Estimation in Networks with Spatiotemporally Correlated Noise

Abstract

In this article, we address the problem of estimating a particular transfer function in a dynamic network where the unknown noise processes are potentially correlated across the nodes. It is assumed that the noise correlations are affine in essence. We model the spatial correlation between the noise processes of two nodes of the network as a new hidden node that influences the two nodes. To be able to apply the notion of $d$-separation from graph theory, we further manipulate the network by adding another fictitious node and slightly altering its structure in a systematic way. The time series generated by a subset of the nodes in this new larger network are equivalent to the time series generated by the original network. In this new larger network, based on the notion of $d$-separation, we formulate sufficient graphical conditions to select a set of predictor inputs. We prove that the selected set of predictor inputs guarantees a consistent estimation of the transfer function of interest using a prediction error method.

I. Introduction

Numerous techniques have been proposed to address the problem of identifying dynamic networks by making use of observational data [1]–[4]. Some recent results have shown that it is possible to obtain a consistent estimate of a particular module in a dynamic network by selecting an appropriate set of additional predictor inputs based on the prediction error method or similar identification techniques [5]–[8]. By consistent identification, it is meant that as the number of data points used increases indefinitely, the estimated parameters converge in probability to their actual values.

These works, however, make the assumption that the random noise processes influencing the nodes of the network are independent across the nodes. Applying such techniques to networks with spatially correlated noise processes will, in general, result in a biased estimate of the module.

Considering networks with spatially correlated noise processes, in [9] a multi-input multi-output (MIMO) identification framework is proposed where the target transfer function is embedded in a MIMO structure. The challenge is the handling of confounders that are created by spatially correlated noise processes and nodes for which no time-series are available. Similarly, in [10] weighted null space fitting and multi-step sequential linear regression techniques are extended to address networks with spatially correlated noise processes, and estimate the noise correlation structure in the full measurement situation.

In [11], the problem of topology identification under spatially correlated noise processes is studied. It is shown that under the assumption that noise processes are affine in essence, networks with spatially correlated noise processes could be transformed to larger networks with hidden nodes where the noise processes are not correlated. Such transformations are not unique and could result in networks with a variety of structures.

In this paper, we address the problem of estimating a module in a network where the noise processes are temporally auto-correlated as well as spatially correlated across the nodes of the network. Unlike [11] where the authors aim to identify the topology of the network, we assume that the topology is partially known. Manipulating the underlying graphical representation and the noise correlation structure of the network, we obtain an augmented graph where the noise processes are spatially uncorrelated, and we can apply the notion of $d$-separation from graph theory. The independence relations we obtain from $d$-separation, enable us to formulate sufficient conditions to select a set of predictor inputs. We prove that the selected set of predictor inputs guarantees consistent estimation of the module of interest using a proposed multi-input single-output prediction error algorithm.

The proposed method can be applied to complex networks involving feedback loops and confounding variables which usually complicate the identification procedure. A main feature of the developed technique is that the proposed conditions for selecting the predictor inputs are graph-theoretic. This enables, unlike other works, design of systematic algorithms to find a set that satisfies the required conditions.

The article is organized as follows. Section II provides some definitions about dynamic networks and graph theory. Section III presents the results for estimation in networks with noise processes that are temporally auto-correlated but spatially independent. Section IV formulates the problem of estimating a module in networks with noise processes that are temporally and spatially correlated. Section V provides a systematic procedure to transform a network with spatially correlated noise processes to a larger network with hidden nodes and uncorrelated noise processes. Section VI presents sufficient conditions to select a set of predictor inputs that guarantees consistent estimation of a module in networks with noise processes that are temporally and spatially correlated. Concluding remarks are given in Section VIII.

II. Preliminaries

In this section, we introduce our notation, the class of models that we are going to consider in this paper, and some concepts from the area of graphical models.

A. A Model Class for Dynamic Networks

The class of dynamic networks considered in this article is similar to the models considered and investigated in [7], [8], [12]–[14].

Definition 1. A network $𝓖$ is a pair $\left(H\left(z\right),n_V\right)$, $V=\left\{1,\dots ,v\right\}$ where $H\left(z\right)$ is a proper rational discrete-time $v\times v$ transfer matrix and $n_V$ is a vector of $v$ zero-mean stochastic processes such that ${\text{Φ}}_{n_V}$ is rational and potentially non-diagonal. The output signals of the network are defined by the relation

$$\begin{array}{cc}{x}_j\left(t\right)=n_j\left(t\right)+\sum _{i\in V}{H}_{ji}\left(z\right)x_i\left(t\right),& forj=1,\dots ,v.\end{array}$$ (1)

We can represent the model in a more compact way as

$$x_V\left(t\right)=n_V\left(t\right)+H\left(z\right)x_V\left(t\right).$$ (2)

We associate two graphs to a network. 1) An undirected graph to represent the structure of correlations across the noise processes. 2) A directed graph to describe the sparsity pattern of $H\left(z\right)$ in model 1 along with some information about the locations of delays in the entries in $H\left(z\right)$.

Definition 2. Let $𝓖=\left(H,n\right)$ be a network with output processes $x_V$ described by (1). The noise correlation graph $G^c=\left(V,E^c\right)$ associated with $𝓖$ is an undirected graph where $V=\left\{1,2,\cdots ,v\right\}$ is the set of nodes and $E^c\subseteq V\times V$ is the set of edges such that $\left(i,j\right)\notin E^c$ implies that $n_i\left(t\right)$ and $n_j\left(t\right)$ are uncorrelated.

In other words, presence of an edge $\left(i,j\right)\in E^c$ implies that $n_i\left(t\right)$ and $n_j\left(t\right)$ are spatially correlated.

Definition 3. Let $𝓖=\left(H,n\right)$ be a network with output processes $x_V$, where $V:=\left\{1,\dots ,v\right\}$, and let $E_1$ and $E_2$ be two disjoint subsets of $V\times V$ such that

1. $i\to j\notin E_1\cup E_2$ implies $H_{ji}\left(z\right)=0$

2. $i\to j\notin E_1$ implies $H_{ji}\left(z\right)$ is strictly proper.

We say that the multi-arrowed graph $G=\left(V,E_1,E_2\right)$ where $E_1$ denotes the set of single-headed edges and $E_2$ denotes the set of double-headed edges, is a graphical representation of the network.

In a graph $G$, node $j$ is a child of node $i$ if the edge $i\to j$ is present in the graph. We also say that $i$ is a parent of $j$. We denote the set containing all children of node $j$ by ${\text{ch}}_G\left(j\right)=\left\{v\in V|j\to v\in E\right\}$ and the set containing all parents of node $i$ by ${\text{pa}}_G\left(i\right)=\left\{v\in V|v\to i\in E\right\}$. Moreover for a set $A\subseteq V$ we define ${\text{ch}}_G\left(A\right)={\cup }_{j\in A}{\text{ch}}_G\left(j\right)$ and ${\text{pa}}_G\left(A\right)={\cup }_{i\in A}{\text{pa}}_G\left(i\right)$. Similarly, node $j$ is a descendant of node $i$ if $j=i$ or if there is a directed path from $i$ to $j$. Equivalently, we say that $i$ is an ancestor of $j$. We denote the set containing all descendants of node $i$ as ${\text{de}}_G\left(i\right)$ and the set containing all ancestors of node $i$ as ${\text{an}}_G\left(i\right)$.

