Modelling of Clock Behaviour
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- June Tate
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1 Mdelling f Clck Behaviur Dnald B. Percival Applied Physics Labratry Bx University f Washingtn Seattle, WA USA Abstract Althugh mdern atmic clcks can keep time with amazing accuracy, there are limits t their capabilities. These limitatins can be assessed with the help f statistical mdels. We present a basic intrductin t the ideas behind the mdelling f clck behaviur and cnsider sme pprtunities fr imprving bth clck mdelling and its use fr interpreting and making efficient use f clck data. 1 Intrductin Mdern atmic clcks can keep time t accuracies that were unimaginable a hundred years ag. Fr systems such as Galile that depend critically n precise time, there is a need t evaluate the perfrmance f individual clcks and ensembles f clcks as accurately as pssible. Any predictable deviatins that an atmic clck makes away frm perfect time culd in principle be crrected fr t imprve the accuracy f the clck; hwever, at sme level f accuracy, all real clcks exhibit unpredictable deviatins frm perfect time. These deviatins are randm in nature, and the descriptin f these deviatins invlves the use f statistical mdels. In Sectin 2 we give a basic intrductin t the ideas behind statistical mdelling f clck behaviur, with an emphasis n what kinds f infrmatin are encapsulated in these mdels. We describe in Sectin 3 a set f five cannical mdels that are cmmnly used as a starting pint fr mdelling clcks. The final sectin (Sectin 4) is devted t a brief discussin f sme pprtunities fr imprving the manner in which clcks are mdelled and the manner in which the mdels are used in practical applicatins such as Galile. 2 What is Clck Mdelling? Let s start with the fllwing thught experiment. We have a clck whse ability t keep time we want t assess. We assume that, whenever we want, we can cmpare the time keep by this clck against sme gd apprximatin t perfect time (in practice this wuld be defined as the time kept by a large ensemble f high precisin clcks). Suppse we set ur clck t agree exactly with perfect time at midnight n a certain day. We then bserve hw ur clck deviates frm perfect time ver the next 24 hurs. The upper panel f Figure 1 shws a plt f these hypthetical deviatins versus elapsed time frm midnight (measured in hurs). We see that ur clck wandered away frm perfect time ver the 24 hur perid dbp@apl.washingtn.edu, Phne: , Fax:
2 1 nansecnds -1 1 nansecnds -1 1 nansecnds hurs frm midnight Figure 1: Simulated deviatins between the time kept by a hypthetical clck and perfect time ver an elapsed perid f a day. The clck is adjusted t agree with perfect time at the beginning f each day. The upper tw plts shw the time deviatins fr the first tw days f this thught experiment, while the lwer plt shws an verlay f the deviatins fr 1 cnsecutive days. A psitive (negative) deviatin means that the clck displayed a time that was ahead (behind) f what a clck keeping perfect time wuld display. and, at the end f the day, had gained abut 5 nansecnds (i.e., a billinth f a secnd; the chice f nansecnds as the unit f measure fr the time difference is entirely arbitrary). Suppse we reset ur clck t agree with perfect time again at midnight and recrd anther set f measurements ver the next 24 hurs. These are shwn in the middle panel f Figure 1. The deviatins are nt the same as what we saw befre. The clck nw ends up lsing abut 3 nansecnds after 24 hurs; hwever, visually the deviatins in the upper and middle plts lk similar in the sense that they bth have a similar bumpiness t them. Suppse we repeat this experiment fr 1 cnsecutive days. The bttm plt shws the 1 series f deviatins that we bserved pltted n tp f each ther. Each individual series tends t have the same bumpiness that we bserved in the first tw days f ur experiment, but each series is unique. Frm this experiment, we learn that we cannt predict exactly the deviatins fr any given day, but the deviatins d have similar characteristics frm day t day. Fr example, the clck never seems t wander mre than abut 1 nansecnds away frm perfect time, but, n the ther hand, it usually wanders at least 1 nansecnds away within 6 hurs f its being reset t perfect time. Under the assumptin that the mechanism
3 1 6 hurs.3 nansecnds cunts/ hurs..3 nansecnds cunts/ day f bservatin -1 1 nansecnds Figure 2: Simulated time deviatins f a clck frm perfect time at tw fixed elapsed times frm midnight ver a 1 day perid, alng with histgrams fr the deviatins and a fitted Gaussian (nrmal) distributin. The upper left-hand plt shws the time deviatins at an elapsed time f 6 hurs (i.e., 6AM) fr 1 cnsecutive days, while the upper right-hand plt shws the crrespnding histgram (staircase curve) and fitted theretical Gaussian distributin (bell-shaped curve). The bttm rw shws the same plts fr the time deviatins at an elapsed time f 24 hurs. that is respnsible fr the bserved deviatins des nt change frm day t day, we can use ur cllectin f measurements t make sme statistical statements abut the nature f the time deviatins within a 24 hur perid. Fr example, the average deviatin after 6 hurs is clse t zer, with 