1 Hypothesis Test Procedures for Detecting Leakage Signals in Water Pipeline Channels

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Hypothesis Test Procedures for Detecting Leakage

Signals in Water Pipeline Channels

Liusha Yang, Matthew R. McKay, Xun Wang

Abstract—We design statistical hypothesis tests for performing

leak detection in water pipeline channels. By applying an appro-

priate model for signal propagation, we show that the detection

problem becomes one of distinguishing signal from noise, with

the noise being described by a multivariate Gaussian distribu-

tion with unknown covariance matrix. We ﬁrst design a test

procedure based on the generalized likelihood ratio test, which

we show through simulations to offer appreciable leak detection

performance gain over conventional approaches designed in an

analogous context (for radar detection). Our proposed method

requires estimation of the noise covariance matrix, which can

become inaccurate under high-dimensional settings, and when the

measurement data is scarce. To deal with this, we present a second

leak detection method, which employs a regularized covariance

matrix estimate. The regularization parameter is optimized for

the leak detection application by applying results from large

dimensional random matrix theory. This second proposed ap-

proach is shown to yield improved performance in leak detection

compared with the ﬁrst approach, at the expense of requiring

higher computational complexity.

Index Terms—Leak detection, hypothesis test, random matrix

theory.

I. INTRODUCTION

Leakage in water supply systems causes wastage of water

and energy resources, and poses public health risk due to

water pollution. Leaks may occur, for example, due to ag-

ing pipelines, corrosion, and excessive steady and/or unsteady

pressures in the system [1]. Thus, an effective leakage detec-

tion method is essential.

Most related research in this area has focused on the prob-

lem of leak estimation, for which the objective is usually to

estimate the location of the leak, assuming that a leak actually

exists in the pipeline. For this purpose, various transient-based

leak location estimation methods have been developed (e.g.,

[1–4]). In this work we address the related (but different)

problem of leak detection, by developing suitable statistical

hypothesis testing procedures. Despite being a natural detec-

tion approach, to our knowledge, hypothesis tests have yet to

be developed for leak detection in water pipeline systems.

Generally speaking, we develop data-driven approaches to

decide between the presence or absence of a leak in the pipeline,

and for the former case, return estimates of the leak param-

eters. The measured data corresponds to primary and sec-

ondary measurements of head differences at different frequen-

cies, taken from multiple sensors deployed at different loca-

tions along the water pipeline. Our tests are developed based

on a linearized transient wave model in the frequency do-

main, as proposed in [3, 4], which has been supported by

experimental data [5]. By applying hypothesis testing theory

to this model, we ﬁnd that, from a technical point of view, the

problem boils down to a binary classiﬁcation problem that dis-

criminates between a “null hypothesis”, corresponding to zero-

mean complex Gaussian noise with non-trivial correlation, and

an “alternative hypothesis”, corresponding to a structured (de-

terministic) signal embedded within the Gaussian noise. For

the latter hypothesis, the deterministic signal is a function of

the leak parameters, including size and location.

Since the signal and noise model parameters (i.e., noise

covariance, leak location and size) are all unknown, we de-

velop test procedures based on the generalized likelihood ratio

test (GLRT) [6], which constructs a likelihood ratio based

on the two hypotheses, and replaces the unknown parameters

in the likelihood functions by appropriate estimates. We ﬁrst

consider a traditional strategy of which replaces the unknown

parameters by their maximum likelihood estimates (MLE),

and develop a suitable test statistic. This statistic exploits the

known structure of the leak signals (under the alternative hy-

pothesis), and is proven to have the desirable property of being

a constant false alarm rate (CFAR) statistic; meaning that a de-

tection threshold can be speciﬁed which achieves a ﬁxed false

alarm probability, regardless of the model parameters. Through

simulations, we demonstrate the good performance of the pro-

posed method in detecting leaks, and show enhancement over

methods that have been developed for related models in the

context of radar detection. This approach is particularly suited

to “data rich” scenarios, where the MLEs provide accurate

parameter estimates.

