Contamination source detection in water distribution networks using belief propagation

Abstract

We present a Bayesian approach for the Contamination Source Detection problem in water distribution networks. Assuming that contamination is a rare event (in space and time), we try to locate the most probable source of such events after reading contamination patterns in few sensed nodes. The method relies on strong simplifications considering binary clean/contaminated states for nodes in discrete time, and therefore focuses on the time structure of the sensed patterns rather than on the concentration levels. As a result, a posterior probability over discrete variables is written, and posterior marginals are computed using belief propagation algorithm. The resulting algorithm runs once on a given observation and reports probabilities for each node being the source and for the contamination patterns altogether. We test it on Anytown model, proving its efficacy even when only a single sensed node is known.

Appendices

Appendix 1: factor graph representation

Algorithm 1 details the construction of a factor graph from the information regarding the water distribution network and the transport times along the pipes. An example of the resulting factor graph can be found in the main text Fig. 2.

Appendix 2: belief propagation

Formally speaking, a consistent description of the full measure in terms of single variables and interacting variables distributions is achieved through a variational representation of the free energy $F = -\log Z$ of the problem in (10) as

$$\begin{aligned} F[b_i(x_i), b_{a}(x_{\partial a})] = \sum _a F_a[b(x_{\partial a})] - \sum _i (d_i-1) F_i[b(x_i)] \end{aligned}$$

The free energy approximation sums the contribution of each factor node free energy, and subtracts the over-counting of the free energies of individual variables ($d_i$ is the number of factors that a variable interacts with). Each free energy term $F = U - T S$ consists, as usual, of an energetic part $U=\sum _{x_{\partial a}} b_a(x_{\partial a}) E_a(x_{\partial a})$ and $S = -\sum _{x_{\partial a}} b_a(x_{\partial a}) \log b_a(x_{\partial a})$, where the energetic term $E_a(\partial a)$ contains only the corresponding additive term in the exponential (10).

Minimization of this free energy over the distributions $b(\cdot )$ needs to respect the consistency among them: if two distributions share a common variable, the marginals should agree:

$$\begin{aligned} \forall _{i,a\in \partial i} \,\, b_i(x_i) = \sum _{x \in a{\setminus } x_i} b_a(x_1,x_2,\ldots ) \end{aligned}$$

(15)

In order to achieve this, we introduce a set of Lagrange multipliers $m_{a \rightarrow i}(x_i)$ that are finally interpreted as messages flowing from every interaction towards every variable in it (Yedidia et al. 2005). The beliefs then are found in terms of these multipliers as in Eq. (12) while the belief over every set of interacting variables in $E_{i,t}$ are given as in Eq. (13)

In Algorithm 2 we outline the main steps of the whole procedure presented in this paper, starting from the WDN graph up to the Belief Propagation fixed point iteration.

Appendix 3: list of symbols

Water network symbols

$C_i(t)$ :

Actual concentration of contaminants at time t in node i of the WDN

$L_{i,j}$ :

Length of the pipe connecting nodes i and node j in the WDN

$v_{i,j}$ :

Velocity of fluid in the pipe connecting nodes i and node j the WDN (considered as constant in time)

$\Delta _{i,j}$ :

Transport time between node i and node j of the WDN, given as an integer number of some time unit

$c_{{\mathrm{th}}}$ :

Concentration threshold for the sensitivity of the sensors placed in the network

Graph symbols

$G(V,E,\Delta )$ :

Weighted directed graph, representing the WDN, where $V=\{i \arrowvert i\in [1,N]\}$ is the set of nodes, E is the set of edges $(i\rightarrow j)$ and $\Delta = \{ \Delta _{i,j} \arrowvert (i\rightarrow j)\in E\}$ is the set of transport times along the edges (pipes)

$G'(V',E')$ :

Time-extended graph representation of the WDN, with a set $V'=\{s_i^t\arrowvert i\in V , t\in [1,T]\}$ of nodes and a set $E'=\{((i,t)-(j,t')) \arrowvert (i\rightarrow j)\in E, t' = t+\Delta _{i, j}\}$ of edges

$\partial ^+_{i,t}$ :

Upstream neighborhood of node $s_i^t\in V'$: subset of nodes $s_j^{t'} \in V'$ of graph $G'$ such that $((j,t')-(i,t)) \in E'$

Model symbols

$o_i^t$ :

Binary (0/1) sensor report for the state of node i at discrete time t, given as input for the inference

$s_i^t$ :

Binary (0/1) state of node i at discrete time t, to be inferred

$y_i^t$ :

Binary (0/1) contaminant-pouring patterns, representing the external addition of contaminants at time t on node i, to be inferred

$\upsilon _i$ :

Binary (1/0) contamination exposure, representing that node i will or will not incorporate contamination coming from the exterior, to be inferred

$\eta$ :

A field penalizing discrepancies between the observed state $o_i^t$ and the real state $s_i^t$ of the observed nodes

$\beta$ :

A temperature-like parameter enforcing the dynamics [Eq. (2)] in the network [see Eqs. (6) and (7)]

$\lambda$ :

A field forcing the contamination to be a rare event (in space), by penalizing the non zero values of $\upsilon _i$ [see Eq. (8)]

$\gamma$ :

A field forcing the contamination to be a short event (in time), by penalizing the non zero values of $y_i^t$ [see Eq. (9)]

$\varphi (s_i^t , \upsilon _i , y_i^t , s_{\partial ^+_i}^{t-\Delta })$ :

Function of soft satisfaction of Eq. 6

$E_{i,t}(s_i^t,\upsilon _i, y_i^t, s_{\partial ^+_i}^{t-\Delta } )$ :

Binary function (7) checking the consistency of the variables with the proposed dynamics (2)

$P(S,V, Y \arrowvert O)$ :

Probability of sources V patterns Y and the states of nodes in time S given the observation O

$P(O \arrowvert S,V, Y)$ :

Probability of an observation O given the active sources V the pouring patterns Y and the and the set of states of the nodes S (actually is only a function $P(O \arrowvert S)$ of O and S)

P(S, V, Y):

a pirori probability of sources V patterns Y and the states of contaminants S in the system

$Z=P(O)$ :

Normalization constant, corresponding in the bayesian setting to the probability of the observation O

F :

Free Energy

b(x):

Inferred probability distribution, also called belief, over any binary variable x, approximating the marginal probability over that variable of the full distribution $P(S,V,Y \arrowvert O)$

$m_{a \rightarrow i}(x_i)$ :

Message sent from the interaction a in the factor node representation to one of its variables i, equivalent to the probability distribution of that variable according to the interaction a and the probabilities of the other variables involved in it. Can also be seen as the Lagrange multiplier enforcing the consistency between the single variable distribution $b_i(x_i)$ and the multivariate factor node distribution $b_a(x_{\partial _a})$

Appendix 4: supplementary data

The following data, models, or code generated or used during the study are available as supplemental material of the submitted manuscript

• File with the directed weighted graph structure of the Anytown network,

• Concentration pattern sensed at node 9 which is all the relevant information required for our inference procedure.

The following data, models, or code generated or used during the study are available from the corresponding author by request

• Belief Propagation code in C++ will be granted upon request to the authors.

• All codes required to generate the input files for the BP code.