Wavelet Analysis

Wavelet Coefficient Plots

Diagnostic plots greatly facilitate the interpretation of a wavelet decomposition. One standard plot is the detail coefficients arranged by level. By using a module included by the WAVINIT macro call, you can produce the plot shown in Figure 20.5 as follows: 

 
      call coefficientPlot(decomp, , , , ,"Quartz Spectrum");

The first argument specifies the wavelet decomposition and is required. All other arguments are optional and need not be specified. You can use the WAVHELP macro to obtain a description of the arguments of this and other wavelet plot modules. The WAVHELP macro is defined in the autocall WAVINIT macro. For example, invoking the WAVHELP macro as follows writes the calling information shown in Figure 20.4 to the SAS log. 

 
      %wavhelp(coefficientPlot);

Figure 20.4: Log Output Produced by %wavhelp(coefficientPlot) Call

 
      coefficientPlot Module                                                    
 
Function: Plots wavelet detail coefficients                                     
 
Usage: call coefficientPlot(decomposition,                                      
                            threshopt,                                          
                            startLevel,                                         
                            endLevel,                                           
                            howScaled,                                          
                            header);                                            
 
Arguments:                                                                      
   decomposition - (required) valid wavelet decomposition produced              
                              by the IML subroutine WAVFT                       
   threshopt     - (optional) numeric vector of 4 elements                      
                              specifying thresholding to be used                
                              Default: no thresholding                          
   startLevel    - (optional) numeric scalar specifying the lowest              
                              level to be displayed in the plot                 
                              Default: start level of decomposition             
   endLevel      - (optional) numeric scalar specifying the highest             
                              level to be displayed in the plot                 
                              Default: end level of decomposition               
   howScaled     - (optional) character: 'absolute' or 'uniform'                
                              specifies coefficients are scaled uniformly       
                              Default: independent level scaling                
   header        - (optional) character string specifying a header              
                              Default: no header                                
  




            

            


            
  

            


Figure 20.5: Detail Coefficients Scaled by Level

               

                  
Detail Coefficients Scaled by Level

               

            

            


            
In this plot the detail coefficients at each level are scaled independently. The oscillations present in the absorbance data
               are captured in the detail coefficients at levels 7, 8, and 9. The following statement produces a coefficient plot of just
               these higher-level detail coefficients and shows them scaled uniformly. 
            


 
      call coefficientPlot(decomp, ,7, , 
                           'uniform',"Quartz Spectrum"); 



The plot is shown in Figure 20.6. 
            

            
  

            


Figure 20.6: Uniformly Scaled Detail Coefficients

               

                  
Uniformly Scaled Detail Coefficients

               

            

            


            
As noted earlier, noise in the data is captured in the detail coefficients, particularly in the small coefficients at higher
               levels in the decomposition. By zeroing or shrinking these coefficients, you can get smoother reconstructions of the input
               data. This is done by specifying a threshold value for each level of detail coefficients and then zeroing or shrinking all
               the detail coefficients below this threshold value. The IML wavelet functions and modules support several policies for how
               this thresholding is performed as well as for selecting the thresholding value at each level. See the section WAVIFT Call for details. 
            

            
An options vector is used to specify the desired thresholding; several standard choices are predefined as macro variables
               in the WAVINIT module. The following statements produce the detail coefficient plot with the "SureShrink" thresholding algorithm
               of Donoho and Johnstone (1995). 
            


 
      call coefficientPlot(decomp,&SureShrink,6,, , 
                           "Quartz Spectrum"); 



The plot is shown in Figure 20.7. 
            

            
  

            


Figure 20.7: Thresholded Detail Coefficients

               

                  
Thresholded Detail Coefficients

               

            

            


            
You can see that "SureShrink" thresholding has zeroed some of the detail coefficients at the higher levels but the larger
               coefficients that capture the oscillation in the data are still present. Consequently, reconstructions of the input signal
               using the thresholded detail coefficients still capture the essential features of the data, but are smoother because much
               of the very fine scale detail has been eliminated.