An introduction to the problem of predicting a signal spatially distributed from observed discrete values

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Donald Maxwell
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1 A troducto to the problem of predctg a sgal spatally dstrbuted from observed dscrete values ) Basc deftos ad otato A sgal s a fucto F of a vector varable t D; to what cocers us the doma D ca be a etre surface or a part of t, e.g. the earth surface or a sphere or a ellpsod. I all the above cases t s the vector of the coordates that detfy a pot P sweepg D. Note that we detfy the pot P wth the vector of ts coordates t. So t could be a vector of Cartesa coordates ( x),( xy, ),( xyz,, ) or of geographc coordates ( λϕ, ),(,, ) deoted also as,( t, t ),( t, t, t ) λϕh etc; such cases t wll be etc. ad, whe approprate, t wll be cosdered as a t algebrac vector, e.g. t =. t Note.: whe D t may be atural to thk of t as a tme varable; ths case the sgal Ft () s also called a process. Otherwse, whe we cosder t as a spatal varable, wthout ay partcular referece to the atural order (past, preset, future) whch tme always mples, we say that Ft () s a feld; the, ths s always so whe t s a vector wth,, dmesos, because these cases D caot have a atural order. Usually F s cosdered to possess some smoothess or regularty property; for stace, we could clam that F s cotuous o D or cotuous wth ts frst dervatves, o D. I all these cases the regularty of the feld F wll be expressed by clamg that t belogs to some lear space of fuctos H, whch some cocept of dstace s troduced. Example.: assume D to be close ad bouded a. whe H s the space of cotuous fuctos we wrte H the dstace δ ( FG), betwee F ad G H as follows: (, ) = C D ad put δ F G Max F t G t, (.) t D b. whe H s made of fuctos cotuous wth ther frst dervatves, we wrte H C ( D) ad put F G δ ( FG, ) = MaxF( t) G( t) + Max t D, t D t t (.) c. whe H s the space of fuctos square tegrable o, we wrte H L D, ad put D that H s a lear space meas that, gve ay two fuctos F, G H ad ay two λ, µ (here we shall use prevalgly, λ µ ) we have λ µ F t + G t H.

2 δ ( FG, ) = F ( t) G( t) dt (.) D d. whe H s the space of fuctos square tegrable ther frst dervatves o D, we wrte H H D ad put F G δ ( F, G) = F G dt + dt t D D t ad so forth. We shall ot pursue the pot of vew of the aalyss of the propertes of such spaces of fuctos; we shall just clam that F H meas that F s a typcal feld wth the regularty propertes that characterze H ad we shall always cosder spaces H that satsfy the followg fudametal assumpto of separablty. Defto: H s sad to be separable f there s H a sequece of fuctos ϕ () t, =,,..., that we shall call a bass of H, such that, take ay F H ad fxed ay ε >, we ca fd a N ad a sequece of umercal coeffcets a, =,,... so that { } (.4) { } δ, N F aϕ < ε. (.5) = Example.: let D [,] ; the a. { ϕ } { } F( t) C( D ) ad ε >, we ca fd N, { } t t s a bass for all four spaces H of Example.; partcular, gve ay a such that [,] N = Max F t a t < ε ; t b. { ϕ } = ( π ) ( π ) {, cos, s ;,,... } t t = s a bass of all four spaces of Example., whch s called Fourer bass; c. cosder the sequece costructed as Fg..

