Fall 2010 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. (n.b. for now, we do not require that k. vectors as a k 1 matrix: ( )

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1 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede Let s nvestgate the effect f a change f varables n the real & symmetrc cvarance matr aa the varance matr aa the errr matr V [ ] ( )( E RR E ) : cv ( ),... cv, cv (, )... cv, V [ ] ( [ ])( [ ]) ( )( E RR E E E E ) cv (, ) cv (, )... Defne:,..., ( ) (,..., ),, (,,..., ) (n.b. fr nw, we d nt reqre that We can genercally defne the cllectn f vectrs as a matr: (,,..., ) (,,..., ) (,,..., ) matr Let s assme that we can epand each (,,..., ) ) vectr n a aylr seres abt and als assme that nd rder (and all hgher rder) terms can (safely) be neglected. hen: (,,..., ) (,,..., ) (,,..., ) + j j j j j j + + R whch we can als symblcally abbrevate as: ( ) ( ) ( ) Eplctly wrtng ths t: (,,..., ) (,,..., ) (,,..., ) (,,..., ) (,,..., ) (,,..., ) + (,,..., ) (,,..., ) (,,..., ) (,,..., ) matr matr matr matr matr P598AEM_Lect06.dc

2 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede he epectatn vale f s als [ ] [ ] ( [ ]) ( ) E E E, as we epect. We can then determne the cvarance matr V asscated wth the set f varables, defned analgsly t thse we defned fr V : cv ( ),... cv, cv (, )... cv, V [ ] ( [ ])( [ ]) ( )( E RR E E E E ) cv (, ) cv (, )... where the -resdal vectr he detals: R s defned analgsly t that fr R,.e. R. R ( ) ( ) E[ ] E[ ] D R / where we have als defned the dervatve matr D / and ts transpse D / as: D / and D / n.b. he dervatve matr D / and ts transpse D / Fr matrces, n general D / D/. hen: n.b. Recall frm lnear algebra that the transpse f the prdct f tw matrces A and B s (AB) B A. P598AEM_Lect06.dc { }{ } [ ] [ ] { ( )}{ ( )} R R E E where: ( ) ( ) ( ) ( ) / / / are nt symmetrc matrces! D R D R D R R D a matr. R D/ R R D/ R RD/ and:.

3 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede he cvarance matr varables can ths be wrtten as: [ ] [ { } ] [ ] [ ] V E R R E D R D R E D R R D D E R R D V asscated wth the / / / / / / ( [ ])( [ ]) ( )( ) E E E E E ( )( ) ( )( ) E cv ( ),... cv, ( ) ( ) cv,... cv, a matr. cv (, ) cv (, )... te that n the last step f the st and 3 rd rws abve, we t advantage f the fact that the dervatve matr D / and ts transpse D / are jst sets f nmbers (cnstants!),.e. they are n lnger fnctns f the randm varables. w, snce the cvarance matr V [ ] ( )( E RR E ) s: V [ ] [ ] [ ] E RR E E E E cv ( ),... cv, ( ) ( ) cv,... cv, cv (, ) cv (, )... he cvarance matr V can then be cmpactly wrtten as: V ERR [ ] ED [ R{ D R} ] ED [ RRD ] D ERR [ ] D D VD / / / / / / / / Imprtant nte: In vars prbablty & statstcs tetbs, the cvarance matr eqatn s nstead wrtten as V D / VD /. A detaled cmparsn f V D / VD / wth the abve dervatn f V D V D shws that the tw seemngly dfferent/cntradctry relatns are ndeed / / eqvalent, becase D / D / and D / D /! he rgn f the tw dfferng cnventns can be traced bac t the respectve defntns f the resdal matrces as a clmn vectr ( R vs. a rw vectr ( R n the frmatn f the ter matr) matr) prdct fr the cvarance matr V E[ R R ] E[ R R ],.e. R R and R R. V P598AEM_Lect06.dc 3

4 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede V Eplctly wrtng t ths relatn, lng-hand, t s: cv ( ),... cv, ( ) ( ) cv,... cv, cv (, ) cv (, )... cv ( ),... cv, cv (, )... cv ( ), cv (, ) cv (, )... Fr the cmmn case f ndependent varables,,,.and. fnctns,,, all f the cvarances vansh, ths thngs smplfy, and we then have, n eplct, lng-hand ntatn: S: V V cv ( ),... cv, ( ) ( ) cv,... cv, cv, cv,... ( ) ( ) P598AEM_Lect06.dc 4

