Relationship between Granger non-causality and network graph of state-space
representations
Jozsa, Monika
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Chapter 3
Granger causality and Kalman representations
in coordinated form
In Chapter2we have shown that the existence of Kalman representations, whose network graphs are directed graphs with two nodes and one edge, is equivalent to Granger non-causality among the components of the output process of these rep-resentations. It has also been shown that Granger non-causality implies the exis-tence of such Kalman representations that can be constructed algorithmically from the covariance sequence of the output process. In this chapter, we step forward to study Kalman representations with more complex network graph. More specif-ically, we study Kalman representations, whose network graphs are star graphs, called Kalman representations in coordinated form. A star graph is a tree graph which has precisely one root and all its other nodes are leaves. The subsystem of a Kalman representation in coordinated form corresponding to the root node of the network graph will be called coordinator and the other subsystems, corresponding to the leaves of the network graph, will be called agents. In such representations, the coordinator sends information to the agents and the agents do not send infor-mation to the coordinator. The existence of Kalman representations in coordinated form is then associated to a collection of conditional and unconditional Granger non-causalities in the output process. We also present algorithms for constructing a Kalman representation in coordinated form in the presence of the appropriate con-ditional and unconcon-ditional Granger non-causalities.
Deterministic LTI–SS representations in coordinated form were already intro-duced in (Kempker, 2012; Kempker et al., 2014a; Ran and van Schuppen, 2014). In (Kempker, 2012) and (Kempker et al., 2014a), a general method was presented to transform deterministic LTI–SS system into coordinated form. In addition, in (Kempker, 2012) and (Pambakian, 2011), Gaussian coordinated systems were stud-ied in the context of LQG control. The cited papers gave the idea and motivated the name of the representations studied in this chapter. However, the cited papers do not study the relation between the coordinated system structure and causality properties of the observed process. The results in (Caines et al., 2003;Caines et al., 2009) are the closest ones to the results in this chapter. The cited papers provide
nec-essary and sufficient conditions for the existence of LTI–SS representations in the so-called conditional orthogonal form which form a subclass of LTI–SS representa-tions in coordinated form. The condirepresenta-tions of (Caines et al., 2003;Caines et al., 2009) for the existence of such systems are much stronger than the conditions proposed in this chapter. Regarding the proofs of the statements in (Caines et al., 2003;Caines and Wynn, 2007;Caines et al., 2009), only the existence of the LTI–SS representation in conditional orthogonal form is proven in (Caines et al., 2003) which completely differs from the idea behind the proofs of this chapter. Note that (Caines et al., 2003;
Caines and Wynn, 2007;Caines et al., 2009) did not provide algorithms to calculate the representations. This chapter, together with Chapter3 is based on the journal paper (Jozsa et al., 2018b).
The structure of this chapter is as follows: First, we introduce Kalman represen-tations in coordinated form. Then, we characterize their existence in terms of condi-tional and uncondicondi-tional Granger causality. This is followed by the presentation of two algorithms for calculating Kalman representations in coordinated form. Finally, we provide an example to illustrate the results. The poofs of the statements can be found in Appendices3.Aand3.B. If not stated otherwise, we assume throughout this chapter that y “ ryT
1, . . . , y T ns
T is a ZMSIR process where n ě 2, y
i P Rri, and
rią 0for i “ 1, . . . , n.
3.1
Kalman representation in coordinated form
In this section, we introduce Kalman representations in coordinated form and dis-cuss their properties. Kalman representations in coordinated form have the star graph as their network graph that defines the communication flow among their subsystems. In order to exclude hidden communication that is inconsistent with this network graph, we also introduce a subclass of these representations, called Kalman representations in causal coordinated form. To begin with, we define these classes of representations:
Definition 3.1. A Kalman representation pA, K, C, I, e “ reT
1, . . . , eTnsT, yq, where
eiP Rri, i “ 1, . . . , n, is called a Kalman representation in coordinated form, if
A “ » — — — — — — – A11 0 ¨ ¨ ¨ 0 A1n 0 A22¨ ¨ ¨ 0 A2n .. . ... . .. ... ... 0 0 ¨ ¨ ¨ Apn´1qpn´1qApn´1qn 0 0 ¨ ¨ ¨ 0 Ann fi ffi ffi ffi ffi ffi ffi fl , K “ » — — — — — — – K11 0 ¨ ¨ ¨ 0 K1n 0 K22¨ ¨ ¨ 0 K2n .. . ... . .. ... ... 0 0 ¨ ¨ ¨ Kpn´1qpn´1qKpn´1qn 0 0 ¨ ¨ ¨ 0 Knn fi ffi ffi ffi ffi ffi ffi fl , (3.1)
3.1. Kalman representation in coordinated form 53 C “ » — — — — — — – C11 0 ¨ ¨ ¨ 0 C1n 0 C22¨ ¨ ¨ 0 C2n .. . ... . .. ... ... 0 0 ¨ ¨ ¨ Cpn´1qpn´1qCpn´1qn 0 0 ¨ ¨ ¨ 0 Cnn fi ffi ffi ffi ffi ffi ffi fl , (3.1)
where Aij P Rpiˆpj, Kij P Rpiˆrj, Cij P Rriˆpj and pi ě 0for i, j “ 1, . . . , n. If, in
addition, for each i “ 1, . . . , n ´ 1 ˆ„Aii Ain 0 Ann ,„Kii Kin 0 Knn ,„Cii Cin 0 Cnn , Iri`rn, „ ei en ˙ (3.2) is a minimal Kalman representation of ryT
i , y T ns
T in causal block triangular form,
then pA, K, C, I, e, yq is called a Kalman representation in causal coordinated form. If n “ 2, then Definition3.1coincides with Definition2.1on Kalman represen-tations in block triangular form. Furthermore, if pA, K, C, I, eq is a Kalman rep-resentation in causal coordinated form, then the dimensions of the block matrices Aij, Kij, Cij, i, j “ 1, . . . , n are uniquely determined by y. Indeed, since (3.2) is a
minimal Kalman representation of ryT
i , yTnsT, i “ 1, . . . , n´1 in causal block
triangu-lar form, the dimensions of Aii, Kii, Ciiand Ain, Kin, Cinare uniquely determined
by ryT i , y
T
nsT, see Section2.1for details. Therefore, all dimensions of the blocks of
A, Kand C in (3.1) are determined by y. Definition3.1is based on the deterministic terminology (Kempker, 2012;Ran and van Schuppen, 2014) and on the definition of Gaussian coordinated systems (Kempker, 2012;Pambakian, 2011) .