3 ϕ () t - ϕ () t ϕ - - ϕ ϕ 4 ϕ 5 ϕ Fg.. The Haar bass { ()} ϕ t s a bass for H = L D but ot for the other spaces cosdered the Example.. Ths { ϕ ()} t s called the Haar bass. Note.: oe should ot beleve that whe we make ε to ted to (.5) we fd a lmt of our approxmato procedure H, such that the ed we ca wrte + = () F = aϕ t (.6) Ths s a much stroger property of the bass { } s verfed we say that { ϕ } s a Resz bass of H. For stace { t } s ot a Resz bass bass C( D ), but t s a Resz bass L ( D ) ad We are ready ow to defe our ma problem. ϕ t, whch does ot always hold true; whe (.6) C D ; The Fourer bass b) of Example. s ot a Resz H D. Problem. (Pb.) assume to have a vector = {, =,,..., M} {, =,,..., } t M, such that ν Y Y of values observed at pots Y = F t +, (.7) o o o o

4 where ν o are observatoal oses sampled from some radom varables, very ofte ormal varates, of whch we kow that, puttg ν { ν, =,,..., } { } = = { } M, ν ν; ν ; νν νν + o E C E (gve); (.8) assume further that we kow F to belog to H, the we wat to predct the value of ay t D, ad we wat to kow the bas of the predctor Fˆ ( Y, t ),.e. { ˆ } b= F() t E F Y, t (.9) ad the varace of the predctor Fˆ ( Y, t ) { } = (, ) { (, )} σ Fˆ E Fˆ Y t E Fˆ Y t (.) order to be able to compute the mea square predcto error F t at { } = E F t Fˆ Y, t = b E + σ F ˆ. (.) Remark: occasoally, we shall cosder a problem slghtly more complcated tha the above oe; F t ad we would lke to for example the observatos could ot smply be pot-wse values of predct ot just F at some pot but rather some quatty (fuctoal) depedg o D F t dt etc. Here follow few examples of problems lke (Pb.), relevat evrometal scece. Example.: a. we have a ra feld o a area ; we have grometers at pots F t, e.g. D D { t ; =,,..., M } measurg, for stace, the daly rafall ad we wat to predct at some other place t the rafall the same day; b. we have M pots a area D, where the presece of a certa pollutat the ar s measured (e.g. the desty of partculate PM per m ); we wat to predct the desty of the pollutat at ay other place; c. by photogrammetrc measuremets, we have derved the heghts of the terra at M pots a area D ad we wat to predct the heght at ay other pot,.e. to buld a dgtal terra model ( DTM ); a aalogous problem s that of measurg the seafloor depth by soar ad buldg a dgtal bathymetry model; d. by drllg, we measure the depth of the water head wth respect to the earth surface, ad we wat to buld a dgtal model for t,.e. predct the depth at ay pot; a aaloguos problem s that of recostructg the thckess of a layer of geologcal terest from drlled values. More complcated are the cases, such as those the followg Example.4, that geeralse the above examples, ether the sese that we observe or wat to predct dfferet fuctoals of the 4

5 feld, or because the feld F tself becomes a vector feld, stead of beg a scalar feld as see up to here. Examples.4: a. geodesy, oe eeds to compute the geod udulato (or better, the geod aomaly, ς ) from gravty observatos, such as free aomales g, here t = ( λϕ,, h ) ad, gve ormal gravty formulas,.e. the two fuctos γ ad γ g are related to the so called aomalous gravty potetal T t g( t) = + γ h γ ( t) T t whle the heght aomaly ς s related to T by ς = T t γ,, γ =, the observables h T t by the equato b. deformato aalyss, for stace motorg a lad slde or the tectoc deformato t that we wat to predct at a pot t s a of a large area, the feld sampled at pots { } vector feld of dsplacemets or dmesos, depedg o the case, F( t ) t y F { x, y } t F( t ) t ( x, y ) t F( t ) x Fg..: a feld of dsplacemets F { x( x, y), y( x, y) } c. oe of the smplest versos of sesmc tomography, oe measures the travel tme of a elastc wave alog patters (straght rays approxmato) as Fg.. order to estmate the slowess, amely the verse of the propagato velocty v 5