95% f the individual curves ranging rughly between 3 and +3 nansecnds. After 24 hurs, the average deviatin is still clse t zer, but nw 95% f the curves range rughly between 6 and +6 nansecnds. By cllecting mre and mre data, we can build up a cmplete statistical descriptin f hw ur clck tends t deviate frm perfect time ver a perid f a day. The purpse f a clck mdel is t summarize the infrmatin cllected in experiments like the abve. As an example, let us fcus again n the deviatins that we bserve at 6AM n each f the 1 days. These 1 deviatins are pltted in the upper left-hand plt f Figure 2 versus the number f the day n which they were recrded, and a histgram fr these deviatins is given by the staircase curve in the upper right-hand plt. The histgram ffers a summary f the distributin f the 1 6AM deviatins and shws that indeed the deviatins are centered arund zer and that mst f them ccur between 3 and 3 nansecnds. We can further summarize the bserved distributin by using a theretical distributin that depends upn a small number f parameters. One theretical distributin that has been used extensively is the Gaussian (r nrmal) distributin, which depends n tw parameters, namely, the mean and the variance. The mean is determined by the average f the 1 6AM deviatins and is clse t zer. The variance is a measure f hw spread ut the deviatins are and quantifies the fact that mst f the values ccur between 3 and
4 1 mean nansecnds -1 1 variance nansecnds hurs frm midnight Figure 3: Sample means and variances fr the 1 time deviatin series shwn in the bttm plt f Figure 1 versus elapsed time. The tp plt indicates that the means are clse t zer fr all elapsed times, whereas the jagged curve in the bttm plt shws that the variances tend t be apprximately prprtinal t elapsed time (the line n this plt is a linear apprximatin t the jagged curve). 3 nansecnds. The larger the variance is, the mre spread ut the assciated histgram wuld be. The fitted Gaussian distributin fr the 1 6AM deviatins is shwn by the bell-shaped curve in the upper right-hand plt. Thus, if we were t cntemplate cntinuing ur experiment beynd 1 days, we cannt predict what value we wuld bserve at 6AM n any given future day, but, under the assumptin that what we will bserve in the future will be similar t what we bserved in the past, the simple tw parameter mdel that we have derived can be used t answer questins abut hw likely any future 6AM measurement is t be, say, within a certain interval (e.g., 1 and 1 nansecnds). We can bviusly use a similar prcedure t mdel the deviatins cllected at elapsed times besides 6 hurs after midnight. The bttm plts f Figure 2 shw crrespnding results fr ur hypthetical experiment after 24 hurs. Nte that the Gaussian distributin is mre spread ut than in the 6 hur case, which is reflected in the fact that the variance is larger. We can create similar mdels fr any given elapsed time between and 24 hurs. Figure 3 shws the mean (upper plt) and variance (jagged curve in the lwer plt) fr the fitted Gaussian distributins as a functin f elapsed time. We see that, while the mean is always clse t zer, the variance tends t increase with time in a linear manner. This result pens up the pssibility f summarizing the means and variances ver all elapsed times between and 24 hurs. The summary cnsists f a Gaussian distributin assciated with each elapsed time, with a mean f zer fr each distributin and a variance that is prprtinal t the elapsed time; i.e., we have biled dwn the means and variances t a single parameter, namely, the cnstant f prprtinality that sets the actual value f the variance (this parameter is essentially the slpe f the line shwn in the bttm plt). The Gaussian distributins that we have cnsidered s far are all univariate distributins;
5 1 nansecnds (7 hurs) nansecnds (6 hurs) Figure 4: Plt f the bserved time deviatin at 7AM versus the crrespnding deviatin at 6AM fr each f the 1 days ver which we cnducted the thught experiment recrded in Figure 1. The tw time deviatins tend t be psitively crrelated; i.e., a large psitive r negative deviatin at 6AM tends t fllwed by a similar deviatin at 7AM. i.e., they can tell us smething abut where we might expect future deviatins t be at a given elapsed time, but they dn t tell us anything abut hw bserved deviatins at distinct elapsed time are related t ne anther. Figure 4 is a first lk at the multivariate prperties f ur experiment. Here we have pltted the time deviatin bserved at 7AM versus the deviatin at 6AM fr all 1 such pairs f data. We see that a large psitive r negative value at 6AM tends t be assciated with a simular value at 7PM; i.e., the time deviatins at these tw elapsed time are psitively crrelated. In additin t univariate prperties, clck mdels are designed t summarize bivariate prperties such as the psitive crrelatin indicated in Figure 4 and als crrespnding higher rder multivariate prperties invlving the statistical relatinship between the time deviatins at three r mre distinct elapsed time. 3 Five Cannical Clcks Mdels Since the 196s, the standard way f frmulating a multivariate clck mdel that can usefully summarize all the salient statistical prperties f data similar t that in ur thught