One limitation of the proposed approach is that for high

dimensional settings when the number of frequency domain

measurements and/or the number of sensors is large, the num-

ber of parameters to estimate is also large. This is particu-

larly the case for the noise covariance matrix, and it is well

known that under high dimensional settings that the MLE –

corresponding to the conventional sample covariance matrix

(SCM) estimate – is particularly inaccurate. This, in turn,

can degrade the performance of the proposed leak detection

algorithm. To deal with this potential problem, we propose

a second detection algorithm that seeks to design a robust

covariance estimation solution which is suitably optimized for

the task of leak detection, under high dimensional settings. The

approach is to replace the SCM with a regularized version

(termed RSCM) in the GLRT statistic, and to optimize the

regularization parameter to maximize the leak detection accu-

racy subject to a prescribed false alarm criteria. The RSCM

is a simple but effective covariance matrix estimator to deal

with problems of sample deﬁciency and high dimensionality

by pulling the spread sample eigenvalues toward their grand

arXiv:2210.13032v1 [eess.SP] 24 Oct 2022

mean [7]. It is used in many ﬁelds, including mathematical

ﬁnance and adaptive array processing [8–12]. Extensions have

also been proposed which replace the SCM with a robust

covariance matrix estimator (such as Tyler’s estimator) to pro-

vide resilience against outliers [13–15]. The main challenge

is generally to develop data-driven methods to optimize the

regularization parameter, which is typically application depen-

dent. In a similar spirit to previous work (e.g., [7, 8, 11, 16–

18]), our solution draws from recent results in the area of

large dimensional random matrix theory. Most speciﬁcally, it

leverages technical results from [17, 18], which considered a

related detection problem, but which considered a different

model to the one in this paper.

The basic idea of the approach is to ﬁrst characterize the

asymptotic behavior of the false alarm and detection probabil-

ities under certain double-limit asymptotics, which we deﬁne,

and subsequently to provide consistent estimators of these

probabilities which are completely data-driven. Based on this,

we can then optimize the regularization parameter in an online

fashion, which maximizes the (estimated) detection probability

while maintaining a prescribed (estimated) false alarm proba-

bility. The performance of this second proposed leak detection

algorithm is demonstrated through simulations, and shown to

outperform the ﬁrst proposed algorithm, particularly under

high-dimensional model settings, at the expense of increased

complexity.

II. SYSTEM MODEL

As shown in Fig. 1, we consider a reservoir-pipe-valve sys-

tem where the pipe of length lmeters is bounded by pU= 0

and pD=l. A total of Mpressure sensors deployed near the

downstream node are used to collect pressure head oscilla-

tions*for leak identiﬁcation. The locations of the Msensors

are pU< x1< x2< . . . < xM< pD. We denote the leak

size and the leak location as sand φ.

𝑥𝐿𝑥1𝑥2

𝑄0

𝐿



Upstream

reservoir

Downstream

reservoir

Fig. 1. Pipeline conﬁguration. Under hypothesis H0, there is no leak in the

water pipe, while under hypothesis H1, a leak of size sis present at location

φ.

By rapidly closing and/or opening the valve at the down-

stream of the pipe, the sensors measure the pressure head oscil-

lations at different frequencies, which are affected by a leak in

the pipe. Let hm(wj)denote the head oscillation at frequency

wjand location xm, and ho

m(wj)the computed head oscil-

lation with no leak, where j= 1, . . . , J and m= 1, . . . , M.

We deﬁne the head difference at frequency wjobserved by the

sensor at xmas zm(wj) = hm(wj)−ho

m(wj). If the pipe is

*The pressure head (in meters) relates the pressure of a ﬂuid to the height

of a column of that ﬂuid having an equivalent static pressure at its base. The

head is deﬁned as h=p/(ρg)where pis the pressure (in Pascals), gdenotes

gravitational acceleration, and ρis the density of the ﬂuid. For example, 50

m of head in a pipe implies that if that pipe bursts, the height of the resulting

water jet would be 50 m.

intact (with no leak), zm(wj) = hm(wj)−ho

m(wj) = nm(wj),

where nm(wj)is the measurement noise, which can be mea-

surement error or environment noise induced by turbulence,

trafﬁc, construction, etc. Otherwise, zm(wj) = sgm(φ, wj) +

nm(wj), in which sgm(φ, wj)is the leak component, which

depends on the leak size sand the leak location φ. The de-

tailed formulas of ho

m(wj)and gm(φ, wj)are provided in the

Appendix A. Assembling zm(wj)into a vector z0∈CNof

length N=J×M, we have

z0= vec[zm(wj), j = 1, . . . , J, m = 1, . . . , M].