The term coordinated is used because the LTI–SS representation at hand can be viewed as consisting of several subsystems; one of which plays the role of a coordi-nator and the others play the role of agents. More precisely, let pA, K, C, I, e, yq be a Kalman representation in coordinated form as in (3.1) and let x “ rxT
1, . . . , x T ns
T be
its state such that xiP Rpi, i “ 1, . . . , n. Then, for i “ 1, . . . , n ´ 1
Sai # xipt ` 1q “ ř j“ti,nuAijxjptq ` Kijejptq yiptq “ ř j“ti,nuCijxjptq ` eiptq (3.3) Sc " xnpt ` 1q “ Annxnptq ` Knnenptq ynptq “ Cnnxiptq ` enptq . (3.4) Notice that subsystem Saigenerates yias output, has xi, eias its state and noise pro-cess and takes xn, enas its inputs, thus takes inputs from subsystem Sc. In contrast,
Sc
Sa1 Sa2 ¨ ¨ ¨ San´1
xn, en xn, en xn, en
Figure 3.1: Network graph of a Kalman representation in coordinated form: Sc is
the coordinator (3.4) and Sai, i “ 1, . . . , n ´ 1 are the agents (3.3).
process but not taking input from subsystems Sai, i “ 1, . . . , n ´ 1 (see Figure3.1). We call Sc the coordinator and Sai with i “ 1, . . . , n ´ 1 the agents. Intuitively, the agents do not communicate with each other, only the coordinator sends information (xnand en) to all agents and does not receive information from them.
Motivation for Kalman representations in causal coordinated form
If we considered a general LTI–SS representation with a network graph like in Fig-ure3.1, then the noise process e could be any process. If e were not the innovation process of y, then it could happen that the agents communicate with each other in an implicit way through e. However, if we assume that pA, K, C, I, e, yq is a Kalman representation in causal coordinated form satisfying (3.1), then reT
i, eTnsT is the
in-novation process of ryT i , y
T
nsT for i “ 1, . . . , n ´ 1 and enis the innovation process
of yn. Hence, the values of eiand en depend only on the past and present values
of yiand yn. Moreover, xndepends only on the past values of ynand therefore xi
depends only on the past values of yi and yn. That is, Kalman representations in
causal coordinated form have the property that there is no communication among the agents or from the agents to the coordinator hidden in the noise process. Note that the lack of communication from the agents to the coordinator is ensured by that (3.2) is a Kalman representation in causal block triangular form.
Kalman representations in causal coordinated form have a number of desirable properties, e.g., they have the smallest possible coordinator. By definition, the sub-system (3.2) of a Kalman representation in causal coordinated form (3.1) is a minimal Kalman representation of ryT
i , ynTsT in causal block triangular form. This implies
that (3.4) is a minimal Kalman representation of ynand thus the coordinator is
min-imal. It assures observability of the coordinator and enables to estimate the states of Kalman representations in causal coordinated form using distributed filters. That is, in order to estimate the state xn of the coordinator using a Kalman filter, only
3.2. Conditional Granger causality and coordinated systems 55 the output yn of the coordinator is necessary. Since (3.3) is also minimal and thus
observable, in order to estimate the state xiof the ith agent using a Kalman filter,
only the output yiof this agent and the output yn of the coordinator are necessary.
Furthermore, from Lemma3.2below, Kalman representations in causal coordinated form are isomorphic (see Definition1.11). Hence, if they represent the same out-put process, their properties are essentially the same. Note that as a consequence of Lemma3.2, if a Kalman representation of a process y in causal coordinated form is not minimal then there does not exist a minimal Kalman representation of y in causal coordinated form. Theproofof Lemma3.2can be found in Appendix3.A.
Lemma 3.2. Any two Kalman representations of y in causal coordinated form are
isomor-phic.
3.2
Conditional Granger causality and coordinated
sys-tems
In this section, we show that the existence of a Kalman representation of y in causal coordinated form can be characterized by conditional Granger non-causalities among the components of y.
To begin with, we define conditional Granger non-causality: the generalization of Granger non-causality between two components of a process in the presence of a third component. The next definition is a particular case of the concept of causality defined in (Granger, 1963), if the latter is applied to ZMSIR processes, and if, using the terminology of (Granger, 1963), there is one external process.
Definition 3.3. Consider a ZMSIR process y “ ryT
1, yT2, y3TsT. We say that y1
condi-tionally does not Granger cause y2with respect to y3, if for all t, k P Z, k ě 0
Elry2pt ` kq | H y2,y3
t´ s “ Elry2pt ` kq | H y1,y2,y3
t´ s.
Otherwise, we say that y1conditionally Granger causes y2with respect to y3.
If y “ ryT 1, y
T 2s
T, then considering y
3 “ y2 Definition3.3coincides with the
unconditional Granger causality defined in Definition2.3. We will be interested in a particular combination of causal dependencies in a process y “ ryT
1, . . . , yTnsT.
Namely, when yidoes not Granger cause ynand yidoes not Granger cause yjwith
respect to ynfor all i, j “ 1, . . . , n ´ 1, i ‰ j. We will show that these causal relations
in y “ ryT
1, . . . , yTnsT hold if and only if y has a Kalman representation in causal
co-ordinated form whose network graph is as in Figure3.1. We will also give condition for when this Kalman representation in causal coordinated form is minimal. Kalman
representations in causal coordinated form are observable, therefore we only need to ensure reachability. For the formulation of the reachability condition, we need to introduce the term of conditionally trivial intersection of two subspaces U, V with respect to a third, closed subspace W .
Definition 3.4. Consider the subspaces U, V, W Ď H such that W is closed. Then U, V have a conditionally trivial intersection with respect to W if
tu ´ Elru|W s | u P U u X tv ´ Elrv|W s | v P V u “ t0u,
i.e., the intersection of the projections of U and V onto the orthogonal complement of W in H is the zero subspace. The conditionally trivial intersection of U and V with respect to W is denoted by U X V |W “ t0u
Now we are ready to state the main result of this chapter:
Theorem 3.5. Consider the following statements for a ZMSIR process y “ ryT1, . . . , y T ns
T:
(i) yidoes not Granger cause yn, i “ 1, . . . , n ´ 1;
(ii) yi conditionally does not Granger cause yj with respect to yn, i, j “ 1, . . . , n ´ 1,
i ‰ j;
(iii) (i)and(ii)hold and for i, j P t1, . . . , n ´ 1u, i ‰ j
ElrHyt`i|Hyt´i,yns X ElrH yj
t`|H yj,yn
t´ s | ElrHyt`n|Ht´yns “ t0u (3.5)
(iv) there exists a minimal Kalman representation of y in causal coordinated form; (v) there exists a Kalman representation of y in causal coordinated form;
(vi) there exists a Kalman representation of y in coordinated form; Then, the following hold:
(a) (iii) ðñ (iv);
(b) (i)and(ii) ðñ (v).
If, in addition, y is coercive, then we have (c) (i)and(ii) ðñ (v) ðñ (vi).