experiment is via a cannical set f five s-called pwer law mdels. The five mdels are classified by a parameter ften dented by the symbl α (this is actually an expnent that determines the manner in which a theretical quantity knwn as the pwer spectral density functin decays as a functin f a variable knwn as Furier frequency). Fr the cannical mdels the parameter α can assume ne f 5 values, namely,, 1, 2, 3 and 4. The crrespnding mdels are knwn as, respectively, white phase nise (α = ), flicker phase nise (α = 1), randm walk phase nise (α = 2), flicker frequency nise (α = 3) and randm walk frequency nise (α = 4). Each f the five mdels in additin depends upn ne additinal parameter, which we will dente as C and refer t as the level parameter. Specificatin f a cannical clck mdel cnsists f determining which f the 5 values f the pwer law parameter α is mst apprpriate and then setting the level parameter C.
6 α = -4 α = -3 α = -2 α = -1 α = hurs frm midnight Figure 5: Like the tp plt f Figure 1, but nw fr a hypthetical clck well mdelled by, frm tp t bttm, white phase nise (α = ), flicker phase nise (α = 1), randm walk phase nise (α = 2), flicker frequency nise (α = 3) and randm walk frequency nise (α = 4). The clck mdel behind ur thught experiment is actually randm walk phase nise (α = 2), s the time deviatins in Figure 1 can be regarded as examples f what we can expect t see frm a clck that is well-mdelled by such nise. Figure 5 shws ne example f what we might see in ne replicatin f ur thught experiment in which the clck under assessment is well-mdelled by ne f the five cannical mdels. Visually, these five mdels yield time deviatins, each f whse characteristic bumpiness is rather distinct frm the ther fur. The deviatins are very erratic lking fr α = and 1 and are much smther fr α = 3 and 4. The usual frmulatin f the cannical mdels is based n a multivariate versin f the Gaussian distributin. The assciated univariate distributins all have means f zer at each elapsed time, but the chice f α sets the way in which the variance changes ver time. When α = 2, we saw frm Figure 3 that the variance grws linearly with elapsed time;
7 fr α =, the variance is a cnstant fr all elapsed time; and, fr α = 4, the variance grws quadratically (fr α = 1 and α = 3, the variance grws with elapsed time, but in a manner that is nt easy t describe). The level parameter C just adjusts these variance curves t have the prper values. In effect, C des nthing mre than t specify the labelling n the vertical axes in plts like thse in Figure 1. It des nt influence the characteristic bumpiness assciated with a particular α, but des determine whether r nt the deviatins after 24 hurs span, e.g., a few micrsecnds, tens f nansecnds r a cuple f picsecnds. The chice f a cannical mdel fr a clck cnsists f setting values fr α and C. Once these tw values are chsen, if the mdel is a gd ne fr a particular clck, we have all the infrmatin that we need t assess the statistical prperties f its time deviatins. This infrmatin can be used t assess hw well a particular clck is ding relative t ther clcks. In particular, if we have mdelled an entire ensemble f clcks, we can use these mdels t determine hw best t cmbine them tgether t generate an ensemble time that is mre accurate than the time generated by any ne clck all this frm knwing just tw values fr each clck! 4 Opprtunities fr Imprving Clcks Mdels and Their Usage In practice, the picture that we have painted s far f clck mdelling has been t simplistic. There are at least fur real-wrld cmplicatins that make it harder t reap the ptential benefits f clck mdelling. 1. It has lng been recgnized that a gd chice fr a cannical mdel fr an actual clck can depend upn the elapsed time; i.e., a mdel that is apprpriate fr a maximum elapsed time f a day is usually nt als apprpriate fr maximum elapsed times f a secnd r a year. T get arund this prblem, the cmmn practice has been t entertain pssibly different cannical mdels fr different elapsed times fr a given clck. In practice, tw numbers are usually nt enugh t adequately mdel an actual clck. 2. There are sme clcks whse time deviatins are nt well described by any ne f the five cannical bumpiness curves in Figure 5. There are n cmpelling physical arguments t supprt use f these particular five mdels, and their prevalence is smewhat f an histrical artifact. In fact, there is a well-defined multivariate Gaussian mdel assciated with all pssible values fr α, and the nes fr which α is between, say, 1 and 2 wuld generate time deviatins whse bumpiness ffers a smth transitin between the secnd and third plts in Figure 5; i.e., there is a big qualitative jump between flicker phase nise (α = 1) and randm walk phase nise (α = 2) that can be bridged if we were t entertain α s between these tw numbers. 3. In ur thught experiment, we were able t cllect enugh data s that we culd reliably make statements abut the statistical prperties f ur hypthetical clck ver the span f a day. If we increase the maximum elapsed time t a year, we culd never cllect a similar amunt f data (i.e., 1 years wrth) fr any actual clck. Faced with a lack f data, determinatin f mdel parameters becmes a very challenging prblem.
