to the Koopman operator theory (KOT) which provides an

inﬁnite dimensional linear representation Kof the nonlinear

system dynamics ϕacting on the ﬁnite-dimensional manifold

M. DMD is an approach that seeks the Amatrix such that

its spectrum approximates the spectrum of the Koopman

operator. The dominant eigenvalues and eigenvectors of this

Amatrix are very informative [18] about the dynamical

attributes of the system such as the frequency, decay, growth,

and ﬂow modes.

Consider a set of pairs of observations xjand yj,j =

1,2,· · · ,nrelated temporally such that

yj(z) = xj(ϕ(z)) (1)

where z∈ M,ϕ:M → M is a dynamical system, and

xj,yj:M → R.yjis one step ahead in time that of xji.e.

if xjdenotes the observation at time t, then yjrepresents the

measurement at time t + ∆t. From these measurement time

series, two data matrices Xand Yare constructed as

X = x1x2· · · xn,Y = y1y2· · · yn(2)

also X,Y∈Rm×n, where mis the dimension of the

manifold M. The objective of the DMD algorithm is to

evaluate the approximation Asuch that

Y = AX(3)

The analytical solution of the above problem is given by

A= YX†(4)

As the input matrices involved in the computation of A

matrix are rectangular the Moore-Penrose pseudo-inverse X†

is used. The solution 4 originates from the least square

problem of minimizing the error

kY− AXkF(5)

k.kFis a Frobenius norm given as

kZkF=v

j=1

k=1

jk (6)

Practically the computation of Ais challenging due to

the very large size of observations i.e. m>nresulting in

an under-determined system. To alleviate this difﬁculty, the

spatial dimensions of the input data are reduced through

Proper Orthogonal Decomposition (POD). The linear sub-

space spanned by the set of rorthogonal modes approximates

the space Rmsufﬁciently to achieve the dimensionality

reduction. The proper orthogonal modes are evaluated via

the Singular Value Decomposition (SVD) of Xgiven as

X = USV∗(7)

with U∈Rm×rwhose each column represents the eigen-

vector of XXT,S∈Rr×ris the diagonal matrix having

eigenvalues of Xin descending order and V∈Rn×ris

the matrix in which each row represents the eigenvector of

XXT. The rproper orthogonal modes correspond to the

dominant left singular vectors (Ur) of SVD associated with

the dominant singular values.

The SVD (7) aids to search for the reduced order subspace

containing the dominant system modes through the projec-

tion of Aonto the POD modes. ˜

A= U∗AU = U∗YVS−1.

The DMD modes and eigenvalues are evaluated from the

eigendecomposition of ˜

A, i.e., ˜

AW = WΛ. The exact

DMD modes Φare evaluated by transforming back to the

original space of higher dimensions, i.e. Φ = YVS−1W.

The temporal evolution of the modes is identiﬁed with the

eigenvalues from the diagonal of matrix Λ.

B. Persistence of excitation (PE)

For the linear time-invariant system which is controllable,

if a component of a state is PE of an adequately higher

order, then the sections of the trajectory span the space

characterizing the behavior of the system [19].

In order to ensure the consistency of the model estimated

during the identiﬁcation experiments [20] [21], the data

sequences used for the subspace identiﬁcation must be PE

of signiﬁcant order. For efﬁciently identifying the system

modes from the deterministic input data sequences [22], the

virtual data sequences analogous to the multi-variable system

fulﬁlling the PE condition are designed with the hankel block

matrix representation of the input data sequence.

Let x∈Rnbe a state vector denoted as x[k,k+S], where

k∈Xis the time instant for a ﬁrst sample and S∈N

are the total samples taken. The state vector in the interval

[k, k +S]∩Xis deﬁned as

x[k,k+S]=







x(k)

x(k+ 1)

x(k+S)







(8)

The state vector x[k,k+S]organized in the Hankel matrix is

represented as







x(k)x(k+ 1) · · · x(k+S−M+ 1)

x(k+ 1) x(k+ 2) · · · x(k+S−M+ 2)

.....

x(k+M−1) x(k+l)· · · x(k+S)







(9)

where k∈X, S, M ∈N. The total number of system

eigenvalues determines the value of M

Deﬁnition 1: A signal trajectory x[k,k+S]is PE of the

order L, if the block hankel matrix in (9) has a full rank

Basically, the deﬁnition indicates that to satisfy the PE

condition, the section of the trajectory of length Lmust

be adequately long enough to excite the controllable sys-

tem modes in the window of Land reproduce them. The

Deﬁnition 1 gives the lower bound on the trajectory length

such that m >= (n+ 1)L−1, the number of rows will

be lesser or equal to the columns i.e. there will be more

temporal samples than the spatial samples [23].

