Sliding-window cross-correlation and mutual information methods in the analysis of solar wind measurements

Abstract

When describing the relationships between two data sets, four crucial aspects must be considered, namely: timescales, intrinsic lags, linear relationships, and non-linear relationships. We present a tool that combines these four aspects and visualizes the underlying structure where two data sets are highly related. The basic mathematical methods used here are cross-correlation and mutual information (MI) analyses. As an example, we applied these methods to a set of two-month's worth of solar wind density and total magnetic field strength data. Two neighboring solar wind parcels may have undergone different heating and acceleration processes and may even originate from different source regions. However, they may share very similar properties, which would effectively "hide" their different origins. When this hidden information is mixed with noise, describing the relationships between two solar wind parameters becomes challenging. Time lag effects and non-linear relationships between solar wind parameters are often overlooked, while simple time-lag-free linear relationships are sometimes insufficient to describe the complex processes in space physics. Thus, we propose this tool to analyze the monotonic (or linear) and non-monotonic (or non-linear) relationships between a pair of solar wind parameters within a certain time period, taking into consideration the effects of different timescales and possible time lags. Our tool consists of two parts: the sliding-window cross-correlation (SWCC) method and sliding-window mutual information (SWMI) method. As their names suggest, both parts involve a set of sliding windows. By independently sliding these windows along the time axis of the two time series, this technique can assess the correlation coefficient (and mutual information) between any two windowed data sets with any time lags. Visualizing the obtained results enables us to identify structures where two time series are highly correlated, while providing information on the relevant timescales and time lags. We applied our proposed tool to solar wind density and total magnetic field strength data. Structures with distinct timescales were identified. Our tool also detected the presence of short-term anti-correlations coexisting with long-term positive correlations between solar wind density and magnetic field strength. Some non-monotonic relationships were also found. The visual products of our tool (the SWCC+SWMI maps) represent an innovative extension of traditional numerical methods, offering users a more intuitive perspective on the data. The SWCC and SWMI methods can be used to identify time periods where one parameter has a strong influence on the other. Of course, they can also be applied to other data, such as multi-wavelength photometric and spectroscopic time series, thus providing a new tool for solar physics analyses.