Theproofcan be found in Appendix3.B. The intuition behind this result is the following. For a coordinator to exist, the outputs of the agents should not influence
3.2. Conditional Granger causality and coordinated systems 57 the output of the coordinator, i.e.,(i)should hold. Moreover, for i ‰ j the output of agent i should not influence the output of agent j, except that information which comes from the output of the coordinator, i.e.,(ii)should hold. Condition(iii)for minimality can be explained as follows. It can be shown that a Kalman represen-tation in causal coordinated form is observable, so for minimality, we only have to ensure its reachability or equivalently, the linear independence of the components of the state x “ rxT
1, . . . , xTnsTat each time. The spaces generated by the components of
xnptqand of rxTi , x T ns
T
ptqare ElrHt`yn|Hyt´nsand ElrHyt`i|Hyt´i,yns, respectively, where
xi and xn are as in (3.4) and (3.3). As a result, condition(iii)is equivalent to the
reachability of a Kalman representation in causal coordinated form.
Our next result helps us in reformulating condition(ii)in Theorem3.5by uncon-ditional Granger causality.
Lemma 3.6. Consider a process y “ ryT
1, yT2, y3TsT and the following statements
(i) y1does not Granger cause y3
(ii) y2does not Granger cause y3
(iii) y1conditionally does not Granger cause y2with respect to y3
(iv) y1does not Granger cause ryT2, yT3sT
Then we have that
(i)and(ii)and(iii) ðñ (ii)and(iv).
Theproofcan be found in Appendix3.A
Remark 3.7(Alternative formulations of(ii)). From Lemma3.6we can reformulate the conditions of Theorem3.5as follows: if(i)holds, then condition(ii)is equivalent to saying that yidoes not Granger cause ryjT, yTnsT, i, j P t1, . . . , n ´ 1u, i ‰ j.
Minimal Kalman representations in causal coordinated form are isomorphic to any other minimal Kalman representation of the same process, see Proposition1.12. Hence, any property of minimal Kalman representations in causal coordinated form that is invariant under isomorphism remains valid for any other minimal Kalman representation. Theorem3.5gives a necessary and sufficient condition for the ex-istence of a minimal Kalman representations in causal coordinated form. From Lemma3.2, we know that any two Kalman representations in causal coordinated form are isomorphic, thus behave as minimal ones among all Kalman representa-tions in coordinated form. Existence of a minimal Kalman representation in coordi-nated form (not causal coordicoordi-nated form) remains a topic of future research.
3.3
Computing Kalman representations in coordinated
form
Next, we describe a procedure to calculate a Kalman representation of y in causal coordinated form. Assume that condition (i)in Theorem 3.5holds. Consider an LTI–SS representation p ¯A, ¯B, ¯C, ¯D, vqof y and the partitions of ¯Cand ¯D
¯ C ““C¯T 1, . . . , ¯CnT ‰T , ¯D ““D¯T 1, . . . , ¯DnT ‰T (3.6) such that ¯Ciand ¯Dihave ri “ dimpyiqrows for all i “ 1, . . . , n. Then, notice that
the tuple p ¯A, ¯B,“C¯T i C¯nT ‰T ,“D¯T i D¯Tn ‰T , vqis an LTI–SS representation of ryT i , y T ns T
for all i “ 1, . . . , n ´ 1. Hence, by Corollary2.6the latter can be transformed into a minimal Kalman representation p ˆAi, ˆKi, ˆCi, Iri`rn, re
T
i, eTnsTqof ryiT, yTnsT in causal
block triangular form, i.e., ˆ Ai “ « ˆAii Aˆin 0 Aˆi,nn ff , ˆKi“ « ˆKii Kˆin 0 Kˆi,nn ff , ˆCi“ « ˆCii Cˆin 0 Cˆi,nn ff , (3.7)
and the process reT
i , eTnsT is the innovation process of ryiT, yTnsT. In addition, p ˆAi,nn,
ˆ
Ki,nn, ˆCi,nn, Irn, enqis a minimal Kalman representation of yn. Since all minimal Kalman representations of yn are isomorphic (Proposition 1.12), there exist
non-singular matrices Tifor i “ 2, . . . , n ´ 1 such that
ˆ
A1,nn “ TiAˆi,nnTi´1, Kˆ1,nn“ TiKˆi,nn, Cˆ1,nn “ ˆCi,nnTi´1. (3.8)
Let T1be the identity matrix and define the matrices A, K and C as in (3.1) such that
for i “ 1, . . . , n ´ 1
Aii “ ˆAii, Kii“ ˆKii, Cii“ ˆCii,
Ain“ ˆAinTi´1, Kin“ ˆKin, Cin“ ˆCinTi´1,
Ann“ ˆA1,nn, Knn“ ˆK1,nn, Cnn“ ˆC1,nn.
(3.9)
Then, for the tuple pA, K, C, I, e, yq, where e “ reT 1, e
T 2, . . . , e
T ns
T, we can state the
following.
Corollary 3.8. The following statements hold:
• If y satisfies conditions(i)and(ii)in Theorem3.5, then pA, K, C, I, e, yq defined by (3.7), (3.8), and (3.9) is a Kalman representation in causal coordinated form.
3.3. Computing Kalman representations in coordinated form 59 Theproofcan be found in Appendix3.B. Note that if condition(ii)in Theorem3.5
does not hold, then A, K, C and e can be calculated as above, but the process e is not necessarily white noise. Hence, if condition(ii)does not hold, then the tuple pA, K, C, I, e, yqdoes not necessarily define an LTI–SS representation.
The procedure above is elaborated in Algorithms6and7. Algorithm6takes an LTI–SS representation as its input and transforms it into a Kalman representation in causal coordinated form. Algorithm7 calculates the same representation from covariances of the output. Hence, by using empirical covariances it can be applied to data.
Algorithm 6Kalman representation in causal coordinated form based on LTI–SS representation
Input t ¯A, ¯B, ¯C, ¯D, Λv
0u: System matrices of an LTI–SS representation
p ¯A, ¯B, ¯C, ¯D, vqof y and variance of v
Output tA, K, Cu: System matrices of (3.1)
for i “ 1 : n ´ 1
Step 1 Consider the partition (3.6) and apply Algorithm 4 with input t ¯A, ¯B, ¯Ci, ¯Di, Λv0u. Denote its output by t ˆAi, ˆKi, ˆCiu, where
p ˆAi, ˆKi, ˆCi, I, ei,n, ryTi , y T ns
T
q is a minimal Kalman representation in block
triangular form.
Step 2If i ą 1, consider the partition (3.7) of ˆAi, ˆKi, ˆCi and define the
non-singular matrix Tias in (3.8).
end for
Step 3 Define A, K and C as in (3.1), such that the submatrices tAii, AinCii, CinKii, Kinui“1,...,n´1satisfy (3.9).
