Propagation of error for multivariable function
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- Percival Gilmore
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1 Proagation o error or mltiariable nction No consider a mltiariable nction (,,, ). I measrements o,,,. All hae ncertaint,,,., ho ill this aect the ncertaint o the nction? L tet) o (Eqation (3.8) ± L ),...,, ( This orks or cases like sstematic errors, hen the errors o most o the ariables hae the same sign. For cases like random errors, this oerestimate and gie an er bond o the actal error: bond o the actal error: W ill t d th d l t i th L We ill std the case o random error later in the corse.
2 Proagation o error or mltiariable nction No the righta: Addition in Qadratre slide Assming to random ariables, according to the Central Limit Theorem, both and ill ollo Normal distribtion. For simlicit, assme their mean eqal to 0, i.e. 0. Pr ob() Pr ob() Pr ob(, ) e π e π - - π e d d - dd
3 Proagation o error or mltiariable nction No the righta: Addition in Qadratre slide No the right a: Addition in Qadratre slide No consider a ne random ariable and e ant to calclate. Strateg: Transorm and into to ne ariables and some other indeendent ariable - - d e Pr ob() d e ) Pr ob(, π - dd e ) Pr ob(, π π - dd e ) ith Pr ob(, Comare this π
4 Proagation o error or mltiariable nction No the righta: Addition in Qadratre slide 3 No the right a: Addition in Qadratre slide 3 ) ( ) ( ) ( ) ( ) ( ) ( ) ( () Comaing coeicient, ) ( ) ( ) ( () 0 ) ( () into (3) : Sbstitte () (3) ()
5 Proagation o error or mltiariable nction No the righta: Addition in Qadratre slide No the right a: Addition in Qadratre slide Sbstitte () into () : ) ( ) ( ) ( this into () Sbstitte ) ( ) (
6 Proagation o error or mltiariable nction No the righta: Addition in Qadratre slide 5 No the right a: Addition in Qadratre slide 5 In smmar, e hae change rom random ariables and (ith ncertaint and ) to random ariables and (ith ncertaint and ) B and ) to random ariables and (ith ncertaint and ). B simle algebra, e sole the onl ossibilit or this to ork: Onl this is imortant becase e care onl the ariable bt not. Thereore, the ncertaint in is (roagation o error). Rle o adding to errors: T T
7 Generalization: Addition in Qadratre Generalization: Addition o to errors T No consider a nction o and, (,) Error in Error in de to error in ( ) de to error in ( ) Error in
8 Proagation o error or mltiariable nction No consider a mltiariable nction (,,, ). I measrements o,,,. All hae ncertaint,,,., ho ill this aect the ncertaint o the nction? the nction? L L Formal eqation or roagation o error.
9 Toics Part Single measrement. Basic st (Chater and ). Proagation o ncertainties (Chater 3) Part Mltile measrements as indeendent reslts. Mean and standard deiation (Chater ). Basic on robabilit distribtion nction (not in tet elicitl) 3. The Binomial distribtion (Chater 0). The Poisson distribtion (Chater ) 5. Normal distribtion (irst hal o Chater 5) 6. χ test ho ell does the data it the distribtion model? (Chater ) Part 3 Mltile measrements as one samle. Central limit theorem (not in tet elicitl). Normal distribtion (second hal o Chater 5) 3. Proagation o error (Chater 3). Rejection o data (Chater 6) 5. Merging to sets o data together (Chater 7) Part - Deendent ariables. Cre itting (Chater 8). Coariance and correlation (Chater 9)
10 Rejection o data h Qestions: Gien a set o data,,., N obtained rom N measrements. There is one or to data oints ling ar aart rom the others. We old like to think this is a reslt o mistake that e did not realize dring the measrement. Sholde reject thisdata oint rom the collection o data? Eamle: Imagine e make si measrement s o the eriod o a endlm and get the reslts (rom. 65 o tet): 38s 3.8s, 35s 3.5s, 3.9s, 39s 3.9s, 39s 3.s, 3s.8s. 8s What are e going to do ith the eriod o.8s?
11 Rejection o data basic idea To anser the qestion qantitatiel, e need to inestigate the sread o that set o data, this is gien b the standard deiation (not the standard deiation o mean). Shold e reject the otling data deends on ho ar it is aa rom the mean in comarison ith the sread o the data.
12 Eamle Imagine e make si measrement s o the eriod o a endlm and get the reslts (rom. 65 o tet): 3.8s, 3.5s, 3.9s, 3.9s, 3.s,.8s Mean 6 Standard deiation 3.38 ( ) SDOM T (3. ± 0.3)s ( ) ( ) ( ) 5 ( ) ( ).8s is ~ rom the mean. Probabilit o obtaining measrements that dier b at least this mch rom the mean P(less than )
13 Chaenet's criterion or discarding otling li data Mltil the nmber o data oints N ith P. I NP is less than 0.5, the orst data can be discarded. d d (Chaenet s criterion). Eamle: Imagine e make si measrement s o the eriod o a endlm and get the reslts (rom. 65 o tet): 3.8s, 3.5s, 3.9s, 3.9s, 3.s,.8s. Probabilit o obtaining measrements that are orse than the.8s data oint 0.05 PV < 0.5, thereore the.8s data oint can be rejected.
14 Eamle We can no re calclate the mean and standard deiation ater rejecting the.8 s data oint: 3.8s, 3.5s, 3.9s, 3.9s, 3.s,.8s Mean 5 Standard deiation 3.7 ( ) ( ) ( ) ( ) ( ) SDOM T (3.7 ± )s 0.
15 A Better Wa Another a is to make more measrements ithot ih rejecting an data. I e do more measrements in the reios eamle : 38s 3.8s, 35s 3.5s, 39s 3.9s, 39s 3.9s, 3s 3.s, 8s.8s, 33s 3.3s, 37s 3.7s, s.s, 38s 3.8s Mean Standard deiation ( ) (.8-3.5) SDOM 0. 0 T (3.5 ± 0.)s ( ) ( ) ( ) ( ) 9 ( ) (.- 3.5) ( ) ( )
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