Definition 4. We say that the multi-arrowed graph $G=\left(V,E_1,E_2\right)$ is recursive if in every directed loop there is at least one double-headed edge.

The class of networks considered in this article are assumed not to have algebraic loops. It is also assumed that we know the location of at least one strictly proper transfer function in each loop. The following definition of graph of instantaneous propagations is an important tool to deal with the presence of direct feed-throughs.

Definition 5. Consider a multi-headed graph $G=\left(V,E_1,E_2\right)$. Its associated graph of instantaneous propagations, denoted as $G$, is the standard directed graph $\left(V,E_1\right)$ obtained from $G$ by removing the double-headed edges.

As shown in [7], there is a strong relationship between signal estimators and graphical representations in a network. Such a relationship will play a central role in the development of our results. For this reason we recall some fundamental notions from estimation theory and introduce our notation. Note that in our notation $x_A\left(t\right)$ denotes a set of processes indexed by the elements of the set $A$.

Definition 6. Given a probability space, for a set of stochastic processes $x_A$ where $A\subseteq V$, we denote the natural filtration generated by the processes $x_A$ up to time $t$ as $I_A\left(t\right)$.

Denoting the set of real-rational causal modules that are analytic and invertible on the unit circle by ${𝓕}^{+}$, we can interpret $I_A\left(t\right)$ as the rational causal transfer span:

$$I_A\left(t\right):=\left\{q=\sum _{i\in A}{P}_i\left(z\right)x_i|P_i\left(z\right)\in {𝓕}^{+}\right\}.$$

We also denote as ${\text{Φ}}_{x_A}$, the power spectral density matrix of processes $x_i\left(t\right)$, $i\in A$. In this article we typically consider the estimate ${\widehat{x}}_j\left(t\right)$ of $x_j\left(t\right)$ from two sets of processes, $x_{D^{+}}$ and $x_{D^{-}}$. The information of the processes in $x_{D^{+}}$ is used up to time $t$ while the information of the processes in $x_{D^{-}}$ is used up to time $t-1$. Using the notation introduced in Definition 6 the mean squared error estimate ${\widehat{x}}_j\left(t\right)$ can be written as

$${\widehat{x}}_j\left(t\right)=\mathbb{E}\left(x_j\left(t\right)|I_{D^{+}}\left(t\right),I_{D^{-}}\left(t-1\right)\right).$$ (3)

In the linear Gaussian case this estimation problem can be solved via Wiener filters, reducing (3) to

$${\widehat{x}}_j=\sum _{k\in D^{+}}{W}_{jk}\left(z\right)x_k+\sum _{k\in D^{-}}{W}_{jk}\left(z\right)x_k$$ (4)

where $W_{jk}\left(z\right)$ for $k\in D^{+}$ are proper transfer functions and for $k\in D^{-}$ are strictly proper transfer functions. So long as ${\text{Φ}}_{x_{\left\{j\right\}\cup D^{+}\cup D^{-}}}$ is the same, the expressions of the Wiener filter components are the same when considering a mean squared error estimation even in the linear non-Gaussian case. Throughout the article, we assume for simplicity that all the processes are jointly Gaussian even though the same results can be easily shown to hold in the linear non-Gaussian case, as well.

B. d-separation

Given a path $\pi$ in a graph $G$ we say that a node $j$ is a fork, when there exist two consecutive edges in the path of the form $i\leftarrow j$ and $j\to k$, a collider (or an inverted fork), when there exist two consecutive edges in the path of the form $i\to j$ and $j\leftarrow k$, a chain link, when there exist two consecutive edges in the path of the form $i\to j$ and $j\to k$. Specifically, the notion of colliders allows one to define if a path $\pi$ is blocked by a set $Z$.

Definition 7. In a directed graph $G$, a path $\pi$ between nodes $i$ and $j$ is blocked by a set of nodes $Z$ if there is a non-collider on $\pi$ that belongs to $Z$; or there is a collider $c$ on $\pi$ such that $d{e}_G\left(c\right)\cap Z=\varnothing$. Otherwise, we say that the path $\pi$ is activated by $Z$.

In the theory of graphical models, a fundamental concept defined over the nodes of a directed graph is $d$-separation.

Definition 8. In a directed graph $G=\left(V,E\right)$ let $A$, $B$, and $C$ be disjoint subsets of $V$. $A$ and $B$ are $d$-separated by $C$ if for all nodes $a\in A$ and $b\in B$, all paths between $a$ and $b$ are blocked by $C$. If $A$ and $B$ are not $d$-separated by $C$ in $G$, we say that they are $d$-connected by $C$ in $G$.

III. Temporally correlated noise

In this section we present some techniques to identify a certain transfer function in networks where the noise processes are spatially uncorrelated. We consider two cases. In the first case, we assume that the the noise processes are temporally and spatially uncorrelated. In the second case, we assume that the noise processes are spatially uncorrelated but temporally correlated.

The following result provides sufficient conditions to consistently estimate a transfer function in a network with noise processes that are temporally and spatially uncorrelated.

Theorem III.1. Consider a dynamic network $𝓖=\left(H,n\right)$ with a recursive graphical representation $G=\left(V,E_1,E_2\right)$. Suppose ${\text{Φ}}_{n_V}$ is real and diagonal. Let $P^{+}=p{a}_{G^{↯}}\left(j\right)$ and $P^{-}=p{a}_G\left(j\right)\backslash P^{+}$. Then, the Wiener filter component $W_{ji}\left(z\right)$ obtained from

$$\mathbb{E}\left(x_j\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}}\left(t-1\right)\right)=\sum _{k\in P^{+}\cup P^{-}}{W}_{jk}\left(z\right)x_k\left(t\right)$$ (5)

is a consistent estimate of $H_{ji}\left(z\right)$ when ${\text{Φ}}_{x_{\left\{i,j\right\}\cup P^{+}\cup P^{-}}}$ is non-singular.

Proof. Since ${\text{Φ}}_{n_V}$ is real and diagonal, we have that $n_a⫫n_b$ for $a\ne b$. First suppose $k\in P^{+}$. Since $G$ is recursive, there is at least a double-headed arrow in every directed path from $j$ to $k$ avoiding algebraic loops. Therefore, we have

$$n_j\left(t\right)⫫I_{P^{+}}\left(t\right).$$ (6)

Now suppose $k\in P^{-}$. Then, by causality we have that

$$n_j\left(t\right)⫫I_{P^{-}}\left(t-1\right).$$ (7)

Combining (6) and (7) we have

$$\mathbb{E}\left(n_j\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}}\left(t-1\right)\right)=0.$$ (8)

On the other hand the output of the target node $j$ is given by

$$x_j\left(t\right)=n_j\left(t\right)+\sum _{k\in {\text{pa}}_G\left(j\right)}{H}_{jk}\left(z\right)x_k\left(t\right).$$ (9)

Therefore, we can write

$$\begin{array}{l}\mathbb{E}\left(x_j\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}}\left(t-1\right)\right)=\hfill \\ \mathbb{E}\left(n_j\left(t\right)+\sum _{k\in P^{+}\cup P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}}\left(t-1\right)\right)=\hfill \\ \mathbb{E}\left(n_j\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}}\left(t-1\right)\right)+\hfill \\ \mathbb{E}\left(\sum _{k\in P^{+}\cup P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}}\left(t-1\right)\right)=\hfill \\ \sum _{k\in P^{+}\cup P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right).\hfill \end{array}$$ (10)

Because ${\text{Φ}}_{x_{\left\{i,j\right\}\cup P^{+}\cup P^{-}}}$ is full rank, comparing (5) and (10) we can conclude that $W_{jk}\left(z\right)=H_{jk}\left(z\right)$ for $k\in P^{+}\cup P^{-}$.

