A local direct method for module identification in dynamic networks with correlated noise

In recent years increasing attention has been given to the development of new tools for the identification of large-scale interconnected systems, also known as dynamic networks. These networks are typically thought of as a set of measurable signals (the node signals) interconnected through linear dynamic systems (the modules), possibly driven by external excitations (the reference signals). Among the literature on this topic, we can distinguish three main categories of research. The first one focuses on identifying the topology of the dynamic network [1, 2, 3, 4, 5]. The second category concerns identification of the full network dynamics [6, 7, 8, 9, 10, 11], including aspects of identifiability, particularly addressed in [12, 13, 14], while the third one deals with identification of a specific component (module) of the network, assuming that the network topology is known (the so called local module identification), see [15, 16, 17, 18, 19, 20].

In this paper we will further expand the work on the local module identification problem. In [15], the classical direct-method [21] for closed-loop identification has been generalized to a dynamic network framework using a MISO identification setup. Consistent estimates of the target module can be obtained when the network topology is known and all the node signals in the MISO identification setup are measured. The work has been extended in [22, 23, 24] towards the situation where some node signals might be non-measurable, leading to an additional predictor input selection problem. A similar setup has also been studied in [18], where an approach has been presented based on empirical Bayesian methods to reduce the variance of the target module estimates. In [16] and [19], dynamic networks having node measurements corrupted by sensor noise have been studied, and informative experiments for consistent local module estimates have been addressed in [20].

A standing assumption in the aforementioned works [15], [18], [20], [23] is that the process noises entering the nodes of the dynamic network are uncorrelated with each other. This assumption facilitates the analysis and the development of methods for local module identification, reaching consistent module estimates using the direct method. However, when process noises are correlated over the nodes, the consistency results for the considered MISO direct method collapse. In this situation it is necessary to consider also the noise topology or disturbance correlation structure, when selecting an appropriate identification setup. Even though the indirect and two-stage methods in [16, 20] can handle the situation of correlated noise and deliver consistent estimates, the obtained estimates will not have minimum variance.

In this paper we particularly consider the situation of having dynamic networks with disturbance signals on different nodes that possibly are correlated, while our target moves from consistency only, to also minimum variance (or Maximum Likelihood (ML)) properties of the obtained local module estimates. We will assume that the topology of the network is known, as well as the (Boolean) correlation structure of the noise disturbances, i.e. the zero-elements in the spectral density matrix of the noise. While one could use techniques for full network identification (e.g., [8]), our aim is to develop a method that uses only local information. In this way, we avoid (i) the need to collect node measurements that are “far away” from the target module, and (ii) the need to identify unnecessary modules that would come with the price of higher variance in the estimates.

Using the reasoning first introduced in [25], we build a constructive procedure that, choosing a limited number of predictor inputs and predicted outputs, builds an identification setup that guarantees maximum likelihood (ML) properties (and thus asymptotic minimum variance) when applying a direct prediction error identification method. In this situation we have to deal with so-called confounding variables (see e.g. [25], [26]), that is, unmeasured variables that directly or indirectly influence both the predicted output and the predictor inputs, and lead to lack of consistency. The effect of confounding variables will be mitigated by extending the number of predictor inputs and/or predicted outputs in the identification setup, thus including more measured node signals in the identification. Preliminary results for the particular “full input” case have been presented in [27]. Here we generalize that reasoning to different node selection schemes, and provide a generally applicable theory that is independent of the particular node selection scheme selected.

This paper is organized as follows. In section II, the dynamic network setup is defined. Section III provides a summary of available results from the existing literature of local module identification related to the context of this paper. Next, important concepts and notations used in this paper are defined in Section IV while the MIMO identification setup and main results are presented in subsequent sections. Sections VII-IX provide algorithms and illustrative examples for three different ways of selecting input and output node signals: the full input case, the minimum input case, and the user selection case. This is followed by Conclusions. The technical proofs of all results are collected in the Appendix.