C. Vector Auto regression (VAR)

A multivariate time series x1, x2,· · · , xm, with each

measurement being a n-dimensional vector consisting the

elements x1t, x2t,· · · , xnt is realised with a vector stochastic

process X1, X2,· · · . This time series can be modeled with

a Vector Autoregressive (VAR) of order prepresented as

Xt=

k=1

AkXt−k+t(10)

Ak∈Rn×nis a regression coefﬁcient matrix and tis

the white noise corresponding to the residuals. VAR models

detect the simultaneous evolution patterns of multivariate

time series. The objective of the VAR model is to ﬁnd the

coefﬁcient matrix demonstrating the temporal correlation be-

tween the multivariate time series. The predictive VAR model

represents the value of xtat any time tas a combination of its

past values. The predictable patterns in the data are captured

with the coefﬁcient matrix whereas the unpredictable part is

accounted for with the residual terms.

GC analysis draws causal inferences among various vari-

ables based on their VAR representation. To obtain a valid

analysis of GC, the coefﬁcients of the VAR model (10)

should be square summable and stable [24] [2]. Square

summability implies that Pp

k=1 kAkk2<∞i.e. even for

the inﬁnite order of the model, the regression coefﬁcients

are not blowing up. The stability is related to the char-

acteristic polynomial of coefﬁcient matrix Ak,ϕ(z) =



I−PPp

k=1 Akzk

where Z∈C. If the characteristic

polynomial of the coefﬁcient matrix is invertible on the unit

disc |z| ≤ 1in the complex plane [2], then the coefﬁcients

of the VAR model are stable.

D. Granger Causality (GC)

Granger causality (GC) is a framework established on

the temporal precedence implying that causes precede their

effects. Temporal precedence being the innate characteristic

of the time series, the GC helps to establish the causal

relationship among the time series inferring that the past

is causing the future. The physical interpretation of the

causality is that the causes are responsible for the unique

changes in the corresponding effects, i.e. the causal series

contains the unique information about the effect series which

is not available otherwise [5]. In fact, GC does not actually

indicate the true causality, rather it examines the inﬂuence

of one series on the forecasting ability of another series.

Considering two time series xtand ytthe GC ﬁnds the

causal relationship amidst xtand ytdepending on whether

the past values of xtassists in the prediction of ytconditional

on having already considered the effect of the past values of

yon the prediction of yt.

Deﬁnition 2: Suppose H<t be representing the history of

all relevant information available up to (t−1),Pr(xt|H<t)

be denoting the prediction of xtgiven H<t, the GC suggests

yto be causal for xif

var [xt−Pr(xt|H<t)] <var [xt−Pr(xt|H<t \y<t)]

(11)

whereas H<t \y<t speciﬁes that the values of y<t are

excluded from H<t. Equation (11) describes that the variance

of the prediction error of xwith the inclusion of the history of

yis reduced [25] consequently inferring that the past values

of yenhance the prediction of x.

The primary argument of GC is based on the identiﬁcation

of a unique linear model from the data. Let xand ybe the

two time series of length N, and described by the uni-variate

vector auto-regressive (VAR) model of order p<N, given

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GrangerCausalityforPredictabilityinDynamicModeDecompositionG.Revati,SyedShadab,K.Sonam,S.R.Wagh,andN.M.SinghAbstractThedynamicmodedecomposition(DMD)tech-niqueextractsthedominantmodescharacterizingtheinnatedynamicalbehaviorofthesystemwithinthemeasurementdata.Forappropriateidenticationofdominantmode...

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Granger Causality for Predictability in Dynamic Mode Decomposition G. Revati Syed Shadab K. Sonam S. R. Wagh and N. M. Singh Abstract The dynamic mode decomposition DMD tech-

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