00footnotetext: †First two authors contributed equally.11institutetext:
Bielefeld University, Bielefeld, Germany 22institutetext: Honda Research
Institute, Offenbach, Germany 33institutetext: University of Cyprus, Nicosia,
Cyprus

# SAM-kNN Regressor for Online Learning in Water Distribution
Networks††thanks: We gratefully acknowledge funding from the VW-Foundation for
the project IMPACT funded in the frame of the funding line AI and its
Implications for Future Society, and funding from from the European Research
Council (ERC) under the ERC Synergy Grant Water-Futures (Grant agreement No.
951424).

Jonathan Jakob† 11 André Artelt † 1133 Martina Hasenjäger 22 Barbara Hammer 11

###### Abstract

Water distribution networks are a key component of modern infrastructure for
housing and industry. They transport and distribute water via widely branched
networks from sources to consumers. In order to guarantee a working network at
all times, the water supply company continuously monitors the network and
takes actions when necessary – e.g. reacting to leakages, sensor faults and
drops in water quality. Since real world networks are too large and complex to
be monitored by a human, algorithmic monitoring systems have been developed. A
popular type of such systems are residual based anomaly detection systems that
can detect events such as leakages and sensor faults. For a continuous high
quality monitoring, it is necessary for these systems to adapt to changed
demands and presence of various anomalies.

In this work, we propose an adaption of the incremental SAM-kNN classifier for
regression to build a residual based anomaly detection system for water
distribution networks that is able to adapt to any kind of change.

###### Keywords:

SAM-kNN Regressor Incremental Anomaly Detection

##  1 Introduction

Water is the foundation of (our) life – we need water for drinking, cooking,
hygiene and farming. Water distribution networks (WDNs), which distribute
water from the supplier to the customers, are therefore considered as critical
infrastructure. A major problem for water utility companies (and society in
general) are anomalies that cause loss or contamination of water – e.g.
leakages such as pipe bursts, sensor faults, pollution, cyber-physical
attacks, etc. [3, 1, 15]. Because of water shortages, among others caused by
climate change, (drinking) water becomes an increasingly valuable resource
that should not be wasted. However, it was estimated that leakages in WDNs
lead to a loss of more than 45 million $m^{3}$ of drinking water in developing
countries – even high developed countries such as the island of Cyprus looses
approx. up to 25% of their drinking water due to leaky pipes [11].

Because of the increasing availability of sensors (e.g. pressure sensors) in
WDNs, water utility companies nowadays use computer systems for (autonomously)
monitoring their networks [14]. These systems are realized using methods from
engineering, statistics and machine learning (ML) [2]. While many different
and successful methods for anomaly detection and localization have been
proposed [17], these methods are usually not able to adapt to an occurring
change or anomaly. These systems might be able to detect anomalies but once
the anomaly is detected, they are “blind” for everything else that happens
while the detected anomaly is present – the systems must be recalibrated or
refitted which becomes challenging because a large amount of data (and
therefore collection time) is needed. Adaptation to changes – in case of WDNs
anomalies or simply changes in the water consumption behavior of the customers
(i.e. changed demand) – can be natural handled by online learning methods.

Online learning [4] can be considered as a sub-field of machine learning which
deals with models that are trained incrementally – i.e. they can learn from a
data stream instead of a fixed training set only. For example, they can be
used for electricity price prediction [20] or electric load forecasting [19].

In this work, we contribute to online learning for regression problems and
water distribution networks as a particular field of application. More
specifically, our contributions are:

  * •

We propose SAM-kNN regression, a memory based online learner for regression
problems.

  * •

We evaluate our proposed SAM-kNN regression method in the context of anomaly
detection in water distribution networks.

The remainder of this paper is structured as follows: After briefly reviewing
related work in Section 2, we introduce the problem setting we are considering
in this work (see Section 3). Next, in Section 4 we propose SAM-kNN regression
for online learning, which we empirically evaluate in the context of anomaly
detection in water distribution networks (see Section 5). Finally, this work
closes with a summary and conclusion in Section 6.