Note that in (5) the Wiener filter components $W_{jk}\left(z\right)$ are strictly proper for $P^{-}$ and proper for $P^{+}$.

For example, consider a network with six nodes in a feedback loop with a graphical representation shown in Figure 1 (a). It is known that the transfer function $H_{32}\left(z\right)$ is strictly proper. This is represented by the double-headed arrow $2\to 3$ in the graph of Figure 1 (a). Suppose that the aim is the identification of the transfer function $H_{21}\left(z\right)$ given the time series data $x_i\left(t\right)$, $i\in V\left\{1,2,\dots ,6\right\}$.

Fig. 1.

(a) The graphical representation $G$ of a six-node network (b) The noise correlation graph $G^c$ associated with $G$.

Assume that $n_i\left(t\right)$, $i\in V\left\{1,2,\dots ,6\right\}$ are mutually independent White Gaussian noise processes. Applying Theorem III.1, for the target node $j=2$ we have $P^{+}=\left\{1\right\}$ and $P^{-}=\left\{\varnothing \right\}$. Then, if we estimate $x_2\left(t\right)$ using the information of $x_1\left(t\right)$ up to time $t$,

$$\mathbb{E}\left(x_2\left(t\right)|I_1\left(t\right)\right)=W_{21}\left(z\right)x_1\left(t\right),$$ (11)

the Wiener filter component $W_{21}\left(z\right)$ corresponding to $x_1\left(t\right)$ will be a consistent estimate of $H_{21}\left(z\right)$.

Now, we consider the case where the noise processes $n_j\left(t\right)$, $j\in V$ are mutually independent but could be potentially colored. That is, $n_j\left(t\right)$ could be temporally auto-correlated with its own past.

Theorem III.2. Consider a dynamic network $𝓖=\left(H,n\right)$ with a recursive graphical representation $G=\left(V,E_1,E_2\right)$. Suppose ${\text{Φ}}_{n_V}$ is diagonal. Let $P^{+}=p{a}_{G^{↯}}\left(j\right)$ and $P^{-}=p{a}_G\left(j\right)\backslash P^{+}$. Then, the quantity ${\widehat{H}}_{ji}\left(z\right)=W_{ji}\left(z\right)/\left(1-W_{jj}\left(z\right)\right)$ where the Wiener filter components $W_{ji}\left(z\right)$ and $W_{jj}\left(z\right)$ were obtained from

$$\mathbb{E}\left(x_j\left(t\right)|I_{P^{+}}\left(t\right),I_{P^{-}\cup \left\{j\right\}}\left(t-1\right)\right)=\sum _{k\in P^{+}\cup P^{-}\cup \left\{j\right\}}{W}_{jk}\left(z\right)x_k\left(t\right)$$ (12)

is a consistent estimate of $H_{ji}\left(z\right)$ when ${\text{Φ}}_{x_{\left\{i,j\right\}\cup P^{+}\cup P^{-}}}$ is non-singular.

Proof. Note that in (12) the Wiener filter components $W_{jk}\left(z\right)$ are strictly proper for $P^{-}\cup \left\{j\right\}$ and proper for $P^{+}$. Since $n_j\left(t\right)=x_j\left(t\right){\sum }_{k\in P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)+{\sum }_{k\in P^{+}}{H}_{jk}\left(z\right)x_k\left(t\right)$, we have that

$$\begin{array}{l}\mathbb{E}\left(x_j\left(t\right)|I_{P^{+}}\left(t\right),I_{\left\{j\right\}\cup P^{-}}\left(t-1\right)\right)=\hfill \\ \mathbb{E}\left(\sum _{k\in \left\{j\right\}\cup P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)|I_{P^{+}}\left(t\right),I_{\left\{j\right\}\cup P^{-}}\left(t-1\right)\right)+\hfill \\ \mathbb{E}\left(\sum _{k\in P^{+}}{H}_{jk}\left(z\right)x_k\left(t\right)|I_{P^{+}}\left(t\right),I_{\left\{j\right\}\cup P^{-}}\left(t-1\right)\right)+\hfill \\ \mathbb{E}\left(n_j\left(t\right)|I_{P^{+}}\left(t\right),I_{\left\{j\right\}\cup P^{-}}\left(t-1\right)\right)=\hfill \\ \sum _{k\in P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)+\sum _{k\in P^{+}}{H}_{jk}\left(z\right)x_k\left(t\right)+W_{jj}\left(z\right)n_j\left(t\right)=\hfill \\ \sum _{k\in P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)+\sum _{k\in P^{+}}{H}_{jk}\left(z\right)x_k\left(t\right)+\hfill \\ W_{jj}\left(z\right)\left[x_j\left(t\right)-\sum _{k\in P^{-}}{H}_{jk}\left(z\right)x_k\left(t\right)+\sum _{k\in P^{+}}{H}_{jk}\left(z\right)x_k\left(t\right)\right]=\hfill \\ W_{jj}\left(z\right)x_j\left(t\right)+\sum _{k\in P^{-}}\left[1-W_{jj}\left(z\right)\right]H_{jk}\left(z\right)x_k\left(t\right)+\hfill \\ \sum _{k\in P^{+}}\left[1-W_{jj}\left(z\right)\right]H_{jk}\left(z\right)x_k\left(t\right).\hfill \end{array}$$ (13)

Because ${\text{Φ}}_{x_{\left\{i,j\right\}\cup P^{+}\cup P^{-}}}$ is full rank, comparing (12) and (13) we can conclude that $W_{jk}\left(z\right)=\left[1-W_{jj}\left(z\right)\right]H_{jk}\left(z\right)$ for $k\in P^{+}\cup P^{-}$. In particular, for $k=i$ we have $W_{ji}\left(z\right)=\left[1-W_{jj}\left(z\right)\right]H_{ji}\left(z\right)$ which completes the proof.

