Pattern Analyses (EOF Analysis) Introduction Definition of EOFs Estimation of EOFs Inference Rotated EOFs

Transcription

1 Patten Analyses (EOF Analyss) Intoducton Defnton of EOFs Estmaton of EOFs Infeence Rotated EOFs

2 . Patten Analyses Intoducton: What s t about? Patten analyses ae technques used to dentfy pattens of the smultaneous tempoal vaatons Gven a m-dmensonal tme sees x ' t, the anomales x t defned as the devatons fom the sample mean can be expanded nto a fnte sees k x ' ˆ α p ˆ t, t wth tme coeffcents αˆ,t and fxed pattens pˆ. Equalty s usually only possble when km The pattes ae specfed usng dffeent mnmzatons ' EOFs: x t s optmally descbed by ' POPs: x s optmally descbed by The pattens can be othogonal t k k ˆ ˆ ' ˆ ˆ α, t p xt α, t p ' ' ' Axt ( xt Axt ) mn! mn!

3 . Patten Analyses Intoducton: What can pattens and the coeffcents descbe? Standng Sgnals A fxed spatal stuctue whose stength vaes wth tme Popagatng Sgnals A stuctue popagatng n space. It has to be descbed by two pattens such that the coeffcent of one patte lags (o leads) the coeffcent of the othe one by a fxed tme lag (often 90 o ) Schematc epesentaton of a lnealy popagatng (left) and clockwse otatng (ght) wave usng two pattens: p and p. If the ntal state of the wave s p, then ts state a quate of peod late wll be p.

4 . Patten Analyses Example: Daly Pofle of Geopotental Heght ove Beln Data: 0-yea data set contanng 0 wnte days tmes 9 vetcal levels between 950 and 300 hpa,.e. 0x0x9600 obsevatons How should we descbe the spatal vaablty? One way s to compute the vaance at each level. Ths howeve does not tell us how the vaatons ae coelated n the vetcal Soluton: descbng spatal coelatons usng a few EOFs Usefulness: To dentfy a small subspace that contans most of the dynamcs of the obseved system To dentfy modes of vaablty The fst two EOFs, labeled z and z, of the daly geopotental heght ove Beln n wnte. The fst EOF epesents 9.% and the second 8.% of the vaance. They may be dentfed wth the equvalent baotopc mode and the fst baoclnc mode of the toposphec cculaton.

5 . Patten Analyses Intoducton: Elements of Lnea Analyss Egenvalues and egenvectos of a eal squae matx Let A be an mxm matx. A eal o complex numbe s sad to be an egenvalue of A, f thee s a nonzeo m-dmensonal vecto e such that Ae e Vecto e s sad to be an egenvecto of A Egenvectos ae not unquely detemned A eal matx A can have complex egenvalues. The coespondng egenvectos ae also complex. The complex egenvalues and egenvectos occu n complex conugate pas Hemtan matces A squae matx A s Hemtan f T A c A A T c whee s the conugate tanspose of A. Hemtan matces have eal egenvalues only. Real Hemtan matces ae symmetc. Egenvalues of a symmetc matce ae non-negatve and egenvectos ae othogonal

6 . Patten Analyses Intoducton: Elements of Lnea Analyss Bases m A collecton of vectos { e, L, e } s sad to be a lnea bass fo an m- dmensonal vecto space V f fo any vecto a V thee exst coeffcents α,,,m, such that a αe The bass s othogonal, when e, e 0 f o othonomal when e, e 0 f and e fo all,..., m whee denotes the nne poduct whch defnes a vecto nom, T x, y x y and x x, x. One has Tansfomatons m If { e, L, e } s a lnea bass and y αe, then α y, e a whee e s the adont of e satsfyng e, e 0 fo and e, e fo a a a

