Another Adaptive Approach to Novelty Detection in Time Series

ANOTHER ADAPTIVE APPROACH TO NOVELTY
DETECTION IN TIME SERIES
Matouš Cejnek, Peter Mark Beneš, Ivo Bukovsky
Department of Instrumentation and Control Engineering,
Czech Technical University, Prague, Czech Republic
{Matous.Cejnek;PeterMark.Benes;Ivo.Bukovsky}@fs.cvut.cz

ABSTRACT
This paper introduces a novel approach to novelty detection of every individual sample of data
in a time series. The novelty detection is based on the knowledge learned by neural networks
and the consistency of data with contemporary governing law. In particular, the relationship of
prediction error with the adaptive weight increments by gradient decent is shown, as the
modification of the recently introduced adaptive approach of novelty detection. Static and
dynamic neural network models are shown on theoretical data as well as on a real ECG signal.

KEYWORDS
Novelty Detection, Time Series, Gradient Descent, Neural Networks, ECG

1. INTRODUCTION
In principle, novelty carried through measured samples of data may be evaluated either via
probability based approaches as exampled in [1] or, via learning system based approaches as in
the work [2]. The first of these streams, i.e. probabilistic, is represented by the statistical
approaches of novelty measures and by probabilistic approaches for evaluation of entropy. The
Sample Entropy (SampEn) and the Approximate Entropy (ApEn) are very typical and very
relevant examples to be mentioned [3]-[4]. These approaches are closely related to the multi-scale
evaluation of fractal measures, where further case studies utilizing SampEn, ApEn, and
Multiscale Entropy (MSE) can be found in [5]-[7]. Further to this, probabilistic entropy approach
to the concept shift (sometimes the concept drift) detection in sensory data is reported in [8]. The
second of the mentioned streams is represented by the utilization of learning systems, such as
neural networks and fuzzy-neural systems, and this is also the main area of focus for the
presented work in this paper. During the last three decades of 20th century, the works that were
focused in regards to learning systems are that of [9]-[12], and for incremental learning approach
can be referenced for example also the work [13]. Then, a particularly focused approach toward
the utilization of learning systems has been rising with works [14]-[17]. Where, nonlinear
estimators and learning algorithm were utilized for the fault detection via the proposed utilization
of a fault function that evaluates behaviour of residuals of a learning system. Currently,
significant research that shall also be referenced is adaptive concept drift detectors, proposed in
[18]. Some readers might also see some analogies of the proposed approach in this paper to the
Adaptive Resonance Theory [19]. Another approach to novelty detection is based on utilization of
adaptive parameters of incrementally learning models (neural networks), i.e. the Adaptation Plot
David C. Wyld et al. (Eds) : CCSIT, SIPP, AISC, PDCTA, NLP - 2014
pp. 341–351, 2014. © CS & IT-CSCP 2014

DOI : 10.5121/csit.2014.4229

342

Computer Science & Information Technology (CS & IT)

[20] that has been recently enhanced with multi-scale approach [21]. A most recent method is the
Learning Entropy, i.e., a multi-scale approach to evaluation of unusual behaviour of adaptive
parameters of a learning model is introduced in [22]. This paper however introduces another,
different approach to novelty detection, that is neither based on statistical approaches, nor is it
based on evaluation of error residual. Contrary to [21], this new approach operates only on
parameter space of incrementally learning systems, and it does not use the multi-scale approach
for detection sensitivity. Here the method introduced shows one of the possibilities how to detect
perturbations within measured data in every new sample.

2. USED METHODOLOGY
The method of detection proposed in this paper, utilises an adaptive prediction model. It is
interesting to note how promising such method of detection works when used with a static linear
adaptive model on non-linear data. Thus, for demonstrational purposes of this method, we will
use a linear based adaptive model. The adaptation technique as applied to the predictive model, is
based on the classical Gradient Descent algorithm. The employed linear adaptive model initially
features a vector of random numbers as initial neural adaptive weights, which over each sample,
are incrementally trained to model the real data, and an input vector comprised from previous
samples of the real data, as follows;
n

y (k + 1) = ∑ wi xi = w.x

(1)

i =0

Where, here, w denotes the row vector of all neural weights and x represents the vector of inputs.
For the scope of this paper, the structure of this input vector for the linear predictive model is as
follows;

x( k ) = 1 yr ( k )


yr ( k − 1) ... yr (k − 10)T 


(2)

