Mathematical Problems of Computer Science 58, 32–41, 2022.

doi:10.51408/1963-0090

UDC 519.218

Comparison of Model-Free Algorithms For Clustering

GARCH Processes

Garik L. Adamyan

Yerevan State University

e-mail: garik.adamyan@ysu.am

Abstract

In this paper, we evaluate several model-free algorithms for clustering time series
datasets generated by GARCH processes. In extensive experiments, we generate
synthetic datasets in different scenarios. Then, we compare K-Means (for Euclidian and
dynamic time warping distance), K-Shape, and Kernel K-Means models with different
clustering metrics. Several experiments show that the K-Means model with dynamic
time warping distance archives comparably better results. However, the considered
models have significant shortcomings in improving the clustering accuracy when the
amount of information (the minimum length of the time series) increases, and in
performing accurate clustering when data is unbalanced or clusters are overlapping.
Keywords: Time series clustering, GARCH process, dynamic time warping, K-Means,
K-Shape.
Article info: Received 02 May 2022; received in revised form 20 July 2022; accepted
29 September 2022.

1. Introduction

Time series clustering has been used in diverse scientific disciplines to discover patterns and
extract valuable information from complex and massive datasets. These algorithms have
a wide range of applications in many research areas, for instance, in finance, biology, and
robotics [1].

Time series clustering approaches can be classified as feature-based, shape-based, and
model-based [1]. It is noteworthy that these methods are based on dissimilarity measures
on time series data, according to which the time series data points are grouped by some
clustering method (for instance, PAM).

In general, shape-based methods use linear and non-linear transformations to align
time series samples and calculate dissimilarity measures on aligned samples. Additionally,
shape-based algorithms process the time series data directly without making any statistical
assumptions about the underlying data generating processes. On the contrary, model-based
methods make statistical assumptions on time series generating processes. In general, model-
based approaches assume that time series samples are generated from specific models (for

32



G. Adamyan 33

instance, ARIMA [2], Mixtures of ARIMAs [3]). Time series samples are transformed into
fitted models, and then a suitable distance and a clustering algorithm are applied to the
estimated model parameters.

Although several benchmarking results on different real-world datasets for non-
parametric clustering methods can be found in ([4], [5], [6]), the comparison of non-
parametric clustering methods on time series data generated from GARCH processes is
not well studied. In this paper, we are interested in non-parametric models evaluation of
time series data generated from the well-known GARCH process, which is the actual choice
for modeling the volatility of returns on financial assets. We simulate multiple GARCH
models with different data generating scenarios and compare several non-parametric time
series clustering models.

Motivated by [4], for comparison we choose well-known partition-based time series
clustering models: K-Means, K-Means with dynamic time warping and DTW barycenter
averaging, K-Shape and Kernel K-Means models. Furthermore, we can find open-source
implementations of these algorithms [7].

Although the main focus in the field of time series clustering comparison remains
clustering accuracy metrics, in this work we also explore a number of other challenges of
model-free methods. In particular, we study the ability of the above-mentioned model-
free methods to cluster GARCH processes with imbalanced, overlapping clusters and also
examine the impact of increasing information on clustering accuracy.

2. Related Work

In time series analysis research, benchmarking and numerical comparison have been
recognized as integral steps to justify theoretical results. The importance of numerical
comparison is emphasized in [8], where the authors reimplemented many time-series
classification algorithms and compared them in 50 real-world datasets. The authors note
that most reported methods have insignificant improvements regarding the variance of the
evaluation metrics. This empirical evidence reclaimed the statement of the importance of
the time series benchmark datasets and the empirical evaluation of the suggested methods.

Among the works that compare time series clustering models based on real-world
datasets, we can mention ([4], [5], [6]) works. In [4], authors compare several partition,
density, and hierarchical clustering methods to cluster all time series datasets available in
the University of California Riverside (UCR) archive [9]. They conclude that the overall
performance of the eight compared algorithms is quite similar with high dependence on the
evaluation dataset.

The method of comparing time series clustering algorithms with synthetic, generated
datasets also attracts a lot of attention among scholars. In addition to the actual clusters
being known, this comparison method gives additional flexibility to examining the behavior
of algorithms in different situations. In particular, scholars discussed the difference between
stationary and non-stationary time series [10], the presence of noise in time series samples
[11], the presence of noise clusters in time series dataset [11].