7 . Patten Analyses Defnton of Empcal Othogonal Functons: The Fst EOF EOFs ae defned as paametes of the dstbuton of an m-dmensonal andom vecto X. The fst EOF s the most poweful sngle patten s epesentng the vaance of X e defned as the sum of vaances of the elements of X. It s obtaned by mnmzng, subected to e, Note : ε E X X, e e Va( X ) Va( X, e ) T T T Va( X, e ) E( X e ) X e ) whch esults n T T e Σe e e Σe e 0 whee s the Langange multple assocated wth the constant e s an egenvecto of covaance matx Σ wth a coespondng egenvalue! Mnmzng ε s equvalent to maxmzng the vaance of X contaned n the - dmensonal subspace spanned by e, Va X,e. ( ) e ε s mnmzed when e s an egenvecto of Σ assocated wth ts lagest egenvalue

8 . Patten Analyses Moe EOFs Havng found the fst EOF, the second s obtaned by mnmzng ε E X X, e e X, e e subected to the constant e e s an egenvecto of covaance matx Σ that coesponds to ts second lagest egenvalue. s othogonal to e e because the egenvectos of a Hemtan matx ae othogonal to each othe EOF Coeffcents o Pncple Components The EOF coeffcents ae gven by α X, e X T e e T X

9 . Patten Analyses Theoem Let X be an m-dmensonal eal andom vecto wth mean µ and covaance matx Σ. m Let L m be the egenvalues of Σ and let e, L, e be the coespondng egenvectos of unt length. Snce Σ s symmetc, the egenvalues ae non-negatve and the egenvectos ae othogonal. The k egenvectos that coespond to,, k mnmze ε k Va k ε k E ( X µ ) k ( X ) X µ, e use of any othe k-dmensonal sunspace wll leads to mean squaed eos at least as lage as ε k Va m ( X ) gves the mean squaed eo ncued when appoxmatng X n a k- dmensonal subspace boken up the total vaance nto m components

10 . Patten Analyses Intepetaton The bulk of the vaance of can often be epesented by a fst few EOFs X The physcal ntepetaton s lmted by the fundamental constant that EOFs ae othogonal. Real wold pocesses do not need to be descbed by othogonal pattens o uncoelated ndces

11 . Patten Analyses Popetes of the EOF Coeffcents The covaances of EOF coeffcents α ae gven by ( ) ( ) ( ) Σ e e e e e XX E e e X e X E Cov T T T T, 0,,,, α α v The EOF coeffcents ae uncoelated

12 . Patten Analyses Vecto Notaton The andom vecto can be wtten as X P α, o α P X T T ( ) X m wth P e e L e, α ( α, L, α m ), whch leads to Σ E PΛP T T ( XX ) PE( α α ) T P T whee Λ s the dagonal mxm matx composed of the egenvalues of Σ.

13 . Patten Analyses Degeneacy It can be shown that the egenvalues ae the m oots of the m-th degee polynomal ( Σ ) p( ) det I whee I s the mxm dentty matx. e If ο s a oot of multplcty and s the coespondng egenvecto, then s unque up to sgn If ο s a oot of multplcty k, the soluton space e e Σ o s unquely detemned n the sense that t s othogonal to the space spanned by the m-k egenvectos of Σ wth o. But any othogonal bass fo the soluton space can be used as EOFs. In ths case the EOFs ae sad to be degeneated. e Bad: pattens whch may epesent ndependent pocesses cannot be dsentangled Good: fo k the pa of EOFs and the coeffcents could epesent a popagatng sgnal. As the two pattens epesentng a popagatng sgnal ae not unquely detemned, degeneacy s a necessay condton fo the descpton of such sgnals

