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CMU SCS 15-826: Multimedia Databases and Data Mining Lecture #23: DSP tools Fourier and Wavelets C. Faloutsos Slide 2 CMU SCS 15-826Copyright: C. Faloutsos (2012)2 Must-read Material DFT/DCT: In PTVF ch. 12.1, 12.3, 12.4; in Textbook Appendix B.PTVF Textbook Wavelets: In PTVF ch. 13.10; in MM Textbook Appendix CPTVFMM Textbook Slide 3 CMU SCS 15-826Copyright: C. Faloutsos (2012)3 Outline Goal: Find similar / interesting things Intro to DB Indexing - similarity search Data Mining Slide 4 CMU SCS 15-826Copyright: C. Faloutsos (2012)4 Indexing - Detailed outline primary key indexing.. multimedia Digital Signal Processing (DSP) tools Discrete Fourier Transform (DFT) Discrete Wavelet Transform (DWT) Slide 5 CMU SCS 15-826Copyright: C. Faloutsos (2012)5 DSP - Detailed outline DFT what why how Arithmetic examples properties / observations DCT 2-d DFT Fast Fourier Transform (FFT) Slide 6 CMU SCS 15-826Copyright: C. Faloutsos (2012)6 Introduction Goal: given a signal (eg., sales over time and/or space) Find: patterns and/or compress year count lynx caught per year Slide 7 CMU SCS 15-826Copyright: C. Faloutsos (2012)7 What does DFT do? A: highlights the periodicities Slide 8 CMU SCS 15-826Copyright: C. Faloutsos (2012)8 Why should we care? A: several real sequences are periodic Q: Such as? Slide 9 CMU SCS 15-826Copyright: C. Faloutsos (2012)9 Why should we care? A: several real sequences are periodic Q: Such as? A: sales patterns follow seasons; economy follows 50-year cycle temperature follows daily and yearly cycles Many real signals follow (multiple) cycles Slide 10 CMU SCS 15-826Copyright: C. Faloutsos (2012)10 Why should we care? For example: human voice! Frequency analyzer http://www.relisoft.com/freeware/freq.html http://www.relisoft.com/freeware/freq.html speaker identification impulses/noise -> flat spectrum high pitch -> high frequency Freq.exe Slide 11 CMU SCS 15-826Copyright: C. Faloutsos (2012)11 Frequency Analyzer Slide 12 CMU SCS 15-826Copyright: C. Faloutsos (2012)12 DFT and stocks Dow Jones Industrial index, 6/18/2001- 12/21/2001 Slide 13 CMU SCS 15-826Copyright: C. Faloutsos (2012)13 DFT and stocks Dow Jones Industrial index, 6/18/2001- 12/21/2001 just 3 DFT coefficients give very good approximation freq Log(ampl) Slide 14 CMU SCS 15-826Copyright: C. Faloutsos (2012)14 DFT: definition Discrete Fourier Transform (n-point): inverse DFT Slide 15 CMU SCS 15-826Copyright: C. Faloutsos (2012)15 How does it work? Decomposes signal to a sum of sine (and cosine) waves. Q:How to assess similarity of x with a wave? 0 1 n-1 time value x ={x 0, x 1,... x n-1 } Skip Slide 16 CMU SCS 15-826Copyright: C. Faloutsos (2012)16 How does it work? A: consider the waves with frequency 0, 1,...; use the inner-product (~cosine similarity) 0 1 n-1 time value freq. f=0 0 1 n-1 time value freq. f=1 (sin(t * 2 n) ) Skip Slide 17 CMU SCS 15-826Copyright: C. Faloutsos (2012)17 How does it work? A: consider the waves with frequency 0, 1,...; use the inner-product (~cosine similarity) 0 1 n-1 time value freq. f=2 Skip Slide 18 CMU SCS 15-826Copyright: C. Faloutsos (2012)18 How does it work? basis functions 0 1 n-1 0 1 0 1 sine, freq =1 sine, freq = 2 0 1 n-1 0 1 cosine, f=1 cosine, f=2 Skip Slide 