EXST Regression Techniques Page 1

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Chad Erick Bates
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1 EXST704 - Rgrssion Tchniqus Pag 1 Masurmnt rrors in X W hav assumd that all variation is in Y. Masurmnt rror in this variabl will not ffct th rsults, as long as thy ar uncorrlatd and unbiasd, sinc thy cancl out. Howvr, w hav assumd that X is masurd without rror, and masurmnt rror in this variabl can caus rror. Sinc all rror is vrtical, w cannot incorporat this masurmnt rror into our modl. As a rsult, this additional rror must, in som way, gt incorporatd into th modl and/or its rror. oftn it is not tru that X is masurd without rror particularly in mristic rlationships g. hight of brothr hight of sistr body lngth scal lngth lngth wight Lt th masurmnt rror in X b dnotd as $ œ X X whr X is th masurd valu and X is th tru valu of th variabl. Thn, whn fitting th supposd modl Y œ X %! w ar actually fitting Y œ (X $ ) %! and, multiplying out and grouping variability ffcts, Y œ X (% $ )!

2 EXST704 - Rgrssion Tchniqus Pag As a rsult, a) X is not fixd (masurd without rror), it is a random variabl b) Th varianc trm is not longr indpndnt of X, sinc $ contains X c) b and b ar biasd (towards zro) and! lack consistncy (i. lim P( )>%)=0 whr % is som arbitrary, 8p positiv ral numbr; so dos not tnd toward probabilistically as n incrass infinitly) d) Thr ar a coupl of cass or aspcts of th variation in X whr variation is not a problm. a) X may b a random variabl, not undr th control of th invstigator. Howvr, this is not a problm as long as th valu of X is masurd without masurmnt rror and is known xactly. b) th Brkson Modl is a spcial cas whr masurmnt rror dos not ffct th rsults, it cancls out. In this modl, th situation for X and X is rvrsd. If X is som fixd valu that th invstigator is shooting for (for xampl, by stting som machin valu; as a thrmostat, adjusting a currnt spd, or som othr machin stting) thn th masurd valu, X is a constant whil th tru valu, X, is th random variabl.

3 EXST704 - Rgrssion Tchniqus Pag What to do? 1) Don't hav masurmnt rror. ) Prtnd you don't hav masurmnt rror. ) if only 1 variabl has rror, thn it must b usd as th dpndnt variabl. Invrs prdiction (fitting Y on X and thn making infrnc about X) is th nxt topic. 4) Masur th masurmnt rror and adjust for it. Thr ar ways to do this. g. Sndcor and Cochran (l980): w want to fit Y =! + X, but w ar using Y = b! + bwx w, whr X w has rror so X w = (X + ) Thn b w is w = x 5 5 whr - =, indicats th magnitud of th bias is rgrssion cofficint for X masurd without rror and is calld th structural rgrssion cofficint. W no longr hav an unbiasd stimat of this paramtr. obtain stimat of 5, rror in X = S obtain stimat of 5, varianc in X = S assum rror in %, and X is normal thn x x - = S (S S ) xw = b = b w (1 + - ) This stimats with rror in Y only (no variability in X) as pr th assumptions.

4 EXST704 - Rgrssion Tchniqus Pag 4 5) anothr solution RICKER (197) approach (limitd utility, but individual applications ar discussd by RICKER) Rickr points out that many prdictiv quations in fishris ar undrstimatd (bias is actually towards 0, thos mntiond hav positiv slops) suggsts solutions (1) Cntral axis or gomtric man axis Brothrs Hight (Y) Y i X i SISTERS HEIGHT (X) (a) rgrss Y on X vrtical rror (b) rgrss X on Y horizontal rror (c) for X and Y bivariat normal th lin which splits th diffrnc such that b for Y on X = is givn by 1 b for X on Y Dy Dx S S y v = É = = = Éb + S (N ) x b r b