For example, consider the problem of estimating the transfer function $H_{21}\left(z\right)$ in the network of Figure 1 (a) given the time series data $x_i\left(t\right)$, $i\in V=\left\{1,2,\dots ,6\right\}$ when the noise processes are spatially uncorrelated but temporally correlated. In this case, $W_{21}\left(z\right)$ obtained from (11) is going to be, in general, a biased estimate of $H_{21}\left(z\right)$. Based on Theorem III.2, however, having $i=1$, $j=2$, $P^{+}=\left\{1\right\}$, and $P^{-}=\left\{\varnothing \right\}$, if we estimate $x_2\left(t\right)$ using the information of $x_1\left(t\right)$ up to time $t$ and the information of $x_2\left(t\right)$ up to time $t-1$,

$$\mathbb{E}\left(x_2\left(t\right)|I_1\left(t\right),I_2\left(t-1\right)\right)=W_{21}\left(z\right)x_1\left(t\right)+W_{22}\left(z\right)x_2\left(t\right),$$ (14)

the quantity ${\widehat{H}}_{21}\left(z\right)=W_{21}\left(z\right)/\left(1-W_{22}\left(z\right)\right)$ where $W_{21}\left(z\right)$ and $W_{22}\left(z\right)$ are obtained from (14), will be a consistent estimate of $H_{21}\left(z\right)$.

Note that, when $n_i\left(t\right)$, $i\in V$ are mutually independent White Gaussian noise processes, Theorem III.2 is reduced to Theorem III.1.

IV. Spatially correlated noise and problem formulation

In this section , we discuss how spatially correlated noise processes complicate the identification of dynamic networks and formally cast the problem that is the main focus of this article.

Similar to the previous section, suppose that the goal is the identification of the transfer function $H_{21}\left(z\right)$ in the network of Figure 1 (a) given the time series data $x_i\left(t\right)$, $i\in V=\left\{1,2,\dots ,7\right\}$. Unlike the scenarios considered in the previous section where the noise processes were assumed to be pairwise independent, now we consider a scenario where the noise processes $n_j\left(t\right)$, $i\in V=\left\{1,2,\dots ,7\right\}$ could potentially be spatially correlated as well as temporally correlated. That is, ${\text{Φ}}_{n_V}$ could potentially be non-diagonal.

Figure 1 (b) shows the undirected noise correlation graph $G^c$ of the network of Figure 1 (a) which characterizes the dependence structure of the noise processes $n_j\left(t\right)$. The presence of the edge $\left(1,3\right)\in V^c$ in the noise correlation graph means that the noise processes $n_1\left(t\right)$ and $n_3\left(t\right)$ are potentially correlated. Similarly, the presence of the edge $\left(2,5\right)\in V^c$ in the noise correlation graph means that the noise processes $n_2\left(t\right)$ and $n_5\left(t\right)$ are potentially correlated.

In this scenario, the estimates we obtain for the transfer function $H_{21}\left(z\right)$ from (11) or (14) will be, in general, biased. This motivates development of a general method to address the problem of estimating a module in networks with spatially correlated noise processes. Formally, we pose this problem as follows.

Problem 1. Consider a dynamic network $𝓖=\left(H\left(z\right),n\right)$ with a graphical representation $G=\left(V,E_1,E_2\right)$ and a noise correlation graph $G^c=\left(V^c,E^c\right)$ with $i$, $j\in V$ . Given the time series data $x_V\left(t\right)$, obtain a consistent estimate of the module $H_{ji}\left(z\right)$.

In the following sections, we provide a solution for Problem 1.

V. Spatial correlation to hidden node transformation

In this section, we show how networks with spatially correlated noise processes could be transformed into larger networks with spatially uncorrelated noise processes.

Assumption 1. In a network $𝓖=\left(H\left(z\right),n\right)$, defined by (1), the noise processes $n_i\left(t\right)$ and $n_j\left(t\right)$ are correlated only via affine interactions.

It is shown in [11] that assuming the correlations are affine and the availability of knowledge of the noise correlation topology, it is possible to transform a network with spatially correlated noise processes to a network with appended agents. These appended agents are represented by hidden nodes for which no associated time-series are available. In the new network, however, all the noise processes are spatially uncorrelated.

Such transformations are not unique. A certain noise correlation structure could be transformed into variety of networks with spatially uncorrelated noise processes with different degrees of complexities. In this article, we consider a particular transformation where the transformed network $\overline{𝓖}$ has $\left|V\right|+\left|E^c\right|$ nodes and $\left|E_1\right|+\left|E_2\right|+2\left|E^c\right|$ edges.

Theorem V.1. Consider a network $𝓖=\left(H\left(z\right),n\right)$ with graphical representation $G=\left(V,E_1,E_2\right)$, noise correlation graph $G^c=\left(V^c,E^c\right)$, and output processes $x_V$ described by (2). Let $\overline{𝓖}=\left(\overline{H},{\overline{n}}_{\overline{V}}\right)$ be a transformed network of $𝓖$ with $\left|V\right|+\left|E^c\right|$ nodes and output processes ${\overline{x}}_{\overline{V}}$, where

$$\overline{V}:=V\cup \underset{{\lambda }_{ab}=\left|V\right|+1}{\overset{\left|V\right|+\left|E^c\right|}{\cup }}\left\{{\lambda }_{ab}|\left(a,b\right)\in E^c\right\}.$$ (15)

Then, under Assumption 1, there is an $F\in {𝓕}^{\left|V\right|\times \left|E^c\right|}$ and mutually uncorrealted processes ${\overline{n}}_{\overline{V}}$ such that

1. ${\overline{x}}_{\overline{V}}\left(t\right)=\left[\begin{array}{cc}H\left(z\right)& F\left(z\right)\\ 0& 0\end{array}\right]{\overline{x}}_{\overline{V}}\left(t\right)+{\overline{n}}_{\overline{V}}$

2. ${\overline{x}}_V\left(t\right)=x_V\left(t\right)$

3. ${\overline{H}}_{ji}\left(z\right)=H_{ji}\left(z\right)$ for $i$, $j\in V$.

Also, the graph $\overline{G}=\left(\overline{V},{\overline{E}}_1,{\overline{E}}_2\right)$ where

$${\overline{E}}_1:=E_1\cup \cup \left\{{\lambda }_{ab}\to a,{\lambda }_{ab}\to b|\left(a,b\right)\in E^c\right\}$$ (16)

$${\overline{E}}_2:=E_2.$$ (17)

is a graphical representation of $\overline{𝓖}$.