19 CMU SCS 15-826Copyright: C. Faloutsos (2012)19 How does it work? Basis functions are actually n-dim vectors, orthogonal to each other similarity of x with each of them: inner product DFT: ~ all the similarities of x with the basis functions Skip Slide 20 CMU SCS 15-826Copyright: C. Faloutsos (2012)20 How does it work? Since e jf = cos(f) + j sin(f) (j=sqrt(-1)), we finally have: Skip Slide 21 CMU SCS 15-826Copyright: C. Faloutsos (2012)21 DFT: definition Discrete Fourier Transform (n-point): inverse DFT Slide 22 CMU SCS 15-826Copyright: C. Faloutsos (2012)22 DFT: definition Good news: Available in all symbolic math packages, eg., in mathematica x = [1,2,1,2]; X = Fourier[x]; Plot[ Abs[X] ]; Slide 23 CMU SCS 15-826Copyright: C. Faloutsos (2012)23 DFT: definition (variations: 1/n instead of 1/sqrt(n) exp(-...) instead of exp(+...) Slide 24 CMU SCS 15-826Copyright: C. Faloutsos (2012)24 DFT: definition (variations: or even Slide 25 CMU SCS 15-826Copyright: C. Faloutsos (2012)25 DFT: definition Observations: X f : are complex numbers except X 0, who is real Im (X f ): ~ amplitude of sine wave of frequency f Re (X f ): ~ amplitude of cosine wave of frequency f x: is the sum of the above sine/cosine waves Slide 26 CMU SCS 15-826Copyright: C. Faloutsos (2012)26 DFT: definition Observation - SYMMETRY property: X f = (X n-f )* ( *: complex conjugate: (a + b j)* = a - b j ) Skip Slide 27 CMU SCS 15-826Copyright: C. Faloutsos (2012)27 DFT: definition Definitions A f = |X f | : amplitude of frequency f |X f | 2 = Re(X f ) 2 + Im(X f ) 2 = energy of frequency f phase f at frequency f XfXf Re Im AfAf ff Slide 28 CMU SCS 15-826Copyright: C. Faloutsos (2012)28 DFT: definition Amplitude spectrum: | X f | vs f (f=0, 1,... n-1) 0 1n-1 SYMMETRIC (Thus, we plot the first half only) Skip Slide 29 CMU SCS 15-826Copyright: C. Faloutsos (2012)29 DFT: definition Phase spectrum | f | vs f (f=0, 1,... n-1): Anti-symmetric (Rarely used) Skip Slide 30 CMU SCS 15-826Copyright: C. Faloutsos (2012)30 DFT: examples flat time freq Amplitude Slide 31 CMU SCS 15-826Copyright: C. Faloutsos (2012)31 DFT: examples Low frequency sinusoid time freq Slide 32 CMU SCS 15-826Copyright: C. Faloutsos (2012)32 DFT: examples Sinusoid - symmetry property: X f = X * n-f time freq Slide 33 CMU SCS 15-826Copyright: C. Faloutsos (2012)33 DFT: examples Higher freq. sinusoid time freq Slide 34 CMU SCS 15-826Copyright: C. Faloutsos (2012)34 DFT: examples examples = + + Slide 35 CMU SCS 15-826Copyright: C. Faloutsos (2012)35 DFT: examples examples Freq. Ampl. Slide 36 CMU SCS 15-826Copyright: C. Faloutsos (2012)36 DFT: Amplitude spectrum year count Freq. Ampl. freq=12 freq=0 Amplitude: Slide 37 CMU SCS 15-826Copyright: C. Faloutsos (2012)37 DFT: Amplitude spectrum year count Freq. Ampl. freq=12 freq=0 Slide 38 CMU SCS 15-826Copyright: C. Faloutsos (2012)38 DFT: Amplitude spectrum year count Freq. Ampl. freq=12 freq=0 Slide 39 CMU SCS 15-826Copyright: C. Faloutsos (2012)39 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? Freq. Slide 40 CMU SCS 15-826Copyright: C. Faloutsos (2012)40 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? A1: compression A2: pattern discovery (A3: forecasting) Slide 41 CMU SCS 15-826Copyright: C. Faloutsos (2012)41 