Clustering Methods for Analysis and Optimization of Time Series Data

1. Introduction

Time series data consists of observations recorded sequentially over time and are characterized by temporal dependencies, trends, seasonality, and noise. In the context of time series data, similarity can be defined in terms of raw values, temporal dynamics, shape, or underlying generative processes. Traditional analytical techniques often focus on forecasting or statistical modeling, assuming a predefined structure or distribution. However, in many real-world scenarios, the underlying patterns are unknown or too complex to be modeled explicitly. This motivates the use of data mining techniques, among which clustering plays a central role [1,2,3,4].

Clustering methods aim to group similar objects such that time series within the same cluster exhibit comparable behavior, while series from different clusters are dissimilar. Unlike static data, time series exhibit strong temporal ordering and autocorrelation. Observations are often non-independent and may vary in length, scale, or sampling frequency. Additional challenges include missing values, noise, temporal misalignment, and concept drift. These properties complicate the direct application of classical clustering algorithms designed for independent and identically distributed data. Another important challenge is the definition of similarity. Two time series may represent the same underlying phenomenon while being shifted in time, scaled in amplitude, or evolving at different speeds. Consequently, the choice of similarity measures and feature extraction has a decisive impact on clustering performance and interpretability. Standard approach for time series clustering includes preprocessing step—similarity measure and feature extraction, while deep approach includes deep learning methods [5,6,7,8,9,10,11].

There is no generally accepted classification for time series clustering methods. [12] considers time series clustering as a conventional technique and a subtype of Multivariate Time Series Anomaly Detection. [13] examined time series clustering methods in three main phases: data representation, similarity measure, and clustering algorithm. [14] explores contemporary clustering algorithms within the machine learning paradigm, focusing on five primary methodologies: centroid-based, hierarchical, density-based, distribution-based, and graph-based clustering. [15] classifies clustering techniques into hard clustering and soft clustering. [16] point out three major groups: depending upon whether they work directly with raw data, indirectly with features extracted from the raw data, or indirectly with models built from the raw data. [17] focuses on deep learning methods and considers 3 main components: network architecture, pretext loss, and clustering loss. All authors point out partitioning methods (k-means, k-medoids, fuzzy C-means, fuzzy C-medoids), hierarchical clustering (bottom-up and top-down), model-based (self-organizing map), and density-based Clustering (DBSCAN—Density-Based Spatial Clustering of Applications with Noise, OPTICS—Ordering Points To Identify the Clustering Structure). Most of clustering methods face significant challenges when dealing with high dimensional, noisy, and large-scale data.

2. Time Series Data Clustering

Clustering approaches provide effective tools for organizing and extracting insights from big and complicated datasets. The most appropriate clustering method is determined by the data properties and the application’s unique needs. It is critical to assess the quality of clustering findings in order to assure the validity and use of the produced clusters [12,13,14].

In this relation Table 1 shows the overviewed research over clustering time series methods, techniques, tasks, applications, results, and limitations.

2.1. Methods

Among the methods considered, the most common are conventional techniques such as k-means, k-medoids, hierarchical agglomerative clustering, spectral clustering, and fuzzy C-means, [13,19,20,23,30,32,36,38,39,40,41,42,43,44,45,46]. Similarity-based methods, such as DTW, its variants, Euclidean distance, Sobolev distance, and hybrid distance measures, improve clustering performance for time-shifted or temporally distorted sequences [21,32,43,45,46]. Feature extraction and subsequence-based approaches, including sliding window techniques, shapelet-based methods, Tsfresh, and spectral feature selection, aim to capture discriminative local patterns in time series data [13,18,22,30,31]. Deep learning-based approaches, such as DAE, DCAT, CNN, convo LSTM, VAE, and hybrid deep architectures are preferred when dealing with complex, multivariate, and nonlinear time series. These models enable automatic feature learning and joint clustering or prediction [13,26,33,37,41,47]. Probabilistic and Bayesian models, including Dirichlet Process Mixture models and Markov Chain Monte Carlo (MCMC)-based approaches, provide flexible frameworks capable of modeling uncertainty and automatically determining the number of clusters [34,35].