Where, x0=1 allows for neural bias in case of the linear predictive model and yr represents
measured values of the time series. The input vector was chosen to feature ten previous samples
of the measured time series. Prior to prediction, the input vector was standardized via equation
(3), in order to acquire better stability of the adaptive algorithm.

yr =

yr − mean( yr )
3 × std ( yr )

(3)

For further improvement of the model stability, we provide adaptation of the learning rate µ via
Normalized Least Squares method [23]. The equation (4) describes how this adaptation is
performed. The Learning Rate adaptation is utilized before calculation of every sample. From
equation (4) it is obvious that the normalized learning rate is smaller than the default learning rate
in dependency on input x(k) for every sample;

η=

µ
1 + x( k )T × x(k )

(4)

Where, η serves as a substitution for µ For novelty detection estimation in every new sample in
discrete time k, we use the product of absolute values of the prediction error and adaptive weight
changes according to equation (5), as follows;

ND( k ) =  e( k ) × ∆wi ( k ) ; i = 0,..,10 



(5)

Equation (5), thus presents the main principle behind this introduced method of novelty detection
of every new sample according to temporary system dynamics. This method is simultaneously
using prediction error and changes of adaptive weights (e(k) and ∆wi(k), respectively). This is the
main theoretical result of this work. In the following sections we will investigate functionality of
this method on real and artificial ECG with sampling of 256Hz frequency.


343

3. EXPERIMENTAL ANALYSIS
In this section, we will demonstrate the newly introduced method via computer simulations
created in programming language Python 2.7 [24].The numerical algorithms were realized via
python library Numpy [25]. The simulation was performed on a personal computer and was faster
than real time measuring of an ECG signal. Thus, this justifies that such implementation is
applicable for online realization.

3.1. Artificial ECG
In this subsection, we will demonstrate usage of the newly introduced method on an artificial
ECG signal both, with and without noise, before eventually testing on a real ECG signal (of 256
Hz frequency), in the following subsection. For demonstrational purposes in this section, noise
was added to the signal to highlight the ability of this introduced method, in detection of
unexpected samples within the data. Especially where, the predictive model features lower
prediction accuracy, comparative to data without noise. The reason why we test this method on an
artificial signal, is to emphasize how well the detection works on perturbed data, if the signal
doesn’t contain any complicated phenomena. The artificial ECG time series used in our paper was
created by a serially repeated pattern of a real ECG signal, sampled over one period. Thus, this
artificial time series is an ideally periodic signal.

Figure 1. Novelty Detection used on artificial ECG without noise (for details refer to Figure 2)

344


Figure 2. Details of prediction in areas of introduced perturbations in artificial ECG without noise (blue –
signal with perturbation, green – predicted signal)

Regarding the length of the pre-training data for the predictive model, we chose to take the first
2000 samples of the artificial data (ref. Figure 1). For sufficient training, 100 epochs was
adequate. Figure 2, depicts the details of all perturbations included in the artificial ECG signal
without noise. The size of the introduced perturbation is 0.03 mV. As we can see on Figure 2,
these perturbations are small in comparison with the amplitude of the signal. Looking at the
behaviour of the used predictive model, here we see the model tries to relearn immediately when
the prediction error and weight adaptation increases (Figure 1). The return of the predictive model
to previous prediction accuracy takes approximately 20 following samples. In Figure 1, it is
possible to see the prediction error in specific places of a single period. These errors are caused
by insufficient prediction ability of the simple, linear predictive model. Furthermore, in Figure.1,
these errors are not detected as new data by the applied novelty detection method. Figure 3,
shows the simulation of the artificially created ECG with the addition of noise. Here again, three
perturbations were introduced to the data. These perturbations are located on the same positions
as the signal without noise. This introduced noise was implemented via a generator of pseudorandom numbers, composed as a vector of random numbers in range from 0 to 0.01, added via the
following equation (6);

yr (k ) = yr (k ) + rand , rand ∈ 〈 0, 0.01〉, k = 1000, 4000, 6000 .

