An Introduction to Wavelet Analysis. with Applications to Vegetation Monitoring

Transcription

1 An Inrducin Wavele Analysis wih Applicains Vegeain Mniring Dn Percival Applied Physics Labrary, Universiy f Washingn Seale, Washingn, USA verheads fr alk available a hp://saff.washingn.edu/dbp/alks.hml

2 Overview: I as a subjec, waveles are relaively new (983 presen) a synhesis f ld/new ideas keywrd in 8, 465+ aricles & bks since 989 (27+ since 22: an inundain f maerial!!!) waveles can help us undersand ime series (i.e., bservains clleced ver ime) images waveles capable f describing hw ime series evlve ver ime n a given scale images change frm ne place he nex n a given scale, where here scale is eiher an inerval (span) f ime (hur, year,... )r a spaial area (square kilmeer, acre,... ) 2

3 Overview: II example: ime series f vegeain areas ver land (5 9 N) (based n mnhly SAT daa frm Climae Research Uni, UK) 2 breal area ( 6 km 2) 4 7 emperae undra year permafrs sme quesins ha waveles can help up address:. are variains hmgeneus acrss ime? 2. are variains frm ne year he nex mre prminen han variains frm ne decade he nex? 3. permafrs is less variable han breal, bu d hey have her saisical prperies ha are significanly differen? 4. hw are any w f hese series relaed n a scale by scale basis (e.g., year year, decade decade)? 3

4 Ouline f Remainder f Talk discuss wha exacly a wavele is discuss wavele analysis (emphasis n physical inerpreain) pin u w fundamenal prperies f he cninuus wavele ransfrm (CWT):. CWT is fully equivalen he ransfrmed ime series 2. CWT ells hw energy in ime series is disribued acrss differen scales and differen imes describe he discree wavele ransfrm (DWT) pin u w analgus fundamenal prperies f DWT lk a DWT f ne f he vegeain area ime series (breal) addresses quesins (hmgeneiy acrss ime) and 2 (prminence f yearly/decadal variains) describe wavele variance addresses quesins 2 and 3 (hw saisical prperies f permafrs & breal cmpare) lk a wavele cvariance beween breal & emperae series addresses quesin 4 (scale by scale relainship f w series) cncluding remarks 4

5 Wha is a Wavele? sines & csines are big waves waveles are small waves hree waveles, including he Haar wavele ψ (H) ( ): ψ (H) ψ (fdg) ψ (Mh) cndiins fr a funcin ψ( ) be a wavele: ψ( ) inegraes ; i.e., ψ( ) balances iself abve/belw ψ 2 ( ) inegraes admissibiliy (mild echnical cndiin) 5

6 Basics f Wavele Analysis: I waveles ell us abu variains in lcal averages quanify, le x( ) be a ime series real-valued funcin f defined ver real axis will refer as ime (bu i need n be such) cnsider average value f x( ) ver[a, b]: b x(u) du α(a, b) b a a (abve nin discussed in elemenary calculus bks) relaed idea f sample mean suppse x( ) is a sep funcin wih N seps in [a, b]: x x x N - a b hen b a b a x(u) du = N b a x j b a N j= = N N j= x j 6

7 Basics f Wavele Analysis: II reparameerize using widh λ and ime f cener f inerval: A(λ, ) α( λ 2,+ λ 2 )= λ + λ 2 λ 2 x(u) du λ b a is called scale =(a + b)/2 is ime assciaed wih cener f inerval A(λ, ) is average value f x( ) ver scale λ a ime average values f ime series are f cnsiderable ineres vegeain area ime series are spaial/empral averages prprin f gaps in ransec ver a fres canpy ec. Q: hw much d averages change frm ne inerval he nex? can quanify changes in A(λ, + 2 ) by cnsidering D(λ, ) A(λ, + λ 2 ) A(λ, λ 2 ) = λ +λ x(u) du λ D(λ, ) fen f mre ineres han A(λ, ) λ x(u) du 7

8 Basics f Wavele Analysis: III can cnnec D(, ) Haar wavele: ne ha D(, ) = if we define x(u) du x(u) du =, <u ; ψ, (u) =, <u ;, herwise ψ, (u)x(u) du cmparing ψ, (u) wih ψ (H) ( ) shws ha ψ, (u) = 2ψ (H) (u) Haar wavele exracs infrmain abu difference beween uni scale averages a = via W (, ) ψ (H) (u)x(u) du D(, ) ψ (H) (u) ψ (H) (u)x(u) x(u) u u u 8