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? A1: (lossy) compression A2: pattern discovery Slide 42 CMU SCS 15-826Copyright: C. Faloutsos (2012)42 DFT: Amplitude spectrum excellent approximation, with only 2 frequencies! so what? A1: (lossy) compression A2: pattern discovery Slide 43 CMU SCS 15-826Copyright: C. Faloutsos (2012)43 DFT: Amplitude spectrum Lets see it in action (defunct now) (http://www.dsptutor.freeuk.com/jsanalyser/FFTSpectrumAnalyser.html)(http://www.dsptutor.freeuk.com/jsanalyser/FFTSpectrumAnalyser.html plain sine phase shift two sine waves the chirp function http://ion.researchsystems.com/ Slide 44 CMU SCS 15-826Copyright: C. Faloutsos (2012)44 Plain sine Slide 45 CMU SCS 15-826Copyright: C. Faloutsos (2012)45 Plain sine Slide 46 CMU SCS 15-826Copyright: C. Faloutsos (2012)46 Plain sine phase shift Slide 47 CMU SCS 15-826Copyright: C. Faloutsos (2012)47 Plain sine phase shift Slide 48 CMU SCS 15-826Copyright: C. Faloutsos (2012)48 Plain sine Slide 49 CMU SCS 15-826Copyright: C. Faloutsos (2012)49 Two sines Slide 50 CMU SCS 15-826Copyright: C. Faloutsos (2012)50 Two sines Slide 51 CMU SCS 15-826Copyright: C. Faloutsos (2012)51 Chirp Slide 52 CMU SCS 15-826Copyright: C. Faloutsos (2012)52 Chirp Slide 53 CMU SCS 15-826Copyright: C. Faloutsos (2012)53 Another applet http://www.falstad.com/fourier/ (seems virus-free but scan, before you install) FFT_applet/index.html Slide 54 CMU SCS Properties Time shift sounds the same Changes only phase, not amplitudes Sawtooth has almost all frequencies With decreasing amplitude Spike has all frequencies 15-826Copyright: C. Faloutsos (2012)54 Slide 55 CMU SCS 15-826Copyright: C. Faloutsos (2012)55 DFT: Parsevals theorem sum( x t 2 ) = sum ( | X f | 2 ) Ie., DFT preserves the energy or, alternatively: it does an axis rotation: x0 x1 x = {x0, x1} Slide 56 CMU SCS 15-826Copyright: C. Faloutsos (2012)56 DFT: Parsevals theorem sum( x t 2 ) = sum ( | X f | 2 ) Ie., DFT preserves the energy or, alternatively: it does an axis rotation: x0 x1 x = {x0, x1} Slide 57 CMU SCS 15-826Copyright: C. Faloutsos (2012)57 DSP - Detailed outline DFT what why how Arithmetic examples properties / observations DCT 2-d DFT Fast Fourier Transform (FFT) Slide 58 CMU SCS 15-826Copyright: C. Faloutsos (2012)58 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? 0 1 time value 1 Skip Slide 59 CMU SCS 15-826Copyright: C. Faloutsos (2012)59 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2 0 / n ) = 1/2 X 1 =? X 2 =? X 3 =? Skip Slide 60 CMU SCS 15-826Copyright: C. Faloutsos (2012)60 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2 0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: does the symmetry property hold? Skip Slide 61 CMU SCS 15-826Copyright: C. Faloutsos (2012)61 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2 0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: does the symmetry property hold? A: Yes (of course) Skip Slide 62 CMU SCS 15-826Copyright: C. Faloutsos (2012)62 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2 0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: check Parsevals theorem Skip Slide 63 CMU SCS 15-826Copyright: C. Faloutsos (2012)63 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2 0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: (Amplitude) spectrum? Skip Slide 64 CMU SCS 15-826Copyright: C. Faloutsos (2012)64 Arithmetic examples Impulse function: x = { 0, 1, 0, 0} (n = 4) X 0 =? A: X 0 = 1/sqrt(4) * 1* exp(-j 2 0 / n ) = 1/2 X 1 = -1/2 j X 2 = - 1/2 X 3 = + 1/2 j Q: (Amplitude) spectrum? A: FLAT! Slide 65 CMU SCS 15-826Copyright: C. Faloutsos (2012)65 Arithmetic examples Q: What does this mean? Slide 66 CMU SCS 15-826Copyright: C. Faloutsos (2012)66 Arithmetic examples Q: What does this mean? A: All frequencies are equally important -> we need n numbers in the frequency domain to represent just one non-zero number in the time domain! frequency leak Slide 67 CMU SCS 15-826Copyright: C. Faloutsos (2012)67 DSP - Detailed outline DFT what why how Arithmetic examples properties / observations DCT 2-d DFT Fast Fourier Transform (FFT) Slide 68 CMU SCS 15-826Copyright: C. Faloutsos (2012)68 Observations DFT of step function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt Slide 69 CMU SCS 15-826Copyright: C. Faloutsos (2012)69 Observations DFT of step function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt f = 0 Slide 70 CMU SCS 15-826Copyright: C. Faloutsos (2012)70 Observations DFT of step function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt f = 0 f=1 Slide 71 CMU SCS 15-826Copyright: C. Faloutsos (2012)71 Observations DFT of step function: x = { 0, 0,..., 0, 1, 1,... 1} t xtxt f = 0 f=1 the more frequencies, the better the approx. ringing becomes worse reason: discontinuities; trends Slide 72 CMU SCS 15-826Copyright: C. Faloutsos (2012)72 Observations Ringing for trends: because DFT sub- consciously replicates the signal t xtxt Slide 73 CMU SCS 15-826Copyright: C. Faloutsos (2012)73 Observations Ringing for trends: because DFT sub- consciously replicates the signal t xtxt Slide 74 CMU SCS 15-826Copyright: C. Faloutsos (2012)74 Observations Ringing for trends: because DFT sub- consciously replicates the signal t xtxt Slide 75 CMU SCS 15-826Copyright: C. Faloutsos (2012)75 original Slide 76 CMU SCS 15-826Copyright: C. Faloutsos (2012)76 DC and 1st Slide 77 CMU SCS 15-826Copyright: C. Faloutsos (2012)77 And 2nd DC and 1st Slide 78 CMU SCS 15-826Copyright: C. Faloutsos (2012)78 And 3rd DC and 1st And 2nd Slide 79 CMU SCS 15-826Copyright: C. Faloutsos (2012)79 And 4th DC and 1st And 3rd And 2nd Slide 80 CMU SCS 15-826Copyright: C. Faloutsos (2012)80 Observations Q: DFT of a sinusoid, eg. x t = 3 sin( 2 / 4 t) (t = 0,..., 3) Q: X 0 = ? Q: X 1 = ? Q: X 2 = ? Q: X 3 = ? Slide 81 CMU SCS 15-826Copyright: C. Faloutsos (2012)81 Observations Q: DFT of a sinusoid, eg. x t = 3 sin( 2 / 4 t) (t = 0,..., 3) Q: X 0 = 0 Q: X 1 = -3 j Q: X 2 = 0 Q: X 3 = 3j check symmetry check Parseval Slide 82 CMU SCS 15-826Copyright: C. Faloutsos (2012)82 Q: DFT of a sinusoid, eg. x t = 3 sin( 2 / 4 t) (t = 0,..., 3) Q: X 0 = 0 Q: X 1 = -3 j Q: X 2 = 0 Q: X 3 = 3j Observations Does this make sense? f AfAf 0 1 2 Slide 83 CMU SCS 15-826Copyright: C. Faloutsos (2012)83 Shifting x in time does NOT change the amplitude spectrum eg., x = { 0 0 0 1} and x = { 0 1 0 0 }: same (flat) amplitude spectrum (only the phase spectrum changes) Useful property when we search for patterns that may slide Property Slide 84 CMU SCS 15-826Copyright: C. Faloutsos (2012)84 DSP - Detailed