Remark 3.9(Correctness of Algorithm6and Algorithm7). Consider a ZMSIR pro-cess y “ ryT
1, . . . , ynTsT with covariance sequence tΛ y
ku8k“0 and an LTI–SS
repre-sentation p ¯A, ¯B, ¯C, ¯D, vqof y. Let e be the innovation process of y and N be any number larger than or equal to the dimension of a minimal LTI–SS representation of y. Assume that y satisfies conditions(i)and(ii)in Theorem3.5and note that Algo-rithms4and5calculate minimal Kalman representations in causal block triangular form (Remark2.7). Then it follows form Corollary3.8that if tA, K, Cu is the output of Algorithm6with input t ¯A, ¯B, ¯C, ¯D, Λv
0 “ ErvptqvTptqsu, then pA, K, C, I, eq is a
Kalman representation of y in causal coordinated form. In addition, pA, K, C, I, eq is minimal if and only if (3.5) holds. Similarly, by Corollary3.8, if tA, K, Cu is the output of Algorithm7with input tΛyku
2N
k“0, then pA, K, C, I, eq is a Kalman
Algorithm 7Kalman representation in causal coordinated form based on output covariances
Input tΛyku 2N
k“0: Covariance sequence of y ““y T 1, .., y
T n
‰T
Output tA, K, Cu: System matrices of (3.1)
for i “ 1 : n ´ 1
Step 1Denote the covariance matrix of yi,n“ ryTi, yTnsT with lag k by Λ yi,n
k “
Eryi,nyTi,ns.
Step 2Calculate the rank of the Hankel matrix formed by tΛyi,n
k u 2N ´1 k“0 and
denote it by Ni. Call Algorithm 5 for tΛ yi,n
k u 2Ni
k“1 and denote its output by
t ˆAi, ˆKi, ˆCiu.
Step 3Step 2 of Algorithm6.
end for
Step 4Step 3 of Algorithm6.
Remark 3.10. In view of Remark1.9and Remark2.8, the computational complexity of Algorithms7and6are polynomial. Algorithm6is polynomial in the dimensions of the state, output, and noise processes of the LTI–SS representation p ¯A, ¯B, ¯C, ¯D, vq. Algorithm7is polynomial in the number and size of the output covariances.
Remark 3.11(Checking(i)–(ii)). Algorithms6and7cannot be directly used to check conditions(i)and(ii)in Theorem3.5. However,(i)consists of unconditional Granger non-causality conditions, and by Remark3.7,(ii)can also be reformulated as uncon-ditional Granger non-causality conditions. Therefore, by Remark2.9, Algorithms4
and5can be used to check these conditions.
Note that Algorithms6 and7 operate in a distributed manner; they combine subsystems belonging to an agent and the coordinator for which the observation of any other agent is not needed. Furthermore, Algorithm7 only uses the covari-ances of the observed process, thus using empirical covaricovari-ances, it is suitable to es-timate Kalman representations in causal coordinated form based on data. Due to its distributed nature, when applied to empirical covariances, Algorithm7is possibly advantageous in terms of estimation error compared to non-distributed procedures.
3.4
Example for coordinated representation
In this section, we adopt a case study from (Kempker, 2012, Section 8.1) to illustrate the results of this chapter in a similar manner as we illustrated the results of Chap-ter2 in Section 2.4. The focus of this study is the dynamics of three underwater
3.4. Example for coordinated representation 61 vehicles that track a reference path in a fixed formation. Among the vehicles there is one acting as a coordinator that tracks a reference path and two others acting as agents that track the coordinator.
In comparison with (Kempker, 2012, Section 8.1) we made the following changes: (1) to ensure stationarity, the coordinator follows the zero position; (2) for conve-nience, we consider the movements of the vehicles along the first coordinate; (3) besides the position disturbance we include measurement noise.
We will show that the relative positions (concerning the formation) of the vehi-cles are ZMSIR processes that can be modeled by a minimal Kalman representation in causal coordinated form. In fact, we reverse engineer the coordinated network topology from the observed process in the following way: We verify that conditions
(i)and(ii)in Theorem3.5hold by calculating Granger non-causal relations based on Remark2.9. Then, we calculate a minimal Kalman representation in causal coor-dinated form using Algorithm7.
Model description Assume that we have three underwater vehicles V1, V2and Vc
where V1,V2act as agents and Vcacts as the coordinator. For j P t1, 2, cu denote the
first coordinate at time t P Z of the position, velocity, acceleration, position distur-bance and measurement noise of Vj by pjptq, sjptq, ajptq, wjptqand ˜wjptq,
respec-tively. Also, denote the first coordinate of the reference position and velocity of Vj
by pR
jptqand sRjptq, respectively. Let
pRcptq “ ´ppcptq ` ˜wcptqq
pRjptq “ ppcptq ` ˜wcptqq ` ∆j, j “ 1, 2.
That is, Vc follows the zero position based on its own measured position and for
j “ 1, 2, Vjfollows Vc in a distance ∆jbased on the same information. To shorten
the expressions, we neglect the dependencies on time. That is, for a process lptq we write l and we use σ to denote the forward time shift operator defined as follows: σlptq “ lpt ` 1q.
The dynamics of rpj, sjsT, j P t1, 2, cu is given by
σ„pj sj “„1 1 0 τ ´1 τ „pj sj `„ 01 τ aj` „1 0 wj, (3.10)
where ajis a control input and a τ is a time constant. The reference signals rpRj, s R js
T
are estimated by an observer with dynamics σ„ ˆp R j ˆ sR j “„1 ´ G p j 1 ´Gsj τ ´1τ „ ˆpRj ˆ sR j `„G p j Gs j pRj (3.11)
where Gpj, G s
jare constant gains. The linear feedback control is then given by
aj “ ” FjpFs j ı„pj´ ˆpR j sj´ ˆsRj .
Combining (3.10) and (3.11), we obtain the closed loop system
σ » — — – pj sj ˆ pR j ˆ sR j fi ffi ffi fl “ » — — — – 1 1 0 0 1 τF p j τ ´1τ ` 1 τF s j ´τ1F p j ´ 1 τF s j 0 0 1 ´ Gpj 1 0 0 ´Gsj τ ´1τ fi ffi ffi ffi fl loooooooooooooooooooooomoooooooooooooooooooooon Aj » — — – pj sj ˆ pR j ˆ sR j fi ffi ffi fl ` » — — – 0 0 Gpj Gs j fi ffi ffi fl loomoon Bj pRj ` » — — – 1 0 0 0 fi ffi ffi fl loomoon E wj. Note that xj :“ rpj ´ ∆j, sj, ˆpRj ´ ∆j, ˆsRjs
T has essentially the same dynamics as
rpj, sj, ˆpRj, ˆs R js
T for j “ 1, 2, namely σx
j “ Ajxj` BjpRc ` Ewj.
Assuming that v :“ rw1, w2, wc, ˜w1, ˜w2, ˜wcsT is a white noise process, we can
define the following LTI–SS representation pA, B, C, D, vq of the process y “ ry1, y2, ycsT, where y1“ p1´ ∆1, y2“ p2´ ∆2and y3“ pc: σ » – x1 x2 xc fi fl“ » – A1 0 B10 0 A2 B20 0 0 Ac´ BcF fi fl loooooooooooomoooooooooooon A » – x1 x2 xc fi fl` » – E 0 0 0 0 B1 0 E 0 0 0 B2 0 0 E 0 0 ´Bc fi fl looooooooooomooooooooooon B v » – y1 y2 yc fi fl“ » – ET 0 0 0 ET 0 0 0 ET fi fl loooooooomoooooooon C » – x1 x2 xc fi fl` » – 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 fi fl looooooomooooooon D v.