##  2 Related Work

Incremental or online learning is a machine learning paradigm in which a model
is updated after each data sample that is fed into it. This paradigm is
especially suited for large data sets, that are too big to be processed in
batch fashion, or in situations where data becomes available only sample after
sample – e.g. in the form of a potentially infinite streams of data.
Incremental learning has made great strides in recent years [5], with most
applications set in a classification environment [18, 10, 12, 9, 7, 21].

One particular system is the SAM-kNN classifier [13]. This incremental
algorithm was created to perform on long data streams, using its internal
memory structure to alleviate the problem of catastrophic forgetting when
frequent concept changes are expected.

On the other hand, only a few approaches that utilize incremental learning for
regression problems exist. In [19] the authors use an incremental variant of
the support vector regressor (SVR) to build models for electricity price
prediction. [20] uses a similar SVR, paired with phase space reconstruction
for time series to facilitate electrical load forecasting. In [6] a wide range
of incremental regression algorithms are compared for their use in exoskeleton
control.

However, all of these publications utilize standard incremental algorithms and
do not explicitly build a new model to work with. We, on the other hand,
propose a reformulation of the SAM-kNN classifier for regression as a
standalone algorithm.

##  3 Problem Setting

Incremental or online regression is the task of predicting a response variable
$y\in\mathbb{R}$ from a stream $S=\\{x_{1},x_{2},x_{3},...\\}$ of variables
$X\in\mathbb{R}^{n}$. Hereby, a new instance of the incremental model is
learned for each incoming sample of the data stream.

We work on water distribution networks that have several internal nodes $n$.
These nodes are equipped with sensors, that measure water pressure and flow
rate. Each sensor provides read outs at specific time intervals $t$. For every
node, this creates a potentially infinite data stream of sensors values
$S=\\{(s_{1},y_{1}),(s_{2},y_{2}),(s_{3},y_{3}),...\\}$, where
$s_{i}\in\mathbb{R}^{n-1}$ represents the sensor values of all but one node in
the network and the predictor variable $y_{i}\in\mathbb{R}$ represents the
sensor value at the remaining node.

This means, that we use the read outs of $n-1$ nodes to predict the value of
the $n$th node using an incremental regression algorithm. Said algorithm
processes the stream $S$ instance after instance by generating a sequence of
models $H=\\{h_{1},h_{2},h_{3},...\\}$, where $h_{i-1}(s_{i})=\hat{y}_{i}$.
After each prediction the true value $y_{i}$ is revealed and a new model
$h_{i}$ is learned.

We use the _Interleaved train test error (ITTE)_ as a cost function to train
the model:

$E(S)=\sqrt{\frac{1}{t}\sum^{t}_{i=1}(h_{i-1}(s_{i})-y_{i})^{2}}$ (1)

This ITTE measures the _Root Mean Squared Error (RMSE)_ over every model
$h_{i}$ up to a given time point $t$.

Whenever the local error $h_{i-1}(s_{i})-y_{i}$ exceeds a certain threshold,
this means, that more water than predicted flows through the observed node.
This is taken as an indication for a water leak and an alarm will be
triggered. Being an incremental algorithm, our model will than automatically
adjust to the new circumstances so that accurate prediction of the water flow
will be maintained throughout the leak. Our model, which will be explained in
detail in the next section, has the capability to remember long term concepts
and therefore, as soon as the water leak is fixed, it will revert back to the
normal circumstances.