Proof. Let $\Lambda$ be the set enumerating $\left|E^c\right|$ new nodes ${\lambda }_{ab},\left(a,b\right)\in E^c$. For every edge $\left(a,b\right)\in E^c$ decompose $n_a\left(t\right)$ and $n_b\left(t\right)$ into three mutually independent processes ${\overline{n}}_a^{k_a}\left(t\right)$, ${\lambda }_{ab}\left(t\right)$, and ${\overline{n}}_b^{k_b}\left(t\right)$, where $k_a$ counts the number of edges connected to node $a$ in $G^c$, such that $n_a\left(t\right)=F_{a,{\lambda }_{ab}}\left(z\right){\lambda }_{ab}\left(t\right)+{\overline{n}}_a^{k_a}\left(t\right)$. This can be done because under Assumption 1 the correlation between $n_a\left(t\right)$ and $n_b\left(t\right)$ is affine. When $k_a$ reaches the number of neighbors of node $a$ in $G^c$, let ${\overline{n}}_a^{k_a}\left(t\right)={\overline{n}}_a\left(t\right)$. Repeating this sequentially, we will have

$$n_V\left(t\right)={\overline{n}}_V\left(t\right)+F\left(z\right){\overline{n}}_{\Lambda }\left(t\right).$$ (18)

By concatenation of processes ${\overline{n}}_V\left(t\right)$ and ${\overline{n}}_{\Lambda }\left(t\right)$ create

$$\tilde{n}\left(t\right)=\left[\begin{array}{l}{\overline{n}}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right].$$ (19)

Note that all processes in $\tilde{n}\left(t\right)$ are mutually independent. Let

$$T\left(z\right)=\left[I_v,F\left(z\right)\right]$$ (20)

where $I_v$ is a $v\times v$ identity matrix. Then we have

$$x_V\left(t\right)=H\left(z\right)x_V\left(t\right)+T\left(z\right)\tilde{n}\left(t\right).$$ (21)

Simple algebraic manipulations leads to

$$\begin{gathered} {\overline{x}}_{\overline{V}}\left(t\right)=\left[\begin{array}{l}{\overline{x}}_V\left(t\right)\\ {\lambda }_{\Lambda }\left(t\right)\end{array}\right]=\left[\begin{array}{l}{x}_V\left(t\right)\\ {\lambda }_{\Lambda }\left(t\right)\end{array}\right]=\left[\begin{array}{c}H\left(z\right)x_V\left(t\right)+T\left(z\right)\tilde{n}\left(t\right)\\ {\lambda }_{\Lambda }\left(t\right)\end{array}\right]= \\ \left[\begin{array}{c}H\left(z\right)x_V\left(t\right)+T\left(z\right)\tilde{n}\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]=\left[\begin{array}{c}H\left(z\right)x_V\left(t\right)+T\left(z\right)\left[\begin{array}{c}{\overline{n}}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]= \\ \left[\begin{array}{c}H\left(z\right)x_V\left(t\right)+\left[I_V,F\left(z\right)\right]\left[\begin{array}{c}{\overline{n}}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]= \\ \left[\begin{array}{c}H\left(z\right)x_V\left(t\right)+F{\overline{n}}_{\Lambda }+{\overline{n}}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]=\left[\begin{array}{c}H\left(z\right)x_V\left(t\right)+F\left(z\right){\overline{n}}_{\Lambda }\\ 0\end{array}\right]+\left[\begin{array}{c}{\overline{n}}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right] \\ =\left[\begin{array}{cc}H\left(z\right)& F\left(z\right)\\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]+\left[\begin{array}{c}{\overline{n}}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]=\left[\begin{array}{cc}H\left(z\right)& F\left(z\right)\\ 0& 0\end{array}\right]{\overline{x}}_{\overline{V}}\left(t\right)+{\overline{n}}_{\overline{V}} \end{gathered}$$ (22)

where ${\overline{x}}_{\overline{V}}\left(t\right)=\left[\begin{array}{c}{x}_V\left(t\right)\\ {\overline{n}}_{\Lambda }\left(t\right)\end{array}\right]$ and ${\overline{n}}_{\overline{V}}\left(t\right)=\tilde{n}\left(t\right)$. It follows from Definition 3 that the graph described by (15) and (16) is the graphical representation of the network $\left(\overline{H},\overline{n}\right)$ with output processes ${\overline{x}}_{\overline{V}}\left(t\right)$ where $\overline{H}=\left[\begin{array}{cc}H\left(z\right)& F\left(z\right)\\ 0& 0\end{array}\right]$.

Theorem V.1 says that for every edge $\left(a,b\right)\in E^c$ in the noise correlation graph $G^c$ we add a new node ${\lambda }_{ab}$ to the transformed network. This new node ${\lambda }_{ab}$ which models the correlation between the noise processes $n_a\left(t\right)$ and $n_b\left(t\right)$, does not have any parents in the new network. It also has nodes $a$ and $b$ as its only children.

For example, Figure 2 shows the graphical representation $\overline{G}$ of the transformed network corresponding to the network with graphical representation $G$ shown in Figure 1 (a) and noise correlation graph $G^c$ shown in Figure 1 (b). Since there is an edge $\left(1,3\right)\in E^c$ in the noise correlation graph graph $G^c$, we have a new node ${\lambda }_{13}$ with two outgoing edges to nodes 1 and 3 in $\overline{G}$. Similarly, since there is an edge $\left(2,5\right)\in E^c$ in the noise correlation graph graph $G^c$, we have a new node ${\lambda }_{25}$ with two outgoing edges to nodes 2 and 5 in $\overline{G}$.

Fig. 2.

Transformed graph $\overline{G}$ corresponding to graph $G$ and noise correlation graph $G^c$ of Figure 1.

Based on Theorem V.1 we have $V\subset \overline{V}$. For these overlapping nodes $V$, we have that the time series ${\overline{x}}_V\left(t\right)$ generated by $V$ in the transformed network are the same with the time series $x_V\left(t\right)$ generated by the original network. Moreover, the overlapping transfer functions ${\overline{H}}_{ab}\left(z\right)$ for $a,b\in V$ in the transformed network $\overline{𝓖}$ are equivalent with their counterparts $H_{ab}\left(z\right)$ for $a,b\in V$ in the original network $𝓖$. What is changed is the unknown noise processes.

The following result guarantees that the transformed network obtained from Theorem V.1 has a recursive graphical representation.

Theorem V.2. Consider a dynamic network $𝓖$ with a graphical representation $G=\left(V,E_1,E_2\right)$ and noise correlation graph $G^c=\left(V^c,E^c\right)$. Let $\overline{𝓖}=\left(\overline{H},{\overline{n}}_{\overline{V}}\right)$ be a transformed network of $𝓖$ with a graphical representation $\overline{G}=\left(\overline{V},{\overline{E}}_1,{\overline{E}}_2\right)$ as described in Theorem V.1. If $G$ is recursive, then $\overline{G}$ is recursive.

Proof. By contradiction, suppose there exists a directed feedback loop $L=\left\{l_1\to \cdots \to l_i\to \cdots \to l_1\right\}$ for $l_i\in \overline{V}$ in ${\overline{G}}^{↯}$. We consider two cases. In the first case, suppose all the nodes involved in $L$ are in $V$. Then, the same directed feedback loop $L$ would exist in $G^{↯}$ which is a contradiction with the fact that $G$ is recursive. In the second case, suppose there is at least a node $l_i$ involved in $L$ such that $l_i\notin V$. Then, based on the way that $\overline{G}$ is constructed (see Theorem V.1), $l_i$ can only be of the form $\pi =\left\{l_{i-1}\leftarrow l_i\to l_{i+1}\right\}$ for some $\left(l_{i-1},l_{i+1}\right)\in E^c$. However, the path $\pi$ cannot be a part of a directed feedback loop because $l_i$ is a fork in $\pi$ and directed feedback loops are comprised of only chain links which is a contradiction.

VI. Module identification with spatiotemporally correlated noise

In this section we present a method to estimate a particular transfer function in a network in which the noise processes are temporally and spatially correlated.