Following the basic setup of [15], a dynamic network is built up out of $L$ scalar internal variables or nodes $w_{j}$, $j=1,\ldots,L$, and $K$ external variables $r_{k}$, $k=1,\ldots K$. Each internal variable is described as:

where $q^{-1}$ is the delay operator, i.e. $q^{-1}w_{j}(t)=w_{j}(t-1)$,

• •

  $G_{jl}$ are proper rational transfer functions, referred to as modules;

• •

  There are no self-loops in the network, i.e. nodes are not directly connected to themselves $G_{jj}=0$;

• •

  $u_{j}(t)$ is generated by the external variables $r_{k}(t)$ that can directly be manipulated by the user and is given by $u_{j}(t)=\sum_{k=1}^{K}R_{jk}r_{k}(t)$ where $R_{jk}$ are stable, proper rational transfer functions;

• •

  $v_{j}$ is process noise, where the vector process $v=[v_{1}\cdots v_{L}]^{T}$ is modelled as a stationary stochastic process with rational spectral density $\Phi_{v}(\omega)$, such that there exists a white noise process $e:=[e_{1}\cdots e_{L}]^{T}$, with covariance matrix $\Lambda>0$ such that $v(t)=H(q)e(t)$, where $H(q)$ is square, stable, monic and minimum-phase. The situation of correlated noise, as considered in this paper, refers to the situation that $\Phi_{v}(\omega)$ and $H$ are non-diagonal, while we assume that we know a priori which entries of $\Phi_{v}$ are nonzero.

We will assume that the standard regularity conditions on the data are satisfied that are required for convergence results of the prediction error identification method¹¹1See [21] page 249. This includes the property that $e(t)$ has bounded moments of order higher than $4$..

When combining the $L$ node signals we arrive at the full network expression

which results in the matrix equation:

It is assumed that the dynamic network is stable, i.e. $(I-G)^{-1}$ is stable, and well posed (see [28] for details). The representation (2) is an extension of the dynamic structure function representation [12]. The identification problem to be considered is the problem of identifying one particular module $G_{ji}(q)$ on the basis of a selection of measured variables $w$, and possibly r.

Let us define $\mathcal{N}_{j}$ as the set of node indices $k$ such that $G_{jk}\neq 0$, i.e. the node signals in $\mathcal{N}_{j}$ are the $w$-in-neighbors of the node signal $w_{j}$. Let $\mathcal{D}_{j}$ denote the set of indices of the internal variables that are chosen as predictor inputs. It seems most obvious to have $\mathcal{D}_{j}\subset\mathcal{N}_{j}$, but this is not necessary, as will be shown later in this paper. Let $\mathcal{V}_{j}$ denote the set of node indices $k$ such that $v_{k}$ has a path to $w_{j}$. Let $\mathcal{Z}_{j}$ denote the set of indices not in $\{j\}\cup\mathcal{D}_{j}$, i.e. $\mathcal{Z}_{j}=\{1,\ldots,L\}\setminus\{\{j\}\cup\mathcal{D}_{j}\}$, reflecting the node signals that are discarded in the prediction/identification. Let $w_{\!{\scriptscriptstyle\mathcal{D}}}$ denote the vector $[w_{k_{1}}\ \cdots\ w_{k_{n}}]^{T}$, where $\{k_{1},\ldots,k_{n}\}=\mathcal{D}_{j}$. Let $u_{\!{\scriptscriptstyle\mathcal{D}}}$ denote the vector $[u_{k_{1}}\ \cdots\ u_{k_{n}}]^{T}$, where $\{k_{1},\ldots,k_{n}\}=\mathcal{D}_{j}$. The vectors $w_{\!{\scriptscriptstyle\mathcal{Z}}}$, $v_{\!{\scriptscriptstyle\mathcal{D}}}$, $v_{\!{\scriptscriptstyle\mathcal{Z}}}$ and $u_{\!{\scriptscriptstyle\mathcal{Z}}}$ are defined analogously. The ordering of the elements of $w_{\!{\scriptscriptstyle\mathcal{D}}}$, $v_{\!{\scriptscriptstyle\mathcal{D}}}$, and $u_{\!{\scriptscriptstyle\mathcal{D}}}$ is not important, as long as it is the same for all vectors. The transfer function matrix between $w_{\!{\scriptscriptstyle\mathcal{D}}}$ and $w_{j}$ is denoted $G_{j\!{\scriptscriptstyle\mathcal{D}}}$. The other transfer function matrices are defined analogously.