outline DFT what why how Arithmetic examples properties / observations DCT 2-d DFT Fast Fourier Transform (FFT) Slide 85 CMU SCS 15-826Copyright: C. Faloutsos (2012)85 DCT Discrete Cosine Transform motivation#1: DFT gives complex numbers motivation#2: how to avoid the frequency leak of DFT on trends? t xtxt Skip Slide 86 CMU SCS 15-826Copyright: C. Faloutsos (2012)86 DCT brilliant solution to both problems: mirror the sequence, do DFT, and drop the redundant entries! t xtxt Skip Slide 87 CMU SCS 15-826Copyright: C. Faloutsos (2012)87 DCT (see Numerical Recipes for exact formulas) Skip Slide 88 CMU SCS 15-826Copyright: C. Faloutsos (2012)88 DCT - properties it gives real numbers as the result it has no problems with trends it is very good when x t and x (t+1) are correlated (thus, is used in JPEG, for image compression) Slide 89 CMU SCS 15-826Copyright: C. Faloutsos (2012)89 DSP - Detailed outline DFT what why how Arithmetic examples properties / observations DCT 2-d DFT Fast Fourier Transform (FFT) Slide 90 CMU SCS 15-826Copyright: C. Faloutsos (2012)90 2-d DFT Definition: Slide 91 CMU SCS 15-826Copyright: C. Faloutsos (2012)91 2-d DFT Intuition: n1 n2 do 1-d DFT on each row and then 1-d DFT on each column Slide 92 CMU SCS 15-826Copyright: C. Faloutsos (2012)92 2-d DFT Quiz: how do the basis functions look like? for f1= f2 =0 for f1=1, f2=0 for f1=1, f2=1 Slide 93 CMU SCS 15-826Copyright: C. Faloutsos (2012)93 2-d DFT Quiz: how do the basis functions look like? for f1= f2 =0 flat for f1=1, f2=0wave on x; flat on y for f1=1, f2=1~ egg-carton Slide 94 CMU SCS 15-826Copyright: C. Faloutsos (2012)94 2-d DFT Quiz: how do the basis functions look like? for f1= f2 =0 flat for f1=1, f2=0wave on x; flat on y for f1=1, f2=1~ egg-carton Slide 95 CMU SCS 15-826Copyright: C. Faloutsos (2012)95 DSP - Detailed outline DFT what why how Arithmetic examples properties / observations DCT 2-d DFT Fast Fourier Transform (FFT) Slide 96 CMU SCS 15-826Copyright: C. Faloutsos (2012)96 FFT What is the complexity of DFT? Skip Slide 97 CMU SCS 15-826Copyright: C. Faloutsos (2012)97 FFT What is the complexity of DFT? A: Naively, O(n 2 ) Skip Slide 98 CMU SCS 15-826Copyright: C. Faloutsos (2012)98 FFT However, if n is a power of 2 (or a number with many divisors), we can make it O(n log n) Main idea: if we know the DFT of the odd time-ticks, and of the even time-ticks, we can quickly compute the whole DFT Details: in Num. Recipes Skip Slide 99 CMU SCS 15-826Copyright: C. Faloutsos (2012)99 DFT - Conclusions It spots periodicities (with the amplitude spectrum) can be quickly computed (O( n log n)), thanks to the FFT algorithm. standard tool in signal processing (speech, image etc signals) Freq. Slide 100 CMU SCS 15-826Copyright: C. Faloutsos (2012)100 Detailed outline primary key indexing.. multimedia Digital Signal Processing (DSP) tools Discrete Fourier Transform (DFT) Discrete Wavelet Transform (DWT) Slide 101 CMU SCS 15-826Copyright: C. Faloutsos (2012)101 Reminder: Problem: Goal: given a signal (eg., #packets over time) Find: patterns, periodicities, and/or compress year count lynx caught per year (packets per day; virus infections per month) Slide 102 CMU SCS 15-826Copyright: C. Faloutsos (2012)102 Wavelets - DWT DFT is great - but, how about