To be able to prove the main result of this section we need a few lemmas. Denoting independence with $\perp$, the following lemma presents decomposition and contraction properties of conditional independence in a graphoid.

Lemma VI.1. Let $A$, $B$, $C$, and $D$ be subsets of $V$ and consider filtrations $I_A$, $I_B$, $I_C$, $I_D$ and $I_V$.

1. $I_A\perp I_{B\cup C}|I_D\iff I_A\perp I_B|I_D$ and $I_A\perp I_C|I_D$

2. $I_A\perp I_B|I_C$ and $I_A\perp I_D|I_{B\cup C}\Rightarrow I_A\perp I_{B\cup D}|I_C$

Proof. The claims could be easily proved using the properties of ordinary orthogonality.

The following lemma provides an equivalency condition for natural filtrations in a dynamic network of related processes.

Lemma VI.2. Suppose $x_k$, $k\in A=\left\{j\right\}\cup B$ is a set of rationally related processes. Let

$$w=\sum _{k\in A}{H}_k\left(z\right)x_k$$ (23)

with $H_k\left(z\right)$ being proper modules that are analytic and invertible on the unit circle $\left|z\right|=1$. Then we have that $H_j^{-1}\left(z\right)$ is proper if and only if

$$I_{\left\{w\right\}\cup B}\left(t\right)=I_A\left(t\right).$$ (24)

Proof. For necessity suppose $H_j^{-1}\left(z\right)$ is proper. It follows from $w={\sum }_{k\in A}{H}_k\left(z\right)x_k$ that $w\in I_A$ which gives us $I_{\left\{w\right\}\cup B}\subseteq I_A$ since $B\subset A$. Now we show $I_{\left\{w\right\}\cup B}\left(t\right)\supseteq I_A\left(t\right)$. Note that $H_j\left(z\right)x_j=w-{\sum }_{k\in B}{H}_k\left(z\right)x_k$. For $H_j\left(z\right)\ne 0$ we get $x_j=H_j^{-1}\left(z\right)\left[w-{\sum }_{k\in B}{H}_k\left(z\right)x_k\right]$. Thus, $x_j\in I_{\left\{w\right\}\cup B}\left(t\right)$ since $H_j^{-1}\left(z\right)$ is proper. This gives us $I_{\left\{j\right\}}\subseteq I_{\left\{w\right\}\cup B}\left(t\right)$ and consequently $I_{\left\{w\right\}\cup B}\left(t\right)\supseteq I_A\left(t\right)$. For sufficiency suppose $I_{\left\{w\right\}\cup B}\left(t\right)=I_A\left(t\right)$. Since $j\in A$, we need to have $x_j\in I_{\left\{w\right\}\cup B}$. It follows from $x_j=H_j^{-1}\left(z\right)\left[w-{\sum }_{k\in B}{H}_k\left(z\right)x_k\right]$ that $H_j^{-1}\left(z\right)$ is proper.

Based on the notion of $d$-separation, the following result gives a criterion to discard some of the predictor inputs when estimating a node of a network.

Lemma VI.3. Let $A$, $B$, and $Z$ be three disjoint subsets of $V$ in a network with a graphical representation $G=\left(V,E_1,E_2\right)$. If $Z$ $d$-separates sets $A$ and $B$ in $G$, then for $a\in A$ and $b\in B$ the following holds.

$$\mathbb{E}\left(x_a\left(t\right)|I_{B,Z}\left(t\right)\right)=\mathbb{E}\left(x_a\left(t\right)|I_Z\left(t\right)\right).$$ (25)

By symmetry of $d$-separation we also have

$$\mathbb{E}\left(x_b\left(t\right)|I_{A,Z}\left(t\right)\right)=\mathbb{E}\left(x_b\left(t\right)|I_Z\left(t\right)\right).$$ (26)

Proof. Combining Lemma 15 and Lemma 16 of [15] leads to (25). (26) follows from the fact that $d$-separation is symmetric.

Lemmas VI.1, VI.2, and VI.3 enable us to formulate sufficient conditions to select a set of predictor inputs $Z$ to estimate $H_{ji}\left(z\right)$.

Theorem VI.4. Consider a dynamic network $𝓖$ with a recursive graphical representation $G=\left(V,E_1,E_2\right)$ and correlation graph $G^c=\left(V^c,E^c\right)$. Let $\overline{𝓖}=\left(\overline{H},{\overline{n}}_{\overline{V}}\right)$ be a transformed network of $𝓖$ with a graphical representation $\overline{G}=\left(\overline{V},{\overline{E}}_1,{\overline{E}}_2\right)$ as described in Theorem V.1. Let $K_1:=p{a}_{\overline{G}}\left(j\right)\backslash i$ and $K_2:=p{a}_{\overline{G}}\left(j\right)\backslash \left\{i\right\}\cup K_1$. Define an augmented graph $G^{\prime }=\left(V^{\prime },E_1^{\prime },E_2^{\prime }\right)$ obtained from $\overline{G}$ as follows.

$$V^{\prime }:=\overline{V}\cup \left\{q\right\}$$

$$E_1^{\prime }:=\left({\overline{E}}_1\cup \left\{q\to j\right\}\cup \left\{k\to q|k\in K_1\right\}\right)\backslash \left\{k\to j|k\in K_1\right\}$$

$$E_2^{\prime }:=\left({\overline{E}}_2\cup \left\{k\to q|k\in K_2\right\}\right)\backslash \left\{k\to j|k\in K_2\right\}.$$

Consider a set $Z\subset V$ such that $Z\cup \left\{q\right\}$ $d$-separates $p{a}_{G^{\prime }}\left(q\right)\backslash Z$ from $\left\{i,j\right\}$ in $G^{\prime }$. Then, the output of Algorithm 1 with $Z$ is a consistent estimate of $H_{ji}\left(z\right)$ when ${\text{Φ}}_{x_{\left\{i,j\right\}\cup Z}}$ is full rank.

Proof. It follows from Theorem V.2 that $\overline{G}$ is recursive. First we show that since $\overline{G}$ is recursive, $G^{\prime }$ is also recursive. By contradiction, suppose there exists a directed feedback loop $L^{\prime }=\left\{l_1\to \cdots \to l_i\to \cdots \to l_1\right\}$ for $l_i\in V^{\prime }$ in ${\overline{G}}^{\prime ↯}$. We consider two cases. In the first case, suppose all the nodes involved in $L^{\prime }$ are in $\overline{V}$. That is, $q$ is not in $L^{\prime }$. Then, the same directed feedback loop $L$ would exist in ${\overline{G}}^{↯}$ which is a contradiction with the fact that $\overline{G}$ is recursive. In the second case, suppose $q$ is involved in $L^{\prime }$. Then, based on the way that $G^{\prime }$ is constructed, $L^{\prime }$ can only be of the form $L^{\prime }=\left\{l_1\to \cdots \to p\to q\to j\to \cdots \to l_1\right\}$ for some $p\in {\text{pa}}_{\overline{G}}\left(j\right)$. However, in that case the path $\overline{L}=\left\{l_1\to \cdots \to p\to j\to \cdots \to l_1\right\}$ would exist in ${\overline{G}}^{↯}$ which is a contradiction with the fact that $\overline{G}$ is recursive. Now let $q$ be a new variable such that $x_q\left(t\right)+H_{ji}\left(z\right)x_i\left(t\right)=x_j\left(t\right)$. Also, define sets $Z^{-}$ and $Z^{+}$ as follows. Let $Z^{-}$ be the set containing all the elements $\lambda$ of $Z$ such that there is no path from $\lambda$ to $j$ in $G^{\prime }$. Let $Z^{+}=Z\backslash Z^{-}$. Suppose $i\in D^{-}$. Since $Z\cup \left\{q\right\}$ $d$-separates $\left\{i,j\right\}$ from ${\text{pa}}_{\overline{G}}\left(q\right)\backslash Z$ in $G^{\prime }$, by Lemma VI.3 and Lemma VI.1 we got

