Wavelet Methods for Time Series Analysis. Quick Comparison of the MODWT to the DWT

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1 Wavelet Methods for Time Series Analysis Part III: MODWT and Examples of DWT/MODWT Analysis MODWT stands for maximal overlap discrete wavelet transform (pronounced mod WT ) transforms very similar to the MODWT have been studied in the literature under the following names: undecimated DWT (or nondecimated DWT) stationary DWT translation (or time) invariant DWT redundant DWT also related to notions of wavelet frames and cycle spinning basic idea: use values removed from DWT by downsampling III Quick Comparison of the MODWT to the DWT unlike the DWT, MODWT is not orthonormal (in fact MODWT is highly redundant) unlike the DWT, MODWT is defined naturally for all samples sizes (i.e., N need not be a multiple of a power of two) similar to the DWT, can form multiresolution analyses (MAs) using MODWT, but with certain additional desirable features; e.g., unlike the DWT, MODWT-based MA has details and smooths that shift along with (if has detail Dj, then T m has detail T m Dj ) similar to the DWT, an analysis of variance (ANOVA) can be based on MODWT wavelet coefficients unlike the DWT, MODWT discrete wavelet power spectrum same for and its circular shifts T m III 2 DWT Wavelet & Scaling Filters and Coefficients recall that we obtain level j = DWT wavelet and scaling coefficients from by filtering and downsampling: H( k N ) 2 W and G( k N ) 2 transfer functions H( ) and G( ) are associated with impulse response sequences {h l } and {g l } via the usual relationships {h l } H( ) and {g l } G( ) V Level j Equivalent Wavelet & Scaling Filters for any level j, rather than using the pyramid algorithm, we could get the DWT wavelet and scaling coefficients directly from by filtering and downsampling: H j ( k N ) 2 j W j and G j ( k N ) 2 j transfer functions H j ( ) &G j ( ) depend just on H( ) &G( ) actually can say just on H( ) since G( ) depends on H( ) note that H ( ) &G ( ) are the same as H( ) &G( )) impulse response sequences {h j,l } and {g j,l } are associated with transfer functions via the usual relationships {h j,l } H j ( ) and {g j,l } G j ( ), and both filters have width L j =(2 j )(L ) + V j III 3 III 4

2 Haar Equivalent Wavelet & Scaling Filters D(4) Equivalent Wavelet & Scaling Filters {h l } {h 2,l } {h 3,l } {h 4,l } {g l } {g 2,l } {g 3,l } {g 4,l } L =2 L 2 =4 L 3 =8 L 4 =6 L =2 L 2 =4 L 3 =8 L 4 =6 {h l } {h 2,l } {h 3,l } {h 4,l } {g l } {g 2,l } {g 3,l } {g 4,l } L =4 L 2 = L 3 =22 L 4 =46 L =4 L 2 = L 3 =22 L 4 =46 L j =2 j is width of {h j,l } and {g j,l } III 5 L j dictated by general formula L j =(2 j )(L ) +, but can argue that effective width is 2 j (same as Haar L j ) III 6 D(6) Equivalent Wavelet & Scaling Filters LA(8) Equivalent Wavelet & Scaling Filters {h l } {h 2,l } {h 3,l } {h 4,l } {g l } {g 2,l } {g 3,l } {g 4,l } L =6 L 2 =6 L 3 =36 L 4 =76 L =6 L 2 =6 L 3 =36 L 4 =76 {h l } {h 2,l } {h 3,l } {h 4,l } {g l } {g 2,l } {g 3,l } {g 4,l } L =8 L 2 =22 L 3 =5 L 4 = 6 L =8 L 2 =22 L 3 =5 L 4 = 6 {h 4,l } resembles discretized version of Mexican hat wavelet III 7 {h j,l } resembles discretized version of Mexican hat wavelet, again with an effective width of 2 j III 8