To illustrate the notation, consider the network sketched in Figure 1, and let module $G_{21}^{0}$ be the target module for identification.

Then $j=2$, $i=1$; $\mathcal{N}_{j}=\{1,4\}$. If we choose the set of predictor inputs as $\mathcal{D}_{j}=\mathcal{N}_{j}$, then the set of remaining (nonmeasured) signals, becomes $\mathcal{Z}_{j}=\{3,5,6\}$.

By this notation, the network equation (2) is rewritten as:

where $G_{\!{\scriptscriptstyle\mathcal{D}}\!{\scriptscriptstyle\mathcal{D}}}$ and $G_{\!{\scriptscriptstyle\mathcal{Z}}\!{\scriptscriptstyle\mathcal{Z}}}$ have zeros on the diagonal.

The following results are available from previous work:

• •

  When $\mathcal{D}_{j}$ is chosen equal to $\mathcal{N}_{j}$ and noise $v_{j}$ is uncorrelated to all $v_{k}$, $k\in\mathcal{V}_{j}$, then $G_{ji}$ can be consistently estimated in a MISO setup, provided that there is enough excitation in the predictor input signals, see [15].

• •

  When $\mathcal{D}_{j}$ is a subset of $\mathcal{N}_{j}$, and disturbance are uncorrelated, confounding variables³³3A confounding variable is an unmeasured variable that has paths to both the input and output of an estimation problem [29]. can occur in the estimation problem, and these have to be taken into account in the choice of $\mathcal{D}_{j}$ in order to arrive at consistent estimates of $G_{ji}$, see [23].

• •

  In [26] relaxed conditions for the selection of $\mathcal{D}_{j}$ have been formulated, while still staying in the context of MISO identification with noise spectrum of $v$ ($\Phi_{v}$) being diagonal. This is particularly done by choosing additional predictor input signals that are not in $\mathcal{N}_{j}$,.i.e. that are no in-neighbors of the output $w_{j}$ of the target module.

• •

  For non-diagonal $\Phi_{v}$, an indirect/two-stage identification method can be used to arrive at consistent estimates of $G_{ji}$ [15, 23, 20]. However the drawback of these methods is that they do not allow for a maximum likelihood analysis, i.e. they will not lead to minimum variance results.

• •

  This latter argument also holds for the method in [22, 24], where Wiener-filter estimates are combined to provide local module estimates, and diagonal $\Phi_{v}$ is considered.

In this paper, we go beyond consistency properties, and address the following problem: How to identify a single module in a dynamic network for the situation that the disturbance signals can be correlated, i.e. $\Phi_{v}$ not necessarily being diagonal, such that the estimate is consistent and asymptotically has Maximum Likelihood, and thus also minimum variance, properties.

Addressing this problem requires a more careful treatment and modelling of the noise that is acting on the different node signals. This can be illustrated through a simple Example that is presented in [25], where a two-node network is considered as given in Figure 2, with $v_{1}$ and $v_{2}$ being dynamically correlated. In this case, a SISO identification using the direct method with input $w_{1}$ and output $w_{2}$ will lead to a biased estimate of $G_{21}$ because of the unmodelled correlation of the disturbance signals on $w_{1}$ and $w_{2}$⁴⁴4In this particular example the bias is caused by the presence of $H_{21}$.. For an analysis of this, see [25]. If both node signals $w_{1}$ and $w_{2}$ are predicted as outputs, then the correlation between the disturbance signals can be incorporated in a $2\times 2$ non-diagonal noise model, thus leading to an unbiased estimate of $G_{21}$. In this way bias due to correlation in the noise signals can be avoided by predicting additional outputs other than the output of the target module. This leads to the following two suggestions:

• •

  confounding variables can be dealt with by modelling correlated disturbances on the node signals, and

• •

  this can be done by moving from a MISO identification setup to a MIMO setup.