2.2. Tasks

The primary problems addressed include time series clustering, similarity detection, pattern discovery, and forecasting [19,21,23,26,33]. Several studies focus on handling noisy, non-stationary, and temporally distorted data, which is common in real-world scenarios such as EEG signals, energy load profiles, and financial time series [23,26,32,43]. Time series clustering is often used as a preprocessing or exploratory step for higher-level tasks such as anomaly detection, classification, and decision support [22,30,42].

2.3. Application

[13] assumes that the proposed method can be used in finance, medical diagnosis, and weather forecasting. [18,33] point out several application domains—financial investment, energy management, weather forecasting, and traffic optimization. [18,23,24,27,30,34,39,40] investigate healthcare data. [22,25,26,32] examine problems in the field of natural phenomena. [20,28,38,41] deal with problems in the field of economics. [13] examines biological data. [19,29,31,35,42,46,47] address energetic issues. [44,45] applied clustering methods over transport data. [37] examines mechatronic issues. [21] provides only a theoretical overview and does not indicate a specific area of application.

2.4. Results

Most studies report significant improvements over baseline methods, such as higher clustering accuracy, reduced forecasting error, or improved separability of clusters [18,19,23,26,29,33,35,39,47]. Deep learning and hybrid approaches frequently achieve state-of-the-art performance, particularly for multivariate and large-scale datasets [26,33,37,41,47]. Some works also highlight improvements in computational efficiency or scalability compared to traditional shapelet-based or distance-based methods [18,38,39].

2.5. Limitations

Despite their effectiveness, many approaches suffer from high computational complexity and resource consumption, especially deep learning, DTW-based, and probabilistic models [13,18,26,32,34,35,47]. Sensitivity to noise and outliers is another common limitation, affecting both classical and advanced clustering techniques [29,37,40,46]. Several studies report challenges related to parameter tuning, scalability, and implementation complexity [18,30,33,35,36]. These limitations indicate that there is still no universally optimal solution for time series clustering, particularly in real-world, large-scale environments [21,24,44].

3. Clustering Transport Time Series: Use Case

3.1. Overview

Transport systems generate large volumes of time series data through traffic sensors, GPS-equipped vehicles, smart ticketing systems, and intelligent transportation infrastructure. Typical transport time series include traffic flow, vehicle speed, travel time, congestion index, passenger demand, and vehicle counts, often recorded at regular intervals across multiple spatial locations. These datasets are inherently multivariate, noisy, seasonal, and non-stationary, making them well-suited for clustering-based analysis [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62].

A key objective in transport time series analysis is to identify segments with similar temporal behavior without assuming predefined traffic models. Clustering enables the discovery of recurring traffic patterns such as peak-hour congestion [44,49,61], off-peak free-flow conditions [50,51,52,55,56], incident-induced disruptions [48], seasonal demand variations [54], anomaly detection [54,60], or cybersecurity [58].

Different clustering paradigms can be applied to transport time series data. Distance-based methods such as DTW combined with partitioning algorithms (e.g., k-means or k-medoids) are effective for handling temporal misalignment between traffic patterns. Feature-based approaches extract statistical, frequency-domain, or shape-based features to reduce dimensionality and improve scalability. Model-based and deep learning approaches, including auto-encoders and recurrent neural networks, are increasingly used for clustering large-scale multivariate transport datasets, capturing complex temporal dependencies and spatial correlations. For example, road segments or sensor locations can be clustered based on daily or weekly traffic flow profiles, allowing transportation authorities to distinguish between stable corridors, bottleneck-prone areas, and highly variable routes.

The outcomes of transport time series clustering include improved traffic state classification, congestion pattern recognition, incident detection, and infrastructure planning support. By grouping similar temporal behaviors, clustering results can guide adaptive traffic signal control, optimize public transport scheduling, and support data-driven decision-making in smart transportation systems.

3.2. Example

The goal of the analysis is to cluster temporal speed profiles (and optionally other variables such as counts or weight aggregates) by sensor/road segment/day to discover typical daily patterns, detect anomalies and incidents, and support planning and adaptive control. For the purposes of the analysis, is selected DTW + k-medoids pipeline, as this combination is practical and robust to temporal misalignment.