3 Squared Gain Functions for Equivalent Filters squared gain functions give us frequency domain properties: H j (f) H j (f) 2 and G j (f) G j (f) 2 example: squared gain functions for LA(8) J = 4 partial DWT f G 4 ( ) H 4 ( ) H 3 ( ) H 2 ( ) H ( ) Definition of MODWT Wavelet & Scaling Filters define MODWT filters { h j,l } and { g j,l } by renormalizing the DWT filters (widths of MODWT & DWT filters are the same): h j,l = h j,l /2 j/2 and g j,l = g j,l /2 j/2 whereas DWT filters have unit energy, MODWT filters satisfy L j L j h 2 j,l = g j,l 2 = 2 j l= let H j ( ) and G j ( ) be the corresponding transfer functions: H j (f) = 2 j/2h j(f) and G j (f) = 2 j/2g j(f) so that { h j,l } H j ( ) and { g j,l } G j ( ) l= III 9 III Definition of MODWT Coefficients: I level j MODWT wavelet and scaling coefficients are defined to be output obtaining by filtering with { h j,l } and { g j,l }: H j ( N k ) W j and G j ( N k ) Ṽj compare the above to its DWT equivalent: H j ( k N ) 2 j W j and G j ( k N ) 2 j DWT and MODWT have different normalizations for filters, and there is no downsampling by 2 j in the MODWT level J MODWT consists of J + vectors, namely, W, W 2,..., W J and ṼJ, each of which has length N V j Definition of MODWT Coefficients: II MODWT of level J has (J +)N coefficients, whereas DWT has N coefficients for any given J whereas DWT of level J requires N to be integer multiple of 2 J, MODWT of level J is well-defined for any sample size N when N is divisible by 2 J, we can write W j,t = L j l= h j,l 2 j (t+) l mod N and W j,t = and we have the relationship L j l= h j,l t l mod N, W j,t =2 j/2 Wj,2 j (t+) and, likewise, V J,t =2 J /2 Ṽ J,2 J (t+) (here W j,t & ṼJ,t denote the tth elements of W j & ṼJ ) III III 2

4 Properties of the MODWT as was true with the DWT, we can use the MODWT to obtain a scale-based additive decomposition (MA) and a scale-based energy decomposition (ANOVA) in addition, the MODWT can be computed efficiently via a pyramid algorithm MODWT Multiresolution Analysis: I starting from the definition W j,t = L j l= h j,l t l mod N, can write W j,t = where { h j,l } is { h j,l } periodized to length N N l= h j,l t l mod N, can express the above in matrix notation as W j = W j, where W j is the N N matrix given by h j, h j,n h j,n 2 h j,n 3 h j,3 h j,2 h j, h j, h j, h j,n h j,n 2 h j,4 h j,3 h j, h j,n 2 h j,n 3 h j,n 4 h j,n 5 h j, h j, h j,n h j,n h j,n 2 h j,n 3 h j,n 4 h j,2 h j, h j, III 3 III 4 MODWT Multiresolution Analysis: II recalling the DWT relationship D j = Wj T W j, define jth level MODWT detail as D j = W j T W j similar development leads to definition for jth level MODWT smooth as S j = ṼT j Ṽj can show that level J MODWT-based MA is given by = J j= D j + S J, which is analogous to the DWT-based MA MODWT Multiresolution Analysis: III if we form DWT-based MAs for and its circular shifts T m, m =,...,N, we can obtain D j by appropriately averaging all N DWT-based details ( cycle spinning ) S 4 D 4 D 3 D 2 D T T S 4 T D 4 T D 3 T D 2 T D t t III 5 III 6

5 MODWT Multiresolution Analysis: IV left-hand plots show D j, while right-hand plots show average of T m D j in MA for T m, m =,,...,5 S 4 D 4 D 3 D 2 D averaged T m S 4 averaged T m D 4 averaged T m D 3 averaged T m D 2 averaged T m D MODWT Decomposition of Energy for any J &N, can show that J 2 = W j 2 + ṼJ 2, j= leading to an analysis of the sample variance of : J ˆσ 2 = W N j 2 + N ṼJ 2 2, j= which is analogous to the DWT-based analysis of variance t t III 7 III 8 MODWT Pyramid Algorithm goal: compute W j & Ṽj using Ṽj rather than letting Ṽ,t t, can show that, for all j, W j,t = L l= h l Ṽ j,t 2 j l mod N and Ṽj,t = inverse pyramid algorithm is given by Ṽ j,t = L l= L h l Wj,t+2 j l mod N + l= L l= g l Ṽ j,t 2 j l mod N g l Ṽ j,t+2 j l mod N algorithm requires N log 2 (N) multiplications, which is the same as needed by fast Fourier transform algorithm Example of J =4LA(8) MODWT oxygen isotope records from Antarctic ice core year T 45 Ṽ 4 T 53 W4 T 25 W3 T W2 T 4 W III 9 III 2