Parameter settings Following the approach of (Kempker, 2012), we take F1p “
F2p “ Fcp and F1s “ F2s “ Fcs as the solution of the linear quadratic problem
minact||z
2 1||
2
` α||a2c||2uwith respect to the dynamics σ„z1 z2 “„1 1 0 τ ´1 τ „z1 z2 `„ 01 τ ac.
For τ “ 2 and α “ 10 the optimal solution is F1p“ F p
2 “ Fcp“ ´0.3and F1s“ F2s“
Fcs “ ´0.5. The gain constants were chosen to be G p 1 “ 1.5, G s 1 “ 0.3, G p 2 “ 1.2, Gs 2 “ 0.1, G p Rc “ 0.9, and G s
Rc “ 0.5for which the matrix A is stable. Finally, the noise process v is chosen to be a normalized Gaussian white noise process.
3.4. Example for coordinated representation 63
Reverse engineering of the network graph Assume that the output process y of the LTI–SS representation S :“ pA, B, C, D, v, yq is observed. Using the result of this chapter, we will calculate a minimal Kalman representation of y in causal coor-dinated form. Note that we do not use prior knowledge of the coorcoor-dinated structure of the network graph. In fact, this representation reconstructs the network graph of S. Note that the network graph of S is a star graph with three nodes: one root node and two leaves where there is two directed edges from the root node to the leaves.
First, we check the Granger non-causal relations among the components of y using the covariance sequence tΛyku
2N
k“0of y where N is larger than or equal to the
dimension of a minimal LTI–SS representation of y. For this, we calculate a Kalman representation pA, K, C, I, eq of y and verify that y is coercive by checking that A ´ KC is invertible (see Section1.2.2). In view of Corollary2.6, a Granger non-causal relation can be verified by observing the output matrix K of Algorithm5: if the left lower block of the matrix K is zero, then an appropriate Granger non-causal relation holds (see Remark2.9). Following this method, we apply Algorithm 5 choosing the coordinator to be y1,y2,yc, ry1, y2sT, ry1, ycsT and ry2, ycsT, thus trying all the
possibilities. We obtain that ry1, y2sT does not Granger cause yc and yj does not
Granger cause ryi, ycsfor all i, j “ 1, 2, where i ‰ j, thus conditions(i)and(ii)in
Theorem3.5hold for the partition y “ ry1, y2, ycsT.
Second, in order to calculate a Kalman representation of y in coordinated form, we apply Algorithm7with the covariance sequence tΛyku
2N
k“0as its input.
Accord-ingly, first the minimal Kalman representations pAk,1, K1, Ck,1, I, e1,c, ry1, ycsTqand
pAk,2, K2, Ck,2, I, e2,c, ry2, ycsTqare calculated in causal block triangular form using
Algorithm4 with the covariances of ry1, ycsT and ry2, ycsT as its input. With our
parameter settings, these matrices are as follows:
Ak,1 “ « Ak,11 Ak,1c 0 Ak,cc ff “ » — — — — — — — — — — – 0.4 ´0.3 ´0.1 0.1 0.0 0.1 0.2 ´0.1 0.2 0.4 0.6 0.2 0.1 0.0 0.2 0.1 0.0 ´0.3 0.4 ´0.1 0.0 0.0 0.0 ´0.4 ´0.2 0 ´0.2 0.7 0.0 ´0.1 0.0 0.0 0 0 0 0 0.2 ´0.9 0.1 0.0 0 0 0 0 0.6 0.3 0.3 0.1 0 0 0 0 ´0.1 0.2 ´0.4 0.4 0 0 0 0 0.1 0.0 ´0.5 0.3 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl K1T “ « KT 11 0 K1cT KccT ff “ „ 0.1 ´0.1 0.0 ´0.3 0 0 0 0 0.2 ´0.1 0.0 ´0.1 0.1 0.2 0.2 ´0.1 Ck,1 “ « Ck,11 Ck,1c 0 Ccc ff “ „ ´0.4 ´0.3 0.4 ´1.8 0.2 ´0.2 0.0 0.2 0 0 0 0 0.5 0.0 ´0.2 0.0
Ak,2 “ « Ak,22 Aˆk,2c 0 Aˆk,cc ff “ » — — — — — — — — — — – 0.5 0.2 0 ´0.1 0 ´0.1 ´0.2 0.2 ´0.2 0.4 0.7 0.0 0.1 0.0 0.1 0.1 0.0 ´0.1 0.4 0.0 0.0 0.0 0.0 ´0.3 0.1 0.0 0.0 0.8 0.0 ´0.1 0.0 0.0 0 0 0 0 0.2 ´0.8 0.1 0.0 0 0 0 0 0.7 0.4 0.3 0.1 0 0 0 0 ´0.1 0.2 ´0.4 0.4 0 0 0 0 0.1 0.1 ´0.3 0.2 fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl K2T “ « KT 22 0 ˆ KT 2c KˆccT ff “ „ ´0.1 0.0 0.0 ´0.3 0 0 0 0 ´0.2 0.0 0.0 ´0.1 0.1 0.2 0.2 ´0.1 Ck,2 “ « Ck,22 Cˆk,2c 0 Cˆcc ff “ „ 0.5 ´0.1 ´0.1 ´1.9 0.2 ´0.1 0.0 0.1 0 0 0 0 0.5 ´0.1 ´0.2 0.0 .
Next, as in Step 3 of Algorithm7, we define a transformation matrix
T :“ ˆ ” Ck,23T Ak,23Ck,23T ıT˙´1” Ck,13T Ak,13Ck,13T ıT
with which, the output matrices tAk, K, Ckuof Algorithm7are calculated as below.
Ak “ » — — – Ak,11 0 Ak,1c 0 Ak,22 T ˆAk,2cT´1 0 0 Ak,cc fi ffi ffi fl K “ » — — – K11 0 A1c 0 K22 T ˆA2c 0 0 Kcc fi ffi ffi fl Ck“ » — — – Ck,11 0 Ck,1c 0 Ck,22Cˆk,2cT´1 0 0 Ck,cc fi ffi ffi fl .
In view of Remark3.9, the tuple Sk :“ pAk, K, Ck, I, e, yq, where e is the innovation
process of y, is a Kalman representation in causal coordinated form. Furthermore, it is easy to check that Skis minimal, which implies that condition(iii)in Theorem3.5
holds. The calculation of Sk only requires the second order statistics of the output
process and does not use prior knowledge of the network topology. Therefore, the construction of Sk shows that the network graph of S can be reverse engineered.
Moreover, by Theorem3.5, the reconstructed representation Sknot only shows the
coordinated structure but also characterizes the causal relations that describe the coordinated relationship in the observed process. The procedure can be repeated based on data, using empirical covariances which provide an estimation of Sk. Note
3.5. Conclusions 65
3.5
Conclusions
This chapter studies the relationship between coordinated state-space representa-tions and (conditional) Granger non-causality. Our results show that a collection of conditional and unconditional Granger non-causalities among the components of a process is equivalent to the existence of an LTI-SS representation with the star graph as its network graph, called Kalman representation in coordinated form. We provided algorithms for calculating this structured representation, in particular, cal-culating it from the covariance sequence of the observed output process. Hence, the results open up the possibility of calculating this representation from output data by using empirical covariances. In addition, the results deal with the minimality of the representations and the so-called coercive property of the output processes.
