A Tutorial on Stochastic Models and Statistical Analysis for Frequency Stability Measurements

Transcription

1 A Tutorial on Stochastic Models and Statistical Analysis for Frequency Stability Measurements Don Percival Applied Physics Lab, University of Washington, Seattle overheads for talk available at

2 Introduction time scales limited by clock noise can model clock noise as stochastic process {X t } set of random variables (RVs) indexed by t X t represents clock noise at time t will concentrate on sampled data, for which will take t Z {..., 1, 0, 1,...} (but sometimes use t Z {0, 1, 2,...}) Q: which stochastic processes are useful models? Q: how can we deduce model parameters & other characteristics from observed data? will cover the following in this tutorial: stationary processes & closely related processes fractionally differenced & related processes two analysis of variance ( power ) techniques spectral analysis wavelet analysis parameter estimation via analysis techniques 1

3 Stationary Processes: I stochastic process {X t } called stationary if E{X t } = µ X for all t; i.e., a constant that does not depend on t cov{x t,x t+τ } = s X,τ, all possible t & t + τ; i.e., depends on lag τ, but not t {s X,τ : τ Z} is autocovariance sequence (ACVS) s X,0 =cov{x t,x t } =var{x t }; i.e., process variance is constant for all t spectral density function (SDF) given by S X (f) = τ= s X,τ e i2πfτ, f 1/2 note: S X ( f) =S X (f) for real-valued processes 2

4 Stationary Processes: II if {X t } has SDF S X ( ), then 1/2 1/2 S X(f)e i2πfτ df = s X,τ, τ Z setting τ = 0 yields fundamental result: 1/2 S 1/2 X(f) df = s X,0 =var{x t }; i.e., SDF decomposes var {X t } across frequencies f if {a u } is a filter, then (with matching condition ) Y t u= a u X t u is stationary with SDF given by S Y (f) =A(f)S X (f), where A(f) u= a u e i2πfu 2 if {a u } narrow-band of bandwidth f about f, i.e., A(f )= 1 2 f, f f 2 f f + f 2 0, otherwise, then have following interpretation for S X (f): var {Y t } = 1/2 1/2 S Y (f ) df = 1/2 1/2 A(f )S X (f ) df S X (f) 3

5 White Noise Process simplest stationary process is white noise {ɛ t } is white noise process if E{ɛ t } = µ ɛ for all t (usually take µ ɛ =0) var {ɛ t } = σɛ 2 for all t cov {ɛ t,ɛ t } = 0 for all t t white noise thus stationary with ACVS and SDF s ɛ,τ =cov{ɛ t,ɛ t+τ } = σɛ 2, τ =0; 0, otherwise, S ɛ (f) = τ= s X,τ e i2πfτ = σ 2 ɛ 4

6 Backward Differences of White Noise consider first order backward difference of white noise: 1, u =0; X t = ɛ t ɛ t 1 = a u ɛ t u with a u 1, u =1; u= 0, otherwise. have S X (f) =A(f)S ɛ (f) = 2 sin(πf) 2 σ 2 ɛ 2πf 2 σ 2 ɛ at low frequencies (using sin(x) x for small x) let B be backward shift operator: Bɛ t = ɛ t 1, B 2 ɛ t = ɛ t 2,(1 B)ɛ t = ɛ t ɛ t 1, etc. consider dth order backward difference of white noise: X t =(1 B) d ɛ t = d d ( 1) k ɛ t k k=0 k = d d! k!(d k)! ( 1)k ɛ t k k=0 = k=0 with δ d, i.e., δ = 1, 2,... SDF given by S X (f) =A(f)S ɛ (f) = Γ(1 δ) Γ(k + 1)Γ(1 δ k) ( 1)k ɛ t k σ 2 ɛ 2 sin(πf) 2δ σ2 ɛ 2πf 2δ 5