These suggestions are being explored in the current paper. Next we will present an example to further illustrate the problem.

In the next sections we will formalize the procedure as sketched in Example 1 for general networks.

We will denote $w_{\!{\scriptscriptstyle\mathcal{Y}}}$ as the node signals in $w$ that serve as predicted outputs, and $w_{\!{\scriptscriptstyle\mathcal{D}}}$ as the node signals in $w$ that serve as predictor inputs. Next we decompose $w_{\!{\scriptscriptstyle\mathcal{Y}}}$ and $w_{\!{\scriptscriptstyle\mathcal{D}}}$ into disjoint sets according to: $\mathcal{Y}=\mathcal{Q}\cup\{o\}\ ;\ \mathcal{D}=\mathcal{Q}\cup\mathcal{U}$ where $w_{\!{\scriptscriptstyle\mathcal{Q}}}$ are the node signals that are common in $w_{\!{\scriptscriptstyle\mathcal{Y}}}$ and $w_{\!{\scriptscriptstyle\mathcal{D}}}$; $w_{o}$ is the output $w_{j}$ of the target module; if $j\in\mathcal{Q}$ then $\{o\}$ is void; $w_{\!{\scriptscriptstyle\mathcal{U}}}$ are the node signals that are only in $w_{\!{\scriptscriptstyle\mathcal{D}}}$. In this situation the measured nodes will be $w_{\!{\scriptscriptstyle\mathcal{D}}\cup\!{\scriptscriptstyle\mathcal{Y}}}$ and the unmeasured nodes $w_{\!{\scriptscriptstyle\mathcal{Z}}}$ will be determined by the set $\mathcal{Z}=\mathcal{L}\backslash\{\mathcal{D}\cup\mathcal{Y}\}$, where $\mathcal{L}=\{1,2,\cdots L\}$.

There can exist two types of confounding variable namely direct and indirect confounding variables. For direct confounding variables the simultaneous paths mentioned in the definition are both direct paths, while in all other cases we refer to the confounding variables as indirect confounding variables. For example, in the network as shown in Figure 3 with $\mathcal{D}=\{2\}$, $\mathcal{Y}=\{1\}$ and $\mathcal{Z}=\{3\}$, for the estimation problem $w_{2}\rightarrow w_{1}$, $e_{2}$ is a direct confounding variable since it has a simultaneous path to $w_{1}$ and $w_{2}$ where both the paths are direct paths. Meanwhile $e_{3}$ is an indirect confounding variable since it has a simultaneous path to $w_{1}$ and $w_{2}$ where one of the path is an unmeasured path⁷⁷7An unmeasured path is a path that runs through nodes in $w_{\!{\scriptscriptstyle\mathcal{Z}}}$ only. Analogously, we can define unmeasured loops through a node $w_{k}$..

On the basis of the decomposition of node signals as defined in the previous section we are going to represent the system’s equations (5) in the following structured form:

where we make the notation agreement that the matrix $H$ is not necessarily monic, and the scaling of the white noise process $e$ is such that $cov(e)=I$. Without loss of generality, we can assume $r=0$ for the sake of brevity.

Our objective is to end up with an an identification problem in which we identify the dynamics from inputs $(w_{\!{\scriptscriptstyle\mathcal{Q}}},w_{\!{\scriptscriptstyle\mathcal{U}}})$ to outputs $(w_{\!{\scriptscriptstyle\mathcal{Q}}},w_{o})$, while our target module $G_{ji}(q)$ is present as one of the scalar transfers (modules) in this identified (MIMO) model. This can be realized by the following steps:

1. 1.