34 Comparison of Model-Free Algorithms For Clustering GARCH Processes

3. Clusters of GARCH

The GARCH process is introduced in [12] for statistical modeling of the volatility of returns
on financial assets. The GARCH model has many extensions such as asymmetric GARCH
[13], threshold GARCH [14]. The GARCH(p,q) model is defined as follows:

yt = µt + ϵt

ϵt = σtet, where et i.i.d E(et) = 0, var(et) = 1,

σ2t = ω +

p∑
i=1

αiϵ
2
t−i +

q∑
j=1

βjσ
2
t−j,

where
ω > 0,

αi ≥ 0, i = 1, 2, ..., p,
βj ≥ 0, i = 1, 2, ..., q.

The GARCH(p, q) model admits a strictly stationary solution with a finite variance if
and only if

p∑
i=1

αi +

q∑
j=1

βj < 1. (1)

Moreover, this strictly stationary solution is also unique. [15]

For the evaluation of non-parametric models, we chose constant zero mean specification
for the GARCH model because it is advised to standardize input data prior to clustering.
In addition, we choose the innovations et as standard Gaussian innovations. So µt = 0 and
et ∼ N(0,1).

In order to measure the clustering accuracy, we need to define the ground truth clusters
of GARCH processes. Let N, K, T ∈ N where K is the number of clusters, N is the number
of samples and T is the time sample size of each series. In this paper, we consider samples
with a fixed time size T, because some of the models (ex. KM-E) support samples with fixed
length. We denote by P i = (ω, α1, α2, ..., αpi, β1, β2, ..., βqi) the vector of all parameters for
the given GARCH(pi,qi) model.

Let {P i}Ki=1 be a family of GARCH process parameters, where K is a number of clusters.
Assume that each P i(i = 1, 2, ..., K) is unique and all the parameters satisfy (1) in order
to provide a strict stationary solution of the corresponding model. We are given N samples
of time series Yi = {yit}Ti=1, where each sample is generated from one of the K GARCH
processes.

Definition 1. We say that Yi and Yj samples are from the same cluster if they are generated
from the same GARCH process.

In other words, a cluster of GARCH processes is a set of samples that are generated with
the same parameters. The uniqueness of the parameters P i and Definition 1 imply that the
given sample belongs to exactly one cluster.



G. Adamyan 35

4. Evaluation Models

For evaluation, we choose well-known non-parametric time series clustering models such as
K-Means with Euclidean (KM-E) and dynamic time warping metrics (KM-DTW), K-Shape,
and Kernel K-Means with Fast Global Alignment Kernel (KKM-GAK) models. KM-E uses
Euclidean distance, for cluster assignment and means averaging for the barycenter (centroid)
computation. It is known that the Euclidean distance metric is not the most accurate metric
for measuring time series similarities. Firstly, to use Euclidean distance, we need to take
into account the order of elements in the time series; secondly, the Euclidean distance does
not consider a phase shift between two curves or a length difference between the series. In
this paper, we consider this model for comparison with more complex approaches.

KM-DTW uses dynamic time warping [16] for cluster assignment and DTW barycenter
averaging (DBA)[17] algorithm for averaging time series within the same cluster.

k-Shape [18] is a partitional clustering algorithm that relies on an iterative refinement
procedure similar to the one used in K-Means. To measure the distance between time series,
K-Shape uses a normalized version of the cross-correlation measure to consider the shapes
of time series while comparing them. During the iterative procedure, this model minimizes
the sum of squared distances between the sequences of time series.

Kernel K-Means[19] is an alternative clustering algorithm that uses kernel functions as
a nonlinear mapping from the input space to a higher dimensional space. By using kernels,
Kernel K-Means can separate clusters in higher dimensional space, even if the input data
is not non-linearly separable in the input space. For treating time series data, practitioners
usually used Global Alignment Kernels [20]. We will refer to this algorithm KKM-GAK.

The problem is to generate synthetic datasets and evaluate non-parametric models for
clustering time series processes generated by the GARCH model.

5. Assessment Metrics

In practice, the use of clustering methods is due to working with unlabeled datasets. As a
result, we can find evaluation metrics that can evaluate clustering models without having
labeled data. These types of metrics are called internal. By the method of our data
generating process, we can use external measures, which assume that ground truth labels
are available. Examples of this type of metrics are the Rand Index (RI) [21], the Adjusted
Rand Index (ARI) [22], the Adjusted Mutual Information (AMI)[23].

Following the evaluation made in [4] in our study, we choose the Adjusted Rand Index,
because the values of this metric are consistently low for random cluster assignments and do
not depend on the number of clusters.