3.2.1. Data

The data is set in the following format:

Fields: Date;Direction;Plate number;Country;Speed;Class;Lane;Length;Weight

Sample rows (semicolon-delimited):

30.9.2023 г. 12:00:27;Center;159785927043;;49;1;L1;5500;1086

30.9.2023 г. 12:00:33;Center;3658603332998141999;66;1;;L1;5200;826

30.9.2023 г. 12:01:16;Ring road;4391747583219136573;66;0;;L1;5200;0

30.9.2023 г. 12:01:17;Ring road;201064482549844772;;66;0;L1;5200;0

30.9.2023 г. 12:02:03;Center;2661655543687701673;;67;1;L1;5600;429

Date: timestamp of the measurement.

Direction: spatial attribute (e.g., “Center”).

Plate number: vehicle identifier (may be noisy or partially anonymized).

Country: extracted from previous attribute—Plate number

Speed: primary temporal variable for profiling.

Class;Lane;Length;Weight: numeric attributes usable for stratification or filtering.

3.2.2. Preprocessing

Preprocessing consists of three steps: construction of time series; normalization and Dynamic Time Warping.

Step 1: Construct time series:

Build time series per entity (e.g., daily speed profile per sensor or number vehicles per time interval).

Parse the CSV, standardize datetime formats.

Filter/segment data by direction, sensor, or road segment; aggregate (e.g., 1, 5, or 15 minutes).

Handle missing speed values (short gaps: linear interpolation; large gaps: exclude or flag).

Extract summary features (mean, std, peak time, autocorrelation).

Separate weekdays vs. weekends or cluster per day-of-week to handle weekly seasonality.

Step 2: Normalization.

To focus the clustering on the shape of daily traffic profiles rather than their absolute magnitude, each time series was normalized independently using z-score normalization. For a given series ${\left\{x_t\right\}}_{t=1}^T$ the normalized values were computed as:

$$x_t^{\prime }={\displaystyle \frac{x_t-{\mu }_x}{{\sigma }_x}},$$

(1)

where ${\mu }_x$ and ${\sigma }_x$ denote the mean value and standard deviation of the series. This transformation ensures zero mean value and unit variance for each profile, allowing DTW to capture relative temporal dynamics while reducing the influence of systematic speed level differences across days or road segments. Raw (unnormalized) series may additionally be retained for downstream analyses where absolute speed levels are of interest.

Step 3: Dynamic Time Warping.

DTW computes a similarity score between two time series by non-linearly aligning them along the temporal axis. Given two sequences

$$X=\left(x_1,..,x_n\right),Y=(y_1,..y_m),$$

DTW defines a local cost function

$$d_{(i,j)}=|x_i-y_i|$$

(2)

and constructs a cumulative cost matrix $D\in R^{nxm}$ according to the recurrence:

$$D(i,j)=d(i,j)+min\left\{\begin{array}{c}D(i-1,j)\\ D(i,j-1)\\ D(i-1,j-1)\end{array}\right.$$

(3)

with boundary condition $D\left(\mathrm{1,1}\right)=d\left(\mathrm{1,1}\right)$. The DTW distance between X and Y is defined as the minimum cumulative cost $D(n,m)$ along a monotonic warping path from (1,1) to (n,m). To prevent unrealistic alignments and reduce computational complexity, a Sakoe–Chiba band constraint was applied.

3.2.3. Clustering

Clustering is performed using the k-medoids algorithm (Partitioning Around Medoids, PAM) applied to the precomputed pairwise DTW distance matrix. In contrast to k-means, k-medoids operates directly on arbitrary distance measures and selects actual observations as cluster representatives (medoids).

Given a set of time series ${\mathrm{X}}_1,..,{\mathrm{X}}_{\mathrm{N}}$ and their DTW distance matrix D, PAM iteratively minimizes the sum of within-cluster dissimilarities by optimizing the choice of medoids. Each time series is assigned to the cluster of the nearest medoid under the DTW distance. The number of clusters k is selected using a combination of internal validity criteria, including the silhouette coefficient computed on DTW distances, the elbow method, and the Davies–Bouldin index.

3.2.4. Evaluation and Interpretation

Cluster quality was assessed using internal validation metrics, including the silhouette score based on DTW distances and the intra-cluster DTW variance.

