ESS 265 Spring Quarter 2005 Time Series Analysis: Error Analysis

Transcription

1 ESS 65 Sprng Qarter 005 Tme Seres Analyss: Error Analyss Lectre 9 May 3, 005

2 Some omenclatre Systematc errors Reprodcbly errors that reslt from calbraton errors or bas on the part of the obserer. Sometmes data can be corrected for these errors bt n other cases we mst estmate these errors and combne them wth errors from statstcal flctatons. Accracy a measre of how close an obseraton comes to the tre ale. How well we compensate for systematc errors. Precson a measre of how a reslt was obtaned, how reprodcble t s. How well we oercome random errors. Uncertanty An error refers to the dfference between a reslt and a tre ale. Often we don't know what the "tre" ale so we mst estmate the error. Repeated measrements of the same thng wll dffer and we can only talk abot the dscrepancy between measrements- there s ncertanty. Probable error- A measre of the magntde of the error we estmate. For two dentcal measrements t s a measre of the probable dscrepancy.

3 Defntons Parent poplaton Set of data ponts from whch epermental data are assmed to be a random sample. Parent dstrbton Probablty dstrbton determnng the choce of sample data from parent poplaton. n Epectaton ale f ( ) lm [ ] f ( ) f ( ) P( ) f ( ) P( ) Medan / P( / )P( / )/ Most probable ale ma - P( ma ) P( ma ). Mean- <> Aerage deaton α Varance -. Standard deaton -. Sample mean -. Sample arance Best estmate of the parent standard arance s ( )

4 Usefl Probablty Dstrbtons: The Bnomal Dstrbton Measres the probablty of obserng sccesses n n tres when the probablty of sccess n each try s p. P (, n, p) The mean s gen by n n! p! n! For a bnomal dstrbton the aerage of the nmber of sccesses approaches the mean ale gen by prodct of the probablty of sccess of each tem tmes the nmber of tems. The arance s gen by B 0 0 p! ( ) n! ( n ) n ( p) For the case of a con toss p/ and the dstrbton s symmetrc abot the mean and the medan and most probable ale are eqal to the mean. The arance s half of the mean.! n ( p) np n n! ( ) p!! ( n ) n ( p) np( p)

5 Usefl Probablty Dstrbtons: The Posson Dstrbton A Posson dstrbton occrs when p<< and <<n. It freqently s sefl for contng eperments sch as partcle detectors. It descrbes the probablty of obserng eents per nt tme ot of n possble eents each of whch has a probablty of p of occrrng. The mean of the Posson dstrbton mst be the parameter n the aboe eqaton. The arance s The standard deaton s the sqare root of the mean. ( ) ( ) e p n P p P B P!,, 0 lm, ( ) 0 0!!! y y y e e e ( ) ( ) 0! e

6 Usefl Probablty Dstrbtons: The Gassan Dstrbton The Gassan dstrbton reslts from the case where the nmber of possble dfferent obseratons (n) s nfntely large and probablty of sccess s fntely large so that np>>. It works for many physcal systems. P G (,, ) ep π The Gassan dstrbton s a contnos fncton descrbng the probablty a random obseraton wll occr from a parent dstrbton wth mean and standard deaton. The probablty fncton s defned so that probablty (dp G (,,) that a random obseraton wll fall n an nteral d abot s dp G (,,)P G (,,) d. The wdth of a Gassan s sally epressed as the fll-wdth at half mamm t s gen by.354. The probable error s defned so that half the obseratons of an eperment are epected to fall wthn ±P.E (the probablty of any deaton s less s eqal to ½). P.E

7 Propagaton of Errors In general we do not know the actal errors n the determnatons of parameters. Instead we se some estmate (e.g. ) of the error n each parameter. Assme that f(, ) and that. The ncertanty n can be fond by consderng the spread n resltng from the spread n the nddal measrements,... The arance s gen by. Epand The frst two terms can be epresses n terms of the arances of and. The thrd term s related to the coarance...), ( f ( ) lm ( ) ( )... ( ) ( ) ( ) ( ) ( )( )... lm... lm ( )( ) [ ] lm

8 Propagaton of Errors The standard deaton of s gen by If and are ncorrelated then 0. Specfc combnatons... ( ) a b a ± : ln b ae b a a a ab b a b a b b ± ± ± ± ± : : : : :