We denote the hypothesis of whether there exists a leak or not

by H1and H0, respectively. Then the problem of detecting a

leak in a noise-contaminated water pipe can be posed in terms

of the following binary hypothesis test:

H0:z0=n0,

H1:z0=sg(φ) + n0(1)

where the noise vector n0= vec[nm(wj)] is assumed to be

Gaussian distributed†with zero mean and covariance matrix

CN, and

g(φ) = vec[gm(φ, wj), j = 1, . . . , J, m = 1, . . . , M].

We assume that Kindependent samples of noise-only data

are available, which are referred to as secondary data:

zk=nk,nk∼CN (0,CN), k = 1, . . . , K.

These may be obtained, for example, by the steady-state pres-

sure measurements when the pipe is newly built.

Thus, the leak detection problem can be recast as the fol-

lowing hypotheses:

H0:z0=n0,zk=nk, k = 1 ...,K

H1:z0=sg(φ) + n0,zk=nk, k = 1 . . . , K.

The joint probability density function (PDF) of the input data

under H0is

f0(z0,...,zK|H0) =

(πNdet(CN))K+1 exp "−

k=1

kC−1

Nzk#exp −zH

0C−1

Nz0

(2)

where det(CN)is the matrix determinant of CN.

Similarly, the joint PDF of the input data under H1is

f1(z0,...,zK|H1) =

(πNdet(CN))K+1 exp "−

k=1

kC−1

Nzk#

×exp−(z0−sg(φ))HC−1

N(z0−sg(φ)).(3)

The most natural approach to detect the presence of a leak

is the likelihood ratio (LR) test, which computes the LR or

its logarithm and compares it with a certain threshold α[20].

Speciﬁcally, the LR test is

L=f1(z0,...,zK|H1)

f0(z0,...,zK|H0)

≷

α.

†The Gaussian noise assumption in water pipes with ﬂow is justiﬁed by

experimental investigations in laboratory pipe systems [19].

Namely, if L > α, we decide H1, and if L≤α, we decide

H0.

The LR test is known to maximize the detection probability

PDat a certain false alarm probability PFA. The PDis deﬁned

as the probability that the detector correctly decides hypothesis

H1:

PD=P[L > α|H1],

and the PFA is deﬁned as the probability that the detector

decides hypothesis H1when the true hypothesis is H0:

PFA =P[L>α|H0].(4)

For leak detection in a water pipeline system, we usually do

not know the parameters s,φand CNin the PDFs f0(z0,...,zK|H0)

and f1(z0,...,zK|H1). In this context, the LR test can not

be employed. The GLRT, which employs the MLEs of the

unknown parameters, is a suitable solution.

III. GENERALIZED LIKELIHOOD RATIO TEST (GLRT)

In this section, we derive a GLRT-based leak detection ap-

proach and demonstrate its desirable CFAR property. The per-

formance of our proposed approach is also assessed by nu-

merical simulations.

A. Derivation of GLRT

We denote the leak component in the data model as p=

sg(φ)and assume that K≥N. By estimating sand φ, we

get the estimate of p. The considered GLRT is

L=maxs,φ maxCNf1(z0,...,zK|H1)

maxCNf0(z0,...,zK|H0)

≷

α. (5)

The MLEs of CNunder H0and H1are equal to the SCM,

which are well known [21]. Namely, the MLE of CNunder

H0is 1

K+1 PK

k=0 zkzH

kand the MLE of CNunder H1is

K+1 h(z0−sg(φ))(z0−sg(φ))H+PK

k=1 zkzH

ki.

