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CPS1857 Nicholas J. et al.
= arg min (7)
2

So that the subset is a subset with the median least squares error and
̂

is the error vector produced by that subset.

5. Calculate the robust standard deviation ( ) that can be formulated as [2]:
0
5 (8)
2
= 1.4826 (1 + ) √
0
−
where the constant 1.4826 is chosen to provide better efficiency for
the clean data with Gaussian noise, and the factor is to compensate the
effect of small sample size [2]. The robust standard deviation calculated
after each network training determines the border between outliers and
majority of the clean data. In the next step the data set is reduced, so the
training process can be more accurate.
6. Calculate the weight of , for example with = 1, | | ≤ ()

0
and = | | .
0

7. Do the data fitting by the weighted least squares method with to get

the final . The Weighted Least Squares (WLS) method is the same as the
̂
Ordinary Least Square (OLS) method. The difference in the WLS method
is that there is a new additional variable which denotes the diagonal
matrix containing .The equation can be written [4]:

= ( ) (9)
̂

−1

where
1 0 … 0 (10)
0 … 0
= [ 2 ⋱ ]
⋮ ⋮ ⋮
0 0 …
8. As the goal of the LMS Regression method is to minimize the outliers, we
will evaluate the models using the Cook’s distance method for detecting
the outliers. The Cook’s distance method can be formulated as [5]:
2 ℎ (11)
= { }

1 − ℎ

where denotes the studentized residuals and ℎ denotes the

diagonal of the Hatt matrix. The Hatt matrix can be formulated as [5]:
= ( ) (12)
−1
′
′

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