METHOD OF MULTIVARIATE TIME-DERIVATIVES: MODELING NEURAL DEGENERATION AND ITS ETIOLOGIES

Abstract

There is a desire to tap into aspects of Magnetic Resonance Imaging (MRI) data of the brain that pertain to the temporal-order of cause and effect. Intuitively, such information can only be derived from longitudinal observations. Current approaches to longitudinal designs, such as repeated-measures ANOVA or mixed effects modeling, can be dauntingly convoluted and often fail to capture the mechanistic changes that describe dynamics between variables in a system. Differentiation, the method of difference ratios between quantities, is used to derive a series of new quantities known as derivatives from the observed variables. The derivatives used in the present demonstration are known as time-derivatives because of how derivatives are calculated with respect to their denominator. Each time-derivative contains information describing the variable's direction and rate of change over a window of time. Integration, the method of summing over the derivatives within a window of the denominator, compacts the information from each derivative into a single quantity. The new quantity includes the components of variance found in raw variables in addition to components of variance contributed from time-derivatives, which contain information pertaining to the mechanics of a system. Time-derivative variables have an edge over the simpler base variables because of the increases in their information content and a relative decrease in their random noise. Our method of multivariate time-derivatives reduces the computational complexity of large, multivariate datasets observed longitudinally. Additionally, a researcher's power to discern is enhanced with time-derivative variables relative to untransformed variables. Variables produced by this method have refined, high-resolution multivariate distribution spaces resulting from having delineated temporal boundaries and differentiating between cases with accreting or depleting quantities. Time-derivative variables maximize the efficiency of a variable, increasing its efficacy, information content, resolution, and ease of deployment in multivariate models. The intricacies of longitudinal big data may appear computationally immutable, but the method of time-derivatives makes quick work of this phenomenon by reducing entire datasets into cross-section-like variables. Importantly, time-derivative variables also incorporate novel information, including components of variance that characterize system dynamics in addition to the standard components available from an analysis of population characteristics.