The Regression Tree (RT) sorts the samples using a specific feature and finds
the split point that produces the maximum variance reduction from a node to its
children. Our key observation is that the best factor to use (in terms of MSE
drop) is always the target itself, as this most clearly separates the target.
Thus using the target as the splitting factor provides an upper bound on MSE
drop (or lower bound on the residual children MSE). Based on this observation,
we define the k-bit lepto-variance ${\lambda}k^2$ of a target variable (or
equivalently the lepto-variance at a specific depth k) as the variance that
cannot be removed by any regression tree of a depth equal to k. As the upper
bound performance for any feature, we believe ${\lambda}k^2$ to be an
interesting statistical concept related to the underlying structure of the
sample as it quantifies the resolving power of the RT for the sample. The max
variance that may be explained using RTs of depth up to k is called the sample
k-bit macro-variance. At any depth, total sample variance is thus decomposed
into lepto-variance ${\lambda}^2$ and macro-variance ${\mu}^2$. We demonstrate
the concept, by performing 1- and 2-bit RT based lepto-structure analysis for
daily IBM stock returns.