6 elationship Between MODWT and DWT Example of J =4LA(8) MODWT MA bottom plot shows W 4 from DWT after circular shift T 3 to align coefficients properly in time top plot shows W 4 from MODWT and subsamples that, upon rescaling, yield W 4 via W 4,t =4 W 4,6(t+) oxygen isotope records from Antarctic ice core S 4 D 4 3 D year T 53 W4 T 3 W year D 2 D III 2 III 22 Example of Variance Decomposition decomposition of sample variance from MODWT ˆσ 2 N N t= ( t ) 2 = 4 j= N W j 2 + N Ṽ4 2 2 LA(8)-based example for oxygen isotope records.5 year changes: N W 2.. =.45 ( = 4.5% of ˆσ 2 ). years changes: N W =.5 ( =5.6%) 2. years changes: N W =.75 ( =23.4%) 4. years changes: N W =.839 ( =26.2%) 8. years averages: N Ṽ4 2 2 =.969. ( =3.2%).. sample variance: =3.24 ˆσ 2 Summary of Key Points about the MODWT similar to the DWT, the MODWT offers a scale-based multiresolution analysis a scale-based analysis of the sample variance a pyramid algorithm for computing the transform efficiently unlike the DWT, the MODWT is defined for all sample sizes (no power of 2 restrictions) unaffected by circular shifts to in that coefficients, details and smooths shift along with (example coming later) highly redundant in that a level J transform consists of (J +)N values rather than just N as we shall see, the MODWT can eliminate alignment artifacts, but its redundancies are problematic for some uses III 23 III 24

7 Examples of DWT & MODWT Analysis: Overview Electrocardiogram Data: I look at DWT analysis of electrocardiogram (ECG) data discuss potential alignment problems with the DWT and how they are alleviated with the MODWT look at MODWT analysis of ECG data, subtidal sea level fluctuations, Nile iver minima and ocean shear measurements discuss practical details choice of wavelet filter and of level J handling boundary conditions handling sample sizes that are not multiples of a power of 2 definition of DWT not standardized ECG measurements taken during normal sinus rhythm of a patient who occasionally experiences arhythmia (data courtesy of Gust Bardy and Per einhall, University of Washington) N = 248 samples collected at rate of 8 samples/second; i.e., t = /8 second.38 seconds of data in all time of taken to be t =.3 merely for plotting purposes III 25 III 26 Electrocardiogram Data: II Electrocardiogram Data: III features include baseline drift (not directly related to heart) intermittent high-frequency fluctuations (again, not directly related to heart) PQST portion of normal heart rhythm provides useful illustration of wavelet analysis because there are identifiable features on several scales P T Haar D(4) LA(8) W W 2 W 3 W 4 V n partial DWT coefficients W of level J = 6 for ECG time series using the Haar, D(4) and LA(8) wavelets (top to bottom) III 27 III 28

8 Electrocardiogram Data: IV Electrocardiogram Data: V elements W n of W are plotted versus n =,...,N = 247 vertical dotted lines delineate 7 subvectors W,...,W 6 & V 6 sum of squares of 248 coefficients W is equal to those of gross pattern of coefficients similar for all three wavelets T 2 V 6 T 3 W 6 T 3 W 5 T 3 W 4 T 3 W 3 T 2 W T 2 W LA(8) DWT coefficients stacked by scale and aligned with time spacing between major tick marks is the same in both plots III 29 III 3 Electrocardiogram Data: VI waves aligned with spikes in W 2 and W 3 intermittent fluctuations appear mainly in W and W 2 setting J = 6 results in V 6 capturing baseline drift Electrocardiogram Data: VII to quantify how well various DWTs summarize, can form normalized partial energy sequences (NPESs) given {U t : t =,...,N }, square and order such that U() 2 U () 2 U (N 2) 2 U (N ) 2 U () 2 is largest of all the U t 2 values while U (N ) 2 is the smallest NPES for {U t } defined as C n nm= U (m) 2 N m= U (m) 2, n =,,...,N III 3 III 32

9 Electrocardiogram Data: VIII Electrocardiogram Data: I plots show NPESs for original time series (dashed curve, plot (a)) Haar DWT (solid curves, both plots) D(4) DWT (dashed curve, plot (b)); LA(8) is virtually identical DFT (dotted curve, plot (a)) with U t 2 rather than Ut 2 S 6 D 4 D 3 D 2 D.. (a) (b) n n C n Haar DWT multiresolution analysis of ECG time series blocky nature of Haar basis vectors readily apparent III 33 III 34 Electrocardiogram Data: Electrocardiogram Data: I S 6 S 6 D 4 D 3 D 4 D 3 D 2 D D(4) DWT multiresolution analysis shark s fin evident in D 5 and D 6 D LA(8) DWT MA (shape of filter less prominent here) note where features end up (will find MODWT does better) D III 35 III 36