Denote SN=PK

k=1 zkzH

k. Following similar derivation

steps in [6], we obtain the MLEs of sand φ:

ˆs=Re{gH(φ)S−1

Nz0}

gH(φ)S−1

Ng(φ),(6)

and the MLE of φis

φ= argmax

φ∈[pU,pD]

Re2{gH(φ)S−1

Nz0}

gH(φ)S−1

Ng(φ).(7)

The statistic in (7) can be seen as a generalization of the

leak location estimator presented in [3, 4], which considered

the problem of leak estimation under white Gaussian noise.

Because of the complicated structure of g(φ), it is not easy

to obtain an explicit formula for ˆ

φfrom (7), unlike for the

other parameters. Thus, we obtain the MLE of φthrough a

grid search in the range [pU, pD]that minimizes λ.

By plugging the MLEs of s,φand CN, the test (5) becomes

1 + zH

0S−1

Nz0

1 + zH

0S−1

Nz0−Re2{gH(ˆ

φ)S−1

Nz0}

gH(ˆ

φ)S−1

Ng(ˆ

φ)

≷

α. (8)

Denote α1= 1 −1

α. The hypothesis test (8) can be further

simpliﬁed as

∆ = Re2{gH(ˆ

φ)S−1

Nz0}

(1 + zH

0S−1

Nz0)gH(ˆ

φ)S−1

Ng(ˆ

φ)

≷

α1.(9)

Similar to the GLRT in [6], the distribution of the test

statistic ∆under H0is independent of CNand g(ˆ

φ). Hence,

the cumulative distribution function (CDF) of ∆under H0, de-

noted as F∆, remains the same for any covariance matrix CN

and nonzero vector g(ˆ

φ). Consequently, although a closed-

form expression of F∆is difﬁcult to derive, it is sufﬁcient to

apply Monte-Carlo simulations to obtain the empirical CDF

F∆based on simulated data by setting g(ˆ

φ) = [1,0,...,0]T,

CN=INand generating zi,i= 0, . . . , K as standard normal

distributed random vectors. The threshold α1for a desired PFA

can then be determined by computing α1=F−1

∆(1 −PFA).

Although the GLRT in (5) is similar to that in [6], we should

point out the main differences between the two GLRTs. Firstly,

in (5), the leak size sis conﬁned to be a real number but in

[6], sis complex, which leads to a different MLE expression

of sas in (6). Additionally, while in [6], the signal vector g

is known, in our case, gis parameterized by unknown leak

location φ, which is estimated in (7).

The detection procedure is summarized in Algorithm 1. As

the detection test (9) uses the SCM as the estimate of CN,

we refer to this leak detection (LD) scheme as LD-SCM.

Algorithm 1 LD-SCM

1) Determine the threshold α1corresponding to the prescribed PFA

and the empirical CDF F∆:

α1=F−1

∆(1 −PFA).

2) Find the optimal estimate of φand thus g(ˆ

φ)by numerically

solving:

φ= argmax

φ∈[pU,pD]

Re2{gH(φ)S−1

Nz0}

gH(φ)S−1

Ng(φ).(10)

3) Compute the test statistic:

∆ = Re2{gH(ˆ

φ)S−1

Nz0}

(1 + zH

0S−1

Nz0)gH(ˆ

φ)S−1

Ng(ˆ

φ).

4) Accept H0(“no leak”), if ∆≤α1; otherwise accept H1(“leak

present”).

5) If H1accepted, set the estimates of φfrom (10) and s:

ˆs=Re{gH(ˆ

φ)S−1

Nz0}

gH(ˆ

φ)S−1

Ng(ˆ

φ).

B. Performance evaluation and comparison

Here we demonstrate the performance of our proposed LD-

SCM scheme, and compare it against alternative detection

methods. The system conﬁguration is shown in Fig. 1. A wa-

ter pipe in a horizontal plane with length l= 2000 m and

diameter D= 0.5m is considered. The locations of upstream

and downstream reservoirs are assumed to be pU= 0 m

and pD= 2000 m, respectively. Two pressure sensors are

situated at x1= 1800 m and x2= 2000 m. The wave speed

is a= 1000 m/s. The utilized frequencies are w=jwth,

j= 1,2,...,32, where wth =aπ/(2l)is the fundamental

frequency (ﬁrst resonant frequency). Thus N= 64. Under

the hypothesis H1, the leak location is φ= 600 m and the

leak size is s= 1.4×10−4m2. Other necessary parameters

required in the system model (see Appendix A) are: f= 0.02,

eL= 0,Q0= 0.0153 m3/s,g= 9.8 m/s2and HL

0= 23.5

m. In the following simulations, we carry out Monte Carlo

simulations using 105runs.