8 4. Our thught experiment cncentrated n the situatin where the time deviatins were truly randm in the sense that there was n tendency fr ur theretical clck t cnsistently gain r lse time ver a given elapsed time; i.e., the average f the deviatins was clse t zer, as demnstrated in the tp plt f Figure 3. In practice, this is nt always the case. Any average deviatin that is nt clse t zer is an indicatin f a trend in the clck. Trend is f interest because, if prperly determined, we can enhance the perfrmance f any clck by merely adjusting its displayed time by subtracting ff an amunt dictated by the trend. Usually trend must be estimated frm the data, and this becmes a particularly challenging prblem when we are faced with a limited amunt f data. In view f the abve cmplicatins, there are a number f ways in which the mdelling f clcks can be imprved upn t bring it mre int line with mdern statistical practice. The first has t d with limiting urselves t just five cannical mdels. Rather than insisting upn setting α t a cannical value, we can estimate it frm a cntinuum f values. Explicit estimatin f α wuld help alleviate a statistical prblem with the cmmn practice f lking at a series f time deviatins and selecting a cannical α by sme means, after which the analysis cntinues under the assumptin that α was knwn in advance; i.e., it is assumed that we d nt need t cnsider the effect f sampling variability n the selectin f α. The cnfidence bunds that are attached t estimates f cmmn clck stability measures (e.g., the Allan variance) invariably are based n the ntin that α is knwn and that all sampling variability is due t uncertainty abut the level parameter C. When we have a limited amunt f data, determinatin f α becmes prblematic and is prne t the vagaries f sampling errrs, but current practice is t ignre this ptentially imprtant surce f variability. In a similar manner, estimatin f trends in time deviatins is usually nt handled in a manner such that the uncertainty in the estimated trend is prpagated thrugh t ther quantities; i.e., the detrended time deviatins are ften treated as if they were the actual measurements, and clck stability measures are estimated frm them. An assessment f uncertainty in these estimated measures des nt take int accunt uncertainties in estimating the trend. While treating α and/r the trend cmpnent as if they were nt subject t sampling errrs is cnvenient frm the pint f view f cmputatins, it is certainly questinable frm a statistical pint f view. Finally let us ffer sme speculatins abut the use f clck mdels in the generatin f time scales frm ensembles f clcks. An elegant mathematical apprach fr frming such scales is based upn the Kalman filter. The usual Kalman filter, hwever, makes the implicit assumptin that a mdel fr particular clck is knwn perfectly; i.e., any sampling variability due t nly having a finite amunt f data t determine α and C is ignred by the Kalman filter. Even thugh the Kalman filter is knwn as an ptimal filtering prcedure, it is nly s under the unrealistic assumptin f perfectly knwn clck mdels. Amngst thse wh have studied the frmatin f time scales, the Kalman filter is regarded as prblematic and inferir t certain ad hc prcedures that appear t frm better time scales than this scalled ptimal prcedure. It is ur cntentin that this dissatisfactin with the Kalman filter is well-funded and is tied t the fact that uncertainty in estimated parameters is nt being handled prperly. This deficiency is particularly evident in an peratinal envirnment in which clcks are added t a time scale after a relatively shrt calibratin perid, during
9 which mdels are determined with a paucity f data and hence have parameter estimates with necessarily nnnegligible sampling errrs. What seems t be called fr is a refrmulatin f the Kalman filter that takes int accunt this imprtant surce f uncertainty. Thus clcks with seemingly very nice statistical prperties wuld be dwnweighted apprpriately if there is large uncertainty in their estimated parameters.
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