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calculates,
given
an
input
signal
and
a
wavelet function, the weights corresponding to each scaling and dilatation
(j,n)
of ψ(t) (Eq. 6.29).
This
produces
a
matrix
cwcx with the j index rep-resenting the scaling dimension and the n index representing dilatation or
(j,n)
time. The cwcx matrix can be interpreted as a spectrogram, although the relationship between scale and frequency, in the Fourier sense, depends on the choice of wavelet.
cwci1 = cwt(i1,ψ) (6.29)
Edges are easy to spot in the dataset, and are usually synonyms with maintenance actions. In order to detect edges, the indicator series is ex-panded
on
the
Haar
[19]
wavelet
using
at
scales
1
through
Jedge. The meaning of a wavelet coe.cient matrix depends on the choice of the wavelet. For the Haar wavelet, the coe.cients signify the numeric derivative of the dataset at
j
di.erent scales. I.e. the vector cwccontains the dataset mean derivative
i1 across a sliding window of 2j points. The Haar wavelet is chosen because it resembles a step. Thus, whenever a step in encountered in the dataset, the wavelet coe.cients will exhibit higher values than if no step is present. Trends are slowly evolving phenomena, and are thus con.ned to the coarser scales of the cwci1 matrix. Random noise is wide band, but its pres-ence in the coarse scales is negligible compared to the energy of the trends. The only component with a signi.cant impact across all scales is the edge. The e.ect of a unit step at a given scale is 2j
2 . Consequently, an edge can be identi.ed by looking for the edge signature
.(j,n)
across the scales. A modi.ed detail matrix cwc i1 (Eq. 6.30)
is
created
to capture the amplitude of the edge. An edge at position n will produce a
.(j,n)
modi.ed detail matrix with coe.cients cwc i1 equal to the edge amplitude for all values of j along the n’th column.
.(j,n)(j,n)
2 i1 i1
cwc =2 .j cwc(6.30)
.(j,n)
Using the above de.nition, an edge is a position in time n where cwc i1 is equal for all values of j. Due to the presence of components w and c,
6.5. NON-PARAMETRIC PROGRESSION ANALYSIS
the values across the scales will not be completely identical. Thus, an edge signature
metric
is
de.ned
(Eq.
6.31).
.(j,n)
|mean[cwc ]|
bp(n)= i1 (6.31)
.(j,n)
std[cwc i1 ]
This represents the degree of edge behavior in each point n along the time line. The functions mean and std are computed across all scales j for each point n in time. Thus, an edge can be de.ned as a point in time n where bp(n) is higher than the threshold Tp. Unfortunately, this will also capture minor transition which also satis.es the above criteria. Thus, an edge
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