We compare the performance of our proposed LD-SCM

scheme against alternative detection methods. First, we con-

sider the “oracle” detector with perfect knowledge of parame-

ters s,φand CN. Although the oracle detector is unachievable

in practice, it provides an upper bound on the performance of

leak detection. We also compare with a classical method used

in radar detection [22], which also uses the SCM as the esti-

mate of CNand is referred to as RD-SCM. Different from the

LD-SCM scheme, this method estimates the leak component

p=sg(φ)as a whole. It ignores the structure of pand does

not estimate sand φseparately. Detailed descriptions of the

oracle detector and the RD-SCM are provided in Appendix B.

In the simulations, we set K= 600,[CN]i,j =ν20.9|i−j|

and deﬁne the signal to noise ratio (SNR) as SNR = kpk2

ν2.

Fig. 2(a) shows the detection probability PDagainst different

SNRs under PFA = 10−3. Our proposed LD-SCM has higher

PDthan that realized by the RD-SCM over different SNRs,

and performs fairly close to the oracle.

To further demonstrate the performance of the LD-SCM,

we plot receiver operating characteristic (ROC) curves for the

different approaches. Fig. 2(b) shows that while the oracle

detector naturally performs the best, the LD-SCM uniformly

outperforms the RD-SCM over the entire span of PFA.

To show the effect of the sample size Kof the secondary

data, we further compare the leak detection performance of

the LD-SCM for ﬁxed N= 64 and different K. As we see

from Fig. 3, the detection probability PDdecreases when K

becomes smaller. This is because the sample size Kis closely

related to the estimation accuracy of the SCM. It is well known

that the estimation error of the SCM becomes large when the

sample size Kis small compared to the data dimension N

[23–25]. This has been demonstrated rigorously using random

matrix theory, which considers the setting when Kand N

are both large, and which has shown that the eigenvalues and

eigenvectors of the SCM behave very differently from those

of CN[26–29]. Thus, the performance degradation of the LD-

SCM is caused in part by the estimation error of the SCM.

To deal with this, a more robust covariance matrix estimate

may help to enhance the leak detection performance when K

is not substantially larger than N. This is the main focus of

the subsequent section.

IV. LEAK DETECTION WITH REGULARIZED SAMPLE

COVARIANCE MATRIX

As shown in the last section, the performance of the LD-

SCM degrades when the sample size Kdoes not greatly ex-

ceed the matrix dimension N. Since the measurements are

collected through Msensors at Jfrequencies, it is possible

-15 -10 -5 0 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Oracle

LD-SCM

RD-SCM

(a) PDagainst SNR with prescribed PFA = 10−3.

10-4 10-3 10-2 10-1 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Oracle

LD-SCM

RD-SCM

(b) ROCs with ﬁxed SNR = -3 dB.

Fig. 2. Performance comparison of the oracle detector, LD-SCM and RD-

SCM when N= 64,K= 600.

-15 -10 -5 0 5 10 15

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

K=80

K=128

K=600

Fig. 3. PDof LD-SCM against SNR with prescribed PFA = 10−3, for

N= 64 and different K.

that the data dimension N=J×Mis large, compared to the

sample size K. Thus it is desirable to design a leak detection

method that yields good performance when the data dimension

is high or the sample size of the secondary data is small. As

the performance degradation is, to some extent, caused by the

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1HypothesisTestProceduresforDetectingLeakageSignalsinWaterPipelineChannelsLiushaYang,MatthewR.McKay,XunWangAbstractWedesignstatisticalhypothesistestsforperformingleakdetectioninwaterpipelinechannels.Byapplyinganappro-priatemodelforsignalpropagation,weshowthatthedetectionproblembecomesoneofdistingui...

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