9 Estmates of Errors: Method of Mamm Lkelhood Snce we don t know the mean of the dstrbton nderlyng an eperment we mst estmate t from data. We want to fgre ot how to dere a mean that wll yeld mamm lkelhood that the parent dstrbton had the same mean. Assme a Gassan dstrbton and hypothesze a tral parent dstrbton wth mean and standard deaton. The probablty of obserng a ale s ' P ( ') ep π Consderng the whole obseratons the probablty for obserng that set s gen by the prodct of the nddal probabltes. P( ') P ( ') P ( ') The probablty P( ) of obtanng obseratons from aros parent poplatons wth dfferent means bt wth s greatest when the data came from a poplaton wth. ' ep π

10 Estmates of Error: Calclatng the Mean We can se the method of mamm lkelhood to calclate the most probable ale for the mean. For a Gassan mamzng the probablty s the same as mnmzng the argment of the eponental. ' X Set the derate of X wth respect to to zero. ' In the preos calclaton we assmed that. Yo can do the same type of deraton for and standard deatons.. Ths ges ( ) ' ( )

11 Estmates of Error: Error of the Mean Each data pont contrbtes to the ncertanty of the mean. Usng the rles for combnng ncertantes we obtan ' If the ncertantes of the data ponts are eqal (.e. ) then If the ncertantes of the data ponts are not eqal then ( ) Instrmental ncertantes (de to fnte precson of nstrments) s ( ) Statstcal flctatons (de to statstcal probablty of takng random samples of fnte nmbers of tems) The ncertanty n the mean s

12 The χ Test When we hae calclated the mean and standard deaton and are confdent of the natre of the parent dstrbton then we can predct the ftre. How do we know that we hae the correct type of dstrbton? Ch sqare (χ ) s defned by χ n [ f( ) P( )] ( f ) where f( ) s the freqency of obseratons,, s the total nmber of measrements, n s the nmber of dfferent measred ales so goes from to n, P( ) s the probablty for obserng, and (f) s the standard deaton assocated wth the freqency, f( ). Usally we don t know what (f) s snce we make only one set of measrements bt the flctatons n f( ) come from the statstcal probabltes of makng random selectons each of fnte nmbers of tems. For a Posson dstrbton, f( )P( ) and χ n [ f( ) P( )] P( )

13 The χ Test Ch sqare s a statstc that characterzes the dsperson of the obsered freqences from epected freqences. The nmerator s a measre of the spread of the obseratons whle the denomnator s a measre of the epected spread. If the freqences agree eactly then χ 0. Larger ales of χ ndcate larger deatons. Ch sqare s sally normalzed by the nmber of degrees of freedom for a Gassan ν n-. χ n χ /ν. Tables ge the probablty that a random sample of data when compared wth ts parent dstrbton wold yeld a ale of normalzed χ as large as the obsered one. The tables are calclated from P χ ( χ, ν ) ( ν ) where Γ s the Gamma fncton. χ z ν Γ e z ( ν ) dz e χ / ( ν ) m 0 ( χ ) m! m

14 The F-Test The χ test s ambgos nless the form of the parent dstrbton s known becase χ measres both the dscrepancy between the estmated fncton and parent fncton bt also deatons between the data and parent fncton. We wold lke a test that separates these. The F-test calclates χ two dfferent ways and compares them to see f the reslts are reasonable. If two ales of ch sqare (χ and χ ) are fond whch follow the χ dstrbton, the rato of normalzed ch-sqares χ ν and χ ν are dstrbted accordng to the F dstrbton, P f (f,ν,ν ) where fχ ν /χ ν The ntegral probablty descrbes the probablty of obserng a large ale of F from a random set of data compared wth the correct fttng fncton. P F ( F, ν, ν ) P ( f ν ν )df f,, F

15 An Eample Dstrbton of poston of Joan bow shock (left) and magnetopase (rght) as a fncton of standoff dstance. Ponts wth error bars are actal obseratons (bns are 4R J ) apart and shapes of bondares were fond from MHD models.) Error bars ge probable error of mean (closest) and mamm range of ales determned by takng 0 sb-samples each wth 0% of the ponts. Reslts hae been ft to sngle Gassan and to bmodal Gassan dstrbtons. Reslts dded nto total data set, solar mamm and solar mnmm. F-test was appled to dstngsh between the sngle Gassan and bmodal dstrbton. For magnetopase the bmodal dstrbton ges a better ft to the 99.9% confdence leel. For the bow shock the bmodal dstrbton s better to the 89.9% leel. Fracton n Solar Wnd Fracton n Solar Wnd Fracton n Solar Wnd a 84 6 b 78 5 c 00 4 Bow Shock All Ma Mn Standoff Dstance (R J ) Fracton n Magnetosphere Fracton n Magnetosphere Fracton n Magnetosphere Magnetopase All Ma Mn d 75 5 e Standoff Dstance (R J ) f 80