10 Effect of Circular Shifts on DWT: I T 5 T 5 Effect of Circular Shifts on DWT: II W 4,4 W 4,5 W 4,6 W 4,7 T 2 V 4 S 4 T 3 W 4 D 4 T 3 W 3 T 2 W 2 T 2 W t t t t bottom row: bump and bump shifted to right by 5 units J = 4 LA(8) DWTs (first 2 columns) and MAs (last 2) III 37 D 3 D 2 D t t t t level J = 4 basis vectors used in LA(8) DWT to produce wavelet coefficients W 4,j, j =4,...,7 (black curves) bump time series (blue curves in top row of plots) shifted bump series T 5 (red curves, bottom row) inner product between plotted basis vector and time series yields labeled wavelet coefficient alignment between basis vectors and time series explains why DWTs for two series are quite different III 38 Effect of Circular Shifts on MODWT T 5 T 5 Electrocardiogram Data: II T 45 Ṽ 4 T 53 W4 S 4 D 4 T 22 W6 T 53 W4 T 89 Ṽ 6 T 9 W5 T 25 W3 T 25 W3 D 3 T W2 T W2 D 2 T 4 W T 4 W t t t t unlike the DWT, shifting a time series shifts the MODWT coefficients and components of MA III 39 D level J = 6 LA(8) MODWT, with W j s circularly shifted vertical lines delineate boundary coefficients (explained later) III 4

11 Electrocardiogram Data: III Electrocardiogram Data: IV T 22 W6 T 3 W 6 comparison of level 6 MODWT and DWT wavelet coefficients, after shifting for time alignment boundary coefficients delineated by vertical red lines subsampling & rescaling W 6 yields W 6 (note aliasing effect) LA(8) MODWT multiresolution analysis of ECG data S 6 D 4 D 3 D 2 D III 4 III 42 Electrocardiogram Data: V Electrocardiogram Data: VI D 6 S 6 D 6 Ṽ MODWT details seem more consistent across time than DWT details; e.g., D6 does not fade in and out as much as D 6 bumps in D 6 are slightly asymmetric, whereas those in D 6 aren t MODWT coefficients and MA resemble each other, with latter being necessarily smoother due to second round of filtering in the above, S6 is somewhat smoother than Ṽ6 and is an intuitively reasonable estimate of the baseline drift III 43 III 44

12 Subtidal Sea Level Fluctuations: I years subtidal sea level fluctuations for Crescent City, CA, collected by National Ocean Service with permanent tidal gauge N = 8746 values from Jan 98 to Dec 99 (almost 2 years) one value every 2 hours, so t =/2 day subtidal is what remains after diurnal & semidiurnal tides are removed by low-pass filter (filter seriously distorts frequency band corresponding to first physical scale τ t =/2 day) Subtidal Sea Level Fluctuations: II years level J = 7 LA(8) MODWT multiresolution analysis S 7 D 6 D 4 D 2 III 45 III 46 Subtidal Sea Level Fluctuations: III Subtidal Sea Level Fluctuations: IV LA(8) picked in part to help with time alignment of wavelet coefficients, but MAs for D(4) and C(6) are OK Haar MA suffers from leakage with J =7, S 7 represents averages over scale λ 7 t =64days this choice of J captures intra-annual variations in S 7 (not of interest to decompose these variations further) III years expanded view of 985 and 986 portion of MA lull in D 2, D3 and D 4 in December 985 (associated with changes on scales of, 2 and 4 days) III 48 S 7 D 6 D 4 D 2

