Wavelet Analysis

(View the complete code for this example.)

Fourier transform infrared (FT-IR) spectroscopy is an important tool in analytic chemistry. The following example demonstrates wavelet analysis applied to an FT-IR spectrum of quartz (Sullivan 2000). The following DATA step creates a data set that contains the spectrum, expressed as an absorbance value for each of 850 wave numbers:

data quartzInfraredSpectrum;
   WaveNumber=4000.6167786 - _N_ *4.00084378;
   input Absorbance @@;
datalines;
  4783  4426  4419  4652  4764  4764  4621  4475  4430  4618
  4735  4735  4655  4538  4431  4714  4738  4707  4627  4523
  4512  4708  4802  4811  4769  4506  4642  4799  4811  4732
  4583  4676  4856  4868  4796  4849  4829  4677  4962  4994

   ... more lines ...   

 43341 41111 36131 35377 34431 31679 29237 26898 24655 22417
 19876 17244 15176 12575 10532  8180  6040  4059  2210   575
;

The following statements produce the line plot of these data, which is displayed in Figure 1:

proc sgplot data=quartzInfraredSpectrum;
   series x=WaveNumber y=Absorbance;
   xaxis reverse min=0;
   yaxis values=(0 to 70000 by 10000);
run;

Figure 1: FT-IR Spectrum of Quartz

These data contain information at two distinct scales, namely a high-frequency oscillation superimposed on a low-frequency curve. Notice that the oscillation is not uniform but occurs in several distinct bands. Wavelet analysis is an appropriate tool for providing insight into this type of data, because it enables you to identify the frequencies present in the absorbance data as the wave number changes. This property of wavelets is known as "time frequency localization"; in this case, the role of time is played by WaveNumber. Also note that the dependent variable Absorbance is measured at equally spaced values of the independent variable WaveNumber. This condition is necessary for the direct use of the discrete wavelet transform that is implemented in the SAS/IML wavelet functions.