(6)

Figure 3 displays the real signal together with the simulated. Here in the first graph, it is possible
to see the difference between the signals within the region of peaks of the amplitude. For the used
predictive model, it is much more difficult to learn the pattern of the signal with added noise in
comparison to that without noise. The errors and absolute values of the weight increments (Figure
3) are not entirely dependent on the periodicity of the signal. In the plot of error on Figure 3, it is
not possible to see the perturbations clearly, as in the plot of prediction error without noise on
Figure 2. In the graph of absolute error of weight increments, we can more evidently see the
location of the perturbations. On the graph of novelty detection (Figure 3) the perturbation
locations are even more evidently seen and, this is because a huge part of the models periodic


345

errors, are filtered. Furthermore, these errors have no impact on the detection of unexpected
samples in the data.

Figure 3. Novelty Detection used on artificial ECG with noise (details on Figure 4)

Figure 4. Details of prediction in areas of introduced perturbations in artificial ECG with noise (blue –
signal, green – predicted signal)

Figure 4 shows the details of three perturbations included into the artificial signal containing
noise. Here, it is clearly shown that the prediction error of the signal with noise and also in the un-

346


perturbed section of the signal, could reach substantially high values. The adaptive model again
immediately reacts to the introduced perturbation and tries to relearn the data signal. The time in
which the model needs for regaining normal accuracy after meeting a perturbation is hard to
estimate, because the prediction error is strongly dependant on the level of noise present in the
data.

3.2. Real ECG Signal
In this paper we also demonstrate the introduced method on an ECG signal. The used time series
was measured by an internal cardio-defibrillator with 256Hz frequency (the measured data was
obtained courtesy of [26]). This signal was chosen because it contains spontaneous ventricular
tachycardia, which is a rare phenomenon to measure. In this subsection, this introduced method of
novelty detection will be demonstrated via a real ECG signal, which features a healthy, sinus like
structure followed, by a ventricular tachycardia (arrhythmia). Later, we will show that this
technique works in both in the sinus like region and also in the arrhythmic region of this signal.
For demonstration, we will choose the same linear neural unit as the one previously used for
prediction of the artificial ECG signal. The size of data chosen for learning of the predictive
model is 1000 samples. In order to achieve sufficient pre-learning of the used neural unit with
such amount of data, less than 500 epochs is adequate for achieving optimal results. Using any
larger size of epochs doesn’t seem to significantly improve the accuracy. Figure. 5 clearly depicts
which part is the healthy ECG signal and which part represents the arrhythmia. In the introduced
novelty detection, it is possible to detect the start of the arrhythmia signal, approximately 1000
samples before the arrhythmia is introduced (the shape of the period before the arrhythmia looks
the same, but the scale of amplitudes starts varying). In the first graph of Figure 5, we can see
how the measured signal is practically equivalent with the predicted signal. The included
perturbations are not clearly seen in this graph. However, these perturbations are located in the
samples of discreet time 1000, 3000, 5000. On the graph of the prediction error, it is possible to
see the perturbations quite well and even more so in the graph of absolute values of adaptive
weights. However, looking on the graph of novelty detection, these perturbations are even more
evidently pronounced. Moreover, the periodic errors are suppressed in region of the healthy ECG
and arrhythmia signal. It is important to notice that the suppressing of the periodic error, is not
that dominant in the onset of arrhythmia, unlike in other parts of the data. On Figure 6, we can see
the prediction models seem to be quite accurate, but in some parts of the ECG period, there is
always some small inaccuracy. These inaccuracies are mostly located in the positions of global
minima’s and maxima’s, for a given period on the measured signal. The included perturbations
are significantly small in size (0.04 – approximately 2% of the amplitude of healthy ECG signal).
Furthermore, the adaptive model immediately reacts with re-learning of the ECG data, where by
achieving the previous model accuracy, takes approximately 5-10 following samples.


347

Figure 5. Novelty Detection used on real measured ECG signal

Figure 6. Details of prediction in areas of introduced perturbations in real measured ECG (blue – real
signal, green – output of neural unit)

348


3. DISCUSSION
As presented in previous chapters, the potentials of this method for novelty detection in
biomedical signals, is quite promising. For demonstration, we chose a predictive model, which
was relatively simple by structure. Here, such model was not capable to properly predict the
measured ECG, yet the introduced novelty detection was able to distinguish the difference
between prediction error caused by the model insufficiency and prediction error caused by
novelty in the samples. This method also seems to be highly useful on artificial signals containing
significant noise, where the model is unable to learn such noise, but can distinguish the noise
from new phenomena in the measured data. Also on artificial data without noise we can certainly
note, that novelty detection distinguishes the prediction error from the novelty in data, also with
even a significantly small, signal perturbation.