13 Subtidal Sea Level Fluctuations: V MA suggests seasonally dependent variability at some scales because MODWT-based MA does not preserve energy, preferable to study variability via MODWT wavelet coefficients cumulative variance plots for W j useful tool for studying time dependent variance can create these plots for LA or coiflet-based W j as follows form T ν(h) j Wj, i.e., circularly shift W j to align with Subtidal Sea Level Fluctuations: VI form normalized cumulative sum of squares: C j,t t W 2 N j,u+ ν (H), t =,...,N ; j mod N u= note that C j,n = T ν(h) j Wj 2 /N = W j 2 /N examples for j = 2 (left-hand plot) and j = 7 (right-hand) C 2,t C 7,t years years III 49 III 5 Subtidal Sea Level Fluctuations: VII easier to see how variance is building up by subtracting uniform rate of accumulation tc j,n /(N ) from C j,t : C j,t C j,t t C j,n N yields rotated cumulative variance plots C 2,t C 7,t years years C 2,t and C 7,t associated with physical scales of and 32 days helps build up picture of how variability changes within a year Subtidal Sea Level Fluctuations: VIII days (from Jan 986) D 5 for D 5 for T 2 D 5 for T 8 comparison of alignment properties of DWT and MODWT details D 5 and D 5, both associated with changes on a physical scale of τ 5 t = 8 days (distance between tick marks) DWT details evidently suffer from alignment effects III 5 III 52

14 Nile iver Minima: I Nile iver Minima: II year time series of minimum yearly water level of the Nile iver data from 622 to 284, but actually extends up to 92 data after about 75 recorded at the oda gauge near Cairo method(s) used to record data before 75 source of speculation oldest time series actually recorded by humans?! III year level J = 4 Haar MODWT MA points out enhanced variability before 75 at scales τ t = year and τ 2 t =2year Haar wavelet adequate (minimizes # of boundary coefficients) III 54 S 4 D 4 D 3 D 2 D Ocean Shear Measurements: I Ocean Shear Measurements: II depth (meters) level J = 6 MODWT multiresolution analysis using LA(8) wavelet of vertical shear measurements (in inverse seconds) versus depth (in meters; series collected & supplied by Mike Gregg, Applied Physics Laboratory, University of Washington) S 6 D j t =. meters and N = 6875 LA(8) protects against leakage and permits coefficients to be aligned with depth J = 6 yields smooth S 6 that is free of bursts (these are isolated in the details D j ) note small distortions at beginning/end of S 6 evidently due to assumption of circularity vertical blue lines delineate subseries of 496 burst free values (to be reconsidered later) since MA is dominated by S 6, let s focus on details alone III 55 III 56

15 Ocean Shear Measurements: III Choice of Wavelet Filter: I depth (meters) D j s pick out bursts around 45 and 975 meters, but two bursts have somewhat different characteristics possible physical interpretation for first burst: turbulence in D 4 drives shorter scale turbulence at greater depths hints of increased variability in D 5 and D 6 prior to second burst D 6 D 5 D 4 basic strategy: pick wavelet filter with smallest width L that yields an acceptable analysis (smaller L means fewer boundary coefficients) very much application dependent LA(8) good choice for MA of ECG data and for time/depth dependent analysis of variance (ANOVA) of subtidal sea levels and shear data D(4) or LA(8) good choice for MA of subtidal sea levels, but Haar isn t (details locked together, i.e., are not isolating different aspects of the data) Haar good choice for MA of Nile iver minima III 57 III 58 Choice of Wavelet Filter: II can often pick L via simple procedure of comparing different MAs or ANOVAs (this will sometimes rule out Haar if it differs too much from D(4), D(6) or LA(8) analyses) for MAs, might argue that we should pick {h l } that is a good match to the characteristic features in hard to quantify what this means, particularly for time series with different features over different times and scales Haar and D(4) are often a poor match, while the LA filters are usually better because of their symmetry properties can use NPESs to quantify match between {h l } and use LA filters if time alignment of {W j,t } with is important (LA filters with even L/2, i.e., 8, 2, 6 or 2, yield better alignment than those with odd L/2) III 59 Choice of Level J :I again, very much application dependent, but often there is a clear choice J = 6 picked for ECG data because it isolated the baseline drift into V 6 and Ṽ6, and decomposing this drift further is of no interest in studying heart rhythms J = 7 picked for subtidal sea levels because it trapped intraannual variations in Ṽ7 (not of interest to analyze these) J = 6 picked for shear data because Ṽ6 is free of bursts; i.e., ṼJ for J < 6 would contain a portion of the bursts J = 4 picked for Nile iver minima to demonstrate that its time-dependent variance is due to variations on the two smallest scales III 6

