Modelling of Clock Behaviour. Don Percival. Applied Physics Laboratory University of Washington Seattle, Washington, USA

Transcription

1 Mdelling f Clck Behaviur Dn Percival Applied Physics Labratry University f Washingtn Seattle, Washingtn, USA verheads and paper fr talk available at 1

2 Overview atmic clcks can keep time t unimaginable precisin but sme peple (clck mdellers!) are never satisfied and insist n fcusing n unimaginable imprecisins will discuss cncepts behind clck mdelling ways in which current mdelling practices can be imprved 2

3 A Thught Experiment suppse we have a clck whse perfrmance we want t evaluate will assume we can cmpare this clck t perfect time at midnight we set the clck t perfect time and measure hw well it des ver the next 24 hurs: 1 nansecnds hurs frm midnight clcks wanders away frm perfect time ver 24 hur perid, ending up abut 5 nansecnds ahead f perfect time n bvius explanatin fr bserved time deviatins 3

4 Secnd Day f Thught Experiment let s d this again! at midnight we reset the clck t perfect time and nw get this: 1 nansecnds hurs frm midnight after 24 hurs, clck is nw abut 3 nansecnds behind again, n bvius explanatin fr bserved time deviatins; hwever, deviatins seem t have the same visual bumpiness 4

5 Thught Experiment After 1 Days we keep n ding this fr 1 days: 1 nansecnds hurs frm midnight can t predict exactly what will happen n any given day can make sme statistical statements abut the nature f the time deviatins ver the 24 hur perid average deviatin after 6 hurs clse t, and 95% f curves are between abut 3 and 3 nansecnds average deviatin after 24 hurs als clse t, but nw 95% f curves are between abut 6 and 6 nansecnds 5

6 Purpse f Clck Mdels clck mdels summarize statistical infrmatin in ur data 1 nansecnds hurs frm midnight let s fcus again n what we bserve each day at 6AM 1 6 hurs.3 nansecnds cunts/ histgram (right-hand plt) ffers sme summary, but we can mre with the help f a theretical distributin 6

7 Gaussian Distributin t the Rescue! ppular theretical distributin is Gaussian (nrmal) bell-shaped curve determined by tw parameters mean, which is set by average ( ) f 1 6AM deviatins variance, which quantifies the fact that mst values ccur between 3 and 3 nansecnds bell-shaped curve shws Gaussian apprximatin: 1 6 hurs.3 nansecnds cunts/ ur simple clck mdel has summarized statistical infrmatin abut 1 6AM measurements using just 2 parameters 7

8 Mdelling Deviatins at 24 Hurs let s nw lk at bservatins after 24 hurs f elapsed time 1 nansecnds hurs frm midnight here are the 1 time deviatins, histgram and Gaussian fit: 1 24 hurs.3 nansecnds cunts/ mean still, but variance is larger, reflecting fact that 95% f the curves are nw rughly between 6 and 6 nansecnds 8

9 Variatin f Gaussian Parameters ver Elapsed Time can repeat the abve prcedure fr all elapsed times frm hurs t 24 hurs let s plt the tw Gaussian parameters versus elapsed time: 1 mean nansecnds -1 1 variance nansecnds hurs frm midnight mean is always clse t, while variance increases apprximately linearly 9

10 Summary Offered by Clck Mdel with help f Gaussian distributin, can summarize univariate statistic prperties bserved in this plt 1 nansecnds hurs frm midnight using just ne variable (a level parameter C determining rate f linear increase in variance)!!! impressive simplificatin, but nly f univariate prperties 1

11 Bivariate Statistical Prperties als interested in relatinship between deviatins at tw distinct elapsed times, e.g., 6 hurs and 7 hurs after midnight: 1 nansecnds hurs frm midnight deviatins at 7 hurs and 6 hurs are psitively crrelated: nansecnds (7 hurs) nansecnds (6 hurs) 11

12 Clck Mdels Incrprating Multivariate Prperties in general, interested in relatinship amngst deviatins at multiple elapsed times, i.e., multivariate statistical prperties interestingly enugh, with help f multivariate Gaussian distributin, can summarize infrmatin with ne mre parameter beynd what we need t summarize univariate prperties!!! additinal parameter α essentially selects a particular pattern fr the increase in variance (this was linear in ur experiment) class f mdels parameterized by C and α knwn as pwer law mdels 12

13 Five Cannical Pwer Law Mdels in practice, parameter α is usually set t ne f five values α = yields mdel knwn as white phase nise α = 1 yields flicker phase nise α = 2 yields randm walk phase nise (ur experiment!) α = 3 yields flicker frequency nise α = 4 yields randm walk frequency nise variance versus elapsed time is cnstant fr α = increases fr α = 1 (but hard t describe exactly!) increases linearly fr α = 2 increases fr α = 3 (again, hard t describe!) increases quadratically fr α = 4 13

14 Deviatins Generated by Cannical Mdels here is ne set f time deviatins frm each mdel: α = -4 α = -3 α = -2 α = -1 α = hurs frm midnight nte: level parameter C merely sets vertical scaling 14

15 Use f Cannical Mdels in Thery when applicable, cannical mdels alng with the multivariate Gaussian distributin ffer a very simple summary f the statistical prperties f time deviatins (nly need C and α) useful fr cmparing statistical prperties f different clcks can use mdels t frm an ensemble time that is better than any individual clck 15

16 Limitatins f Cannical Mdels in Practice picture painted s far f clck mdelling is t simplistic real-wrld cmplicatins make it harder t reap benefits need different cannical mdels fr different elapsed times spacing f α is t carse (mdels exist fr all α) α = -2 α = hurs frm midnight lack f data fr decent parameter estimatin prblem f trend and its estimatin 16

17 Opprtunities fr Imprvements regard α as parameter t be estimated rather than identified (typically preselectin f α nt accunted fr, but this can be an imprtant surce f sampling variability) sampling variability in trend estimates usually nt prpagated prperly lack f attentin t sampling variability in α/trend estimates might explain failure f ptimal prcedures fr frming ensemble time scales much has been dne, but still mre t d! use f prper statistical prcedures will hpefully lead t better perfrmance f systems depending n clcks, but wrk must be dne t see if this is s 17

18 Final Wrd thanks t rganizers fr making this presentatin pssible!!! 18