2.1 Introduction
Temporal data are available at various resolutions depending on the context. Social and economic data are often collected and reported at coarse temporal scales such as monthly, quarterly or annually. With recent advancements in technology, more and more data are recorded at much finer temporal scales. Energy consumption may be collected every half an hour, energy supply may be collected every minute, and web search data might be recorded every second. As the frequency of data increases, the number of questions about the periodicity of the observed variable also increases. For example, data collected at an hourly scale can be analyzed using coarser temporal scales such as days, months or quarters. This approach requires deconstructing time in various possible ways called time granularities (Aigner et al. 2011).
It is important to be able to navigate through all of these time granularities to have multiple perspectives on the periodicity of the observed data. This aligns with the notion of exploratory data analysis (EDA) (Tukey 1977) which emphasizes the use of multiple perspectives on data to help formulate hypotheses before proceeding to hypothesis testing. Visualizing probability distributions conditional on one or more granularities is an indispensable tool for exploration. Analysts are expected to comprehensively explore the many ways to view and consider temporal data. However, the plethora of choices and the lack of a systematic approach to do so quickly can make the task overwhelming.
Calendar-based graphics (Wang, Cook, and Hyndman 2020b) are useful in visualizing patterns in the weekly and monthly structure and are helpful when checking for the effects of weekends or special days. Any temporal data at sub-daily resolution can also be displayed using this type of faceting (Wickham 2016) with days of the week, month of the year, or another sub-daily deconstruction of time. But calendar effects are not restricted to conventional day-of-week or month-of-year deconstructions. There can be many different time deconstructions, based on the calendar or on categorizations of time granularities.
Linear time granularities (such as hours, days, weeks and months) respect the linear progression of time and are non-repeating. One of the first attempts to characterize these granularities is due to Bettini et al. (1998). However, the definitions and rules defined are inadequate for describing non-linear granularities. Hence, there is a need to define some new time granularities, that can be useful in visualizations. Cyclic time granularities can be circular, quasi-circular or aperiodic. Examples of circular granularities are hour of the day and day of the week; an example of a quasi-circular granularity is day of the month; examples of aperiodic granularities are public holidays and school holidays.
Time deconstructions can also be based on the hierarchical structure of time. For example, hours are nested within days, days within weeks, weeks within months, and so on. Hence, it is possible to construct single-order-up granularities such as second of the minute, or multiple-order-up granularities such as second of the hour. The lubridate package (Grolemund and Wickham 2011) provides tools to access and manipulate common date-time objects. However, most of its accessor functions are limited to single-order-up granularities.
The motivation for this work stems from the desire to provide methods to better understand large quantities of measurements on energy usage reported by smart meters in households across Australia, and indeed many parts of the world. Smart meters currently provide half-hourly use in kWh for each household, from the time they were installed, some as early as 2012. Households are distributed geographically and have different demographic properties as well as physical properties such as the existence of solar panels, central heating or air conditioning. The behavioral patterns in households vary substantially; for example, some families use a dryer for their clothes while others hang them on a line, and some households might consist of night owls, while others are morning larks. It is common to see aggregates (see Goodwin and Dykes 2012) of usage across households, such as half-hourly total usage by state, because energy companies need to plan for maximum loads on the network. But studying overall energy use hides the distribution of usage at finer scales, and makes it more difficult to find solutions to improve energy efficiency. We propose that the analysis of smart meter data will benefit from systematically exploring energy consumption by visualizing the probability distributions across different deconstructions of time to find regular patterns and anomalies. Although we were motivated by the smart meter example, the problem and the solutions we propose are practically relevant to any temporal data observed more than once per year. In a broader sense, it could be even suitable for data observed by years, decades, and centuries as might be the case in weather or astronomical data.
This work provides tools for systematically exploring bivariate granularities within the tidy workflow (Wickham and Grolemund 2016). In particular, we
- provide a formal characterization of cyclic granularities;
- facilitate manipulation of single- and multiple-order-up time granularities through cyclic calendar algebra;
- develop an approach to check the feasibility of creating plots or drawing inferences for any two cyclic granularities.
The remainder of the paper is organized as follows: Section provides some background material on linear granularities and calendar algebra for computing different linear granularities. Section formally characterizes different cyclic time granularities by extending the framework of linear time granularities, and introducing cyclic calendar algebra for computing cyclic time granularities. The data structure for exploring the conditional distributions of the associated time series across pairs of cyclic time granularities is discussed in Section . Section discusses the role of different factors in constructing an informative and trustworthy visualization. Section examines how systematic exploration can be carried out for a temporal and non-temporal application. Finally, we summarize our results and discuss possible future directions in Section .