7 Fractional Differences of White Noise for δ not necessary an integer, X t = k=0 Γ(1 δ) Γ(k + 1)Γ(1 δ k) ( 1)k ɛ t k makes sense as long as δ<1/2 k=0 {X t } stationary fractionally differenced (FD) process SDF is as before: S X (f) = σ 2 ɛ 2 sin(πf) 2δ σ2 ɛ 2πf 2δ {X t } said to obey power law at low frequencies if S X (f) lim f 0 C f =1 α for C>0; i.e., S X (f) C f α at low frequencies FD processes obey above with α = 2δ note: FD process reduces to white noise when δ =0 a k (δ)ɛ t k 6

8 ACVS & PACS for FD Processes for δ<1/2 &δ 0, 1,..., ACVS given by s X,τ = σɛ 2 sin(πδ)γ(1 2δ)Γ(τ + δ) ; πγ(1 + τ δ) when δ =0, 1,...,haves X,τ = 0 for τ > δ & s X,τ = σɛ 2 ( 1) τ Γ(1 2δ), 0 τ δ Γ(1 + τ δ)γ(1 τ δ) for all δ<1/2, have s X,0 =var{x t } = σɛ 2 Γ(1 2δ) Γ 2 (1 δ), and rest of ACVS can be computed easily via τ + δ 1 s X,τ = s X,τ 1, τ Z + {1, 2,...} τ δ (for negative lags τ, recall that s X, τ = s X,τ ). for all δ < 1/2, partial autocorrelation sequence (PACS) given by φ t,t δ t δ, t Z+ (useful for constructing best linear predictors) FD processes thus have simple and easily computed expressions for SDF, ACVS and PACS 7

9 Simulating Stationary FD Processes for 1 δ<1/2, can obtain exact simulations via circulant embedding (Davies Harte algorithm) given s X,0,...,s X,N, use discrete Fourier transform (DFT) to compute S k N τ=0 s X,τ e i2πfkτ + 2N 1 τ=n+1 s X,2N τ e i2πf kτ, k =0,...,N given 2N independent Gaussian deviates ε t with mean zero and variance σɛ 2, compute ε 0 2NS0, k =0; (ε Y k 2k 1 + iε 2k ) NS k, 1 k<n; ε 2N 1 2NSN, k = N; Y2N k, N < k 2N 1; (asterisk denotes complex conjugate) use inverse DFT to construct the real-valued sequence Y t = 1 2N 1 Y k e i2πfkt, t =0,...,2N 1 2N k=0 Y 0,Y 1,...,Y N 1 is exact simulation of FD process implication: can represent X 0,X 1,...,X N 1 as X t = 2N 1 c t,k (δ)ε k rather than X t = a k (δ)ɛ t k k=0 k=0 8

10 Nonstationary FD Processes: I suppose X t (1) is FD process with parameter δ (s) such that 1/2 δ (s) < 1/2 define X t,t Z, as cumulative sum of X (1) t,t Z : (for l<0, let X t 0) since, for t Z, X t t l=0 X (1) l X (1) t = X t X t 1 & S X (1)(f) = filtering theory suggests using relationship S X (1)(f) = 2 sin(πf) 2 S X (f) to define SDF for X t, i.e., S X (f) = σ 2 ɛ 2 sin(πf) 2δ(s), S X (1) (f) 2 sin(πf) 2 = σ 2 ɛ 2 sin(πf) 2δ with δ δ (s) + 1 (Yaglom, 1958) 9

11 Nonstationary FD Processes: II X t has stationary 1st order backward differences 1 sum defines FD processes for 1/2 δ<3/2 2 sums define FD processes for 3/2 δ<5/2, etc X t has stationary 2nd order backward differences, etc if X (1) t is white noise (δ (s) =0)soS X (1)(f) =σ 2 ɛ, then X t is random walk (δ = 1) with S X (f) = σ 2 ɛ 2 sin(πf) 2 σ2 ɛ 2πf 2 if X (2) t is white noise and if X (1) t t l=0 X (2) l & X t t l=0 then X t is random run (δ = 2), and S X (f) σ2 ɛ 2πf 4 X (1) l, t Z, 10