   Firstly, we write the system’s equations for the measured variables as

   $$\underbrace{\begin{bmatrix}w_{\!{\scriptscriptstyle\mathcal{Q}}}\\ w_{o}\\ \hline\cr w_{\!{\scriptscriptstyle\mathcal{U}}}\end{bmatrix}}_{w_{m}}\!=\!\underbrace{\left[\begin{array}[]{c|c}\bar{G}&0\\ \hline\cr\\[-8.53581pt] \bar{G}_{\!{\scriptscriptstyle\mathcal{U}}\!{\scriptscriptstyle\mathcal{D}}}&\bar{G}_{\!{\scriptscriptstyle\mathcal{U}}o}\end{array}\right]}_{\bar{G}_{m}}\!\begin{bmatrix}w_{\!{\scriptscriptstyle\mathcal{Q}}}\\ w_{\!{\scriptscriptstyle\mathcal{U}}}\\ \hline\cr w_{o}\end{bmatrix}+\underbrace{\left[\begin{array}[]{c|c}\bar{H}&0\\ \hline\cr\\[-8.53581pt] 0&\bar{H}_{\!{\scriptscriptstyle\mathcal{U}}\!{\scriptscriptstyle\mathcal{U}}}\end{array}\right]}_{\bar{H}_{m}}\!\underbrace{\begin{bmatrix}\xi_{\!{\scriptscriptstyle\mathcal{Q}}}\\ \xi_{o}\\ \hline\cr\xi_{\!{\scriptscriptstyle\mathcal{U}}}\end{bmatrix}}_{\xi_{m}}$$

   (7)

   with $\xi_{m}$ a white noise process, while $\bar{H}$ is monic, stable and stably invertible and the components in $\bar{G}$ are zero if it concerns a mapping between identical signals. This step is made by removing the non-measured signals $w_{\!{\scriptscriptstyle\mathcal{Z}}}$ from the network, while maintaining the second order properties of the remaining signals. This step is referred to as immersion of the nodes in $w_{\!{\scriptscriptstyle\mathcal{Z}}}$ [23].

2. 2.

   As an immediate result of the previous step we can write an expression for the output variables $w_{\!{\scriptscriptstyle\mathcal{Y}}}$, by considering the upper part of the equation (7), as

   $$\underbrace{\begin{bmatrix}\!w_{\!{\scriptscriptstyle\mathcal{Q}}}\!\\ \!w_{o}\!\end{bmatrix}}_{w_{\!{\scriptscriptstyle\mathcal{Y}}}}\!\!=\!\!\underbrace{\begin{bmatrix}\!\bar{G}_{\!{\scriptscriptstyle\mathcal{Q}}\!{\scriptscriptstyle\mathcal{Q}}}\!&\!\bar{G}_{\!{\scriptscriptstyle\mathcal{Q}}\!{\scriptscriptstyle\mathcal{U}}}\!\\ \!\bar{G}_{o\!{\scriptscriptstyle\mathcal{Q}}}\!&\!\bar{G}_{o\!{\scriptscriptstyle\mathcal{U}}}\!\end{bmatrix}}_{\bar{G}}\underbrace{\begin{bmatrix}\!w_{\!{\scriptscriptstyle\mathcal{Q}}}\!\\ \!w_{\!{\scriptscriptstyle\mathcal{U}}}\!\end{bmatrix}}_{w_{\!{\scriptscriptstyle\mathcal{D}}}}+\underbrace{\begin{bmatrix}\bar{H}_{\!{\scriptscriptstyle\mathcal{Q}}\!{\scriptscriptstyle\mathcal{Q}}}&\bar{H}_{\!{\scriptscriptstyle\mathcal{Q}}o}\\ \bar{H}_{o\!{\scriptscriptstyle\mathcal{Q}}}&\bar{H}_{oo}\end{bmatrix}}_{\bar{H}}\underbrace{\begin{bmatrix}\xi_{\!{\scriptscriptstyle\mathcal{Q}}}\\ \xi_{o}\end{bmatrix}}_{\xi_{\!{\scriptscriptstyle\mathcal{Y}}}}$$

   (8)

   with $cov(\xi_{\!{\scriptscriptstyle\mathcal{Y}}}):=\bar{\Lambda}$.

