IndepDist

Variable-wise aggregate of multivariate kernel or distance function.

A common baseline method to turn a univariate time series distance or kernel into a multivariate time series distance or kernel.

Also sometimes known as “independent distance” in the special case where aggfun is the sum or mean and the pairwise transformer is a time series distance.

Formal details (for real valued objects, mixed typed rows in analogy): Let \(d: \mathbb{R}^n \times \mathbb{R}^n\rightarrow \mathbb{R}\) be the pairwise function in dist, when applied to univariate series of length \(n\). This class represents the pairwise function \(d_g: \mathbb{R}^{n\times D} \times \mathbb{R}^{n\times D}\rightarrow \mathbb{R}\) defined as \(d_g(x, y):= g(d(x_1, y_1), \dots, d(x_D, y_D))\), where \(x_i\), \(y_i\) denote the \(i\)-th column, and \(x\), :math:``y` are interpreted as multivariate time series with :math:`D` variables, and where :math:`g` is a function :math:`g: \mathbb{R}^D\times \mathbb{R}^D \rightarrow \mathbb{R}`, representing the input ``aggfun.

In particular, if aggfun="sum" (or default), then \(g(x) = \sum_{i=1}^D x_i\), and \(d_g(x, y):= \sum_{i=1}^D d(x_i, y_i)\), which corresponds to the usual terminology “independent distance”.