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CHAPTER 7. USAGE EXAMPLES
7.1.3
Example 3: difference between ’main’ and ’total’ scores in
FAST
In this example we will consider FAST performance for the function:
f (x1 , x2 , x3 ) = x21 + 2x1 x2 + x23 , xi ∈ [−1, 1], i = 1, 2, 3,
(7.3)
that on the one hand is still simple enough to form some expectations of what true scores
should be, but on the other hand it already has some feature interactions.
So in this example one may expect to see x1 having the largest score, x2 be on the second
place and x3 be the least important feature.
We will use this example to demonstrate the difference between main and total FAST
scores. Main scores take into account only isolated variable contribution to the variance of
output, meaning that main scores would ignore influence of the x1 · x2 term. Total scores on
the other side should account all feature interactions. In the manual dependency analysis
comparison of these two indices allows for some investigation of the dependency nature.
We’ve estimated these scores using 500 and 1000 points samples to show the difference in
the results.
Total scores are presented in the Table 7.7.
Sample size
x1
x2
x3
500
0,4449
0,3985
0,1503
1000
0,4965
0,4125
0,0869
Table 7.7: Example 2. FAST (total) scores
Main scores are presented in the Table 7.8.
Sample size
x1
x2
x3
500
0,3353
0,0019
0,6604
1000
0,5016
0,0033
0,4910
Table 7.8: Example 2. FAST (main) scores
Let ST 1 , ST 2 , ST 3 be total indexes of variables and SM 1 , SM 2 , SM 3 be main indices.
One may see that SM 2 ≈ 0, ST 2 SM 2 , it gives one a hint that x2 feature appears only
in interaction with some other. Also one may remember that ST i = SM i +interaction terms,
i.e. say ST 1 = SM 1 + S12 + S13 , where for the example S12 - is a term accounting for x1 and
x2 interaction. Notice also that SM 1 ≈ SM 3 , SM 2 ≈ 0 and ST 1 ≈ ST 2 + ST 3 ⇒ S12 + S13 ≈
S12 + S23 + S13 + S23 ⇒ S23 ≈ 0. As a result we can make an educated guess that our function
has the following form f (x1 , x2 , x3 ) = f1 (x1 ) + f2 (x3 ) + f3 (x1 , x2 ) + f4 (x1 , x3 ).
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