6 (, ) s x z = v x z, (, ) assumg that the medum s cotaed a layer of kow thckess d. x ( x, T ) x ( x, T ) ( x, T ) P α P P x d ( x, z ) dl P z P P Fg..: geometry of straght rays reflectos a layer of thckess d, T the arrval tme x the abscssa, Notg that d d ta α = =, cosα =, x x + ta α dx 4d dl = = + cosα x dx, z = x taα = d x x the observato equatos are T P P x / dl 4d dx = dt = = + v x d v x, x x ad what we would lke to predct s the fucto s( x, z ) (or (, ) layer. v x z ) at every pot of the I these lecture otes we shall maly deal wth problems of the type Pb.., or at most wth ther vector geeralzato. 6

7 ) A emprcal classfcato of dfferet compoets of a feld F Sometmes the feld F we try to predct s kow to be a soluto of some rgorous equato ad ts kowledge depeds oly o a small umber of parameters. I such cases a small umber of observatos together wth a least squares estmato wll solve the problem. Ths happes for stace whe we study the structural behavour of buldgs, whch, subjected to exteral forces, follow well kow mechacal laws, ad furthermore are kow terms of geometrcal shape ad materal costtuto. However, atural felds are sometmes subject to ot perfectly kow laws ad ther geometry ad materal structure are usually varable a way that caot be precsely defed. F t at may pots wll leave room for As a result, eve a very detaled ad precse kowledge of a upredctable varato whe we move to a ew pot t, so that the use of statstcal models ad statstcal tools s just atural ths matters. Therefore, very much as statstcs, we shall adopt a F t behaves. emprcal atttude, tryg to lear from data how the feld To ths purpose, we shall set up a classfcato of possble compoets of F, whch, though ot uvocal or exhaustve, provdes a useful template to the emprcal aalyss of our data. Assume that the doma D, o whch data are gve, has a dameter, defed as = Max t t. (.) t, t D Note that here we mplctly assume a Eucldea dstace betwee pots, whle sometmes o surfaces S t s coveet to use a dfferet dstace. For stace o a sphere of radus R we ca use a sphercal dstace ψ defed by s ψ = t t R R. (.) Assume further that data are gve at M pots {, =,,..., } s a graularty dex whch ca be more precsely defed as follows. Defto.: gve { t } the earest eghbour of t M wth a mea dstace d. Ths d t s t () such that t t t t j ; (.) () j the the mea dstace betwee data pots s d = t (). (.4) M M t = 4 For example, we ca thk of a case R, where we have M > pots, where s some value betwee km ad km ad d somewhere betwee m ad m. Naturally, each case wll have the two scales, d defed some way, but we shall typcally assume that 7

8 . (.5) d The compoets formula: the most geeral case, we suggest F beg splt to 4 compoets F( t) = T( t) + P + R( t) + S. (.6) Defto.: T s the tred compoet ad t represets a sgfcat varato of F o the scale of. T t s represeted by a polyomal of some order p ; examples are Typcally () p t R T t = a + at apt, (.7) t R T t = a + at+ at + at + att + at ( p+ )( p+ ). (.8) p p at... + a+ p+ t = Typcally T represets the polyomal behavour of F o spatal scale betwee ad. Defto.: P s the perodc compoet, o scales comparable to. To better uderstad, let us take the case t R. The we could represet the perodc compoet of F by meas of a Fourer bass up to some maxmum frequecy N () = + N t t F t a acos π + bs π. (.9) = Note that (.9) the wavelegths of the harmoc compoets are ; smlarly to the prevous case we shall try to avod to push (.9) to very short wavelegths, although there mght be cases wth a marked, clear perodcty eve at frequeces hgher tha (scales smaller tha ). The aalogous of (.9) R s () = + N t t t t F t a k ak cos π cos π + bk cos π s π +., (.) t t t t + ck s π cos π + dk s π s π Defto.4: the regoal compoet R represets localzed effects, wth a resoluto that ca be aroud. They are typcally recostructed by combg so called sples,.e. fuctos wth lmted support, traslated to the cetres of a regular grd. The dea s llustrated R wth the Haar sples (see Fg..). 8