To support interpretability, cluster medoids are visualized as representative daily speed profiles, with individual cluster members overlaid to reveal characteristic temporal patterns such as peak timing, congestion duration, and recovery dynamics.

Spatial analysis is conducted by mapping cluster assignments to corresponding sensors or road segments, enabling the identification of recurrent traffic regimes, bottlenecks, and structurally similar corridors.

Time series exhibiting unusually large DTW distances to their assigned medoids are flagged as potential anomalies, indicative of incidents, sensor malfunctions, or atypical traffic conditions.

4. Discussion

The analysis of the reviewed literature reveals several important trends and open challenges in time series clustering research. A clear progression can be observed from classical clustering techniques toward more advanced similarity-based, feature-driven, and deep learning approaches. While conventional methods such as k-means, k-medoids, hierarchical clustering, and fuzzy clustering remain widely used due to their simplicity and interpretability, their effectiveness is often limited when applied to noisy, high-dimensional, or large-scale time series data.

Centroid-based methods, for instance, perform well on small- to medium-scale datasets with low noise, whereas DTW-based or deep learning approaches are better suited for multivariate, non-linear, and temporally misaligned series. In the context of transport datasets, temporal misalignments due to varying congestion patterns make similarity-based clustering, particularly DTW or hybrid deep learning models, more effective than classical approaches.

Nevertheless, these advanced methods introduce a trade-off: improved accuracy and pattern capture often come at the cost of reduced interpretability and increased computational requirements. Feature-based and subsequence-oriented techniques partially alleviate these limitations by reducing dimensionality and capturing local discriminative patterns, yet they require careful feature selection, window sizing, and parameter tuning. Deep learning frameworks further enhance performance on complex multivariate time series but demand substantial computational resources and large datasets, which can hinder practical adoption.

Across all categories, a recurring issue is the balance between clustering accuracy, scalability, and interpretability. No single method consistently outperforms others across all datasets and application domains, highlighting the importance of application-driven method selection. This underscores the need for hybrid frameworks that combine simplicity, scalability, and robust handling of temporal dynamics, particularly for large-scale, noisy, and spatially dependent transport datasets.

In the context of transport time series data, the challenges are further amplified by strong seasonality, spatial dependencies, and non-stationary behavior. The reviewed transport-related studies demonstrate that clustering can successfully uncover recurring traffic patterns, congestion regimes, and anomalous events. However, existing approaches often fail to jointly capture temporal dynamics, spatial correlations, and scalability requirements, indicating a clear research gap for integrated and efficient clustering frameworks tailored to intelligent transportation systems.

5. Conclusions

This paper presented a review of time series clustering techniques, analyzing a wide range of methods, application domains, achieved results, and inherent limitations. The reviewed studies confirm that time series clustering is a powerful tool for exploratory data analysis, pattern discovery, and decision support in complex temporal datasets.

Classical clustering techniques remain relevant due to their simplicity and interpretability, but they are insufficient for modern large-scale and multivariate time series. Similarity-based and feature-driven methods improve clustering quality but often suffer from high computational complexity and parameter sensitivity. Deep learning and probabilistic models achieve state-of-the-art performance in many scenarios, yet their practical adoption is constrained by resource demands, scalability issues, and reduced interpretability.

In transport applications, effective time series clustering has the potential to significantly enhance traffic monitoring, congestion management, anomaly detection, and infrastructure planning. As transportation systems continue to evolve toward data-driven and intelligent solutions, robust and scalable clustering methods will play a crucial role in supporting smart mobility and decision-making processes.

Future work will focus on practical extensions of existing time series clustering methods with an emphasis on transport-related data. Rather than developing highly complex models, the goal is to evaluate and adapt established clustering techniques for traffic flow, speed, and travel time data. A key direction is the comparison of distance-based and feature-based approaches to identify recurrent traffic patterns such as peak-hour congestion and off-peak conditions.

Additionally, lightweight hybrid solutions combining classical clustering algorithms with basic dimensionality reduction techniques can be explored to ensure scalability and applicability to large transport datasets. The clustering results will be assessed in practical cases, such as traffic state classification and anomaly detection, to support data-driven decision-making in intelligent transportation systems.