Page 14 - Special Topic Session (STS) - Volume 4

STS556 Mohd Bakri A. et al.
The running median of even span size is computed by taking the average of
two subsequent points in the middle using arithmetic mean. This value is
better than running median of odd span size in the sense that it preserves the
significant spike in the data series. The smooth value produce by running
median span size four and re-centered by running median of span size two is
a combination of Equation (1) and (2) and can be expressed as follows:
1
M 2 (X t ) = median (X − t 2 , X 1 − t , X t , X + t 1 ) median+ (X 1 − t , X t , X 1 + t , X + t 2 )
4 (7)
= 1 (X * + X * + X ' + X ' )
4 1 − t t t 1 + t
where X is the ordered observation from window in X − t 2 , X 1 − t , X t , X 1 + t and
*
X is the ordered observation from window in X 1 − t , X t , X 1 + t , X + t 2 .
'
Some adjustments are proposed by applying different type of mean. The type
of means involved are geometric, quadratic, harmonic and contra harmonic.
The modification of running median span size 42 are as follows:

Geometric Mean
1
M (X ) = (X *  X *  X '  X ' ) (8)
4
2 t − t 1 t t + t 1

Quadratic Mean
1
 X 2 * + X 2 * + X 2 ' + X 2 '  2
M 2 (X t ) =   1 − t t t + t 1  (9)

 4 
Harmonic Mean

M 2 (X t ) = 1  1 + 1 + 1 + 1   (10)


4 X * t −1 X * t X t ' X t ' +1 

Contra harmonic Mean
 X 2 * + X 2 * + X 2 ' + X 2 ' 
M 2 (X t ) =   t −1 * t * t ' ' t +1  (11)

 X t −1 + X t + X t + X t +1 

The types of mean that produce smaller value than arithmetic mean;
geometric and harmonic are expected to be more resistant to negative
impulse or block pulse. On the other hand, quadratic and contra harmonic are
more responsive to positive changes in the data series. Some of the
modifications are not working if the observations consist of zero or negative
values. Hence, a constant point should be added to the data to ensure the
smoothed value can be computed. Modification of 4253HT involves the
running median of span size 42 only. For smoothing the rough part, original
algorithm is maintained where the middle points for running median of span
size four and two are computed by arithmetic mean. The evaluation process

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