9 S S ( t ) a) support of S () t b) k Fg.. a) the mother sple S () t ad ts traslate by ; b) a combato of type as ( t k ) k As we see, startg from () k k S t the combatos of the type as ( t k ) are starcase fuctos, clearly dscotuos. If we wat a smoother approxmato to F, maybe because we kow a pror that F s cotuous, we could use the order sple preseted, the case Fg... t R, the 9

10 S () t S t ( ) a) support of S () t b) 4 5 k Fg.. a) the mother sple S () t ad ts traslate by ; b) a combato of type as ( t k ) k I a smlar way oe ca costruct sples of hgher order S l t, possessg more smoothess propertes,.e. cotuty of st, d dervatves ad so forth. The same smoothess the s herted by the combatos of type as ( t k l k ). k Fally, we wat to geeralze the regoal (sple) models, for stace to R, as follows: frst of all we defe a mother sple = S t S t S t ; (.) l l l the we traslate the pole of l ( l kh l l S t to the kots of a grd of pots t Q ( k, h), by wrtg S t t = S t k S t h); (.) ad fally, we create models of F by combatos of the form = kh l kh kh ) ( R t a S t t. (.) Note.: f oe wats to traslate the above theory to a regoal compoet R adapted to a grd of sze δ, t s eough to use = R t akhs kh l t t kh. (.4) δ It s fact easy to verfy that, for stace, kh kh

11 t kδ S t k = S δ δ δ δ s a Haar sple wth cetre k δ ad wth support kδ, k δ +, ad the same reasog geeralzes to ad to hgher orders l. So the choce of R t. R δ regulates the resoluto of Defto.5: S s the stochastc or radom feld compoet of F. S t s cosdered as a sample from a radom varable defed by some probablty dstrbuto o a sutable space cotag all the possble sgals S. Ths radom fucto however s characterzed by kowg the dstrbuto of -dmesoal varables { S}, t ; of -dmesoal varables { S, S( t )}, t, t ; of -dmesoal varables { } S t, S t, S t, t, t, t ad so forth. Yet, we do t usually kow all these dstrbutos; so that we develop a so called lear predcto theory, where the oly thgs we eed to kow are: ad { } t, E S t (.5) { } t, t, C t, t E S t S t. (.6) Furthermore, we shall lear how to estmate C( t, t ) from data, by troducg several ad hoc hypotheses o S ad ts stochastc behavour. Note.: aturally, the represetato (.6) does ot eed to be appled to all cases. There mght be cases whch oe or more compoets are ot preset, or where there s o et separato betwee oe ad the other. For stace, a sutable polyomal ca have a very smlar behavour to that of a se or cose o a closed terval. Ths suggest us to apply a crtero of parsmoy costructg our emprcal model; a smpler model s always preferable to a rcher oe eve f t leaves larger resduals. The correct balace betwee the two factors s a matter tha ca be solved sometme wth the help of testg theory (resduals that dsplay the correct statstcal behavour are more acceptable tha smaller resduals but wth a upredcted behavour), sometmes oly o the bass of the modeller s experece. Aother crtcal pot s that, although a ufed aalyss of all the 4 factors of the (.6) (plus the observatoal ose) s certaly theoretcally more appealg ad rgorous, t s usually almost mpossble. Therefore, we shall pursue a stepwse strategy, where oe compoet a tme s estmated va least squares; the, the resduals are cosdered as ew observatos where the ext compoet s to be sought. So, we shall frst determe the tred, the we shall study the presece of perodc sgals the resduals, the we shall possbly elmate regoal sgals ad fally we shall try to aalyse the last resduals terms of a radom feld wth some pecular characterstc of homogeety ad sotropy. I dog so, we mght ot be rgorous, but we are certaly effectve ad we probably avod stablty of the model that typcally arse whe we try to estmate altogether.

Summary of the lecture in Biostatistics

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