##  4 Model

Our proposed model is an adaption of the Self Adjusting Memory (SAM) [13], an
incremental classifier, to regression. This approach is based on two distinct
internal memories, the Short-Term (STM) and the Long-Term memory (LTM).
Hereby, the STM is a dynamic sliding window over the last $m$ samples, that is
supposed to only hold the most recent concept of the data stream:

$M_{ST}=\\{(x_{i},y_{i})\in\mathbb{R}^{n}\times\mathbb{R}\ |\
i=t-m+1,...,t\\}$ (2)

The LTM, on the other hand, is a collection of $p$ samples, which hold older
concepts, that do not contradict the STM and might still be of use in the
future:

$M_{LT}=\\{(x_{i},y_{i})\in\mathbb{R}^{n}\times\mathbb{R}\ |\ i=1,...,p\\}$
(3)

Additionally, there is the combined memory (CM), which is a simple union of
the STM and the LTM:

$M_{C}=M_{ST}\cup M_{LT}$ (4)

Each memory induces a kNN regressor that can be used independently from the
others. To determine which kNN is used for every new incoming data sample, the
ITTE (see section 3) is tracked for all sub-models and the one with the lowest
current ITTE is chosen.

###  4.1 Model Parameters

The proposed model has three parameters that are continuously adapted during
deployment:

  1. 1.

The size $m$ of the STM sliding window

  2. 2.

The data samples in the LTM

  3. 3.

The ITTEs of the three sub-models

Additionally, there are three hyperparameters that can be chosen robustly and
are set before deployment:

  1. 1.

The number of neighbours $k$

  2. 2.

The minimum size $L_{min}$ of the STM

  3. 3.

The maximum size $L_{max}$ of the LTM

###  4.2 Model Adaption

Whenever a new data sample arrives, it is added to the STM, which means that
this memory grows continuously. However, since it is supposed to hold only the
most recent concept, a reduction of the STM window size is performed on a
regular basis. This is facilitated by testing smaller window sizes at every
iteration and choosing the one that is optimizing the ITTE. Tested windows
are:

$M_{l}=\\{(x_{t-l+1},y_{t-l+1}),...,(x_{t},y_{t})\\}$ (5)

where $l\in\\{m,m/2,m/4...\\}$ and $l\geq L_{min}$.

$M_{ST_{t+1}}=\underset{S\in\\{M_{m},M_{m/2},...\\}}{\mathrm{argmin}}E(S)$ (6)

Whenever the STM is shrunk in size, the data samples $O_{t}$ that fall out of
the sliding window are not discarded.

$O_{t}=M_{ST_{t}}\setminus M_{ST_{t+1}}$ (7)

Instead, they undergo a cleaning process, and those, that are still consistent
with the new STM are added to the LTM. Afterwards, the whole of the LTM is
cleaned as well, to ensure consistency with the STM at all times. When the LTM
reaches its maximum size, samples get discarded in a way that ensures minimal
information loss.

###  4.3 Cleaning Process

The process to clean a set of samples with respect to the STM is defined in
the following way:

A set $A$ is cleaned by another set $B$ regarding an example $(x_{i},y_{i})\in
B$

$clean:(A,B,(x_{i},y_{i}))\mapsto\hat{A}$ (8)

where $A,B,\hat{A}\subset\mathbb{R}^{n}\times\mathbb{R}$ and $(x_{i},y_{i})\in
B$.

$\hat{A}$ is defined in five steps:

  1. 1.

Determine the $k$ nearest neighbours of $x_{i}$ in $B\setminus(x_{i},y_{i})$
and find the maximum distance

$\Delta_{x}^{\ast}=\max\big{\\{}d(x_{i},x)\ |\ x\in
N_{k}(x_{i},B\setminus(x_{i},y_{i}))\big{\\}}$ (9)

  2. 2.

Compute the maximum weighted difference of $y_{i}$ and $y\in
N_{k}(x_{i},B\setminus(x_{i},y_{i}))$

$\Delta_{y}^{\ast}=\max\Bigg{\\{}\Bigg{(}\frac{y_{i}-y}{e^{\frac{x_{i}-x}{\Delta_{x}^{\ast}}}}\Bigg{)}\
|\ y\in N_{k}(x_{i},B\setminus(x_{i},y_{i}))\Bigg{\\}}$ (10)

  3. 3.