3. 3.

   Thirdly, we will provide conditions to guarantee that $\bar{G}_{ji}(q)=G_{ji}(q)$, i.e the target module appearing in equation (8) is the target module of the original network (invariance of target module). This will require conditions on the selection of node signals in $w_{\!{\scriptscriptstyle\mathcal{Q}}},w_{o},w_{\!{\scriptscriptstyle\mathcal{U}}}$.

4. 4.

   Finally, it will be shown that, on the basis of (8), under fairly general conditions, the transfer functions $\bar{G}(q)$ and $\bar{H}(q)$ can be estimated consistently, and with maximum likelihood properties. A pictorial representation of the identification setup with the classification of different sets of signals in (8) is provided in Figure 4. The figure also contains set $\mathcal{A},\mathcal{B},\mathcal{F}_{n}$ which will be introduced in the sequel.

The combination of steps 3 and 4 will lead to a consistent and maximum likelihood estimation of the target module $G_{ji}(q)$. It has to be noted that an identification setup results, in which signals can simultaneously act as input and as output (the set $w_{\!{\scriptscriptstyle\mathcal{Q}}}$). Because $\bar{G}_{\!{\scriptscriptstyle\mathcal{Q}}\!{\scriptscriptstyle\mathcal{Q}}}$ is restricted to be hollow, this does not lead to trivial transfers between signals that are the same. A related situation appears when identifying a full network, while using all node signals as both inputs and outputs, as in [8].

The steps 1)-4) above will require conditions on the selection of node signals, based on the known topology of the network and an allowed correlation structure of the disturbances in the network. Specifying these conditions on the selection of sets $w_{\!{\scriptscriptstyle\mathcal{Q}}},w_{o},w_{\!{\scriptscriptstyle\mathcal{U}}}$, will be an important objective of the next section.

The first algorithm to be presented is based on the strategy that for any node signal that is selected as output, we have access to all of its $w$-in-neighbors, that are to be included as predictor inputs. This strategy will lead to an identification setup with a maximum use of measured node signals that contain information that is relevant for modeling our target module $G_{ji}$. The following strategy will be followed:

• •

  We start by selecting $i\in\mathcal{D}$ and $j\in\mathcal{Y}$;

• •

  Then we extend $\mathcal{D}$ in such a way that all $w$-in-neighbors of $w_{\!{\scriptscriptstyle\mathcal{Y}}}$ are included in $w_{\!{\scriptscriptstyle\mathcal{D}}}$.

• •

  All node signals in $w_{\!{\scriptscriptstyle\mathcal{D}}}$ that have noise terms $v_{k}$, $k\in\mathcal{D}$ that are correlated with any $v_{\ell}$, $\ell\in\mathcal{Y}$ (direct confounding variables for $w_{\!{\scriptscriptstyle\mathcal{D}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$), are included in $\mathcal{Y}$ too. They become elements of $\mathcal{Q}$.

• •

  With $\mathcal{A}:=\mathcal{D}\backslash\mathcal{Q}$ it follows that by construction there are no direct confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$.

• •

  Then we choose $w_{\!{\scriptscriptstyle\mathcal{B}}}$ as a subset of nodes that are not in $w_{\!{\scriptscriptstyle\mathcal{Y}}}$ nor in $w_{\!{\scriptscriptstyle\mathcal{A}}}$. This set needs to be introduced to block the indirect confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$, and will be chosen to satisfy Condition 2a and 2c of Theorem 1.