16 Choice of Level J :II as J increases, there are more boundary coefficients to deal with, which suggests not making J too big if application doesn t naturally suggest what J should be, an ad hoc (but reasonable) default is to pick J such that circularity assumption influences < 5% of W J or D J (next topic of discussion) Handling Boundary Conditions: I DWT and MODWT treat time series as if it were circular circularity says N is useful surrogate for (sometimes this is OK, e.g., subtidal sea levels, but in general it is questionable) first step is to delineate which parts of W j and D j are influenced (at least to some degree) by circular boundary conditions by considering W j,t =2 j/2 Wj,2 j (t+) and W j,t L j l= h j,l t l mod N, can determine that circularity affects W j,t, t =,...,L j with L j (L 2) ( 2 ) j III 6 III 62 Handling Boundary Conditions: II can argue that L = L 2 and L j = L 2 for large enough j circularity also affects the following elements of D j : t =,...,2 j L j and t = N (L j 2 j ),...,N, where L j =(2 j )(L )+ for MODWT, circularity affects W j,t, t =,...,min{l j 2,N } circularity also affects the following elements of D j : t =,...,L j 2 and t = N L j +,...,N Handling Boundary Conditions: III T 2 V 6 T 3 W 6 T 3 W 5 T 3 W 4 examples of delineating LA(8) DWT boundary coefficients for ECG data and of marking parts of MA influenced by circularity S 6 D 6 D 5 D 4 III 63 III 64

17 Handling Boundary Conditions: IV Handling Boundary Conditions: V boundary regions increase as the filter width L increases for fixed L, boundary regions in DWT MAs are smaller than those for MODWT MAs for fixed L, MA boundary regions increase as J increases (an exception is the Haar DWT) these considerations might influence our choice of L and DWT versus MODWT III 65 Haar D(4) LA(8) Haar D(4) LA(8) comparison of DWT smooths S 6 (top 3 plots) and MODWT smooths S 6 (bottom 3) for ECG data using, from top to bottom within each group, the Haar, D(4) and LA(8) wavelets III 66 S 6 S 6 Handling Boundary Conditions: VI Handling Boundary Conditions: VII just delineating parts of W j and D j that are influenced by circular boundary conditions can be misleading (too pessimistic) effective width λ j =2τ j =2 j of jth level equivalent filters can be much smaller than actual width L j =(2 j )(L )+ arguably less pessimistic delineations would be to always mark boundaries appropriate for the Haar wavelet (its actual width is the effective width for other filters) {h l } {h 2,l } {h 3,l } {h 4,l } {g l } {g 2,l } {g 3,l } {g 4,l }. L = 8 vs. 2.. L... 2 = 22 vs. 4 L.. 3 = 5 vs. 8 L = 6 vs. 6 L... = 8 vs. 2 L... 2 = 22 vs. 4 L.. 3 = 5 vs. 8 L... 4 = 6 vs. 6 plots of LA(8) equivalent wavelet/scaling filters, with actual width L j compared to effective width of 2 j III 67 III 68

18 Handling Boundary Conditions: VIII Handling Boundary Conditions: I to lessen the impact of boundary conditions, we can use tricks from Fourier analysis, which also treats as if it were circular extend series with (similar to zero padding) polynomial extrapolations use reflection boundary conditions by pasting a reflected (time-reversed) version of to end of year note that series so constructed of length 2N has same sample mean and sample variance as original series year comparison of effect of reflection (red/blue) and circular (black) boundary conditions on LA(8) DWT-based MA for oxygen isotope data S 4 D 4 D 3 D 2 D III 69 III 7 Handling Non-Power of Two Sample Sizes Lack of Standard Definition for DWT: I not a problem with the MODWT, which is defined naturally for all sample sizes N partial DWT requires just N = M2 J rather than N =2 J can pad with sample mean etc. can truncate down to multiple of 2 J truncate at beginning of series & do analysis truncate at end of series & do analysis combine two analyses together can use a specialized pyramid algorithm involving at most one special term at each level our definition of DWT matrix W based upon convolutions rather than inner products odd indexed downsampling rather than even indexed using ( ) l+ h L l to define g l rather than ( ) l h l ordering coefficients in resulting transform from small to large scale rather than large to small choices other than the above are used frequently elsewhere, resulting in DWTs that can differ from what we have presented III 7 III 72

19 Lack of Standard Definition for DWT: II two left-hand columns: D(4) DWT matrix W as defined here two right-hand columns: S-Plus Wavelets D(4) DWT matrix (after reordering of its row vectors) only the scaling coefficient is guaranteed to be the same!!! t t t t III 73