Determine all samples in $A$ that are within $\Delta_{x}^{\ast}$ of $x_{i}$

$C=\big{\\{}(x,y)\in A\ |\ d(x_{i},x)<\Delta_{x}^{\ast}\big{\\}}$ (11)

  4. 4.

Compute the weighted differences of $y_{i}$ and $y\in C$

$\Delta_{y}=\Bigg{\\{}\Bigg{(}\frac{y_{i}-y}{e^{\frac{x_{i}-x}{\Delta_{x}^{\ast}}}}\Bigg{)}\
|\ y\in C\Bigg{\\}}$ (12)

  5. 5.

Discard samples from $C$ that have a larger weighted difference than
$\Delta_{y}^{\ast}$

$\hat{A}=A\setminus\\{(x,y)\in C\ |\ \Delta_{y}(x)>\Delta_{y}^{\ast}\\}$ (13)

Furthermore, the cleaning operation for the full set $B$

$clean:(A,B)\mapsto\hat{A}_{|B|}$ (14)

is defined by iteratively applying the former cleaning for all
$(x_{i},y_{i})\in B$

$\displaystyle\hat{A}_{0}$ $\displaystyle=A$ $\displaystyle\hat{A}_{t+1}$
$\displaystyle=clean(\hat{A}_{t},B,(x_{t+1},y_{t+1}))$

In summary, when the STM is shrunk in size, the process to clean the discarded
set $O_{t}$ is described by the operation:

$clean(O_{t},M_{ST_{t+1}})$ (15)

After that, the LTM is cleaned as well:

$clean(M_{LT_{t}},M_{ST_{t+1}})$ (16)

###  4.4 Compression of the LTM

When new samples are added to the LTM while it reaches maximum capacity, old
samples need to be discarded. To avoid a significant information loss, samples
are discarded in an iterative process one after another until
$|M_{LT}|<L_{max}$ again. Hereby, the data sample with the lowest distance to
any other two samples is chosen for every iteration of the discarding process.

###  4.5 Final Model

The complete pseudocode of our proposed model is given in Algorithm 1.

Algorithm 1 SAM-kNN Regression

Input: Data stream $S$, one $s_{i}$ at a time
Output: $\hat{y}_{i}$ for every $s_{i}$

1:$M_{ST}$, $M_{LT}$ = $\\{s_{0},..,s_{L_{min}}\\}$ $\triangleright$
Initialize STM and LTM

2:$E_{ST},E_{LT},E_{C}=0$ $\triangleright$ Initialize tracked errors

3:for $s_{i}\in S\setminus\\{s_{0},..,s_{L_{min}}\\}$ do $\triangleright$ Loop
over the remaining data stream

4: $BM=argmin(E_{ST},E_{LT},E_{C})$ $\triangleright$ Find best memory with
lowest error

5: $\hat{y_{i}}=kNN_{BM}(s_{i})$ $\triangleright$ Predict with kNN of best
memory

6: Update $E_{ST},E_{LT},E_{C}$

7: $M_{ST}=M_{ST}\ \cup\ \\{s_{i}\\}$ $\triangleright$ Add current sample to
STM

8: Evaluate smaller STM sizes

9: if STM is reduced then

10: $O_{t}=M_{ST_{t}}\setminus M_{ST_{t+1}}$ $\triangleright$ Take discarded
samples from STM

11: $clean(O_{t},M_{ST_{t+1}})$ $\triangleright$ Clean discarded samples with
respect to new STM

12: $M_{LT}=M_{LT}\ \cup\ clean(O_{t},M_{ST_{t+1}})$ $\triangleright$ Add
cleaned samples to LTM

13: $clean(M_{LT},M_{ST_{t+1}})$ $\triangleright$ Clean new LTM with respect
to new STM

14: end if

15:end for

##  5 Experiments

We empirically evaluate our proposed method in an online scenario for
detecting leakages and sensor faults in water distribution networks – all
experiments are implemented in Python111Implementation is available on GitHub:
https://github.com/andreArtelt/SAM-kNN-Regressor_OnlineLearning_WDNs.