14 . Patten Analyses Coodnate Tansfomatons Consde two m-dmensonal andom vectos and elated though Z LX whee L s an nvetble matx. If the tansfomaton s othogonal (.e. L - L T ), the egenvalue of the covaance matx of, Σ XX, s also the egenvalue of the covaance matx of Z X, Σ ZZ, and the EOFs of X,, ae elated to those of Z X e Z, e, va Z X e Le X Z Poof: Snce Σ ZZ Le X T Σ ZZ LΣ XX L, Σ T X LΣ L Le LΣ XX XX XX e e X X e Le X X Consequence of usng an othogonal tansfomaton: The EOF coeffcents ae nvaant, snce α X Z T T ( LPX ) Z Z Z α Z T T T X PX PX L P

15 . Patten Analyses Estmaton of Empcal Othogonal Functons Appoach I Estmate the covaance matx and use the egenvectos and the egenvalues of the estmated covaance matx as estmatos of the EOFs and the coespondng egenvalues Appoach II Use a set of othogonal vectos that epesent as much as the sample vaance as possble as estmatos of EOFs The two appoaches ae equvalent and lead to the followng theoem

16 . Patten Analyses Theoem Let Σˆ be the estmated covaance matx deved fom a sample x, L, epesentng n ealzaton of X. Let ˆ be the egenvalues of and ˆ, L, Σˆ m the coespondng egenvectos of unt length. Snce Σˆ s symmetc, the egenvalues ae non-negatve and the egenvectos ae othogonal The k egenvectos coespondng to,, ˆ ˆ L m mnmze { } x n e, ˆ ˆ, L e m n ˆk x ε k x, e eˆ ˆ ε Va ˆ k Va ˆ k ( X ) m ( X ) ˆ ˆ The EOF estmates epesent the sample vaance n the same way as the EOFs do wth the andom vaable

17 . Patten Analyses Popetes of the Coeffcents of the Estmated EOFs As wth the tue EOFs, the estmated EOFs span the full m-dmensonal vecto space. The andom vecto X can be wtten as m X ˆ α e wth ˆ α X, eˆ When X s multvaate nomal, the dstbuton of the m-dmensonal vecto of EOF coeffcents, condtonal upon the sample used, s multvaate nomal wth mean and covaance matx E ( ) Pˆ T, ( ˆ, ˆ,, ) Pˆ T α x, L, x µ Cov α α x L x Pˆ ˆ m m Σ whee Pˆ has ê n th column The vaance of the EOF coeffcents computed fom the sample s n α α ˆ ˆ ˆ n The sample covaance of a pa of EOF coeffcents computed fom the sample s zeo Two ntepetatons of ˆ as an estmate of the vaance of the tue α as an estmate of the vaance of αˆ

18 . Patten Analyses The Vaance of EOF Coeffcents of a Gven Set of Estmated EOFs Gven a set of egenvalues and EOFs deved fom a fnte sample, any andom vecto X can be epesented n the space spanned by these estmated EOFs usng the tansfomaton X P ˆ ˆ, α ˆ α P ˆ X T Queston: s the vaance of the tansfomed andom vaables ˆ α X, e ˆ equal the tue EOF coeffcent (.e. s the egenvalue of the estmated covaance matx equal to the tue egenvalue)? The answe s no Snce the fst EOF mnmzes ε E X X, e e one has Va( X ) Va( α) E X X, e e E X X, eˆ ˆ < e Va ( X ) Va( ˆ α ) Va( α ˆ ) > Va( α ) fo the fst few EOFs Snce the total vaance s estmated wth nealy zeo m bas by Va( X ) ˆ, t follows that Va α ) < Va( ˆ α ) ( fo the last few EOFs

19 . Patten Analyses The Bas n Estmatng Egenvalues The bas can be assessed usng the followng asymptotc fomulae that apply to egenvalue estmates computed fom samples that can be epesented by n d nomal andom vectos (Lawley) ( ) ( ) ( ) ( ) 3 ˆ ˆ O n n n Va O n n E m m The egenvalue estmatos ae consstent: ( ) 0 ˆ lm n E The estmatons of the lagest and the smallest egenvalues ae based ( ) < > fo the smallest fo the lagest ˆ E ( ) ( ) ( ) ( ) ( ) ( ) fo the smallest ˆ ˆ fo the lagest ˆ ˆ α α α α Va Va E Va Va E < < > >

