One-sided MedianΒΆ

One-sided Median algorithm computes the median

\[m_n^Y = \text{Median}\{Y_{n-2k}, \ldots, Y_{n-1}\}\]

and the median

\[m_n^Z = \text{Median}\{Z_{n-2k}, \ldots, Z_{n-1}\}\]

with

\[Z_n = Y_n - Y_{n-1}.\]

\(2k\) is the size of the neighborhood window. Defining

\[\hat{m}_n = m_n^Y + k m_n^Z \, \text{,}\]

\(Y_n\) is treated as a value outside the region of interest if \(|Y_n - \hat{m}_n| \geq \tau\).

Input Parameters

Parameter Type Constraint Description Remarks
\(Y\) \(Y \in \mathbb R^N\) \(N \in \mathbb{N}\) Input data sequence of length \(N\) None
\(\tau\) \(\tau \in \mathbb R^+\) None User-specified threshold None

Output Parameters

Parameter Type Constraint Description Remarks
\(\hat{Y}\) \(\hat{Y} \in \mathbb R^N\) None Values in the \(Y\) list which are outside the region of interest are marked None

Single Steps using the Algorithm

References

    1. Basu, M. Meckesheimer, Automatic outlier detection for time series: an application to sensor data, Knowledge and Information Systems, vol. 11, Issue 2, pp 137-154, 2007.