###  5.1 Data

Figure 1: L-Town network [16] – we only use “Area A” where we have $29$
pressure sensors.

We use a version of the L-Town water distribution network as used in [16], as
a prominent realistic benchmark for anomaly detection – we use Area A only
which consists of $661$ nodes, $766$ links, and $29$ (optimally placed)
pressure sensors. We build and simulate $10$ scenarios where the first $5$
scenarios each contain a single leakage – we vary position, time and size of
the leakage – and the remaining $5$ scenarios each contain a single different
sensor fault (position is varied):

  * •

Scenario 6: Sensor measurement is overflowing over time – i.e. it is going to
infinity over time.

  * •

Scenario 7: Gaussian noise is added to the sensor measurement.

  * •

Scenario 8: A constant value is added to the sensor measurement.

  * •

Scenario 9: Sensor measurement is set equal to zero.

  * •

Scenario 10: Sensor measurement is shifted by a small amount.

Each scenario is simulated (using WNTR [8]) for $3$ month and pressure sensors
are sampled every $5$ minutes.

The data stream of sensor measurements is post processed by using a sliding
window of size $4$ – we average all samples dimensional wise in this time
window, so that we end up with $29$ dimensional samples.

###  5.2 Setup

We compare our proposed SAM-kNN regressor (see Section 4) to several other
online learning regressors. In order to justify the introduced overhead of the
SAM architecture, we compare its performance to vanilla regressors (kNN
regression and linear regression) wrapped as online learners by using the
river toolbox222https://github.com/online-ml/river.

For each pressure sensor, we build a corresponding virtual sensor based on all
other pressure sensors – i.e. we try to predict (using a regressor) the
pressure based on the past pressure values of all other pressure sensors (see
Section 3). These virtual sensors are then used for a residual based anomaly
detection – i.e. an alarm (detected anomaly) is raised when the predicted
pressure value deviates to much from the observed pressure measurement. For
each scenario, the processed data stream is fed as batches of $200$ samples to
the regressors (realizing the virtual sensors).

For each regressor, we evaluate the performance of the resulting anomaly
detector – since we are interested in the detection of a single anomaly
(leakage and sensor fault), we report true positives (TP), false positives
(FP) and false negatives (FN). Note that the true positives and false
negatives are always either $0$ or $1$ because we only check whether an alarm
was raised when the single anomaly was present or not – however, for the false
positives, we count every single false alarm.

###  5.3 Results

The results for leakage detection, for each regressor and each scenario, are
shown in Table 1. Likewise, the results for sensor fault detection are shown
in Table 2.

Table 1: Leakages: Evaluation of residual based anomaly detection in water distribution networks – note that each scenario consists of approx. $23000$ samples. Method | Metric | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 5
---|---|---|---|---|---|---
SAM-kNN | TP | $1$ | $1$ | $1$ | $1$ | $1$
FP | $48$ | $20$ | $3$ | $20$ | $17$
FN | $0$ | $0$ | $0$ | $0$ | $0$
kNN | TP | $1$ | $1$ | $1$ | $1$ | $1$
FP | $17057$ | $19216$ | $11146$ | $19082$ | $18751$
FN | $0$ | $0$ | $0$ | $0$ | $0$
Linear regression | TP | $0$ | $0$ | $0$ | $0$ | $0$
FP | $0$ | $0$ | $0$ | $0$ | $0$
FN | $1$ | $1$ | $1$ | $1$ | $1$
Table 2: Sensor faults: Evaluation of residual based anomaly detection in water distribution networks – note that each scenario consists of approx. $23000$ samples. Method | Metric | Scenario 6 | Scenario 7 | Scenario 8 | Scenario 9 | Scenario 10
---|---|---|---|---|---|---
SAM-kNN | TP | $1$ | $1$ | $1$ | $1$ | $1$
FP | $155$ | $20$ | $157$ | $96$ | $156$
FN | $0$ | $0$ | $0$ | $0$ | $0$
kNN | TP | $1$ | $1$ | $1$ | $1$ | $1$
FP | $18596$ | $18596$ | $18596$ | $18596$ | $18596$
FN | $0$ | $0$ | $0$ | $0$ | $0$
Linear regression | TP | $1$ | $0$ | $0$ | $0$ | $0$
FP | $0$ | $0$ | $0$ | $0$ | $0$
FN | $0$ | $1$ | $1$ | $1$ | $1$

