Testing and Estimation for Functional Data with Applications to Magnetometer Records

• Main Author: Maslova, Inga
• Format: Others
• Published: DigitalCommons@USU 2009
• Subjects: Functional data; functional linear model; independence test; magnetic storms; substorms; Mathematics
• Online Access: https://digitalcommons.usu.edu/etd/384; https://digitalcommons.usu.edu/cgi/viewcontent.cgi?article=1384&context=etd

The functional linear model, $Y_n = Psi X_n + varepsilon_n$, with functional response and explanatory variables is considered. A simple test of the nullity of $Psi$ based on the principal component decomposition is proposed. The test statistic has asymptotic chi-squared distribution, which is also an excellent approximation in finite samples. The methodology is applied to data from terrestrial magnetic observatories. In recent years, the interaction of the auroral substorms with the equatorial and mid-latitude currents has been the subject of extensive research. We introduce a new statistical technique that allows us to test at a specified significance level whether such a dependence exists, and how long it persists. This quantitative statistical technique, relying on the concepts and tools of functional data analysis, uses directly magnetometer records in one minute resolution, and it can be applied to similar geophysical data which can be represented as daily curves. It is conceptually similar to testing the nullity of the slope in the straight line regression, but both the regressors and the responses are curves rather than points. When the regressors are daily high latitude $H$--component curves during substorm days and the responses are daily mid-- or low latitude $H$--component curves, our test shows significant dependence (the nullity hypothesis is rejected), which exists not only on the same UT day, but also extends into the next day for strong substorms. We propose a novel approach based on wavelet and functional principal component analysis to produce a cleaner index of the intensity of the symmetric ring current. We use functional canonical correlations to show that the new approach more effectively extracts symmetric global features. The main result of our work is the construction of a new index, which is an improved version of the existing wavelet-based index (WISA) and the old Dst index, in which a constant daily variation is removed. Here, we address the fact that the daily component varies from day to day and construct a ``cleaner'' index by removing non-constant daily variations. A wavelet-based method of deconvoluting the solar quiet variation from the low and mid-latitude H-component records is proposed. The resulting daily variation is non--constant, and its day--to--day variability is quantified by functional principal component scores. The procedure removes the signature of an enhanced ring current by comparing the scores at different stations. The method is fully algorithmic and is implemented in the statistical software R. R package for space physics applications is developed. It consists of several functions that compute indices of the storm activity and estimate the daily variation. Storm indices are computed automatically without any human intervention using the most recent magnetometer data available. Functional principal component analysis techniques are used to extract day-to-day variations. This package will be publicly available at Comprehensive R Archive Network (CRAN).