Mdelling f Clck Behaviur Dn Percival Applied Physics Labratry University f Washingtn Seattle, Washingtn, USA verheads and paper fr talk available at http://faculty.washingtn.edu/dbp/talks.html 1
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
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 -1 6 12 18 24 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
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 -1 6 12 18 24 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
Thught Experiment After 1 Days we keep n ding this fr 1 days: 1 nansecnds -1 6 12 18 24 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
Purpse f Clck Mdels clck mdels summarize statistical infrmatin in ur data 1 nansecnds -1 6 12 18 24 hurs frm midnight let s fcus again n what we bserve each day at 6AM 1 6 hurs.3 nansecnds cunts/1.2.1-1. histgram (right-hand plt) ffers sme summary, but we can mre with the help f a theretical distributin 6
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/1.2.1-1. ur simple clck mdel has summarized statistical infrmatin abut 1 6AM measurements using just 2 parameters 7
Mdelling Deviatins at 24 Hurs let s nw lk at bservatins after 24 hurs f elapsed time 1 nansecnds -1 6 12 18 24 hurs frm midnight here are the 1 time deviatins, histgram and Gaussian fit: 1 24 hurs.3 nansecnds cunts/1.2.1-1. mean still, but variance is larger, reflecting fact that 95% f the curves are nw rughly between 6 and 6 nansecnds 8
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 2 6 12 18 24 hurs frm midnight mean is always clse t, while variance increases apprximately linearly 9
Summary Offered by Clck Mdel with help f Gaussian distributin, can summarize univariate statistic prperties bserved in this plt 1 nansecnds -1 6 12 18 24 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
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 -1 6 12 18 24 hurs frm midnight deviatins at 7 hurs and 6 hurs are psitively crrelated: nansecnds (7 hurs) 1-1 -1 1 nansecnds (6 hurs) 11
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
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
Deviatins Generated by Cannical Mdels here is ne set f time deviatins frm each mdel: α = -4 α = -3 α = -2 α = -1 α = 6 12 18 24 hurs frm midnight nte: level parameter C merely sets vertical scaling 14
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
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 = -1 6 12 18 24 hurs frm midnight lack f data fr decent parameter estimatin prblem f trend and its estimatin 16
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
Final Wrd thanks t rganizers fr making this presentatin pssible!!! 18