Theoretical properties of distance distributions and novel metrics for nearest-neighbor feature selection

The performance of nearest-neighbor feature selection and prediction methods depends on the metric for computing neighborhoods and the distribution properties of the underlying data. Recent work to improve nearest-neighbor feature selection algorithms has focused on new neighborhood estimation methods and distance metrics. However, little attention has been given to the distributional properties of pairwise distances as a function of the metric or data type. Thus, we derive general analytical expressions for the mean and variance of pairwise distances for L(q) metrics for normal and uniform random data with p attributes and m instances. The distribution moment formulas and detailed derivations provide a resource for understanding the distance properties for metrics and data types commonly used with nearest-neighbor methods, and the derivations provide the starting point for the following novel results. We use extreme value theory to derive the mean and variance for metrics that are normalized by the range of each attribute (difference of max and min). We derive analytical formulas for a new metric for genetic variants, which are categorical variables that occur in genome-wide association studies (GWAS). The genetic distance distributions account for minor allele frequency and the transition/transversion ratio. We introduce a new metric for resting-state functional MRI data (rs-fMRI) and derive its distance distribution properties. This metric is applicable to correlation-based predictors derived from time-series data. The analytical means and variances are in strong agreement with simulation results. We also use simulations to explore the sensitivity of the expected means and variances in the presence of correlation and interactions in the data. These analytical results and new metrics can be used to inform the optimization of nearest neighbor methods for a broad range of studies, including gene expression, GWAS, and fMRI data.