5 EXST704 - Rgrssion Tchniqus Pag 5 (d) Rickr calls th lin th GM (gomtric man axis) THIS LINE IS NOT A BISECTOR, though it will always fall btwn th othr two lins It is th lin is that which minimizs th product of th horizontal and vrtical distancs of th point from th lin. Y i X i () onc th nw slop is obtaind, th intrcpt can b calculatd as usual Y bwx 6) MAJOR AXIS Th lin that minimizs th SS (prpndicular distanc) of obsrvd points to th fittd lin z = Dy Dx + È( Dx Dy) + 4( Dxy) Dxy if is th masurmnt rror in Y and is th masurmnt rror in X # thn this quation prsums that = #

6 EXST704 - Rgrssion Tchniqus Pag 6 A MORE GENERAL EQUATION availabl in softwar SUPERCARP (Wayn Fullr, Stat Dpt., Iowa) mploys an xprssion $ =, thn # È = D y $D x + ( $Dx D y) + 4( $ Dxy) Dxy if $ Á 1, thn th distanc minimizd is not prpndicular NOTE: In all of th abov cass, onc th slop has bn calculatd, th intrcpt is obtaind by b = Y bx! For our purposs, Gnrally a last squars fit with traditional assumptions will b adquat W will considr a corrction to som quations, particularly whn w xpct som thortical valu g. = Not: Rickr (197) suggsts spcific applications of ithr gomtric man axis or major axis and provids quations for th varianc of ach

7 EXST704 - Rgrssion Tchniqus Pag 7 Invrs Prdiction : Somtims it is ncssary to mak prdictions of th indpndnt variabl, X, instad of th dpndnt variabl, Y. This may occur bcaus w hav only th rgrssion quation availabl, and not th original data. Or it may b that w ar intrstd in prdicting X, which is masurd without rror, from Y, which is masurd with rror. Th procss of prdicting th indpndnt variabl from th dpndnt variabl is calld invrs prdiction. Invrs prdiction starts th sam as any SLR Th population modl is Y =! + X + % And th usual last squars analysis is don on a sampl of n obsrvations from th parnt population. Y = b! + X In ordr to stimat X for som valu of Y, w thn solv th quation for X. Y b b! X œ whr b Á 0 (i a rlationship must xist) a confidnc intrval for th nw obsrvation X is givn by MSE 1 b s X = 1 n + (X X) D(X X) which is a transformation of th sam quation w usd bfor. This is an approximat valu, which your book points out is appropriat if t! *MSE # ß8 # b* D(X X) is small (i < 0.1) in our cas (Vial xampl), rcommndd.06 * * œ œ , within th

8 EXST704 - Rgrssion Tchniqus Pag 8 For our vial brakag xampl; Analysis of Varianc Sum of Man Sourc DF Squars Squar F Valu Prob>F Modl Error C Total Root MSE R-squar Dp Man Adj R-sq C.V Paramtr Estimats Paramtr Standard T for H0: Variabl DF Estimat Error Paramtr=0 Prob > T INTERCEP X Th quation to prdict X is givn by, suppos w wish to prdict how may transfrs would caus 0 vials to b brokn. Y b Y b 4 4 4! X œ œ œ œ œ.45 a confidnc intrval for th nw obsrvation X =.45 is givn by MSE 1 b s X = 1 n + (X X) D(X X). 1 (.45 1) = 1 + = = 0.175*0.105 = s = È X œ

9 EXST704 - Rgrssion Tchniqus Pag 9 sinc t!.06, thn # ß).0 œ P(X t s E(X) X! Ÿ Ÿ t! s ) œ 1-! Y=0 X # ß8 # Y=0 # ß8 # X P(.45.06*0.065 Ÿ E(X) Ÿ.45.06*0.065 ) œ 0.95 P(1.978 Ÿ E(X) Ÿ.96 ) œ 0.95 so it appars most likly that transfrs would b involvd in this damag, though transfrs is not out of th qustion.

Data Assimilation 1. Alan O Neill National Centre for Earth Observation UK

Data Assimilation 1. Alan O Neill National Centre for Earth Observation UK Data Assimilation 1 Alan O Nill National Cntr for Earth Obsrvation UK Plan Motivation & basic idas Univariat (scalar) data assimilation Multivariat (vctor) data assimilation 3d-Variational Mthod (& optimal