Page 387 - Contributed Paper Session (CPS) - Volume 2

CPS1891 Tzee-Ming Huang
For comparison purpose, two other knot selection methods: Method 2 and
Method 3, are applied to the same 100 data sets, and the results are also
reported in Table 1, where the details of Method 2 and Method 3 are given
below. Method 2 is a modified version of the knot selection scheme in Zhou
and Shen (2001). There are four steps in the scheme in Zhou and Shen (2001).
To save computation time, the number of iterations between Step 2 and Step
3 is limited to three, and in Step 4, the algorithm does not go back to Step 2
for refinement. The initial knot set is {0:5} and the spline order is 3.
For Method 3, five sets of equally spaced knots: , … , are considered,
5
1
where
= {ℓ/2 : ℓ = 1, … , 2 }

for = 1, … , 5, and then BIC is used to choose one of the five sets as the
selected knot set. All knots in the selected knot set are used.
The mean and median values for mean squared errors are given in Table 1.
The median values are in parentheses.

Algorithm 1 Method 2 Method 3
1 5.05 × 10 (4.44 × 10 ) 1.77 × 10 (9.68 × 10 ) 3.14 × 10 (3.01 × 10 )
−4
−3
−4
−4
−4
−4
2 2.16 × 10 (1.80 × 10 ) 1.37 × 10 (1.28 × 10 ) 1.77 × 10 (1.87 × 10 )
−3
−3
−3
−3
−3
−3
3 4.06 × 10 (3.49 × 10 ) 1.24 × 10 (1.07 × 10 ) 4.73 × 10 (4.61 × 10 )
−3
−4
−3
−4
−4
−4
4 3.85 × 10 (3.28 × 10 ) 1.15 × 10 (5.07 × 10 ) 2.29 × 10 (2.22 × 10 )
−4
−3
−3
−3
−4
−4
Table 1: Mean and median values for mean squared errors

The frequencies that one method outperforms the other two for the 100
data sets are reported in Table 2.

Algorithm 1 Method 2 Method 3
1 24 3 73
2 27 60 13
3 75 9 16
4 81 19 0
Table 2: Frequencies of being the outperforming method
From Tables 1 and 2, when the regression function is a smooth function
like or , Algorithm 1 and Method 3 perform better than Method 2. When
1
3
= , Method 3 outperforms the other two methods. When = ,
1
3
Algorithm 1 performs slightly better than Method 3. When f is a function with
sharp change like or , Algorithm 1 and Method 2 perform better than
4
2
Method 3. When = , Method 2 outperforms Algorithm 1 since it allows for
2
_ner adjustment of knot locations. When = , Algorithm 1 outperforms
4
Method 2, which is not quite as expected. In such case, Method 2 sometimes
selects extra knots near 1, which makes the mean squared error a little larger
than Algorithm 1.
To examine the stablity of the performance for each of the three methods,
the standard deviations of the mean squared errors are reported in Table 3.
From the results in Table 3, for most cases, Method 3 is the most stable
method and Algorithm 1 is the second most stable method.
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