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MovingMedian

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

MovingMedian[list, r]
    gives the moving median of list, computed using spans of r elements.
MovingMedian returns a list of length Length[list] - r + 1; for matrix input the medians are taken column-wise within each row-window.
MovingMedian requires real-valued numeric data and stays unevaluated when r < 1 or r > Length[list].

Examples

All examples below are verified against the current Mathilda build.

In[1]:= MovingMedian[{1, 2, 5, 6, 1, 4, 3}, 3]
Out[1]= {2, 5, 5, 4, 3}

In[2]:= MovingMedian[{{1, 2}, {5, 3}, {1, 4}, {3, 2}, {5, 5}}, 2]
Out[2]= {{3, 5/2}, {3, 7/2}, {2, 3}, {4, 7/2}}

In[3]:= MovingMedian[N[{1, 5, 7, 3, 6, 2}], 3]
Out[3]= {5.0, 5.0, 6.0, 3.0}

In[4]:= MovingMedian[{1, 2, 3, 4}, 2]
Out[4]= {3/2, 5/2, 7/2}

In[5]:= MovingMedian[{2^100, 2^101, 2^102, 2^103}, 2]
Out[5]= {1901475900342344102245054808064, 3802951800684688204490109616128, 7605903601369376408980219232256}

In[6]:= MovingMedian[{a, b, c}, 2]
Out[6]= MovingMedian[{a, b, c}, 2]

Implementation notes

Algorithm. builtin_moving_median takes (list, r) with r a positive integer window (EXPR_INTEGER/EXPR_BIGINT); output length is n - r + 1 and the call is unevaluated unless 1 <= r <= n. It auto-detects vector vs. matrix mode by whether the first element is a List, and validates every leaf with is_real_numeric (matrices must be rectangular); invalid data prints MovingMedian::arg1 and leaves the call unevaluated. It then slides the window, builds an r-element sublist, and delegates each window to Median (so a matrix window yields a column-wise median vector). Window results are assembled into the output List. ATTR_PROTECTED.

  • Protected.
  • Output length is Length[list] - r + 1.
  • Operates on real-valued vectors and matrices. For matrix input, each window of r consecutive rows is reduced via Median, yielding a column-wise median vector per window.
  • Exact rationals, bignums (arbitrary-precision integers), machine-precision reals, and NumericQ-real symbolic constants (Pi, E, ...) are all supported. Even-window medians yield exact rational midpoints when the data is exact.
  • Stays unevaluated when r < 1, when r > Length[list], when r is non-integer, or when the first argument is not a List.
  • Non-numeric data triggers the MovingMedian::arg1 message and the expression remains unevaluated.

Attributes: Protected.

Implementation status

Stable — documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

References

Notes & additional examples

Worked examples

In[1]:= MovingMedian[{1, 3, 2, 8, 5, 4, 9}, 3]
Out[1]= {2, 3, 5, 5, 5}

In[1]:= MovingMedian[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, 5]
Out[1]= {3, 4, 5, 6, 7, 8}

In[1]:= MovingMedian[Table[Mod[n^2, 11], {n, 1, 12}], 4]
Out[1]= {9/2, 9/2, 4, 4, 4, 9/2, 9/2, 5/2, 1}

Notes

MovingMedian[list, r] returns a list of length Length[list] - r + 1: the median of each window of r consecutive elements as the window slides across list. Even-width windows average the two central order statistics, so results may be exact rationals (e.g. 9/2) rather than integers. The robust median makes it less sensitive to outliers than MovingAverage.