Wavelets Earl F. Glynn Scientific Programmer Bioinformatics 9 July 2003 1 Wavelets History Background Denoising Yeast Cohesins Binding Data Possible Applications of Wavelets Wavelet References and Resources 2 1
Wavelets: History Fourier analysis (~1807): classic, but limited, tool for analyzing signals (problems with spikes, transients) Wavelets have ~15 distinct roots that trace back to 1930s Denis Gabor: windowed Fourier transform (1946) Jean Morlet: cycle-octave transform (1975) Mallat and Meyer: mathematical structure multiresolution analysis (1987) Sweldens: lifting technique (1994) ( 2 nd generation wavelets used in JPEG2000, MPEG-4) 3 Fourier Analysis Wavelets: History Windowed Fourier Wavelets www.maths.leeds.ac.uk/~kisilv/courses/wavelets-wt4-20.png www.science.org.au/nova/029/029img/wave1.gif www.engineering.uiowa.edu/~ahod/aip/hw2/hw2_files/image011.gif www.cerm.unifi.it/eucourse2001/gunther_lecturenotes.pdf 4 2
Wavelets: History Fourier Windowed-Fourier Wavelet Wavelets chop up data into frequency components, and analyze each frequency component with a resolution matched to its scale. From www.cerm.unifi.it/eucourse2001/gunther_lecturenotes.pdf, p. 10 5 Wavelets: History From ATMS 552 Notes, by D. L. Hartmann, p. 204 6 3
Fourier Transform Background Time Series Fourier Transform Filtered Series Remove High Frequency Component Decomposition 0 Reconstruction www.vectorsite.net/ttdsp1.html selected 7 Wavelet Transform Background Data Series Wavelet Transform Filtered Data Decomposition Reconstruction Mother Wavelet Coefficients measure the variations of the field f(t) about the point b, with the scale given by a. From ATMS 552 Notes, by D. L. Hartmann, p. 204 8 4
Wavelet Transform Background Mother Wavelet Dilations and translations of mother wavelet define an orthogonal basis set. Any function may be represented as a linear combination of these basis functions. Haar wavelet From ATMS 552 Notes, by D. L. Hartmann, p. 204 and Wavelets for Kids 9 Continuous Wavelet Transform User s Guide MatLab Wavelet Toolbox P 1-13 10 5
Discrete Wavelet Transform Choose scales and positions based on powers of two, i.e., dyadic scales and positions, for efficiency. Mallat developed a Discrete Wavelet Transform algorithm using filters in 1988, which is called the Fast Wavelet Transform. Approximation Detail User s Guide, MatLab Wavelet Toolbox, p. 1-19 11 Discrete Wavelet Transform User s Guide, MatLab Wavelet Toolbox, p. 1-20 12 6
Approximation / Detail Given: x and y Approximation: A = (x + y)/2 (low pass filter averaging or smoothing) Detail: D = y - x (high pass filter) Points x and y can be recovered if you know A and D: X = A D/2 Y = A + D/2 13 Approximation / Detail For Haar Wavelet: 14 7
Discrete Wavelet Transform User s Guide, MatLab Wavelet Toolbox, p. 1-21 and 2-33 15 Denoising Yeast Cohesins Binding Data Sample Cohesins Binding Data for Yeast Chromosome III from Jennifer Gerton s Lab 16 8
Denoising of Yeast Cohesins Binding Data How many decomposition levels? How much detail to keep? Type of thresholding to use? Which wavelet basis set? 17 Denoising of Yeast Cohesins Binding Data 18 9
Denoising of Yeast Cohesins Binding Data 19 Denoising of Yeast Cohesins Binding Data 20 10
Controlled Experiment 21 Controlled Experiment 22 11
Controlled Experiment 23 Controlled Experiment 24 12
Controlled Experiment 25 Conclusions Wavelet denoising seems to be a powerful technique Several aspects of wavelets need further study - exact algorithms used in Matlab analysis? - choice of wavelet basis function? - levels of decomposition (statistical test? entropy?) - thresholding options: hard vs. soft? - statistics behind detail noise to remove? Choice of wavelet basis function affects compactness of wavelet decomposition 26 13
Wavelets: Possible Applications Revolutionary or just another analysis tool? Data Approximation Denoising Data compresssion Time-frequency analysis Image analysis Detecting patterns in DNA sequences Protein structure investigation Microarray data analysis Modeling biological microstruture 27 Wavelet References and Resources [1] Fischer, Patrick, et al, WavePred: A Wavelet-Based Algorithm for the Prediction of Transmembrane Proteins, Comm. Math. Sci., Vol 1, No. 1, 2003, pp. 44-56 [2] Graps, Amara, An Introduction to Wavelets, IEEE Computational Science and Engineering, Summer 1995, Vol 2, No. 2. www.amara.com/ieeewave/ieeewavelet.html. Also, Amara s Wavelet Page: www.amara.com/current/wavelet.html [3] Hubbard, Barbara B., The World According to Wavelets: The Story of a Mathematical Technique in the Making (2 nd ed), Natick, MA: A K Peters, 1998. [4] Liò, Pietro, Wavelets in bioinformatics and computational biology: state of art and perspectives, Bioinformatics, Vol 19, No. 1, 2003, pp. 2-9. [5] Vidaković, Brani and Peter Müller, Wavelets for Kids A Tutorial Introduction, Duke University, 1991, unpublished. http://www.isye.gatech.edu/~brani/wp/kidsa.pdf [6] Wavelet Digest Site, www.wavelet.org (hosted by Swiss Federal Institute of Technology) 28 14