6. Experiments

To evaluate non-parametric models, we simulate random datasets with different setups. In
the first experiment, we measure the ability of the models of clustering different numbers
of clusters. For this purpose, we generate datasets for 2, 4, 8, and 10 clusters, respectively.
For each number of clusters, we generate random parameter families, which satisfy (1) for
guaranteeing a unique and stationary solution of processes. For the purpose of generating
a family of parameters, we constrain the maximum length of p and q by 5. This constraint
is inherited from the common choice of GARCH models with fewer parameters. For every



36 Comparison of Model-Free Algorithms For Clustering GARCH Processes

parameter vector P i (cluster), we generate samples for the given cluster and separate them
into training and testing parts (30% testing) and repeat this process for averaging purposes.
In Table 1, we present the results of the first experiment evaluated with the AMI metric. We
can see that the KM-DTW model outperforms other models. In the second experiment,

Table 1: AMI score for different N clusters

N clusters KM-E KM-DTW k-Shape KKM-GAK

2 0.003+-0.001 0.325+-0.403 0.004+-0.009 0.003+-0.002

4 0.004+-0.001 0.463+-0.129 0.02+-0.007 0.002+-0.001

6 0.018+-0.016 0.578+-0.151 0.043+-0.021 0.001+-0.0005

8 0.006+-0.003 0.498+-0.077 0.005+-0.011 0.001+-0.0005

10 0.005+-0.01 0.624+-0.03 0.062+-0.022 0.0001+-0.00005

we measure the clustering quality in scenarios when the amount of information increases.
We generate datasets with 5 clusters and 100 samples in each cluster. We set T = 1000 and
consider 5 intervals on the time axis. We train and evaluate models in the first interval and
consequently add information. From the second experiment, we can see that the KM-DTW
model outperforms other models, but we do not observe increased accuracy as a result of
adding information. There is a significant increase in the accuracy of the KM-DTW model
when the number of samples increases from 200 to 400, but further increases in the number
of samples do not improve the accuracy of the model. The K-Shape model also shows a slight
improvement in accuracy when the number of samples increases from 800 to 1000. Given
that model-based methods rely on ML/Quasy ML estimates of the parameters of GARCH
models and also the asymptotic properties of these estimates, this experiment may suggest
that model-based methods have the potential to increase clustering accuracy as information
increases. The results of the second experiment are displayed in Fig. 1.

Fig. 1. AMI for different time intervals.



G. Adamyan 37

Fig. 2 shows the results of the third experiment. In this experiment, we measure the
ability of the KM-DTW model to cluster an imbalanced dataset. For the fairness of the
experiment, we generate time series samples with the GARCH(1,1) process and ensure that
parameters satisfy (1). In addition, we constrain the L2 norm of generated parameters
to obtain non-overlapping clusters. We generate a dataset with different sample ratios and
increase the ratio to 1. In the figure, we can observe that the best model for other experiments
KM-DTW is dependent on cluster imbalance. This experiment shows that the claim made
in [24] that centroid-based methods should be adapted to unbalanced scenarios also holds in
the domain of time series clustering.

Fig. 2. Results for clustering imbalanced dataset.

Moreover, we measure the effect of the L2 norm of generated parameters in clustering
accuracy. We generate parameters for GARCH(1,1) process so that the parameters satisfy
the current restriction on the L2 norm. Throughout the experiment, we increase the bounds
of the L2 norm. During each step, we generate a balanced dataset with T = 500, C = 2, and
100 samples per cluster. We train models ten times for averaging purposes. We can observe
that the KM-DTW model depends on clusters overlapping and increasing the bounds of
parameters L2 norm results in improvement of AMI. This problem is directly related to the
ability of the similarity measure used in the KM-DTW algorithm to distinguish realizations
of the GARCH process with parameters that are close to each other with the L2 norm.
7. Conclusion and Future Work

In this work, several non-parametric clustering algorithms for clustering time series datasets
generated by GARCH processes are evaluated. We generate multiple datasets and conduct
multiple experiments to evaluate the K-Means (with Euclidean and dynamic time warping
distance), K-Shape, and Kernel K-Means models. In the first experiment, we evaluate the
ability of models to cluster different numbers of clusters. The results of the first experiment



38 Comparison of Model-Free Algorithms For Clustering GARCH Processes

Fig. 3. GARCH parameters vector L2 norm versus AMI score.

are displayed in Table 1. In the second experiment, we measure the clustering quality in the
scenarios when the amount of information increases. We generate a dataset with 1000 time
length and increase the information set. The results of the second experiment are shown in
Fig. 1. During both experiments, the KM-DTW model shows better results. In the third
experiment, we measure the ability of the KM-DTW model to cluster imbalanced datasets
by generating multiple datasets with imbalanced samples in the cluster. The results are
provided in Fig. 2. In the fourth experiment, we measure the ability of the KM-DTW model
to cluster overlapping clusters. We constrain the norm of the parameters of the GARCH(1,1)
model and evaluate the KM-DTW model. The experiment shows that KM-DTW is highly
dependent on the norm of the generated parameters. The results of the fourth experiment
are shown in Fig. 3.