3.A
Proof of Lemmas
3.2
and
3.6
Proof of Lemma3.2. Consider a process y “ ryT
1, . . . , yTnsT where yi P Rri, for rią
0, i “ 1, . . . , n. Let pA, K, C, I, eq and p ˆA, ˆK, ˆC, I, eqbe two Kalman representations of y in causal coordinated form (3.1) with blocks Aij P Rpiˆpj, Kij P Rpiˆrj, Cij P
Rriˆpj and ˆA
ij P Rpiˆpj, ˆKij P Rpiˆrj, ˆCij P Rriˆpj for i, j “ 1, . . . , n. Let Si be
the Kalman representation (3.2), and let ˆSi be the counterpart of (3.2), obtained
by replacing Aij, Kij, Cij by the matrices ˆAij, ˆKij, ˆCij for j “ i, n, i “ 1, . . . , n.
From Definition3.1, it follows that Siand ˆSiare minimal Kalman representations of
ryTi , ynTsT in block triangular form, thus there exists an isomorphism Tifrom ˆSito
Si, i “ 1, . . . , n ´ 1. We will show that Tiis of the form
Ti“
„Tii Tin
0 Tnn
, Tij P Rpiˆpj for j P ti, nu.
This then implies that p ˆA, ˆK, ˆC, I, eqand pA, K, C, I, eq are isomorphic such as A “ T ˆAT´1, K “ T ˆKand C “ ˆCT´1with the matrix T defined by
T “ » — — — — — — – T11 0 ¨ ¨ ¨ 0 T1n 0 T22¨ ¨ ¨ 0 T2n .. . ... . .. ... ... 0 0 ¨ ¨ ¨ Tpn´1qpn´1qTpn´1qn 0 0 ¨ ¨ ¨ 0 Tnn fi ffi ffi ffi ffi ffi ffi fl .
Consider the partition Ti“
„ Tii Tin
TinTnn
, TklP Rpkˆplfor k, l P ti, nu.
By reordering the rows of the observability matricesOiand ˆOiof Siand ˆSiwe obtain
that Oi“ „OiOin 0 On , Oˆi“ « ˆOi Oˆin 0 Oˆn ff ,
where On and ˆOn are the observability matrices of pAnn, Cnnqand p ˆAnn, ˆCnnq,
re-spectively. SinceOiTi “ ˆOi, it follows that OnTin “ 0. Since Si is a Kalman
rep-resentation in causal block triangular form, pAnn, Cnnqis observable and therefore,
3.A. Proof of Lemmas3.2and3.6 67
Proof of Lemma3.6. To help the readability, we recall the conditions that the state-ments were formulated on:
(i) y1does not Granger cause y3
(ii) y2does not Granger cause y3
(iii) y1conditionally does not Granger cause y2with respect to y3
(iv) y1does not Granger cause ryT2, yT3sT
We first prove that(i)-(ii)-(iii)implies(iv). By definition,(iv)means that
El „„y2pt ` kq y3pt ` kq | Hy1,y2,y3 t´ “ El „„y2pt ` kq y3pt ` kq | Hy2,y3 t´
for all t, k P Z, k ě 0. Looking at the equation above by the components, by the first component it is equivalent to(iii)and by the second component it is equivalent to El“y3pt ` kq | Hyt´1,y2,y3
‰
“ El“y3pt ` kq | Hyt´2,y3‰. Since(iii)is assumed to hold, we
only need to see the latter. Define α “ y3pt ` kq ´ Elry3pt ` kq|H y3
t´s. From(i)and
from(ii)we respectively obtain that
α “ y3pt ` kq ´ Elry3pt ` kq|H y1,y3 t´ s, α “ y3pt ` kq ´ Elry3pt ` kq|H y2,y3 t´ s. Therefore, α is orthogonal to Hy1,y3 t´ and to H y2,y3
t´ , thus also to their closed union,
Hy1,y2,y3
t´ . Hence, Elrα|H y1,y2,y3
t´ s “ 0, which is equivalent to that
Elry3pt ` kq|H y1,y2,y3 t´ s “ Elry3pt ` kq|H y3 t´s, since Hy3 t´Ď H y1,y2,y3
t´ . Then, from(ii)we obtain that Elry3pt ` kq|H y3 t´s “ Elry3pt ` kq|Hy2,y3 t´ sand thus Elry3pt ` kq|H y1,y2,y3 t´ s “ Elry3pt ` kq|H y2,y3
t´ s. With this,(iv)
follows.
Next, we show that(ii)-(iv)implies(i)and(iii). As we saw above,(iv)is equiva-lent to the condition(iii)and that El“y3pt ` kq | H
y1,y2,y3 t´ ‰ “ El“y3pt ` kq | H y2,y3 t´ ‰.
Therefore, we only have to see that(ii)-(iv)implies(i). From(iii)and from(ii)we know that El“y3pt ` kq | Hyt´1,y2,y3 ‰ “ El“y3pt ` kq | Ht´y2,y3 ‰ El“y3pt ` kq | H y2,y3 t´ ‰ “ El“y3pt ` kq | H y3 t´ ‰
and thus El“y3pt ` kq | H y1,y2,y3
t´
‰
“ El“y3pt ` kq | H y3
t´‰. Taking the projection of
both part of the latter equation onto Hy1,y3
t´ , we obtain that El“y3pt ` kq | Hyt´1,y3
‰ “ El“y3pt ` kq | Hyt´3‰ which, by definition, gives(i).
3.B
Proof of Theorem
3.5
and Corollary
3.8
To prove Theorem3.5, we need an auxiliary result. For the sake of simplicity, a ZM-SIR process ryT j1, . . . , y T jks T is shortened by y j1,...,jkor by yJ where J “ tj1, . . . , jku and the Hilbert spaces generated by the present, past, and future values of yJ are
written by Hyj1,...,jk t , H yj1,...,jk t´ and H yj1,...,jk t` or by H yJ t , H yJ t´and H yJ t`, respectively.
Lemma 3.12. Consider a ZMSIR process y “ ryT
1, yT2, yT3, y4TsT. Then y1and y2
condi-tionally do not Granger cause y3with respect to y4if and only if ryT1, y2TsT conditionally
does not Granger cause y3with respect to y4.
Proof. Sufficiency: By definition, the joint process ryT
1, yT2sT conditionally does not
Granger cause y3with respect to y4if Elry3pt ` kq|H y3,4
t´ s “ Elry3pt ` kq|Hyt´sfor all
t, k P Z, k ě 0. By projecting both sides onto Hy1,3,4
t´ and to H y2,3,4 t´ we have that Elry3pt ` kq|H y3,4 t´ s “ Elry3pt ` kq|H y1,3,4 t´ s Elry3pt ` kq|H y3,4 t´ s “ Elry3pt ` kq|H y2,3,4 t´ s, (3.12) which implies that y1and y2conditionally does not Granger cause y3with respect
to y4.