20 . Patten Analyses Relablty of EOF estmates I The elablty s often assessed usng so-called selecton ules. The basc supposton s full space sgnal-subspace (EOFs) + nose-subspace (degeneated) Thus, the dea s to dentfy the sgnal-subspace as the space spanned by the EOFs that ae assocated wth lage, well-sepaated egenvalues. Ths s done by consdeng the egenspectum Poblems The detemnaton of sgnal- and nose-subspace s vague. Geneally, the shape of the egenspectum s not necessaly connected to the pesence o absence of dynamcal sgnal No consdeaton of the elablty of the estmated pattens, snce the selecton ules ae focused on the egenvalues

21 . Patten Analyses Relablty of EOF estmates II: Noth s Rule-of-Thumb Usng a scale agument, Noth et al. obtaned an appoxmaton fo typcal eo of the estmated EOFs, whch n combnaton wth a smplfed veson of Lawley s fomula, eads eˆ n closest whee c and c ae constants, n s the numbe of ndependent samples, ~(/n) / the typcal eo n ˆ, closest the closest egenvalue to m c e c' e The fst-ode eo s of the ode of (/n) /. Thus convegence to zeo s slow The fst-ode eo s othogonal to the tue -th EOF The estmate of the -th EOF s most stongly contanmnated by the pattens of those othe EOFs that coespond to the egenvalues closest to. The smalle the dffeence between and, the moe sevee the contamnaton Noth s Rule-of-Thumb If the samplng eo of a patcula egenvalue s compaable to o lage than the spacng between and a neghbong egenvalue, then the samplng eo of the -th EOF wll be compaable to the sze of the neghbong EOF EOFs ae mxed

22 . Patten Analyses Noth et al. s Example Noth et al. constucted a synthetc example n whch the fst fou egenvalues and the typcal eos fo the estmated egenvalues ae 4,.6, 0.7, 0.4, -.4, - 3, , fo n300, 0.6 fo n000 The fst two EOFs ae mxed when n300. The thd and fouth EOFs ae mxed fo both n300 and n000

23 . Patten Analyses Examples The fst EOF epesents ENSO, whose coeffcent s shown as cuve D The second EOF may epesent tend, as suggested by ts coeffcent shown as cuve A. The fst two EOFs of the monthly mean sea suface tempeatue of the global ocean between 40S and 60N

24 . Patten Analyses Examples The fst EOF of the toposphec zonal wnd between 45S and 45N at 850, 700, 500, 300 and 00 hpa The analyss s pefomed n two steps by fst estmatng EOF at each level and etanng coeffcents epesentng 90% of the vaance and secondly pefomng EOF analyss wth a vecto composng EOF coeffcents selected fo fve levels The coeffcent tme sees (cuve B) exhbts a tend paallel to that found n the coeffcent of the second SST EOF Does ths tend ognate fom a natual low-fequency vaaton o fom some othe cause? 00hPa 300hPa 500hPa 700hPa 850hPa %

25 . Patten Analyses

26 . Patten Analyses Rotaton of EOFs Why otated EOFs? One hopes that the otated EOFs can be moe easly ntepeted than the EOFs themselves The dea of otaton Gven a subspace that contans a substantal facton of the total vaance, t s sometmes nteestng to look fo a lnea bass of the subspace wth specfed popetes, such as Bass vectos that contan smple geometcal pattes, e.g. pattens whch ae egonally confned o have two egons, one wth lage postve and the othe wth negatve values Bass vectos that have tme coeffcents wth specfc types of behavo, such as havng nonzeo values only dung some compact tme epsodes The esult depends on the numbe o the length of the nput vectos, and on the measue of smplcty Po: a means fo dagnosng physcally meanngful and statstcally stable pattens