9 Basics f Wavele Analysis: IV exrac infrmain a her s, jus shif ψ (H) (u) frm ψ (H), (u) ψ(h)(u ) ψ (H), 3( ) ψ,( ) (H) ψ,3( ) (H) u -5 5 u -5 5 u exrac infrmain abu her λ s, frm ψ (H) λ, (u) ( ) u ψ λ (H) λ ψ (H) 2 ψ(h),( ) 2,( ) ψ 4,( ) (H) u -5 5 u -5 5 u can check ha ψ (H) λ, ( ) are indeed waveles 9

10 Basics f Wavele Analysis: V use ψ (H) λ, ( ) bain W (λ, ) ψ (H) λ, (u)x(u) du D(λ, ) lef-hand side is Haar cninuus wavele ransfrm (CWT) can d he same wih her waveles: W (λ, ) ψ λ, (u)x(u) du, where ψ λ, (u) λ ψ lef-hand side is CWT based n ψ( ) ( ) u λ ψ (H) ψ (fdg) ψ (Mh) ψ (fdg) ( ) &ψ (Mh) ( ) yield differences f adjacen weighed averages

11 Tw Fundamenal Prperies f CWT. can recver x( ) frm is CWT: x() = [ W (λ, u) ( ) u ψ C ψ λ λ here C ψ is a cnsan depending n jus ψ( ) cnclusin: W (λ, ) equivalen x() ] dλ du λ 2 2. can decmpse energy in ime series using CWT: x 2 W 2 (λ, ) () d = d dλ C ψ λ 2 lef-hand side called energy in x( ) pl f x 2 () versus gives energy disribuin acrss ime righ-hand inegrand gives energy disribuin acrss ime & scale

12 The Discree Wavele Transfrm (DWT) when dealing wih samples x, x,... x N frm x( ), mre cnvenien deal wih DWT han CWT can regard DWT as slices hrugh CWT resric λ dyadic scales τ j 2 j, j =, 2,...,J resric imes inegers =,,..., N ne: cnsidering maximal verlap DWT (MODWT) (can resric imes furher ge rhnrmal DWT) yields wavele cefficiens W j, W (τ j,) als ge scaling cefficiens ṼJ, relaed averages ver a scale f λ =2τ J summary f infrmain in W (λ, ) aλ 2τ J =2 J cllec W j, in vecr W j fr levels j =, 2,...,J als cllec ṼJ, in vecr ṼJ W,..., W J and ṼJ frm he DWT f X [x,...,x N ] T 2

13 Tw Fundamenal Prperies f DWT. can recver X perfecly frm is DWT; i.e., W,..., W J & Ṽ J are equivalen X 2. energy in X preserved in is DWT: X 2 N = x 2 = J j= W j 2 + ṼJ 2 3

14 Example: DWT f Breal Time Series year Ṽ 4 W 4 (scale 8) W 3 (scale 4) W 2 (scale 2) W (scale ) X large value fr a wavele cefficien indicaes large variain a a paricular scale & ime variains fairly hmgeneus acrss scales (i.e., sainariy migh be a reasnable assumpin here) smaller scales seem be dminan 4

15 Wavele Variance: I energy preservain leads analysis f sample variance: ˆσ x 2 N (x x) 2 = J W j 2 + ṼJ N N 2 x 2, = where x x /N N W j 2 prin f ˆσ 2 X due changes in averages ver scale τ j; i.e., scale by scale analysis f variance example: subidal sea levels and assciaed wavele variances N W j 2 versus scales τ j =2 j fr j =,..., j= O 4 O O O O O O O scale 5

16 Wavele Variance: II wavele variances fr vegeain ime series 2 breal emperae undra permafrs au (years) - 2 au (years) sum f wavele variances is equal sample variance 95% cnfidence inervals based n saisical hery cnfirms ha uni scale is dminan excep fr undra, rll-ffs similar & cnsisen wih whie nise undra pssibly pssesses lng memry (rlls ff mre slwly) 6

17 Wavele Cvariance recnsider breal and emperae ime series: 2 breal area ( 6 km 2) 4 7 emperae year sample crss-crrelain is.79; i.e., series are anicrrelaed can decmpse crss-crrelain acrss differen scales: >8 scale w series anicrrelaed a all scales, bu sample crss-crrelain mainly due w smalles scales (75%) 7

18 Cncluding Remarks waveles decmpse ime series wih respec w variables: ime (lcain) scale (exen) CWT & DWT have w fundamenal prperies:. fully equivalen riginal ime series 2. energy in ime series is preserved wavele variance gives scale-based analysis f variance (naural mach fr many gephysical prcesses) echniques exends naurally images many her uses fr waveles (have barely scrached he surface!) apprximaely decrrelae cerain ime series can assess sampling prperies f cerain saisics signal exracin ( wavele shrinkage ) edge idenificain in images cmpressin f ime series/images fas simulain f ime series/images cme jin he fun aricle #8,467 is begging be wrien! big hanks sympsium rganizers fr inviain speak! 8