Density Estimation Using Wavelets
This section takes you through the features of 1-D wavelet density estimation using one of the Wavelet Toolbox™ specialized tools.
The toolbox provides Wavelet Analyzer app to estimate the density of a sample and complement well known tools like the histogram (available from the MATLAB® core) or kernel based estimates.
For the examples in this section, switch the extension mode to symmetric padding, using the command
dwtmode('sym')
1-D Estimation Using the Wavelet Analyzer App
Start the Density Estimation 1-D Tool.
From the MATLAB prompt, type
waveletAnalyzer
.The Wavelet Analyzer appears.
Click the Density Estimation 1-D menu item. The discrete wavelet analysis tool for 1-D density estimation appears.
Load data.
At the MATLAB command line, type
In the Density Estimation 1-D tool, choose File > Import from Workspace.load ex1cusp1
When the Import from Workspace dialog box appears, select
ex1cusp1
. Click OK to import the noisy cusp data.The sample, a 64-bin histogram, and the processed data obtained after a binning are displayed. In this example, we'll accept the default value for the number of bins (250). The binned data, suitably normalized, will be processed by wavelet decomposition.
Perform a Wavelet Decomposition of the binned data.
Select the
sym6
wavelet from the Wavelet menu and select 4 from the Level menu, and click the Decompose button. After a pause for computation, the tool displays the detail coefficients of the decomposition of the binned data.Perform a density estimation.
Accept the defaults of global soft thresholding. The sliders located on the right of the window control the level dependent thresholds, indicated by dashed blue lines running horizontally through the graphs on the left of the window.
Continue by clicking the Estimate button.
You can see that the estimation process delivers a very irregular resulting density. The density estimate (in purple) is the normalized sum of the signals located below it: the approximation
a4
and the reconstructed details after coefficient thresholding.Perform thresholding.
You can experiment with the various predefined thresholding strategies by selecting the appropriate options from the menu located on the right of the window or directly by dragging the dashed blue lines with the left mouse button. Let's try another estimation method.
From the menu Select thresholding method, select the item By level threshold 2. Next, click the Estimate button.
The estimated density is more satisfactory. It correctly identifies the smooth part of the density and the cusp at 0.7.
Importing and Exporting Information from the Wavelet Analyzer App
The tool lets you save the estimated density to disk. The toolbox creates a MAT-file in the current folder with a name you choose.
To save the estimated density, use the menu option File >
Save Density. A dialog box appears that lets you specify a folder and
filename for storing the density. Type the name dex1cusp
. After
saving the density data to the file dex1cusp.mat
, load the
variables into your workspace:
load dex1cusp whos
Name | Size | Bytes | Class |
---|---|---|---|
ex1cusp1 | 1x1000 | 8000 | double array |
thrParams | 1x4 | 544 | cell array |
wname | 1x4 | 8 | char array |
xdata | 1x250 | 2000 | double array |
ydata | 1x250 | 2000 | double array |
The original noisy cusp data ex1cusp1
has 1000 samples. The
variables thrParams
, wname
,
xdata
, and ydata
are stored in
dex1cusp.mat
. The estimated density is given by
xdata
and ydata
. The length of these
vectors is equal to the number of bins you choose in step 4. In addition, the
parameters of the estimation process are given by the wavelet name in
wname
.
wname wname = sym6
and the level dependent thresholds contained in thrParams
,
which is a cell array of length 4 (the level of the decomposition). For i from 1 to
4, thrParams{i}
contains the lower and upper bounds of the
interval of thresholding and the threshold value (since interval dependent
thresholds are allowed).
For
example, for level 1,
thrParams{1} ans = 0.0560 0.9870 2.1179
Note
When you load data from a file using the menu option File > Load Data for Density Estimate, the first 1-D variable encountered in the file is considered the signal. Variables are inspected in alphabetical order.
At the end of this section, turn the extension mode back to zero padding using
dwtmode('zpd')