12 Summary of FD Processes X t said to be FD process if its SDF is given by S X (f) = σ 2 ɛ 2 sin(πf) 2δ σ2 ɛ 2πf 2δ at low frequencies well-defined for any real-valued δ FD process obeys power law at low frequencies with exponent α = 2δ if δ<1/2, FD process stationary with ACVS given by s X,0 = σɛ 2 Γ(1 2δ) Γ 2 (1 δ) & s τ + δ 1 X,τ = s X,τ 1, τ Z + τ δ PACS given by φ t,t δ t δ, t Z+ if δ 1/2, FD process nonstationary but its dth order backward difference is stationary FD process with parameter δ (s), where d δ +1/2 and δ (s) δ d (here x is largest integer x) 11

13 Alternatives to FD Processes: I fractional Brownian motion (FBM) B H (t), 0 t<, has SDF given by S BH (t)(f) = σ2 XC H f 2H+1, <f<, where σx 2 > 0, C H > 0&0<H<1 (H called Hurst parameter; C H depends on H) power law with 3 <α< 1 discrete fractional Brownian motion (DFBM) B t,t Z +, is DFBM if B t = B H (t) B t has SDF given by S Bt (f) =σ 2 XC H j= 1 f + j 2H+1, f 1/2 power law at low frequencies with 3 <α< 1 reduces to random walk if H =1/2 12

14 Alternatives to FD Processes: II fractional Gaussian noise (FGN) X t,t Z +,isfgnifx t = B t+1 B t X t has SDF given by S X (f) =4σXC 2 H sin 2 1 (πf) j= f + j 2H+1, f 1/2 power law at low frequencies with 1 <α<1 X t is stationary, with ACVS given by s X,τ = σ2 ( X τ +1 2H 2 τ 2H + τ 1 2H ), τ Z, 2 where σx 2 =var{x t } reduces to white noise if H =1/2 discrete pure power law (PPL) process SDF given by S X (f) =C S f α, f 1/2 if α > 1, stationary, but ACVS takes some effort to compute if α = 0, reduces to white noise α 1, nonstationary but backward differences of certain order are stationary 13

15 FD Processes vs. Alternatives FD processes cover full range of power laws FBMs, DFBMs and FGNs cover limited range PPL processes also cover full range differencing FD process yields another FD process; differencing alternatives yields new type of process FD process has simple SDF; if stationary, has simple ACVS & PACS FBM has simple SDF DFBM has complicated SDF FGN has simple ACVS, complicated SDF & PACS PPL has simple SDF, complicated ACVS & PACS FD, DFBM, FGN and PPL model sampled noise might be problematic to change sampling rate FBM models unsampled noise Fig. 1: comparison of SDFs for FGN, PPL & FD Fig. 2: comparison of realizations 14

16 Extensions to FD Processes: I composite FD processes S X (f) = M σm 2 m=1 2 sin(πf) ; 2δ m i.e., linear combinations of independent FD processes autoregressive, fractionally integrated, moving average (ARFIMA) processes idea is to replace ɛ t in X t = k=0 with ARMA process, say, U t = p k=1 yields process with SDF S X (f) = a k (δ)ɛ t k φ k U t k + ɛ t q σ 2 ɛ 2 sin(πf) 2δ k=1 θ k ɛ t k 1 q k=1 θ k e i2πfk 2 1 p k=1 φ k e i2πfk 2 ARMA part can model, e.g., high-frequency structure in noise 15

17 Extensions to FD Processes: II can define time-varying FD (TVFD) process via X t = k=0 as long as δ t < 1/2 for all t can use representation X t = 2N 1 k=0 a k (δ t )ɛ t k c t,k (δ t )ε k, t =0, 1,...,N 1, to extend definition to handle arbitrary δ t Fig. 3: realizations from 4 TVFD processes can also make σɛ 2 time-varying 16

