tlpipe.utils.robust_stats

Robust statistical utilities.

This implements the Median Absolute Deviation (MAD) and some Winsorized statistical methods.

A sample \(x_1, \cdots, x_n\) is sorted in ascending order. For the chosen \(0 \le \gamma \le 0.5\) and \(k = [\gamma n]\) winsorization of the sorted data consists of setting

\[\begin{split}W_i = \left \{ \begin{array}{lll} x_{(k+1)}, & \mbox{ if } & x_{(i)} \le x_{(k+1)} \\ x_{(i)}, & \mbox{ if } & x_{(k+1)} \le x_{(i)} \le x_{(n-k)} \\ x_{(n-k)}, & \mbox{ if } & x_{(i)} \ge x_{(n-k)}. \end{array} \right.\end{split}\]

The winsorized sample mean is \(\hat{\mu}_w = \frac{1}{n} \sum_{i=1}^{n} W_i\) and the winsorized sample variance is \(D_w = \frac{1}{n-1} \sum_{i=1}^{n} (W_i - \hat{\mu}_w)^2\).

For this implementation, the statistics is computed for winsorized data with \(\gamma = 0.1\).

Functions

MAD(a)	Median absolute deviation divides 0.6745.
mad(a)	Median absolute deviation.
winsorized_mean_and_std(a)
winsorized_mode(a)