Necessity: Define the process αpt ` kq :“ y3pt ` kq ´ Elry3pt ` kq|H y3,4
t´ s for
t, k P Z, k ě 0 . Then, αpt ` kq is orthogonal to Hy3,4
t´ and from the Granger
non-causality conditions we also know that αpt`kq is orthogonal to Hy1,3,4
t´ and to H
y2,3,4
t´ .
Therefore, αpt ` kq is orthogonal to the sum of the subspaces Hy1,3,4
t´ ` H y2,3,4 t´ “ Hy1,y2,y3,y4, thus to Hy t´. By projecting αpt ` kq onto H y t´we obtain that Elrαpt ` kq|Hyt´s “ 0thus Elry3pt ` kq|H y t´s “ Elry3pt ` kq|H y3,4
t´ s, which by definition is that
ryT1, y2TsT conditionally does not Granger cause y3with respect to y4.
Proof of Theorem3.5. To start with, any Kalman representation in causal coordi-nated form is a Kalman representation in coordicoordi-nated form, hence(v) ùñ(vi) fol-lows. We assume now that y “ ryT
1, . . . , yTnsT is a ZMSIR process where yi P Rri,
rią 0, i “ 1, . . . , n and we continue with the proof of the remaining implications.
(i)and(ii) ùñ(v): Condition(i)and Theorem2.5imply the existence of minimal Kalman representations p ˆAi, ˆKi, ˆCi, I, ei,nqof yi,n “ ryTi, y
T
nsT, i “ 1, . . . n ´ 1 in
causal block triangular form. Note that ei,n“ reTi , eTnsT, where eiP Rri, en P Rrn, is
the innovation process of yi,n. Furthermore, enis the innovation process of yn. By
using(i),(ii)and Lemma3.12, we obtain that yt1,2,...,n´1uztiudoes not Granger cause
yiwith respect to ynand yt1,2,...,n´1udoes not Granger cause yn, i.e.,
eiptq “ yiptq ´ Elryiptq|H yi,n
t´ s “ yiptq ´ Elryiptq|Hyt´s
3.B. Proof of Theorem3.5and Corollary3.8 69
It then follows that e “ reT
1, . . . , eTnsT is the innovation process of y. Consider the
partitions of ˆAifor ˆKi, ˆCi, i “ 1, . . . , n ´ 1 ˆ Ai “ « ˆAii Aˆin 0 Aˆi,nn ff ˆ Ki“ « ˆKii Kˆin 0 Kˆi,nn ff ˆ Ci“ « ˆCii Cˆin 0 Cˆi,nn ff
as in (3.7). Let Ti, i “ 2, . . . , n ´ 1 be the matrix as in (3.8) that transforms the
Kalman representation p ˆAi,nn, ˆKi,nn, ˆCi,nn, I, enqof yninto the isomorphic Kalman
representation p ˆA1,nn, ˆK1,nn, ˆC1,nn, I, enq of yn and define T1 “ I. Then define
Ain, Cin, Kin for i “ 1, . . . , n ´ 1 as in (3.9) and A, K, C as in (3.1). Note that the
stability of ˆAi, i “ 1, . . . , n ´ 1 implies the stability of A. Then, pA, K, C, I, eq is a
Kalman representation of y in coordinated form. Finally, since p ˆAi, ˆKi, ˆCi, I, ei,nqis
isomorphic with ˆ„Aii Ain 0 Ann ,„Kii Kin 0 Knn ,„Cii Cin 0 Cnn , I, ei,n ˙ (3.13)
by the isomorphism defined by the transformation matrix ”I 0
0 Ti
ı
, it follows that the LTI-SS representation (3.13) is also a minimal Kalman representation of yi,n in
causal block triangular form for all i “ 1, . . . , n ´ 1. As a result, pA, K, C, I, eq is a Kalman representation of y in causal coordinated form.
(iii) ùñ (iv): Consider the Kalman representation pA, K, C, I, eq of y in causal coordinated form which was constructed in the proof of(i)and(ii) ùñ(v). First, we show that pA, Cq is an observable pair thus rCT
pA ´ λIqTsT is full column rank for all λ P C. Minimality of (3.13) implies that pAii, Ciiqare observable pairs for
i “ 1, . . . , n so that the matrices rCT
ii pAii ´ λIqTsT are full column rank for all
λ P C. Notice that rCT
pA ´ λIqTsT can be transformed, by permuting the rows of rCT pA ´ λIqTsT, into an upper block triangular form such that the diagonal blocks are rCT
ii pAii´ λIqTsT for i “ 1, . . . , n. Hence, rCT pA ´ λIqTsT is full column rank
for all λ P C, which implies that pA, Cq is observable.
Below, we prove that if condition (iii)holds, then pA, Kq is controllable which is equivalent to that the components of the state process x “ rxT
1, . . . , x T ns
T, which
are consistent with (3.4) and (3.3), are linearly independent. Notice that the repre-sentations (3.13) are minimal Kalman representations. Hence, the components of xi,n “ rxTi, xTnsT are linearly independent and H
xi
t X H xn
t “ t0u for all t P Z and
i “ 1, . . . , n ´ 1. Therefore, for the linear independence of x it is enough to show that dimpHx
tq “
řn
i“1dimpH xi
t q, where dimpHztqdenotes the number of scalar
com-ponents of a basis in the Hilbert space Hz
tgenerated by the random variable zptq.
Re-call that the orthogonal complement of B Ď Hx
by A a B. The linear independence of the components of xi,nand HxtiX H xn
t “ t0u
imply that Hxi,n
t “ pH xi t a H xn t q ‘ H xn t and dimpH xi t a H xn t q “ dimpH xi t q. Next, by
using (3.5) in Theorem3.5, we show that Hx
t can be decomposed as Hx t “ H xn t ‘ ` pHx1 t a H xn t q 9` . . . 9`pH xn´1 t a H xn t q˘ , (3.14)
from which it follows that dimpHx tq “
řn
i“1dimpH xi
t q, i.e., the components of xptq
are linearly independent. Define the process Yiptq :“ ryTi ptq, . . . , yTipt ` N ´ 1qsT for
i “ 1, 2, . . . , nand notice that Yiptqspans Hyt`i. Consider the observability matrix
ONi :““CT
ii pCiiAiiqT . . . pCiiAN ´1ii qT
‰T
of pAii, Ciiq, i “ 1, . . . , n where N ě dimpxq and notice that because of the causal
coordinated form of pA, K, C, I, eq we have the following equations with an appro-priate M matrix: ElrYnptq|H yn t´s “ O N nxnptq, ElrYiptq|H yi,yn t´ s “ O N i xiptq ` M xnptq.
for i “ 1, . . . , n ´ 1. It implies that ElrHyt`n|H yn t´s Ď H xn t and ElrHyt`i|H yi,n t´ s Ď H xi,n t .