• •

  Every node signal $w_{k}$, $k\in\mathcal{A}$ for which there are only indirect confounding variables and cannot be blocked by a node in $w_{\!{\scriptscriptstyle\mathcal{B}}}$, is

  • –

    moved to $\mathcal{B}$ if Conditions 2a and 2c of Theorem 1 are satisfied and $k\neq i$; (else)

  • –

    included in $\mathcal{Y}$ and moved to $\mathcal{Q}$;

• •

  Finally, we define the identification setup as the estimation problem $w_{\mathcal{D}}\rightarrow w_{\mathcal{Y}}$, with $\mathcal{D}=\mathcal{Q}\cup\mathcal{A}\cup\mathcal{B}$ and $\mathcal{Y}=\mathcal{Q}\cup\{o\}$.

Note that because all $w$-in-neighbors of $w_{\!{\scriptscriptstyle\mathcal{Y}}}$ are included in $w_{\!{\scriptscriptstyle\mathcal{D}}}$, we automatically satisfy the parallel path and loop condition 1. In order for the selection of node signals $w_{\!{\scriptscriptstyle\mathcal{B}}}$ to satisfy the conditions of Theorem 1, we will specify the following Property 1.

Property 2a) ensures that, after including $w_{\!{\scriptscriptstyle\mathcal{B}}}$ in the set of measured signals, there are no indirect confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$, and Property 2b) guarantees that there are no confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{B}}}$. Together we satisfy Condition 2a) of Theorem 1. Also, Property 2c) guarantees condition 2c) of Theorem 1 to be satisfied. Finally, as per the algorithm, $w_{i}$ can be either in $w_{\!{\scriptscriptstyle\mathcal{A}}}$ or $w_{\!{\scriptscriptstyle\mathcal{Q}}}$. Therefore at the end of the algorithm, we will obtain sets of signals that satisfy the conditions in Theorem 1 for target module invariance.

Rather than measuring all node signals that are $w$-in-neighbors of the output $w_{j}$ of our target module $G_{ji}$, we now focus on an identification setup that uses a minimum number of measured node signals, according to the following strategy:

• •

  We start by selecting $i\in\mathcal{D}$ and $j\in\mathcal{Y}$;

• •

  Then we extend $\mathcal{D}$ with a minimum number of node signals that satisfies the parallel path and loop Condition 1.

• •

  Every node signal $w_{k}$ in $w_{\!{\scriptscriptstyle\mathcal{D}}}$ for which there is a direct or indirect confounding variable for the estimation problem $w_{k}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$ is included in $\mathcal{Y}$ and $\mathcal{Q}$.

• •

  With $\mathcal{A}:=\mathcal{D}\backslash\mathcal{Q}$ and $\mathcal{B}=\emptyset$ it follows that by construction there are no confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$.

• •

  Finally, we define the identification setup as the estimation problem $w_{\mathcal{D}}\rightarrow w_{\mathcal{Y}}$, with $\mathcal{D}=\mathcal{Q}\cup\mathcal{A}$.

As we can observe, the algorithm does not require selection of set $\mathcal{B}$. This is attributed to the way we handle the indirect confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$. Instead of tackling these confounding variables by adding blocking node signals $w_{\!{\scriptscriptstyle\mathcal{B}}}$ (as in full input case) to be added as predictor inputs, we deal with them by moving the concerned $w_{k},k\in\mathcal{A}$ to $w_{\!{\scriptscriptstyle\mathcal{Q}}}$ and thus to the set of predicted outputs. We choose this approach in order to minimize the required number of measured node signals. In this way, by construction, there will be no direct or indirect confounding variables for the estimation problem $w_{\!{\scriptscriptstyle\mathcal{A}}}\rightarrow w_{\!{\scriptscriptstyle\mathcal{Y}}}$. From this result, we can guarantee that the conditions in Theorem 1 will be satisfied since $\mathcal{B}=\emptyset$. Thus at the end of the algorithm we obtain a set of signals that provides target module invariance.