Applying Analytics

120 Applying Analytics: A Practical Introduction

where n is the number of elements in the data set, i is a position index, ∆x

is the spacing of the elements in x-space, and M

∗

is the complex conjugate

of the normalized mother wavelet (Torrence and Compo 1998). Without or-

thogonality, the wavelet transform can be performed on an arbitrary series of

scales, though they won’t be indep e ndent. It is often convenient to have the

resolution in scale s pace vary exponentially, such as:

b

k

= b

0

2

k∆k

, k = 0, 1, . . . , K, (5.47)

where b

0

is the smallest scale that can be sampled, 2∆x, analogous to the

Nyquist limit for frequency a nalysis discussed in Section