Exact box-counting and temporal sampling algorithms for fractal dimension estimation with applications to animal...

Fractal dimension (FD) is a widely recognized metric in mathematics and physics for quantifying the complexity of intricate objects and processes. In this study, we propose novel algorithms to estimate the FD of animal movement using high-resolution spatial and temporal data. To enhance estimation accuracy, we developed an oversampling technique that linearly interpolates between adjacent points on movement paths. Furthermore, we introduced an exact box-counting algorithm tailored for piecewise linear paths, ensuring accurate fractal dimension estimations. Considering that animal behavior is typically recorded at fixed frame rates, we also propose a temporal sampling method for calculating FD using temporal domain scales. Recognizing that FD varies with scale, we employ a dual total least squares method to identify the optimal scale for FD comparisons across different genotypes. Through large-scale experiments on the movement of Drosophila melanogaster larvae, we show that mutations in the schizophrenia-associated gene Dysbindin significantly increase FD, suggesting potential impairments in motor function. These findings highlight FD as a robust and quantitative metric for assessing the complexity of movement behavior.