14.1 Quantlib twave
You can use
XploRe
to become more familiar with wavelets.
We load the
twave
library by entering the command
library("twave")
on the
XploRe
command line.
XploRe
loads the
twave
quantlib and the interactive menu appears. Quantlib
twave
is an interactive introduction to wavelets consisting of a set
of examples with explanations. The quantlib
twave
uses
the library
wavelet
which was designed for advanced users who
are already familiar with wavelets.
By choosing an item in the interactive menu you can learn more
about the corresponding topic.
Some choices in the menus are the same in all
sections, e.g. the Print menu item. The common menus are
described in the following subsections.
14.1.1 Change Basis
- Haar -- which could be used for a discontinuous function or
a signal with jumps
- Daubechies 4 -- provides a more smooth basis
- Coiflet 2 -- provides the ``smoothest'' basis in our teachware
program
If you are familiar with wavelets you will know that much more
bases are available. To concentrate on the important problems with
wavelet estimation we have restricted ourselves to three typical bases.
14.1.2 Change Function
In some lessons it is useful not just to study the example we
provide as default, but also other functions. Try for example the
lesson about data compression with the Doppler function. We offer
the following functions:
- Jump -- function which has a jump between 0.25 and
0.75. Until the jump the function is the identity function and
after the jump
.
- Up-down -- function with a jump between 0.25 and
0.75. The function is 0 till the jump and then 1.
- Sine -- sine function
- Freq. sine -- sine function which changes its frequency
at 0.5.
- Doppler -- sine function with a decreasing frequency
and an increasing amplitude.
14.1.3 Change View
We provide four different views to the mother wavelet coefficients.
Three of the four views are used in appropriate lessons so that
they deliver a maximum of information.
- Standard view -- shows the mother wavelet coefficients as
vertical bars, where the bar length depends on the
magnitude of the coefficient. The coefficients will be shown
at a place which corresponds to the position of the basis
function in the time-scale plane. See Section 14.3,
function approximation.
- Ordered coefficients -- shows the mother wavelet
coefficients sorted by the absolute magnitude. This allows a
fast judgment about the compression properties for chosen
basis and function. See Section 14.4, data
compression.
- Circle coefficients -- circles are used instead
of vertical bars. The radius of the circle
corresponds to the magnitude of the coefficient. Additionally
the circles are drawn in different colors, red if the coefficient
is used in the approximation, blue if not; see Subsection 14.7.1,
hard thresholding.
- Partial sum -- shows for each resolution scale how well the
function or signal is approximated, if we include all
coefficients up to this resolution scale.