We observe the same/similar effects for both types of anomalies (leakages and
sensor faults): We observe that linear regression completely fails to detect
the anomaly – this indicates that a linear model is not sufficient to model
the hydraulic dynamics in the water distribution network and hence fails to
detect any anomalies. There is only one exception: For scenario 6 the linear
model is able to detect the sensor fault – recall that in this particular
scenario the sensor fault is characterized by a slowly overflowing sensor
measurements (i.e. the pressure value goes to infinity), which is expected to
be easily detected because it is a very “loud” fault. The kNN model shows good
performance in detecting the anomaly but has a huge number of false positives
– i.e. it is too sensitive and raises lots of false alarms. Our proposed SAM-
kNN shows the best performance – it is able to consistently detect the
anomalies while having a small numbers of false positives only. The huge
reduction of the false positives in comparison to vanilla kNN indicates that
the overhead introduced by the SAM architecture actually pays off.

##  6 Summary & Conclusion

Inspired by the SAM-kNN classifier, we proposed SAM-kNN regressor as an online
learner for regression. In contrast to other online learners, our proposed
method comes with a memory component which allows it to remember past concepts
quite easily. We empirically evaluated our proposed online learner in an
anomaly detection scenario for a realistic water distribution network – our
proposed online learner consistently outperforms other standard online
learners.

Although our proposed method shows good performance for leakage and sensor
fault detection (two very common anomalies), it is unclear, how well it
performs for other (more complex) types of anomalies such as cyber-physical
attacks, etc. Furthermore, another challenge is high-dimensional data – in
this work we had $29$ dimensional which is already somewhat high but can be
still managed by our kNN based method. However, in case of really high
dimensional data, kNN will encounter performance problems – e.g some kind of
integrated dimensionality reduction might be required. We leave these aspect
as future work.

## Acknowledgment

We acknowledge the bachelor thesis by Augustin Harter (University of
Bielefeld) and Yannik Sander (University of Bielefeld) which served as a
mental starting point for this work.

