Theory of Lehmer transform and its applications in identifying the electroencephalographic signature of major depressive disorder

We propose a novel transformation called Lehmer transform and establish a theoretical framework used to compress and characterize large volumes of highly volatile time series data. The proposed method is a powerful data-driven approach for analyzing extreme events in non-stationary and highly oscillatory stochastic processes like biological signals. The proposed Lehmer transform decomposes the information contained in a function of the data sample into a domain of some statistical moments. The mentioned statistical moments, referred to as suddencies, can be perceived as the moments that generate all possible statistics when used as inputs of the transformation. Besides, the appealing analytical properties of Lehmer transform makes it a natural candidate to take on the role of a statistic-generating function, a notion that we define in this work for the first time. Possible connections of the proposed transformation to the frequency domain will be briefly discussed, while we extensively study various aspects of developing methodologies based on the time-suddency decomposition framework. In particular, we demonstrate several superior features of the Lehmer transform over the traditional time-frequency methods such as Fourier and Wavelet transforms by analyzing the challenging electroencephalogram signals of the patients suffering from the major depressive disorder. It is shown that our proposed transformation can successfully lead to more robust and accurate classifiers developed for discerning patients from healthy controls.