Homework 1: Solutions

Transcription

1 Howo : Solutos No-a Fals supposto tst but passs scal tst lthouh -f th ta as slowss [S /V] vs t th appaac of laty alty th path alo whch slowss s to b tat to obta tavl ts ps o th ol paat S o V as a cosquc of Fat s tho! a Passs both scal a supposto tsts Data [ ] z Γ s a la fucto of th ol paats a Passs both scal a supposto tsts Data a ρ ρ π π ρ Γ wh Γ s th la tal opato s a la fucto of th ol paat ρ v otoally a f ata [ ] X U ' ' ' 5 ' ' ~ Γ th Ữ passs both supposto a scal tsts Howv X s stll a la fucto of th ol paats spctv of th aat bcaus as w hav s abov t passs both tsts oc w subtact a costat b a ata that w alay ow! Pa of 6 Rav Kaa

2 aty of Fowa ol : t us assu that aps a -vcto th ol spac to a N-vcto th ata spac N : R R N Epa th ol vcto ts of th staa colu bass: [ th t ] wh So h N ata vctos ca ow b pst as [ ] Γ Scal Supposto & a colu vctos of lth N a thfo Γs a at of so N whos lts p o th pobl paatzato a oty Each ow of Γ ts how to wh th ol paats o to obta th cospo copot of th ata vcto Pa of 6 Rav Kaa

3 3 Fo bacou foato o ths pobl f to of st t al p-pt to v 3v ooss of ft s pss ts of th su of squas of vual ata os whch a assu to follow a aussa stbuto hs su thfo follows th χ -stbuto a Fst stat th ub of s of fo DoF of th pobl: ν #Data - #ol paats N- b h stat th p-valu th pobablty of obta a χ valu as la o la tha th obsv valu fo th abov ub of DoFs: p χ χ obs f χ ν χ obs c W wat ths ub to b th too clos to o too clos to #DoF ν N NOE: th %-valus v ths pobl a actually p-valus χ : p-valu s aly o h s a aly % chac of that χ obs coul b ay wos - th ol fts th ata alost actly! Eth th ata os hav b ovstat o th s th pottal of ata-fau χ 49: p-valu s ouhly 5 h s a 5% chac that χ obs coul b ay wos So ths s a accptabl sult χ : p-valu s aly zo h s a lbl chac that χ obs coul b ay wos w hav th wost possbl ft to th ata So th th fowa ol s coct o ata os hav b ustat o o ot follow a aussa a la-z vs pobl wth aussa ata os th ol paats ca b pss as a la cobato of th ata plus os hfo th ol paats thslvs a oally stbut wth th a pct valu b v by th last squas soluto So fo a sl paat th 95% cofc tval tval fo whch th aa u th aussa 95 fo ol paat s v by ±96σ wh σ s th squa-oot of th aoal t of th ol covaac at cospo to that ol paat So ± 96σ ± σ ± h facto fot of th σ wll b ouhly fo th 68% cofc tval a 3 fo th 99% cofc tval ths cas w wat to f th poctos of th 3D o-llpso o th plas f by ach pa of ol paats Eq 4 of st t al 3 ca b -wtt as: Δ DoF Substtut Eq 43 Evalu coposto of - PΛP to Eq 4 a cos oly ol paats at a t vs: Δ DoF Δ DoF λ λ wh λ s th th valu of th vs of th ol covaac at a χ DoF s th χ stbuto wth ν So th abov quato fs th o s a llps wth as Δ χdof / λ So th stat valus of th ol paats wll b ± Δ χdof / λ Pa 3 of 6 Rav Kaa

4 4 φ 4 z φ wt th ol paats vs us th ol that zs th sual btw th fowa ol a th ata s also th ost lly ol v th ata 4 W wat to laz th fowa pobl th hbohoo of a fc ol : o Hh O s "HO" wh s a N at that vs th sstvty of th fowa pobl to th ol paats at : 4v th ptubato ; t s so fc ol wh whch s th laz vs pobl fo th ptubato of about 4v Us th sa otato as abov Hc φ H w hav o HO lso sc & a all N vctos a - s a NN at whos ts a pt of th ol ptubatos th RHS of th abov appoato s a quaatc psso ts of th copots of Pa 4 of 6 Rav Kaa

5 4v ths cas z φ wt th ol paat ptubatos vs us th ol that zs th sual btw th fowa ol ptubato a th ata ptubato : ssu that th pobl s ov-t w ca v th last squas soluto by stt [ ] [ ] [ ] [ ] φ wh w hav us th vcto calculus latoshps: a B B B B Not that - s sytc w ca splfy th abov to t: [ ] [ ] o Whch s th last squas soluto Now lt a so o a a th a So w stat wth as th tal uss a coput succssv tatos of th ptubato: a 4v azato wos oly wh w ca o th hh o ts HO abov hus f th hh o vatvs a ot lbl as th cas of a vy ouh φ ths appoato fals So ths appoato wos wll fo sooth o waly-ola vs pobls ato w also a oo fst uss to th tu ol that s th fc ol ust b clos ouh to th tu ol fo th tatos to cov to th lobal u Pa 5 of 6 Rav Kaa

6 4v t th covaac of th ata vcto b qual to W a v that - W wat to f th covaac of th ol paats Fo splcty w wt th abov latoshp as Now ach t of th ol covaac at ca b pst as: ov E[ E[ ] E[ ] E[ ] E[ ] E[ E[ ] E[ ] E[ ] o Substtut th actual valu of to th last quato vs E[ ov ] E[ E[ E[ ]] s also a sytc postv ft at f w oalz th off-aoal ts of ach ow of w obta a solss colato at ρ whch vs us th colatos ao ol paats f th off-aoal ts of ρ a clos to zo th all ol paats ca b ptly stat us th assu fowa ol a th v ata f ay of th off-aoal ts a clos to say th t th th cospo ol paats th a th paats a hhly pt o ach oth a caot b ptly stat fo th v ata a ou assu ol 4 h ol soluto at R s f fo th follow: R st obs tu tu Fo th last squas soluto ov-t pobl all ol paats a wll solv thy ca b t pt of ach oth a R Usually R s a fucto of th vs opato - a ts to hav o-zo off-aoal ts cat a statstcal pc btw th lts of th ol vcto So R ts what cobatos of ol paats ca b solv by ou assu ol v th ata ] ] ] 4 Fo a ut syst coas paatzato th ol soluto has a boa scal lth that s th vual ol paats a pooly solv - but th ol covaacs a sall f w choos to cas th soluto f paatzato t cos at th cost of cas ol covaac as llustat blow s also class ots p-6 ow soluto sall vaac Hh soluto la vaac Pa 6 of 6 Rav Kaa