5 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede Fr the case f ndependent randm varables,,,.and. fnctns,,, the cvarance matr V has dagnal elements f the frm: ( ) cv, var fr,,..., V whch agrees wth the crrespndng aylr seres dervatn n p. 5 f P598AEM Lect. tes 5. In addtn, the cvarance matr V als has nn-zer ff-dagnal elements f the frm: V cv, l fr,,..., whch need nt be zer! he transfrmatn frm has ndced crrelatns between the new randm varables even thgh the rgnal varables were ndependent!!! A smple eample f V D/ VD/ V : Let and be ndependent randm varables. Let ( ) and ( ) 0 Assme that we are gven the -bass cvarance matr V 0 hen: D / And: S: +. / D here D 0 V D/ VD/ V 0 ( ) + + cv ( ), + ( ) ( + ) cv (, ) hs n general and are nt ndependent {b/c ( ) ( ) cv, 0 } nless. P598AEM_Lect06.dc 5

6 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede A mercal Eample: Sppse the -sgma ncertanty n s he V 5 and the -sgma ncertanty n s 3. cvarance matr asscated wth the transfrmatn ( ), ( ) ( + ) + s: cv ( ), cv ( ), hs the -sgma ncertanty n ( ) and the -sgma ncertanty n ( ) + s: s: Hwever, even thgh (here) 7 4.3, the randm varables and are nt ndependent they are (pstvely) crrelated wth each ther, becase: ( ) ( ) ( ) cv, cv, 8 > 0 r, eqvalently: cv (, ) 8 ρ (, ) Undng the Errr Prpagatn: Invertng ths transfrmatn, we have: ( + ) and ( ) +. Let s (temprarly) pretend t gnre crrelatns,.e. treat and as f they were ndependent. We wld then btan: hese reslts are clearly wrng!!! (Snce we ntally stated that 5 and 3!!!) he (crrect) nverse transfrmatn, sng matrces s: V D/ V D/ V where: D D D / / / P598AEM_Lect06.dc 6

7 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede hen: 7 8 V D/ V D/ V hs, we see va the se f the abve matr frmalsm, that we ndeed (crrectly) recver the rgnal 5 and 3,.e. the randm varables and are ndependent. he abve smple eample(s) f a change f (rthnrmal) varables can be easly nderstd as a smple change f rthnrmal bass vectrs frm the ndependent randm varables and t the nn-ndependent randm varables and va the rthnrmal + and/r the nverse rthnrmal transfrmatn ( ), transfrmatn ( + ), ( ) +. (n.b. bth f these rthnrmal transfrmatns are frm RH RH crdnate systems ) he rthnrmal transfrmatn s a cnseqence f applyng a ϕ 45 (CW) rtatn n the plane, as shwn by the red ellpse n the fgre belw: ϕ ϕ 45 n.b. we als see frm the ellptcal symmetry asscated wth the abve fgre that 4. smply becase f the specfc chce f the ϕ 45 (CW) rtatn n the -D plane, resltng n the bass vectrs each havng eqal prjectns nt the bass vectrs. Had we nstead chsen an arbtrary ϕ -rtatn n the -D plane, then n general. P598AEM_Lect06.dc 7

8 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede Fr an arbtrary ϕ -rtatn f -bass vectrs: csϕ + snϕ snϕ + csϕ csϕ + sn ϕ csϕ snϕ snϕ + csϕ snϕ csϕ he nverse ϕ -rtatn transfrmatn f -bass vectrs s (als) gven by: R R csϕ + snϕ snϕ + csϕ csϕ + sn ϕ csϕ snϕ snϕ + csϕ snϕ csϕ Bt nte that ϕ ϕ( 45 here) R R +, ths we can re-wrte the nverse ϕ -rtatn transfrmatn as: csϕ snϕ snϕ + csϕ wth: csϕ snϕ csϕ snϕ snϕ + csϕ sn ϕ csϕ R R R R csϕ snϕ csϕ snϕ 0 csϕ sn ϕ csϕ snϕ R R snϕ csϕ sn ϕ csϕ 0 sn ϕ csϕ snϕ csϕ Graphcally, the nverse rthnrmal transfrmatn fr a ϕ + 45 (CCW) rtatn s shwn by the red ellpse n the fgre belw: ϕ ϕ P598AEM_Lect06.dc 8

9 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede he eqatn fr the red ellpse, epressed n terms f the ndependent randm varable rthnrmal bass vectrs s gven by: te that the matr 0 s the nverse f the -bass varance matr V 0 0 : Whch can be wrtten cmpactly as: VV V V where s the nt matr, and ths we can als wrte the abve -bass ellpse eqatn cmpactly as: V he eqatn fr the red ellpse, epressed n terms f the nn-ndependent randm varable rthnrmal bass vectrs s gven by: (, ) + ( ) ρ (, ) ρ (, ) ρ (, ) where ρ (, ) cv, ( ) ρ, Here, the matr V s the nverse f the ρ (, ) ρ(, ) cv ( ), ρ, -bass cvarance matr V : cv (, ) ρ, ρ P598AEM_Lect06.dc 9