compressing a spike? value time Slide 103 CMU SCS 15-826Copyright: C. Faloutsos (2012)103 Wavelets - DWT DFT is great - but, how about compressing a spike? A: Terrible - all DFT coefficients needed! Freq. Ampl. value time Slide 104 CMU SCS 15-826Copyright: C. Faloutsos (2012)104 Wavelets - DWT DFT is great - but, how about compressing a spike? A: Terrible - all DFT coefficients needed! value time Slide 105 CMU SCS 15-826Copyright: C. Faloutsos (2012)105 Wavelets - DWT Similarly, DFT suffers on short-duration waves (eg., baritone, silence, soprano) time value Slide 106 CMU SCS 15-826Copyright: C. Faloutsos (2012)106 Wavelets - DWT Solution#1: Short window Fourier transform (SWFT) But: how short should be the window? time freq time value Slide 107 CMU SCS 15-826Copyright: C. Faloutsos (2012)107 Wavelets - DWT Answer: multiple window sizes! -> DWT time freq Time domain DFT SWFT DWT Slide 108 CMU SCS 15-826Copyright: C. Faloutsos (2012)108 Haar Wavelets subtract sum of left half from right half repeat recursively for quarters, eight-ths,... Slide 109 CMU SCS 15-826Copyright: C. Faloutsos (2012)109 Wavelets - construction x0 x1 x2 x3 x4 x5 x6 x7 Slide 110 CMU SCS 15-826Copyright: C. Faloutsos (2012)110 Wavelets - construction x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... level 1 Slide 111 CMU SCS 15-826Copyright: C. Faloutsos (2012)111 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 level 2 Slide 112 CMU SCS 15-826Copyright: C. Faloutsos (2012)112 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 etc... Slide 113 CMU SCS 15-826Copyright: C. Faloutsos (2012)113 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 Q: map each coefficient on the time-freq. plane t f Slide 114 CMU SCS 15-826Copyright: C. Faloutsos (2012)114 Wavelets - construction d2,0 x0 x1 x2 x3 x4 x5 x6 x7 s1,0 + - d1,0 s1,1d1,1....... s2,0 Q: map each coefficient on the time-freq. plane t f Slide 115 CMU SCS 15-826Copyright: C. Faloutsos (2012)115 Haar wavelets - code #!/usr/bin/perl5 # expects a file with numbers # and prints the dwt transform # The number of time-ticks should be a power of 2 # USAGE # haar.pl my @vals=(); my @smooth; # the smooth component of the signal my @diff; # the high-freq. component # collect the values into the array @val while(){ @vals = ( @vals, split ); } my $len = scalar(@vals); my $half = int($len/2); while($half >= 1 ){ for(my $i=0; $i< $half; $i++){ $diff [$i] = ($vals[2*$i] - $vals[2*$i + 1] )/ sqrt(2); print "\t", $diff[$i]; $smooth [$i] = ($vals[2*$i] + $vals[2*$i + 1] )/ sqrt(2); } print "\n"; @vals = @smooth; $half = int($half/2); } print "\t", $vals[0], "\n" ; # the final, smooth component Slide 116 CMU SCS 15-826Copyright: C. Faloutsos (2012)116 Wavelets - construction Observation1: + can be some weighted addition - is the corresponding weighted difference (Quadrature mirror filters) Observation2: unlike DFT/DCT, there are *many* wavelet bases: Haar, Daubechies- 4, Daubechies-6, Coifman, Morlet, Gabor,... Slide 117 CMU SCS 15-826Copyright: C. Faloutsos (2012)117 Wavelets - how do they look like? E.g., Daubechies-4 Slide 118 CMU SCS 15-826Copyright: C. Faloutsos (2012)118 Wavelets - how do they look like? E.g., Daubechies-4 ? ? Slide 119 CMU SCS 15-826Copyright: C. Faloutsos (2012)119 Wavelets - how do