4.1 Introduction

Time series clustering is the process of unsupervised partitioning of \(n\) time series data into \(k\) (\(k<n\)) meaningful groups such that homogeneous time series are grouped together based on a certain similarity measure. The time series features, length of time series, representation technique, and, of course, the purpose of clustering time series all influence the suitable similarity measure or distance metric to a meaningful level. The three primary methods to time series clustering (Liao (2005)) are algorithms that operate directly with distances or raw data points in the time or frequency domain (distance-based), with features derived from raw data (feature-based), or indirectly with models constructed from raw data (model-based). The efficacy of distance-based techniques is highly dependent on the distance measure utilized. Defining an appropriate distance measure for the raw time series may be a difficult task since it must take into account noise, variable lengths of time series, asynchronous time series, different scales, and missing data. Commonly used distance-based similarity measures as suggested by a decade review of time series clustering approaches (Aghabozorgi, Seyed Shirkhorshidi, and Ying Wah (2015)) are Euclidean, Pearson’s correlation coefficient and related distances, Dynamic Time Warping (DTW), Autocorrelation, Short time series distance, Piecewise regularization, cross-correlation between time series, or a symmetric version of the Kullback–Liebler distances (Liao (2007)) but on a vector time series data. Among these alternatives, Euclidean distances have high performance but need the same length of data over the same period, resulting in information loss regardless of whether it is on raw data or a smaller collection of features. DTW works well with time series of different lengths (Corradini (2001)), but it is incapable of handling missing observations. Surprisingly, probability distributions, which may reflect the inherent temporal structure of a time series have not been considered in determining time series similarity.

This work is motivated by a need to cluster a large collection of residential smart meter data, so that households can be grouped into similar energy usage patterns. These can be considered to be univariate time series of continuous values which are available at fine temporal scales. These time series data are long (with more and more data collected at finer resolutions), are asynchronous, with varying time lengths for different houses and sporadic missing values. Using probability distributions is a natural way to analyze this types of data because they are robust to uneven length, missing data, or noise. This paper proposes two approaches for obtaining pairwise similarities based on Jensen-Shannon distances between probability distributions across a selection of cyclic granularities. Cyclic temporal granularities (Gupta, Hyndman, and Cook 2021), which are temporal deconstructions of a time period into units such as hour-of-the-day, work-day/weekend, can measure repetitive patterns in large univariate time series data. The resulting clusters are expected to group customers that have similar repetitive behaviors across cyclic granularities. The benefits of this approach are as follows.

  • When using probability distributions, data does not have to be the same length or observed during the exact same time period (unless there is a structural pattern).

  • Jensen-Shannon distances evaluate the distance between two distributions rather than raw data, which is less sensitive to missing observations and outliers than other conventional distance methods.

  • While most clustering algorithms produce clusters similar across just one temporal granularity, this technique takes a broader approach to the problem, attempting to group observations with similar distributions across all interesting cyclic granularities.

  • It is reasonable to define a time series based on its degree of trend and seasonality, and to take these characteristics into account while clustering it. The modification of the data structure by taking into account probability distributions across cyclic granularities assures that there is no trend and that seasonal variations are handled independently. As a result, there is no need to de-trend or de-seasonalize the data before applying the clustering method. For similar reasons, there is no need to exclude holiday or weekend routines.

The primary application of this work is data from the Smart Grid, Smart City (SGSC) project (2010–2014) available through the Department of the Environment and Energy. Half-hourly measurements of usage for more than 13,000 household electricity smart meters is provided from from October 2011 to March 2014. Households vary in size, location, and amenities such as solar panels, central heating, and air conditioning. The behavioral patterns differ amongst customers due to many temporal dependencies. Some households use a dryer, while others dry their clothes on a line. Their weekly usage profile may reflect this. They may vary monthly, with some customers using more air conditioners or heaters than others, while having equivalent electrical equipment and weather circumstances. Some customers are night owls, while others are morning larks. Daily energy usage varies depending on whether customers stay home or work away from home. Age, lifestyle, family composition, building attributes, weather, availability of diverse electrical equipment, among other factors, make the task of properly segmenting customers into comparable energy behavior complex. The challenge is to be able to cluster consumers into these type of expected patterns, and other unexpected patterns, using only their energy usage history (Ushakova and Jankin Mikhaylov (2020)). There is a growing need to have methods that can examine the energy usage heterogeneity observed in smart meter data and what are some of the most common power consumption patterns.

There is a growing body of literature focused on time series clustering related to smart meter data. Tureczek and Nielsen (2017) conducted a systematic study of over \(2100\) peer-reviewed papers on smart meter data analytics. The most often used algorithm is \(k\)-means (Rhodes et al. 2014). \(k\)-means can be made to perform better by explicitly incorporating time series features such as correlation or cyclic patterns rather than performing it on raw data. To reduce dimensionality, several studies use principal component analysis (PCA) or factor analysis to pre-process smart-meter data before clustering (Ndiaye and Gabriel (2011)). PCA eliminates correlation patterns and decreases feature space, but loses interpretability. Other algorithms utilized in the literature include \(k\)-means variants, hierarchical clustering, and greedy \(k\)-medoids. Time series data, such as smart meter data, are not well-suited to any of the techniques mentioned in Tureczek and Nielsen (2017). Only one study (Ozawa, Furusato, and Yoshida 2016) identified time series characteristics by first conducting a Fourier transformation, to convert data from time to frequency domain, followed by \(k\)-means to cluster by greatest frequency. Motlagh, Berry, and O’Neil (2019) suggests that the time feature extraction is limited by the type of noisy, patchy, and unequal time series common in residential customers and addresses model-based clustering by transforming the series into other objects such as structure or set of parameters which can be more easily characterized and clustered. (Chicco and Akilimali 2010) addresses information theory-based clustering such as Shannon or Renyi entropy and its variations. Melnykov (2013) discusses how outliers, noisy observations and scattered observations can complicate estimating mixture model parameters and hence the partitions. None of these methods focuses on exploring heterogeneity in repetitive patterns based on the dynamics of multiple temporal dependencies using probability distributions, which forms the basis of the methodology reported here.

This paper is organized as follows. Section~ provides the clustering methodology. Section~ shows data designs to validate our methods. Section~ discusses the application of the method to a subset of the real data. Finally, we summarize our results and discuss possible future directions in Section~.