## References

  * [1] Alexander, A., Julius, T., Andrew, T., Ezera, A., Christine, A.: Contamination potentials of household water handling and storage practices in kirundo subcounty, kisoro district, uganda (2019)
  * [2] Chan, T.K., Chin, C.S., Zhong, X.: Review of current technologies and proposed intelligent methodologies for water distributed network leakage detection. IEEE Access 6, 78846–78867 (2018)
  * [3] Farley, M., Trow, S.: Losses in water distribution networks. IWA publishing (2003)
  * [4] Gama, J.: A survey on learning from data streams: current and future trends. Progress in Artificial Intelligence 1(1), 45–55 (2012)
  * [5] Gomes, H.M., Read, J., Bifet, A., Barddal, J.P., Gama, J.: Machine learning for streaming data: state of the art, challenges, and opportunities. In: ACM SIGKDD Explorations Newsletter. pp. 6–22 (2019)
  * [6] Jakob, J., Hasenjäger, M., Hammer, B.: On the suitability of incremental learning for regression tasks in exoskeleton control. In: IEEE Symposium on Computational Intelligence in Data Mining (CIDM). IEEE (December 2021)
  * [7] Jodelet, Q., Liu, X., Murata, T.: Balanced softmax cross-entropy for incremental learning. In: Farkaš, I., Masulli, P., Otte, S., Wermter, S. (eds.) Artificial Neural Networks and Machine Learning – ICANN 2021. pp. 385–396. Springer International Publishing, Cham (2021)
  * [8] Klise, K.A., Murray, R., Haxton, T.: An overview of the water network tool for resilience (wntr). (2018)
  * [9] Lei, C.H., Chen, Y.H., Peng, W.H., Chiu, W.C.: Class-incremental learning with rectified feature-graph preservation. In: Ishikawa, H., Liu, C.L., Pajdla, T., Shi, J. (eds.) Computer Vision – ACCV 2020. pp. 358–374. Springer International Publishing, Cham (2021)
  * [10] Li, H., Dong, W., Hu, B.G.: Incremental concept learning via online generative memory recall (2019). https://doi.org/10.48550/ARXIV.1907.02788, https://arxiv.org/abs/1907.02788
  * [11] Liemberger, R., Marin, P., et al.: The challenge of reducing non-revenue water in developing countries–how the private sector can help: A look at performance-based service contracting (2006)
  * [12] Liu, Y., Su, Y., Liu, A.A., Schiele, B., Sun, Q.: Mnemonics training: Multi-class incremental learning without forgetting. In: 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). IEEE (jun 2020). https://doi.org/10.1109/cvpr42600.2020.01226
  * [13] Losing, V., Hammer, B., Wersing, H.: Knn classifier with self adjusting memory for heterogeneous concept drift. In: 2016 IEEE 16th International Conference on Data Mining (ICDM). pp. 291–300 (2016). https://doi.org/10.1109/ICDM.2016.0040
  * [14] Makropoulos, C., Savić, D.: Urban hydroinformatics: past, present and future. Water 11(10), 1959 (2019)
  * [15] Nikolopoulos, D., Moraitis, G., Bouziotas, D., Lykou, A., Karavokiros, G., Makropoulos, C.: Cyber-physical stress-testing platform for water distribution networks. Journal of Environmental Engineering 146(7), 04020061 (2020)
  * [16] Vrachimis, S.G., Eliades, D.G., Taormina, R., Ostfeld, A., Kapelan, Z., Liu, S., Kyriakou, M., Pavlou, P., Qiu, M., Polycarpou, M.M.: Battledim: Battle of the leakage detection and isolation methods (2020)
  * [17] Wu, Y., Liu, S.: A review of data-driven approaches for burst detection in water distribution systems. Urban Water Journal 14(9), 972–983 (2017)
  * [18] Wu, Y., Chen, Y., Wang, L., Ye, Y., Liu, Z., Guo, Y., Zhang, Z., Fu, Y.: Incremental classifier learning with generative adversarial networks (2018). https://doi.org/10.48550/ARXIV.1802.00853, https://arxiv.org/abs/1802.00853
  * [19] Yan, J., Tian, C., Wang, Y., Huang, J.: Online incremental regression for electricity price prediction. In: Proceedings of 2012 IEEE International Conference on Service Operations and Logistics, and Informatics. pp. 31–35. IEEE. https://doi.org/10.1109/SOLI.2012.6273500
  * [20] Yang, Y., Che, J., Li, Y., Zhao, Y., Zhu, S.: An incremental electric load forecasting model based on support vector regression. Energy 113, 796–808 (2016). https://doi.org/https://doi.org/10.1016/j.energy.2016.07.092, https://www.sciencedirect.com/science/article/pii/S0360544216310167
  * [21] Zhu, Q., He, Z., Ye, X.: Incremental classifier learning based on pedcc-loss and cosine distance (2019). https://doi.org/10.48550/ARXIV.1906.04734, https://arxiv.org/abs/1906.04734