We hope that our findings can motivate scholars to examine the discussed issues related
to clustering accuracy, cluster overlapping, and available information effect. We think that
already designed GARCH-based clustering methods have the potential to overcome these
problems, so it is important to conduct similar experiments to show this. Moreover, as a
direct application of our findings, it is worth applying clustering algorithms to the real-world
financial dataset.

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G. Adamyan 39

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4 0 Comparison of Model-Free Algorithms For Clustering GARCH Processes

[2 2 ] J. M. S a n t o s a n d M. E m b r e c h t s , \ On t h e u s e o f t h e a d ju s t e d r a n d in d e x a s a m e t -
r ic fo r e va lu a t in g s u p e r vis e d c la s s i¯ c a t io n ," Arti¯cial Neural Networks ICANN 2009,
p . 1 7 5 1 8 4 , 2 0 0 9 .

[2 3 ] S . R o m a n o , N . t h e V in h , J. C. B a ile y, a n d K . M. V e r s p o o r , \ A d ju s t in g fo r c h a n c e
c lu s t e r in g c o m p a r is o n m e a s u r e s ," J ournal of M achine L earning (J M L R ), p p . 4 6 3 5 { 4 6 6 6 ,
vo l. 1 7 , n o . 1 , 2 0 1 6 .

[2 4 ] B . K r a wc z yk, \ L e a r n in g fr o m im b a la n c e d d a t a : Op e n c h a lle n g e s a n d fu t u r e d ir e c t io n s ,"
P rogress in Arti¯cial Intelligence, vo l. 5 , n o . 4 , p . 2 2 1 { 2 3 2 , 2 0 1 6 .

GARCH åñáó»ëÝ»ñÇ Ïɳëï»ñǽ³ódzÛÇ Ñ³Ù³ñ Ùá¹»ÉÝ»ñÇó
³ÝÏ³Ë ³É·áñÇÃÙÝ»ñÇ Ñ³Ù»Ù³ïáõÃÛáõÝ

¶³ñÇÏ È. ²¹³ÙÛ³Ý

ºñ¨³ÝÇ å»ï³Ï³Ý ѳٳÉë³ñ³Ý, ºñ¨³Ý, г۳ëï³Ý

e-mail: garik.adamyan@ysu.am

²Ù÷á÷áõÙ

Ðá¹í³ÍáõÙ Ù»Ýù ·Ý³Ñ³ïáõÙ »Ýù ÙÇ ù³ÝÇ Ùá¹»ÉÝ»ñÇó ³ÝÏ³Ë Ïɳëï»ñǽ³ódzÛÇ
³É·áñÇÃÙÝ»ñÇ GARCH åñáó»ëÝ»ñáí ·»Ý»ñ³óí³Í ųٳݳϳÛÇÝ ß³ñù»ñÇ ïíÛ³ÉÝ»ñÇ
Ïɳëï»ñ³íáñÙ³Ý áõݳÏáõÃÛáõÝÁ: ȳÛݳͳí³É ÷áñÓ»ñÇ ÁÝóóùáõÙ Ù»Ýù ·»Ý»ñ³óÝáõÙ
»Ýù ëÇÝûïÇÏ ïíÛ³ÉÝ»ñÇ Ñ³í³ù³ÍáõÝ»ñ ï³ñµ»ñ ëó»Ý³ñÝ»ñáí: ²ÛÝáõÑ»ï¨,
Ù»Ýù ѳٻٳïáõÙ »Ýù K-Means Ùá¹»ÉÝ»ñÁ (¾íÏÉǹ»ëÛ³Ý ¨ ųٳݳÏÇ ¹ÇݳÙÇÏ
÷á˳ϻñåÙ³Ý Ù»ïñÇϳݻñáí), K-Shape ¨ Kernel K-Means Ùá¹»ÉÝ»ñÇ ï³ñµ»ñ
Ïɳëï»ñ³ÛÇÝ ã³÷ÇãÝ»ñáí: ØÇ ù³ÝÇ ÷áñÓ»ñÁ óáõÛó »Ý ï³ÉÇë, áñ K-Means Ùá¹»ÉÁ
ųٳݳÏÇ ¹ÇݳÙÇÏ ÷á˳ϻñåÙ³Ý Ù»ïñÇϳÛáí óáõÛó ¿ ï³ÉÇë ѳٻٳﳵ³ñ ³í»ÉÇ
É³í ³ñ¹ÛáõÝùÝ»ñ: ²ÛÝáõ³Ù»Ý³ÛÝÇí, ¹Çï³ñÏí³Í Ùá¹»ÉÝ»ñÝ áõÝ»Ý ½·³ÉÇ Ã»ñáõÃÛáõÝÝ»ñ
ÇÝýáñÙ³ódzÛÇ (ųٳݳϳÛÇÝ ß³ñùÇ Ýí³½³·áõÛÝ »ñϳñáõÃÛáõÝÁ) ù³Ý³ÏÇ ³í»É³óÙ³Ý
Ñ»ï Ïɳëï»ñ³íáñÙ³Ý ×ß·ñïáõÃÛ³Ý µ³ñÓñ³óÙ³Ý Ñ»ï ϳåí³Í, ÇÝãå»ë ݳ¨
ïíÛ³ÉÝ»ñÇ ³Ýѳí³ë³ñ³ÏßéáõÃÛ³Ý Ï³Ù Ïɳëï»ñÇ Ñ³ÙÁÝÏÝÙ³Ý ¹»åùáõÙ ×ß·ñÇï
Ïɳëï»ñ³íáñáõÙ Çñ³Ï³Ý³óÝ»Éáõ ѳñóáõÙ:

´³Ý³ÉÇ µ³é»ñ` ųٳݳϳÛÇÝ ß³ñù»ñÇ Ïɳëï»ñǽ³ódz, GARCH åñáó»ëÝ»ñ,
ųٳݳÏÇ ¹ÇݳÙÇÏ ÷á˳ϻñåáõÙ, K-Means, K-Shape.



G. Adamyan 4 1

Ñðàâíåíèå áåçìîäåëüíûõ àëãîðèòìîâ êëàñòåðèçàöèè
GARCH-ïðîöåññîâ

Ãàðèê Ë. Àäàìÿí

Åðåâàíñêèé ãîñóäàðñòâåííûé óíèâåðñèòåò, Åðåâàí, Àðìåíèÿ
e-mail: garik.adamyan@ysu.am

Àííîòàöèÿ

 ýòîé ñòàòüå ìû îöåíèâàåì íåêîòîðûå áåçìîäåëüíûå àëãîðèòìû êëàñòåðèçàöèè
íàáîðîâ äàííûõ âðåìåííûõ ðÿäîâ, ñãåíåðèðîâàííûõ GARCH ïðîöåññàìè. Â
îáøèðíûõ ýêñïåðèìåíòàõ ìû ãåíåðèðóåì ñèíòåòè÷åñêèå íàáîðû äàííûõ äëÿ
ðàçëè÷íûõ ñöåíàðèÿõ. Çàòåì ìû ñðàâíèâàåì ìîäåëè K-Means (ñ ìåòðèêàìè
åâêëèäîâîé è äèíàìè÷åñêîé òðàíñôîðìàöèè âðåìåííîé øêàëû), ìîäåëè K-
Shape è Kernel K-Means ñ ðàçëè÷íûìè ìåòðèêàìè êëàñòåðèçàöèè. Íåñêîëüêî
ýêñïåðèìåíòîâ ïîêàçûâàþò, ÷òî ìîäåëü K-Means ñ ìåòðèêîé äèíàìè÷åñêîé
òðàíñôîðìàöèè âðåìåííîé øêàëû äàåò ñðàâíèòåëüíî ëó÷øèå ðåçóëüòàòû.
Îäíàêî ðàññìîòðåííûå ìîäåëè èìåþò ñóùåñòâåííûå íåäîñòàòêè â ïîâûøåíèè
òî÷íîñòè êëàñòåðèçàöèè ïðè óâåëè÷åíèè êîëè÷åñòâà èíôîðìàöèè (ìèíèìàëüíîé
äëèíû âðåìåííîãî ðÿäà), à òàêæå ïðè íåñáàëàíñèðîâàííîñòè äàííûõ èëè
ïåðåêðûòèè êëàñòåðîâ.

Êëþ÷åâûå ñëîâà: êëàñòåðèçàöèÿ âðåìåííûõ ðÿäîâ, ïðîöåññ GARCH,
äèíàìè÷åñêàÿ äåôîðìàöèÿ âðåìåíè, K-Means, K-Shape.


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