10 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede ( ) ( ) ρ (, ) ( ) ρ, ρ, ρ, ρ, (, ) (, ) + (, ) ρ (, ) ρ (, ) ρ (, ) ρ (, ) ρ ρ ρ ( ) ρ, 0 0 ρ (, ) 0 ρ (, ) 0 whch can be wrtten cmpactly as: V V V V and ths we can als wrte the abve -bass ellpse eqatn cmpactly as: ( ) ρ, + If the P.D.F. asscated wth the ndependent randm varables and s the Gassan/nrmal dstrbtn,.e. V G(, ) G G e e e π π π + he 3-D srface asscated wth the -D Gassan/nrmal prbablty dstrbtn (, ) shwn n the fgre belw (fr the specal/lmtng case f ): G s P598AEM_Lect06.dc 0

11 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede We see that cntrs f cnstant/eqal prbablty densty are (n general) ellpses n the -D plane, where the argment f the epnental s eqal t a cnstant,.e. + cnstant At the pnts (, ) ( ±,0) and/r (, ) ( 0, ) plane, we see that cnstant / (tre fr any vale f (, ) ± n the ellpse crve n the -D lyng n ths partclar ellpse) and ths we see that the eqatn f an ellpse fr whch + asscated wth the G,, crrespnds t a -sgma ( standard devatn) abve -D Gassan/nrmal P.D.F. cntr (aa -sgma eqptental ) f cnstant/eqal prbablty densty. Smlarly, e.g. fr an arbtrary # f (nteger) sgma/standard devatns,.e. n,,3,4,5,... ths crrespnds t cntrs asscated wth eqatn f ellpses fr whch s satsfed, as shwn n the fgre belw: n P598AEM_Lect06.dc

12 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede Fr the Gassan/nrmal P.D.F. asscated wth the dependent randm varables + : ( ) and (, ) G π ( ) ρ, e ρ(, ) + ρ (, ) where ρ (, ) ( ) cv, {hs s the general frm f a -D Gassan dstrbtn n tw varables, ncldng crrelatns} hen we see that cntrs f cnstant/eqal prbablty densty are ndeed ellpses n the -D plane, where the argment f the epnental s eqal t a cnstant,.e. ( ) ρ, + ρ (, ) At the pnts (, ) ( ±,0) and/r (, ) ( 0, ) plane, we see that cnstant / (tre fr any vale f (, ) cnstant ± n the ellpse crve n the -D lyng n ths partclar ellpse) and ths we see that the eqatn f an ellpse fr whch ρ (, ) + asscated wth the abve -D ρ (, ) Gassan/nrmal P.D.F. G(, ), crrespnds t a -sgma ( standard devatn) cntr (aa -sgma eqptental ) f cnstant/eqal prbablty densty fr ths dstrbtn. Smlarly, e.g. fr an arbtrary # f (nteger) sgma/standard devatns,.e. n,,3,4,5,... ths crrespnds t cntrs asscated wth eqatn f ellpses fr whch ρ (, ) n + s satsfed, as shwn n the fgre belw: ρ (, ) P598AEM_Lect06.dc

13 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede te frm the abve dscssn(s), fr ndependent randm varables that we can als wrte the Gassan P.D.F. n matr ntatn as: (,,... )... G G G G e e V ( π) ( π ) Fr dependent randm varables, the Gassan P.D.F. n matr ntatn s: ( π) (, ) ρ j j V j jρ(, ) j +, j (, j) ρ j j G(,,... ) e e π V, j V In the fgre belw, fr a gven n ellpse cntr, we shw the effect f varyng the crrelatn ceffcent ρ ( ) ellpse eqatn: cv, + between ts pper/lwer lmts n the general n, ρ, n + ρ (, ) P598AEM_Lect06.dc 3

14 Fall 00 Analyss f Epermental Measrements B. Esensten/rev. S. Errede e.g. fr the (, ) G P.D.F., when and n general are dependent randm varables: ρ.0 ρ 0.8 ρ ρ 0.0 ρ +.0 n n n n te that each f the n ellpses n the abve fgre tch/are tangent t the sdes f an enclsng rectanglar b f dmensns n n. he abve n ellpse eqatn(s), sed n cnjnctn wth the 4 straght-lne eqatns that descrbe each f the fr sdes f the enclsng rectanglar b can be slved smltanesly t determne the tangent/ntersectn pnt(s) f a gven ellpse wth the enclsng rectanglar b. P598AEM_Lect06.dc 4