$$I_i\left(t-1\right)\perp x_q\left(t\right)|I_{Z^{-}\cup \left\{q\right\}}\left(t-1\right),I_{Z^{+}}\left(t\right).$$ (27)

Thus, when we estimate $x_q\left(t\right)$ using $\left(I_{Z^{-}\cup \left\{i,q\right\}}\left(t-1\right),I_{Z^{+}}\left(t\right)\right)$, the Wiener filter component associated with $x_i$ will be zero:

$$\begin{array}{l}\mathbb{E}\left(x_q\left(t\right)|I_{Z^{-}\cup \left\{i,q\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)\right)=\hfill \\ W_{qq}\left(z\right)x_q\left(t\right)+\sum _{k\in Z^{-}}{W}_{qk}\left(z\right)x_k\left(t\right)+\sum _{l\in Z^{+}}{W}_{ql}\left(z\right)x_l\left(t\right),\hfill \end{array}$$ (28)

where $W_{qq}\left(z\right)$ and $W_{qk}\left(z\right)$, $k\in Z^{-}$ are strictly proper modules, and $W_{ql}\left(z\right)$, $l\in Z^{+}$ are proper modules. From Lemma VI.2 we have that $I_{Z^{-}\cup \left\{i,q\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)=I_{Z^{-}\cup \left\{i,j\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)$. Therefore, we can write

$$\begin{array}{l}\mathbb{E}\left(x_j\left(t\right)|I_{Z^{-}\cup \left\{i,j\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)\right)=\hfill \\ \mathbb{E}\left(x_q\left(t\right)+H_{ji}\left(z\right)x_i\left(t\right)|I_{Z^{-}\cup \left\{i,j\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)\right)=\hfill \\ H_{ji}\left(z\right)x_i\left(t\right)+\mathbb{E}\left(x_q\left(t\right)|I_{Z^{-}\cup \left\{i,j\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)\right)=\hfill \\ H_{ji}\left(z\right)x_i\left(t\right)+\mathbb{E}\left(x_q\left(t\right)|I_{Z^{-}\cup \left\{i,q\right\}}\left(t-1\right)\cup I_{Z^{+}}\left(t\right)\right)=\hfill \\ H_{ji}\left(z\right)x_i\left(t\right)+T_{qq}\left(z\right)x_q\left(t\right)+\sum _{k\in Z^{-}}{T}_{qk}\left(z\right)x_k\left(t\right)+\sum _{l\in Z^{+}}{T}_{ql}\left(z\right)x_l\left(t\right)=\hfill \\ H_{ji}\left(z\right)x_i\left(t\right)+T_{qq}\left(z\right)x_q\left(t\right)-H_{ji}\left(z\right)x_i\left(t\right))+\hfill \\ \sum _{k\in Z^{-}}{T}_{qk}\left(z\right)x_k\left(t\right)+\sum _{l\in Z^{+}}{T}_{ql}\left(z\right)x_l\left(t\right)=\hfill \\ T_{qq}\left(z\right)x_j\left(t\right)+\left[H_{ji}\left(z\right)-T_{qq}\left(z\right)H_{ji}\left(z\right)\right]x_i\left(t\right)\hfill \\ +\sum _{k\in Z^{-}}{T}_{qk}\left(z\right)x_k\left(t\right)+\sum _{l\in Z^{+}}{T}_{ql}\left(z\right)x_l\left(t\right),\hfill \end{array}$$ (29)

where $H_{ji}\left(z\right)$ is a strictly proper module. In (29), the third equality follows from Lemma VI.2. Since ${\text{Φ}}_{x_{\left\{i,j\right\}\cup Z}}$ is non-singular, we get

$$\begin{array}{ccc}{W}_{jk}\left(z\right)=T_{qk}\left(z\right)& \text{for}\text{all}& k\in Z^{-}\cup \end{array}{Z}^{+},$$ (30)

$$W_{jj}\left(z\right)=T_{qq}\left(z\right),$$ (31)

and

$$W_{ji}\left(z\right)=H_{ji}\left(z\right)-T_{qq}\left(z\right)H_{ji}\left(z\right)=H_{ji}\left(z\right)-W_{jj}\left(z\right)H_{ji}\left(z\right),$$ (32)

where $H_{ji}\left(z\right)$ is strictly proper. This completes the proof for the scenario $i\in D^{-}$. The scenario where $i\in D^{+}$ can be proven similarly.

$$\overline{\underset{¯}{\begin{array}{l}\underset{¯}{Algorithm1\text{Transfer}\text{function}\text{identification}}\\ \text{1:}Input:\text{Augmented}\text{graph}{G}^{\prime },\text{Set}Z\\ \text{2:}Output:{\widehat{H}}_{ji}\left(z\right)\\ \text{3:}\text{Partition}Z\cup \left\{i,j\right\}\text{into}{D}^{+}\text{and}{D}^{-}:\\ D^{+}:={\text{an}}_{G^{\prime ↯}}\left(j\right)\cap \left(Z\cup \left\{i\right\}\right)\\ D^{-}:=\left(Z\cup \left\{i,j\right\}\right)\backslash D^{+}\\ \text{4:}\text{Compute}\text{the}\text{coefficients}{w}_{j,k}\left(d\right)\text{minimizing}\text{the}\text{least}\\ \text{square}\text{error}\text{of}{\widehat{x}}_j\left(t\right)\text{from}{x}_j\left(t\right)\\ {\widehat{x}}_j\left(t\right)=\sum _{k\in D^{+}}\sum _{d\ge 0}{w}_{j,k}\left(d\right)x_k\left(t-d\right)+\sum _{k\in D^{-}}\sum _{d\ge 1}{w}_{j,k}\left(d\right)x_k\left(t-d\right)\\ \text{5:}\text{Compute}\\ \begin{array}{cc}{W}_{ji}\left(z\right)=\sum _{d\ge 0}{w}_{j,i}\left(d\right)z^{-d}& W_{jj}\left(z\right)=\sum _{d\ge 1}{w}_{j,j}\left(d\right)z^{-d}\end{array}\\ \text{6:}\text{Compute}{\widehat{H}}_{ji}\left(z\right):=\frac{w_{ji}\left(z\right)}{1-W_{jj}\left(z\right)}\end{array}}}$$

Theorem VI.4 creates an augmented graph $G^{\prime }$ by manipulating the graphical representation $\overline{G}$ of the transformed network of $𝓖$ obtained from Theorem V.1 by adding a new fictitious node $q$. In $G^{\prime }$, the target node $j$ has only two parents, nodes $i$ and $q$. All the parents of $j$ in $\overline{G}$ except $i$ are parents of $q$ in $G^{\prime }$. This graphical manipulation, enables application of the notion of $d$-separation in the augmented graph $G^{\prime }$ and consistent estimation of $H_{ji}\left(z\right)$ using Algorithm 1.

