Page 13 - Special Topic Session (STS) - Volume 4

STS556 Mohd Bakri A. et al.
is compound smoother. compound smoother is known as a powerful tool to
smooth a data series without excessively disrupting the details of a data series.
Despite this good traits, the compound smoother does not respond well to
oscillated trend, Tothmeresz and Erdei (1995) and Jin and Xu (2013). The
number observations of compound smoother should be at least seven,
otherwise it will converge to constant root, Janosky et al. (1997). The
Velleman's compound smoother as indicated by Sargent and Bedford (2010)
has been revised when possible combinations of multiple step of running
median, Hanning and resmooth the rough are tested out. Improvement on the
existing compound smoother in comparison has yet been explored.

2. Methodology
The 4253HT is one of the non-linear smoothing techniques that combined
running median, weighted moving average and re-smoothed the rough. This
technique was first introduces by (Tukey, 1977) and described detail in
different version by (Velleman et al., 1981). Let X be a doubly-infinite sequence
of real data X ,..., X , X , X ,..., X t+ n  . A smoother M is defined as an
t
t+1
t−1
t−
n
algorithm that works on X to generate a new series M (X t ) , smoothed values.
The algorithm of 4253HT is as follows:
Step 1: Perform running median of span size two
M 1 (X t ) = median (X − t 2 , X 1 − t , X t , X + t 1 ) (1)

Step 2: Re-centered the equation (1)
M (X ) = median (M (X ),M (X ) ) (2)
2 t 1 t 1 1 + t

Step 3: Next, equation (2) are smoothed again by applying running median of
span size three
M 3 (X t ) = median (M 2 (X − t 2 ),M 2 (X 1 − t ),M 2 (X t ),M 2 (X 1 + t ),M 2 (X + t 2 ) ) (3)

Step 4: Perform running median of span size three
M 4 (X t ) = median (M 3 (X 1 − t ),M 3 (X t ),M 3 (X 1 + t ) ) (4)

1
1
Step 5: Apply weighted moving average or Hanning with coefficients ,
4 2
1
and
4
1 1 1
M (X ) = M (X ) + M (X ) + M (X ) (5)
5 t 4 − t 1 4 t 4 + t 1
4 2 4
Step 6: Re-smooth the rough and added to the smoothed values in (5)
M 6 (X t ) = M 5 (X t ) M+ 5 X − M 5 (X t  ) (6)
t

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