27 . Patten Analyses The Mathematcs of the Rotaton Rotaton conssts of a tansfomaton and a constant The tansfomaton A set of nput vectos P ( p L p K ) s tansfomed nto anothe set of vectos Q ( q Lq K ) by means of an nvetble K x K matx R( ): QPR o fo each vecto q : K q p The constant The matx R s chosen fom a class of matces, such as othogonal (R - R T ), subected to the constant that a functonal V(R) s mnmzed

28 . Patten Analyses Consequence of a Othogonal Tansfomaton A andom vecto whch s epesented by the K nput vectos can be wtten, because of the otaton, as X Pα - whee ( PR)( R α) Qβ α and β R α ae K-dmensonal vecto of andom expanson coeffcents fo the nput and the otated pattens, espectvely. If R s othonomal Q T QR T P T PRR T DR Thus, gven othogonal nput vectos, the otated vectos wll be othogonal only f DI, o, f the nput vectos ae nomalzed to unt length Σ ββ Cov T T ( R α, T R α ) R Σ R αα Thus, gven uncoelated expanson coeffcents of the nput vectos, the coeffcents of the otated pattens ae also pa wse uncoelated only f coeffcents α have unt vaance

29 . Patten Analyses Consequence of a Othogonal Tansfomaton The otated EOFs deved fom nomalzed EOFs ae also othogonal, but the tme coeffcents ae not uncoelated The otated EOFs deved fom non-nomalzed EOFs (.e. the vaance of EOF coeffcents equal one) ae no longe othogonal, but the coeffcents ae pawse uncoelated The esult of the otaton depends on the lengths of the nput vectos. Dffeently scaled but dectonally dentcal sets of nput vectos lead to sets of otated pattens that ae dectonally dffeent fom one anothe The otated vectos ae a functon of the nput vectos athe than the space spanned by the nput vectos The otated EOFs and the coeffcents ae not othogonal and uncoelated at the same tme. Consequently, the pecentage of vaance epesented by the ndvdual pattens s no longe addtve

30 . Patten Analyses An Example of the Smplcty Functonal: The Vamax Method Vamax s a wdely used othogonal otaton that mnmzes the smplcty functonal wth ( ) ( ) K V K q f q q V L,, ( ) m m V K s q m s q m q f p q 4, The functonal f V can be vewed as the spatal vaance of the nomalzed squaes (q /s ),.e. f V measues the weghted squae ampltude vaance of the otated EOF The constants s can be chosen feely. One deals wth a aw vamax otaton when s a nomal vamax otaton when ( ) K s p

31 . Patten Analyses Example I: Repoducble Identfcaton of Teleconnecton Pattens Bazon and Lvzey used a vamax otaton of nomalzed EOFs to solate the domnant cculaton pattens n the Nothen Hemsphee: EOFs ae computed fo each calenda month usng a 35-yea data set of monthly mean 700 hpa heghts Rotaton s pefomed on the fst 0 EOFs epesentng 80% of the total vaance n wnte and 70% n summe NAO n wnte NAO n summe PNA n wnte

32 . Patten Analyses Example II: Weak Effect of Rotaton EOFs and otated EOFs of Noth Atlantc monthly mean SLP n wnte: EOFs Rotated EOFs deved fom K5 nomalzed EOFs Rotated EOFs deved fom K0 non-nomalzed EOFs The dffeence between the unotated and the otated EOF s not lage If the EOFs have smple stuctues, the effect of otaton s neglgble

33 . Patten Analyses Example III: Rotaton could splt featues nto dffeent pattens even though they ae pat of the same physcal patten EOFs and otated EOFs of Noth Atlantc monthly mean SST n DJF: EOFs Rotated EOFs deved fom K5 nomalzed EOFs Rotated EOFs deved fom K5 non-nomalzed EOFs The otated EOFs tend to epesent the thee acton centes n the fst EOF sepaately n dffeent EOFs