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# WAVELETS STAT 530

UW

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This 23 page Class Notes was uploaded by Providenci Mosciski Sr. on Wednesday September 9, 2015. The Class Notes belongs to STAT 530 at University of Washington taught by Staff in Fall. Since its upload, it has received 48 views. For similar materials see /class/192518/stat-530-university-of-washington in Statistics at University of Washington.

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Date Created: 09/09/15

Examples of DWT amp MODWT Analysis Overview 0 look at DWT analysis of electrocardiogram ECG data a discuss potential alignment problems with the DWT and how they are alleviated with the MODWT a look at MODWT analysis of ECG data 85 3 other time series subtidal sea level uctuations Nile River minima ocean shear measurements 0 discuss practical details choice of wavelet lter and of level J0 handling boundary conditions handling sample sizes that are not multiples of a power of 2 de nition of DWT not standardized V1171 Electrocardiogram Data I 715 I I I I I I I I I I I I 0 2 4 6 8 10 12 t seconds 0 ECG measurements X taken during normal sinus rhythm of a patient who occasionally experiences arhythmia data courtesy of Gust Bardy and Per Reinhall University of Washington a N 2048 samples collected at rate of 180 samples second ie At 1180 second a 1138 seconds of data in all a time of X0 taken to be 250 031 merely for plotting purposes V1172 Electrocardiogram Data 05 705 715 I I I I 6 8 t seconds 0 D 4 0 features include II baseline drift not directly related to heart intermittent high frequency uctuations again not directly related to heart PQRST portion of normal heart rhythm 0 provides useful illustration of wavelet analysis because there are identi able features on several scales V1173 Electrocardiogram Data III W1 W2 W3 W4 V6 Hear a partial DWT coef cients W of level J0 6 for ECG time series using the Haar D4 36 and LA8 wavelets top to bottom V1174 Electrocardiogram Data IV 0 elements Wn of W are plotted versus n 07 N 1 2047 0 vertical dotted lines delineate 7 subvectors W1 W6 85 V6 0 sum of squares of 2048 coef cients W is equal to those of X a gross pattern of coef cients similar for all four wavelets Vl175 Electrocardiogram Data V quotIll WW IllI39IIII7II T BWG 39 I 39 73W5 quot 7 3W4 7 3W3 T2W2 T2W1 t seconds a LAlt8gt DWT coef cients stacked by scale and aligned with time a spacing between major tick marks is the same in both plots V1176 Electrocardiogram Data VI 0 R waves aligned with spikes in W2 and W3 0 intermittent uctuations appear mainly in W1 and W2 0 setting J0 6 results in V6 capturing baseline drift V1177 Electrocardiogram Data VII 0 to quantify how well various DWT s summarize X can form normalized partial energy sequences NPESs a given U75 t 07 N 1 square and order such that 2 2 o o o 2 2 U ZWDZ ZWM2gtWMm o U30 is largest of all the U152 values while U 2N1 is the smallest o NPES for U75 de ned as n 2 WW0quot 0 1 On N 127 ZmOUmgt V1178 Electrocardiogram Data VIII 0 plots show NPESs for original time series dashed curve plot a Haar DWT solid curves both plots D4 DWT dashed curve plot b LAlt8gt is Virtually iden tical DFT dotted curve plot a with U752 rather than U752 007 I07 0 1024 2048 n V1179 Electrocardiogram Data IX 56 D4 D3 D2 D1 15 05 X 705 715 I 0 2 4 6 8 10 12 t seconds o Haar DWT multiresolution analysis of ECG time series a blocky nature of Haar basis vectors readily apparent V11710 Electrocardiogram Data X 15 05 705 715 I I 0 2 4 6 13 seconds a D4 DWT multiresolution analysis 0 Cshark s n evident in D5 and D6 V1171 1 56 D4 D3 D2 731 Electrocardiogram Data XI 705 715 u 13 seconds 0 06 DWT multiresolution analysis a pyramids evident in D6 V1171 2 56 D4 D3 D2 731 Electrocardiogram Data XII 56 D4 D3 D2 Dr 15 05 705 X 715 I I I I I I 0 2 4 6 8 10 12 t seconds a LAlt8gt DWT MRA shape of lter less prominent here a note where features end up will nd MODWT does better V1171 Effect of Circular Shifts on DWT I X 75X 75X 7 2W2 quotIquot l 39II I I D2 T ZW 1 l l I l I l I D1 0 6394 12839 0 6394 12839 0 6394 12839 0 6394 12839 a bottom row bump X and bump shifted to right by 5 units 0 J0 4 LAlt8gt DWTs rst 2 columns and MRAs last 2 V11714 Effect of Circular Shifts on DWT II W44 W45 W45 W47 41 min AW 1W M VAL 1A7 LA7 0 64 128 0 64 128 0 64 128 0 64 128 t t t t a level J0 4 basis vectors used in LAlt8gt DWT to produce wavelet coefficients W4jj j 47 7 black curves 0 bump time series X blue curves in top row of plots 0 shifted bump series 75X red curves bottom row a inner product between plotted basis vector and time series yields labeled wavelet coefficient a alignment between basis vectors and time series explains why DWTs for two series are quite different V11715 Effect of Circular Shifts 0n MODWT r l vAv l vAv l vnv I will 7 4W1 l1 l1 l1 l1 151 0 64 128 0 64 128 0 64 128 0 64 128 t t t o unlike the DWT shifting a time series shifts the MODWT CO ef eients and components of MBA V11716 Electrocardiogram Data XIII 7 189 26 722211716 7 109 5 39 53VV4 N T 25WB 741692 7 4661 0 2 4 6 8 10 12 t seconds a level J0 6 LAlt8gt MODWT with Wj s circularly shifted a vertical lines delineate boundary coef cients explained later V1171 Electrocardiogram Data XIV Ill IIIIIIII39IllllI 73W6 15 R 705 715 0 2 4 6 8 10 12 t seconds a comparison of level 6 MODWT and DWT wavelet coefficients after shifting for time alignment a boundary coef cients delineated by vertical red lines o subsampling amp rescaling W6 yields W6 note aliasing effect Vl1718 Electrocardiogram Data XV 0 2 4 6 8 10 12 t seconds a LAlt8gt MODWT multiresolution analysis of ECG data V11719 Electrocardiogram Data XVI t seconds 0 MODWT details seem more consistent across time than DWT details cg D6 does not fade in and out as much as D6 0 bumps in D6 are slightly asymmetric whereas those in 156 aren t Vl I72 0 Electrocardiogram Data XVII M g6 I I I I I ML 76 15 I I I I I I R SiZiWWMW x 0 2 4 6 8 10 12 t seconds o MODWT coef cients and MBA resemble each other with lat ter being necessarily smoother due to second round of ltering a in the above g6 is somewhat smoother than 76 and is an intuitively reasonable estimate of the baseline drift Vl I721 Subtidal Sea Level Fluctuations I 720 5 5 5 5 5 5 l 740 i i i i 1980 1984 i i i 1988 1991 years a subtidal sea level uctuations X for Crescent City CA col lected by National Ocean Service with permanent tidal gauge a N 8746 values from Jan 1980 to Dec 1991 almost 12 years a one value every 12 hours so At 1 2 day a subtidal is what remains after diurnal amp semidiurnal tides are removed by low pass lter lter seriously distorts frequency band corresponding to rst physical scale 71 At 12 day Vlli22

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