Foundations of Data Science

one needs to find the c i such that
⎛ ⎞ ⎛
3 1 1 1 0 1 0 0 0
1
1 1 1 0 −1 0 0 0
4
1 1 −1 0 0 1 0 0
8
3
=
1 1 −1 0 0 −1 0 0
1 −1 0 1 0 0 1 0
⎜ 5
⎟ ⎜ 1 −1 0 1 0 0 −1 0
⎝ 7 ⎠ ⎝ 1 −1 0 −1 0 0 0 1
9 1 −1 0 −1 0 0 0 −1
⎞ ⎛
⎟ ⎜
⎠ ⎝
⎞
c 1
c 2
c 3
c 4
c 5
c 6
⎟
c 7
⎠
c 8
The first column represents the scale function φ(x) and subsequent columns the ψ’s.
The tree in Figure 11.2 illustrates an efficient way to find the coefficients representing
the vector ( 3 1 4 8 3 5 7 9 ) in the basis. Each vertex in the tree contains the
average of the quantities of its two children. The root gives the average of the elements in
the vector, which is 5 in this example. This average is the coefficient of the basis vector
in the first column of the above matrix. The second basis vector converts the average
of the eight elements into the average of the first four elements, which is 4, and the last
four elements, which is 6, with a coefficient of -1. Working up the tree determines the
coefficients for each basis vector.
.
3 1 4 8 3 5 7 9
2 6 4 8
4 6
5
⎛
⎜
⎝
3
1
4
8
3
5
7
9
⎞
⎛
= 5
⎟ ⎜
⎠ ⎝
1
1
1
1
1
1
1
1
⎞
⎛
−1
⎟ ⎜
⎠ ⎝
Figure 11.2: Tree of function averages
1
1
1
1
−1
−1
−1
−1
⎞
⎛
−2
⎟ ⎜
⎠ ⎝
1
1
−1
−1
0
0
0
0
⎞
⎛
−2
⎟ ⎜
⎠ ⎝
0
0
0
0
1
1
−1
−1
⎞
⎛
+1
⎟ ⎜
⎠ ⎝
1
−1
0
0
0
0
0
0
⎞
⎛
−2
⎟ ⎜
⎠ ⎝
0
0
1
−1
0
0
0
0
⎞
⎛
−1
⎟ ⎜
⎠ ⎝
0
0
0
0
1
−1
0
0
⎞
⎛
−1
⎟ ⎜
⎠ ⎝
0
0
0
0
0
0
1
−1
⎞
⎟
⎠
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