Since pAii, Ciiq is an observable pair, ONi has left inverse and we also have that
ElrHyt`n|Ht´yns Ě Htxnand ElrHyt`i|H yi,n t´ s ` H xn t Ě H xi t , i “ 1, . . . , n. Hence, Hxn t “ ElrH yn t`|H yn t´s, H xi,n t “ ElrH yi t`|H yi,n t´ s ` ElrH yn t`|H yn t´s. (3.15) Notice that ElrHyt`i|H yi,n t´ sXElrH yj t`|H yj,n t´ s|ElrHyt`n|H yn t´s “ t0u ðñ `ElrHyt`i|H yi,n t´ s`ElrHyt`n|Hyt´ns ˘ X`ElrH yj t`|H yj,n t´ s`ElrHyt`n|Hyt´ns ˘ |ElrHyt`n|Hyt´ns “ t0u
and similarly, that Hxi
t X H xj t |H xn t “ t0u ðñ H xi,n t X H xj,n t |H xn
t “ t0u. By using the
equations (3.15), we obtain that the condition (3.5) in Theorem3.5is equivalent to Hxi
t X H xj
t |H xn
t “ t0uwhich implies that Htxcan be decomposed as in (3.14). Hence,
the components of x are linearly independent and thus pA, Kq is controllable. By Proposition1.10, the observability of pA, Cq and the controllability of pA, Kq implies the minimality of the Kalman representation pA, K, C, I, e, yq.
(v) ùñ (i)and(ii): Assume that pA, K, C, I, eq is a Kalman representation of y in causal coordinated form where the matrices A, K, C are as in (3.1). Then, by defini-tion, (3.2) is a minimal Kalman representation of ryT
i , ynTsT in causal block triangular
form. Hence, by Theorem2.5, yidoes not Granger cause ynfor i “ 1, . . . , n ´ 1 and
3.B. Proof of Theorem3.5and Corollary3.8 71
not Granger cause ryT i , y
T
nsT for all i, j “ 1, . . . , n ´ 1, i ‰ j. By Lemma3.12, this
is equivalent to saying that ryT
1, . . . , yTi´1, yTi`1, . . . , yn´1T sT does not Granger cause
ryTi , ynTsT, which is further equivalent to reTi, eTnsT being the innovation process of
ryTi , ynTsT due (Dufour and Renault, 1998, Proposition 2.3), where e “ reT1, . . . , eTnsT
such that ei P Rri, i “ 1, . . . , n. Since (3.2) is a minimal Kalman representation of
ryTi , ynTsT, by Definition3.1, reTi, eTnsT is indeed the innovation process of ryiT, yTnsT
and thus condition(ii)follows.
(iv)ùñ(iii): Let pA, K, C, I, eq be a minimal Kalman representation of y in causal coordinated form and assume that A, K, C satisfy (3.1). Then for k ě 0
Elrynpt ` kq|H yn t´s “ CnnAknnxnptq Elryipt ` kq|H yi,n t´ s “ CiiAkiixiptq ` CinAknnxnptq, (3.16)
from which ElrHyt`n|Hyt´ns Ď Hxtnand ElrHyt`i|H yi,n
t´ s Ď H xi,n
t . Since the Kalman
rep-resentation pAnn, Knn, Cnn, I, en, ynqis minimal, the pair pAnn, Cnnqis observable.
Define the (finite) observability matrix of pAnn, Cnnqby
ONn :“ “ CT nnpCnnAnnqT . . . pCnnAN ´1nn qT ‰T ,
where N ě dimpxnq. Then from (3.16) we obtain that Elr
“ Ynptq ‰ |Hyn t´s “ O N nxnptq,
where Ynptq :“ rynTptq, . . . , ynTpt ` N ´ 1qsT. From the observability of pAnn, Cnnqwe
know that ON
n has left inverse and thus ElrH yn
t`|H yn
t´s “ H xn
t . Since the Kalman
rep-resentation pA, K, C, I, e, yq is minimal, the components of xptq are linearly indepen-dent for each t P Z. In particular, this means that Hxi,j
t X H xn t “ t0uand H xi t X H xj t “
t0ufor i, j “ 1, . . . , n´1, i ‰ j. In turn, this implies that Hxi,n
t X H xj,n t |H xn t “ t0u. By combining it with ElrH yi t`|H yi,n t´ s Ď H xi,n t and ElrH yn t`|H yn t´s “ H xn t we can conclude
that condition (3.5) in Theorem3.5holds.
(vi) ùñ (i)and(ii)if y is coercive: Assume that pA, K, C, I, eq is a Kalman repre-sentation of y in coordinated form satisfying (3.1). Since y is coercive,
eptq “ yptq ´
8
ÿ
k“1
CpA ´ KCqk´1Kypt ´ kq.
From (3.1) it is easy to see that for any k ě 1,
CpA ´ KCqk´1K “ » — — — — — – Mk,11 0 ¨ ¨ ¨ 0 Mk,1n 0 Mk,22¨ ¨ ¨ 0 Mk,2n .. . ... . .. ... ... 0 0 ¨ ¨ ¨ Mk,pn´1qpn´1q Mk,pn´1qn 0 0 ¨ ¨ ¨ 0 Mk,nn fi ffi ffi ffi ffi ffi fl ,
where Mk,ii“ CiipAii´ KiiCiiqk´1Kii, i “ 1, . . . , n and Mk,inare suitable matrices for i “ 1, . . . , n ´ 1. Hence, enptq “ ynptq ´ 8 ÿ k“1 Mk,nnynpt ´ kq eiptq “ yiptq ´ 8 ÿ k“1 Mk,iiyipt ´ kq ` Mk,inynpt ´ kq, where eiptq “ yiptq ´ Elryiptq | H y
t´sfor i “ 1, . . . , n. This implies that Elrynptq |
Hyt´s Ď Hyt´n and Elryiptq | Ht´y s Ď H yi,n t´ . Therefore, Elrynptq | Hyt´s “ Elrynptq | Hyn t´sand Elryiptq | Ht´y s “ Elryiptq | H yi,n
t´ s. It follows from (Dufour and Renault,
1998, Proposition 2.3) that yidoes not Granger cause yn, and yj does not Granger
cause yi,nfor all i, j P t1, . . . , n ´ 1u, i ‰ j. In view of Remark3.7this is equivalent
to(i)and(ii).
Proof of Corollary3.8. Consider the LTI-SS representation pA, K, C, I, eq defined by (3.7), (3.8), and (3.9) before Corollary3.8. Then pA, K, C, I, eq coincides with the Kalman representation defined in the proof of(i) and(ii) ùñ (v)of Theorem 3.5. Hence, the first statement of Corollary3.8 is a consequence of the implication(i)
and(ii) ùñ(v)of Theorem3.5. Similarly, the second statement of Corollary3.8is a direct consequence of the implication(iii)ùñ(iv)of Theorem3.5.
