ABSTRACT
Commonly used dependence measures, such as linear correlation, cross-correlogram, or Kendall's tau , cannot capture the complete dependence structure in data unless the structure is restricted to linear, periodic, or monotonic. Mutual information (MI) has been frequently utilized for capturing the complete dependence structure including nonlinear dependence. Recently, several methods have been proposed for the MI estimation, such as kernel density estimators (KDEs), k -nearest neighbors (KNNs), Edgeworth approximation of differential entropy, and adaptive partitioning of the XY plane. However, outstanding gaps in the current literature have precluded the ability to effectively automate these methods, which, in turn, have caused limited adoptions by the application communities. This study attempts to address a key gap in the literature-specifically, the evaluation of the above methods to choose the best method, particularly in terms of their robustness for short and noisy data, based on comparisons with the theoretical MI estimates, which can be computed analytically, as well with linear correlation and Kendall's tau . Here we consider smaller data sizes, such as 50, 100, and 1000, and within this study we characterize 50 and 100 data points as very short and 1000 as short. We consider a broader class of functions, specifically linear, quadratic, periodic, and chaotic, contaminated with artificial noise with varying noise-to-signal ratios. Our results indicate KDEs as the best choice for very short data at relatively high noise-to-signal levels whereas the performance of KNNs is the best for very short data at relatively low noise levels as well as for short data consistently across noise levels. In addition, the optimal smoothing parameter of a Gaussian kernel appears to be the best choice for KDEs while three nearest neighbors appear optimal for KNNs. Thus, in situations where the approximate data sizes are known in advance and exploratory data analysis and/or domain knowledge can be used to provide a priori insights into the noise-to-signal ratios, the results in the paper point to a way forward for automating the process of MI estimation.
ABSTRACT
BACKGROUND: Conventional (rigid) fusion instrumentation is believed to accelerate the degeneration of adjacent discs by increasing stresses caused by motion discontinuity. Fusion instrumentation that employs reduced rod stiffness and increased axial motion, or dynamic instrumentation, may partially alleviate this problem, but the effects of this instrumentation on the stresses in the adjacent disc are unknown. We used a finiteelement model to calculate and compare the stresses in the adjacent-level disc that are induced by rigid and dynamic posterior lumbar fusion instrumentation. METHODS: A 3-dimensional finite-element model of the lumbar spine was obtained that simulated flexion and extension. The L5-S1 segment of this model was fused, and the L4-L5 segment was fixed with rigid or dynamic instrumentation. The mechanical properties of the dynamic instrumentation were determined by laboratory testing and then used in the finite-element model. Peak stresses in the lumbar discs were calculated and compared. RESULTS: The reduced-stiffness component of the dynamic instrumentation was associated with a 1% to 2% reduction in peak compressive stresses in the adjacent-level disc (at 45° flexion), and the increased axial motion component of this instrumentation reduced peak disc stress by 8% to 9%. Areas of disc tissue exposed to 80% of peak stresses of 6.17 MPa were 47% less for discs adjacent to dynamic instrumentation than for those adjacent to rigid instrumentation. CONCLUSIONS: Reduced stiffness and increased axial motion of dynamic posterior lumbar fusion instrumentation designs result in an approximately 10% cumulative stress reduction for each flexion cycle. The effect of this stress reduction over many cycles may be substantial. CLINICAL RELEVANCE: The cumulative effect of this reduced amplitude and distribution of peak stresses in the adjacent disc may partially alleviate the problem of adjacent-level disc degeneration.