Figure 7. Correlation coefficients between adaptive weight increments and error, for ∆w1 and ∆w9 in the
artificial ECG signal without noise (upper pair), artificial ECG signal with noise (middle pair) and real
ECG signal (lower pair)

A further remark may be seen in Figure 7. Where, the correlation coefficients between the
adaptive weight increments and error, over various samples of lag are analysed. For


349

demonstrative purposes, Figure 7 only includes an analysis over the ∆w1 and ∆w9 adaptive weight
increments with the error. Here, we may note that in all cases of the corresponding correlation
coefficients, statistically insignificant correlation between the adaptive weight increments and
error is seen and the same was observed for all other weight increments. This is justified by the
notion that the prediction error of the model carries the information about the size of inaccuracy,
while the adaptive weight increments carry information about how much the model tried to adapt
on new data. Thus, even when the model error may be high, this may not necessarily be related
with the adaptive weight increments (for more on this principle, please refer to the works of [20]–
[22]). In case of a high error value, caused by common phenomena (what is out of the models
capability to learn), novelty detection doesn’t mark those particular samples, because the model
has already recognized its inability to learn, which also further justifies the functionality of this
method for novelty detection.
Another remark is in regards to the small time consumption in detection of this newly introduced
method. This is due to the nature of the model being merely multiplication of values, which was
calculated during the prediction of the data. Novelty detection doesn’t slow down the process of
prediction. This fact is a big advantage especially in processing the signal in real time, what could
be a crucial feature in the case of ECG or other biomedical signals. Furthermore, as we can see on
the results obtained by application of novelty detection on real ECG (Fig. 5), this technique of
detection of unexpected data highlights the onset of arrhythmia from the other data. According to
this method it is possible to see the onset of arrhythmia before it appears in the measured signal or
prediction error. The first few samples show that there is inconsistency in the data, which appears
approximately 1000 samples before the start of the arrhythmia signal. This model could be used
for the detection of arrhythmia after proper optimizing and, it is an objective for further research.

4. CONCLUSION
Unusual behaviour of adaptable parameters and actual prediction error together, establishes novel
learning-system-based measure of novelty that is introduced in this paper. The method presented
in this paper is different from other methods and from the methods in [20] & [21], because this
method is using actual error of the predictive model, together with cognitive information in
changes of adaptive weights of the model. This is an interesting approach, because the prediction
accuracy and behaviour of a learning model are not necessarily correlated with each other [21];
however, both values together provide us with important information about consistency at every
sample with temporary system dynamics. These facts are the reason why this and similar models
of novelty detection are important for future research and development.

ACKNOWLEDGEMENTS
The authors would like to thank their colleagues from Tohoku University, namely from the
Yoshizawa-Sugita Lab (formerly Yoshizawa-Homma Lab) and from the Department of
Radiological Imaging and Informatics, for their vital, continuous cooperation and support. The
authors would also like to acknowledge the SGS12/177/OHK2/3T/12 grant, for support of this
work.

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Short Biography of Authors
Matouš Cejnek received his Bachelor’s Degree at Czech Technical University in Prague in 2012.
Currently he is a Master’s student, with expected PhD studies to follow in 2014 at CTU. His research
focuses on non-conventional neural networks for adaptive systems and novelty detection in time series
and biomedical applications. Matous’s work has been awarded in local and international student
competitions in 2013.
Peter Mark Beneš received his Bachelor’s degree with honours at Czech Technical University in Prague
(CTU) in 2012. Currently, he is a Master’s student, with expected PhD studies to follow at CTU in 2014.
His research focuses on non-conventional neural networks for adaptive identification and control of
industrial systems including hoist mechanisms and skew control of rail-based mechanisms. Peter’s work
has been awarded in local and international student competitions also with an industrial BOSCH award in
2013.
Ivo Bukovsky graduated from Czech Technical University in Prague where he received his Ph.D. in the
field of Control and System Engineering in 2007 and became associate professor since 2013. His research
interests include higher-order neural networks, adaptive evaluation of time series and systems, multiscale-analysis approaches, control and biomedical applications. He was a visiting researcher at the
University of Saskatchewan (2003), at the University of Manitoba in Canada (2010), and he was a
visiting professor at Tohoku University in 2011.

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