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of the decomposition illustrate the broad trends in the data, whereas the
more localized trends are captured by the lower order coecients.
We assume for ease in description that the length
q
of the series is
a power of 2. This is without loss of generality, because it is always
possible to decompose a series into segments, each of which has a length
that is a power of two. The Haar Wavelet decomposition defines 2
k−
1
coecients of order
k
.Eachofthese2
k−
1
coecients corresponds to a
contiguous portion of the time series of length
q/
2
k−
1
.The
i
th of these
2
k−
1
coecients corresponds to the segment in the series starting from
position (
i
q/
2
k−
1
. Let us denote this
coecient by
ψ
k
and the corresponding time series segment by
S
k
.At
the same time, let us define the average value of the first half of the
S
k
by
a
i
k
and the second half by
b
i
k
. Then, the value of
ψ
k
is given by
(
a
i
k
−
q/
2
k−
1
+ 1 to position
i
−
1)
·
∗
b
i
k
)
/
2. More formally, if Φ
i
k
denote the average value of the
S
k
,
then the value of
ψ
k
can be defined recursively as follows:
ψ
k
=(Φ
2
·i−
1
Φ
2
·i
−
k
+1
)
/
2
(6.1)
k
+1
The set of Haar coecients is defined by the Ψ
i
k
coecients of order 1
to log
2
(
q
). In addition, the global average Φ
1
is required for the purpose
of perfect reconstruction. We note that the coecients of different order
provide an understanding of the major trends in the data at a particular
level of granularity. For example, the coecient
ψ
k
is half the quantity
by which the first half of the segment
S
k
is larger than the second half of
the same segment. Since larger values of
k
correspond to geometrically
reducing segment sizes, one can obtain an understanding of the basic
trends at different levels of granularity. We note that this definition of
the Haar wavelet makes it very easy to compute by a sequence of av-
eraging and differencing operations. In
Table 6.1
, we have illustrated
how the wavelet coecients are computed for the case of the sequence
(8
,
6
,
2
,
3
,
4
,
6
,
6
,
5). This decomposition is illustrated in graphical form
in
Figure 6.1
. We also note that each value can be represented as a sum
of log
2
(8) = 3 linear decomposition components. In general, the entire
decomposition may be represented as a tree of depth 3, which represents
the hierarchical decomposition of the entire series. This is also referred
to as the
error tree
. In
Figure 6.2
,
we have illustrated the error tree for
contain the values of the wavelet coecients, except for a special
super-
root
node which contains the series average. This super-root node is not
necessary if we are only considering the relative values in the series, or
the series values have been normalized so that the average is already
zero. We further note that the number of wavelet coecients in this
series is 8, which is also the length of the original series. The original
