In recent years, with the rise of mobile devices and the Internet of Things, the use of sensors has surged, leading to an exponential growth in the generation of time series data 1. Along with this trend, there is an increasing demand for analyzing and processing time series data 2,3. Time series data analysis is an important branch in the field of data analysis, which mainly focuses on the analysis, modeling, prediction, and other aspects of time series data 4–7. Time series data can be applied in many fields such as finance, economics, weather forecasting, signal processing, and so on 8–11. However, time series data often exhibit non-smoothness, which can affect the accuracy and reliability of data analysis and modeling 12. Therefore, measuring the non-smoothness of time series data has become an important issue in time series analysis.
The non-smoothness of time series data is typically manifested as the instability of statistical characteristics of the data over time 13. For example, the mean, variance, auto correlation, and other statistical characteristics of the data may vary greatly at different moments. In time series data analysis, we typically use some common indicators and methods to measure the non-smoothness of time series data, to better understand the properties and characteristics of time series data. One of the most commonly used indicators is the Sample Autocovariance Function (SACF), which can be used to measure the covariance of time series data at different moments, thus helping us to assess the smoothness of time series data 14. If the time series data is smooth, then its SACF at different moments should be very similar, otherwise non-smoothness exists. In addition, Autocorrelation Function (ACF) and Partial Autocorrelation Function (PACF) are also used to measure the non-smoothness of time series data. ACF and PACF can help us determine the correlation of time series data at different moments, further indicating the smoothness of time series data 15,16. If the values of ACF and PACF fluctuate greatly over time, then non-smoothness may exist. In addition to statistical indicators, we can also use some graphical methods to measure the non-smoothness of time series data, such as ACF Plot, PACF Plot, Stationarity Plot, etc. These graphical methods can help us more intuitively judge the non-smoothness of time series data. In general, the non-smoothness of time series data is an important issue in time series data analysis. Measuring the non-smoothness of time series data can help us better understand the properties and characteristics of time series data and choose appropriate models and analysis methods. In practical applications, we can choose suitable indicators and methods based on specific analysis purposes and data characteristics to ensure the accuracy and reliability of analysis and modeling.
Based on graph theory and matrix theory, this paper proposes a Dirichlet mean energy function method for measuring the non-smoothness of time series data. This method measures the non-smoothness of time series data with different lengths and sampling interval by giving a standardized metric. The typhoon wind speed data collected by the sensor is used to validate the validity of the proposed method.
The structure of the remaining parts of this paper is as follows: Section 2 provides a detailed description of the graph model for time series data and presents the Dirichlet mean energy function as a mathematical tool for measuring non-smoothness of time series data. In Section 3, we analyze and validate the validity of the proposed Dirichlet mean energy function in measuring this non-smoothness using time series data collected by sensors. Section 4 presents the conclusion of this paper and provides prospects for future research work.