they look like? E.g., Daubechies-4 Slide 120 CMU SCS 15-826Copyright: C. Faloutsos (2012)120 Wavelets - Drill#1: Q: baritone/silence/soprano - DWT? t f time value Slide 121 CMU SCS 15-826Copyright: C. Faloutsos (2012)121 Wavelets - Drill#1: t f time value Q: baritone/silence/soprano - DWT? Slide 122 CMU SCS 15-826Copyright: C. Faloutsos (2012)122 Wavelets - Drill#2: Q: spike - DWT? t f Slide 123 CMU SCS 15-826Copyright: C. Faloutsos (2012)123 Wavelets - Drill#2: t f Q: spike - DWT? 0.00 0.00 0.71 0.00 0.00 0.50 -0.35 0.35 Slide 124 CMU SCS 15-826Copyright: C. Faloutsos (2012)124 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f Slide 125 CMU SCS 15-826Copyright: C. Faloutsos (2012)125 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f Slide 126 CMU SCS 15-826Copyright: C. Faloutsos (2012)126 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f Slide 127 CMU SCS 15-826Copyright: C. Faloutsos (2012)127 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f Slide 128 CMU SCS 15-826Copyright: C. Faloutsos (2012)128 Wavelets - Drill#3: Q: weekly + daily periodicity, + spike - DWT? t f Slide 129 CMU SCS 15-826Copyright: C. Faloutsos (2012)129 Wavelets - Drill#3: Q: DFT? t f t f DWT DFT Slide 130 CMU SCS 15-826Copyright: C. Faloutsos (2012)130 Wavelets - Drill: Lets see it live: http://dsp.rice.edu/software/dsp-teaching-tools delta; cosine; cosine2; chirp Haar vs Daubechies-4, -6, etc Slide 131 CMU SCS 15-826Copyright: C. Faloutsos (2012)131 Delta? x(0)=1; x(t)= 0 elsewhere t f Slide 132 CMU SCS 15-826Copyright: C. Faloutsos (2012)132 Delta? x(0)=1; x(t)= 0 elsewhere t f Slide 133 CMU SCS 15-826Copyright: C. Faloutsos (2012)133 2 cosines? x(t) = cos( 2 * pi * 4 * t / 1024 ) + 5 * cos( 2 * pi * 8 * t / 1024 ) t f Slide 134 CMU SCS 15-826Copyright: C. Faloutsos (2012)134 2 cosines? x(t) = cos( 2 * pi * 4 * t / 1024 ) + 5 * cos( 2 * pi * 8 * t / 1024 ) t f Which one is for freq.=4? Slide 135 CMU SCS 15-826Copyright: C. Faloutsos (2012)135 2 cosines? x(t) = cos( 2 * pi * 4 * t / 1024 ) + 5 * cos( 2 * pi * 8 * t / 1024 ) t f Which one is for freq.=4? f~4 f~8 Slide 136 CMU SCS 15-826Copyright: C. Faloutsos (2012)136 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f Slide 137 CMU SCS 15-826Copyright: C. Faloutsos (2012)137 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f Slide 138 CMU SCS 15-826Copyright: C. Faloutsos (2012)138 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f SWFT? Slide 139 CMU SCS 15-826Copyright: C. Faloutsos (2012)139 Chirp? x(t) = cos( 2 * pi * t * t / 1024 ) t f SWFT Slide 140 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)140 BGP-lens: Patterns and Anomalies in Internet Routing Updates B. Aditya Prakash et al, SIGKDD 2009 time #updates Slide 141 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)141 freq. Low freq.: omitted t f Slide 142 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)142 freq. ?? Slide 143 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)143 Prolonged spike t f freq. Slide 144 CMU SCS More examples (BGP updates) 15-826Copyright: C. Faloutsos (2012)144 freq. 15K msgs, for several hours: 6pm-4am Slide 145 CMU SCS 15-826Copyright: C. Faloutsos (2012)145 Wavelets - Drill Or use R, octave or matlab R: install.packages("wavelets") library("wavelets") X1