As inputs, Algorithm 1 takes the augmented graph $G^{\prime }$ and the set of predictor inputs $Z$, which is characterized by Theorem VI.4. Set $D^{+}$ defined in line 3 of Algorithm 1 contains the nodes in $Z\cup \left\{i,j\right\}$ that are used in the prediction of $x_j\left(t\right)$ up to time $t$. Similarly, set $D^{-}$ defined in line 3 of Algorithm 1 contains the nodes in $Z\cup \left\{i,j\right\}$ that are used in the prediction of $x_j\left(t\right)$ up to time $t-1$. Note that this partitioning of $Z\cup \left\{i,j\right\}$ into $D^{+}$ and $D^{-}$ is crucial. If we mistakenly used a node that should have been in $D^{+}$ in the prediction of $x_j\left(t\right)$ up to time $t-1$ instead of up to time $t$, our estimate of $H_{ji}\left(z\right)$ would be, in general, biased. Lines 5 and 6 of Algorithm 1 show how we can compute the Wiener filter components $W_{jk}\left(z\right)$, $k\in D^{+}\cup D^{-}$ using a prediction error method. Algorithm 1 returns an estimate ${\widehat{H}}_{ji}\left(z\right)$ of $H_{ji}\left(z\right)$ which is consistent if the set of predictor inputs $Z$ satisfies the conditions of Theorem VI.4.

For example, Figure 3 shows the augmented graph $G^{\prime }$ corresponding to the transformed graph $\overline{G}$ of Figure 2 when the goal is to estimate the transfer function $H_{21}\left(z\right)$. By Theorem VI.4, a set $Z$ such that $Z\cup \left\{q\right\}$ $d$-separates {1, 2} and ${\lambda }_{25}$ guarantees consistent estimation of $H_{21}\left(z\right)$ using Algorithm 1. As can be seen in Figure 3, however, the choice of predictor inputs set $Z$ is not unique. For instance, if we were looking for a set of predictor inputs with minimal cardinality, four different sets, {4, 5}, {3, 5}, {3, 6}, and {4, 6} satisfy conditions of Theorem VI.4 ( $d$-separating {1, 2} and ${\lambda }_{25}$). Therefore, application of Algorithm 1 using any of theses sets leads to a consistent estimate of the module $H_{21}\left(z\right)$

Fig. 3.

The augmented graph $G^{\prime }$ corresponding to the transformed graph $\overline{G}$ of Figure 2 when the goal is to estimate the transfer function $H_{21}\left(z\right)$. By Theorem VI.4, a set $Z$ such that $Z\cup \left\{q\right\}$ $d$-separates {1, 2} and ${\lambda }_{25}$ guarantees consistent estimation of $H_{21}\left(z\right)$.

VII. Simulations

This section aims to investigate our variable selection method’s estimation performance when dealing with finite data. We illustrate the consistency properties of our theoretical results through a numerical example.

Consider the network $𝓖$ with a graphical representation shown in Figure 1 (a) and a correlation graph shown in Figure 1 (b). Figure 4 (a) shows, as an example, a realization of noise processes $n_2\left(t\right)$ and $n_5\left(t\right)$ which are correlated. Figure 4 (b) shows processes ${\overline{n}}_2\left(t\right)$, ${\overline{n}}_5\left(t\right)$, and ${\lambda }_{25}\left(t\right)$ which are mutually independent (see Theorem V.1 and spatial correlation to hidden nodes transformation described in Section V).

Fig. 4.

(a) A realization of noise processes $n_2\left(t\right)$ and $n_5\left(t\right)$ which are correlated. (b) Processes ${\overline{n}}_2\left(t\right)$, ${\overline{n}}_5\left(t\right)$, and ${\lambda }_{25}\left(t\right)$ which are mutually independent.

The goal is the estimation of the transfer function $H_{21}\left(z\right)$ using Algorithm 1. In order to verify the consistency property of the proposed estimation method when selecting a set of predictor inputs that satisfies the conditions of Theorem VI.4, we generated time-series data by numerically simulating the network $𝓖$. We considered a predictor inputs set and computed the variance and the bias of the estimated modules, based on the generated time-series data.

Considering a parameterization $H\left(z,\theta \right)$ of $𝓖$, we indicate the subset of parameters corresponding to the module $H_{21}\left(z\right)$ by ${\theta }_{21}$ and the estimated parameters by ${\widehat{\theta }}_{21}$.

We selected a predictor inputs set $Z_1=\left\{3,5\right\}$ that satisfies the graphical conditions of Theorem VI.4 and ran Algorithm 1 for time-series with different lengths. We simulated the network $𝓖$ and performed a linear regression method to compute ${\widehat{\theta }}_{21}$ for each and every time-series length and for each set of predictor inputs sets. Repeating this 10³ times, we estimated $\mathbb{E}\left({\widehat{\theta }}_{21}\right)$ and the covariance matrix of ${\widehat{\theta }}_{21}$.

Figure 5 shows the results of the Monte Carlo simulations for $Z_1$. The horizontal axis represents different lengths of time-series. Red squares indicate the estimates of $\mathbb{E}{‖{\theta }_{21}-{\widehat{\theta }}_{21}‖}_1$. We can see from Figure 5 that for larger number of measurements this quantity approaches zero which numerically confirms that the bias of estimated parameters ${\widehat{\theta }}_{21}$ approaches zero asymptotically. The semi-amplitude of the interval defined by the blue candlesticks is equal to the square root of the trace of the estimate of the covariance matrix of ${\widehat{\theta }}_{21}$. As can be seen in Figure 5, for larger number of measurements the amplitudes of these intervals approaches zero which numerically confirms that our estimate is consistent.

Fig. 5.

Estimation performance of the predictor set $Z_1=\left\{3,5\right\}$ for different number of measurements.

VIII. Conclusion

Most techniques developed in the literature to estimate a module in a dynamic network assume that the unknown noise processes influencing the nodes of the network are independent. Potential correlations across the noise processes could introduce bias in the estimate of the module. We investigated this problem in scenarios with different noise structures: 1) when the noise processes were independent temporally and spatially, 2) when the noise processes were independent spatially but correlated temporally, 3) and when the noise processes were spatiotemporally correlated in an affine way. More complex techniques were required as the structural complexity of noise processes increased. The proposed techniques were based on the prediction of the output node using the information of the input node along with the information of a set of additional predictor inputs selected from the nodes of the network. Sufficient graphical conditions to select the set of additional predictor inputs were formulated guaranteeing consistent estimation of the module of interest.