Wavelet Methods in Statistics with R

WhenZhouEnlai,PremierofthePeople’sRepublicofChina(1949–1976),was asked his opinion of the French Revolution (1789–1799) he replied “It’s too early to tell”, see Rosenberg (1999). I believe that the same can be said about wavelets. Although particular wavelets were discovered many years ago, the substantial body of literature that we might today call ‘wavelet theory’ began to be established during the 1980s. Wavelets were introduced into statistics during the late 1980s and early 1990s, and they were initially popular in the curve estimation literature. From there they spread in di?erent ways to many areassuchassurvivalanalysis,statisticaltimeseriesanalysis,statisticalimage processing, inverse problems, and variance stabilization. The French Revolution was also the historical backdrop for the introd- tion of Fourier series which itself raised considerable objections from the s- enti?c establishment of the day, see Westheimer (2001). Despite those early objections, we ?nd that, 200 years later, many new Fourier techniques are regularly being invented in many di?erent ?elds. Wavelets are also a true s- enti?c revolution. Some of their interesting features are easy to appreciate: e.g., multiscale, localization, or speed. Other important aspects, such as the unconditional basis property, deserve to be better known. I hope that this book, in some small way, enables the creation of many new wavelet methods.