18 FD Process Parameter Estimation Q: given realization (clock noise) of X 0,...,X N 1 from FD process, how can we estimate δ & σ 2 ɛ? many different estimators have been proposed! (area of active research) will concentrate on estimators based on spectral analysis (frequency-based) wavelet analysis (scale-based) advantages of spectral and wavelet analysis physically interpretable both are analysis of variance techniques (useful for more than just estimating δ & σɛ 2 ) can assess need for models more complex than simple FD process (e.g., composite FD process) provide preliminary estimates for more complicated schemes (maximum likelihood estimation) 17

19 Estimation via Spectral Analysis recall that SDF for FD process given by and thus S X (f) = σ 2 ɛ 2 sin(πf) 2δ log (S X (f)) = log (σ 2 ɛ ) 2δ log ( 2 sin(πf) ); i.e., plot of log (S X (f)) vs. log ( 2 sin(πf) ) linear with slope of 2δ for 0 <f<1/8, have sin(πf) πf, so log (S X (f)) log (σ 2 ɛ ) 2δ log (2πf); i.e., plot of log (S X (f)) vs. log (2πf) approximately linear at low frequencies with slope of 2δ = α basic scheme estimate S X (f) via ŜX(f) fit linear model to ŜX(f) vs. log (2πf) over low frequencies use estimated slope ˆα to estimate δ via ˆα/2 use estimated intercept to estimate σɛ 2 18

20 The Periodogram: I basic estimator of S(f) is periodogram: Ŝ (p) (f) 1 N N 1 t=0 X t e i2πft 2, f 1/2; represents decomposition of sample variance: 1/2 1/2 Ŝ(p) (f) df = 1 N N 1 t=0 X 2 t for stationary processes & large N, theory says Ŝ (p) (f) = d S(f) χ2 2 2, 0 <f<1/2, approximately, implying that E{Ŝ(p) (f)} E{S(f)χ 2 2/2} = S(f) var {Ŝ(p) (f)} var {S(f)χ 2 2/2} = S 2 (f) (in above d = means equal in distribution, and χ 2 2 is chi-square RVwith 2 degrees of freedom) additionally, cov {Ŝ(p) (f j ), Ŝ(p) (f k )} 0 for f j j/n &0<f j <f k < 1/2 19

21 The Periodogram: II taking log transform yields log (Ŝ(p) (f)) = d log S(f) χ2 2 = log (S(f))+log 2 Bartlett & Kendall (1946): E log χ 2 η η = ψ(η) log (η) & var log χ 2 η η χ 2 2 where ψ( ) &ψ ( ) are di & trigamma functions yields E{log (Ŝ(p) (f))} = log (S(f)) + ψ(2) log (2) = log (S(f)) γ var{log (Ŝ(p) (f))} = ψ (2) = π 2 /6 (γ. = is Euler s constant) 2 = ψ (η) 20

22 The Periodogram: III define Y (p) (f j ) log (Ŝ(p) (f j )) + γ can model Y (p) (f j )as Y (p) (f j ) log (S(f j )) + ɛ(f j ) log (σɛ 2 ) 2δ log (2πf j )+ɛ(f j ) over low frequencies indexed by 0 <j<j error ɛ(f j ) in linear regression model such that E{ɛ(f j )} =0&var{ɛ(f j )} = π 2 /6 (known!) if {X t } Gaussian & Ŝ(p) (f j ) s uncorrelated, then ɛ(f j ) s pairwise uncorrelated ɛ(f j ) d = log (χ 2 2) markedly non-gaussian least squares procedure yields estimates ˆδ and ˆσ 2 ɛ for δ and σ 2 ɛ estimates of variability in ˆδ and ˆσ 2 ɛ 21

23 Multitaper Spectral Estimation: I warnings about periodogram: approximations might require N to be very large! approximations of questionable validity for nonstationary FD processes Fig. 4: periodogram can suffer from leakage tapering is technique for alleviating leakage: Ŝ (d) (f) N 1 t=0 a t X t e i2πft 2 {a t } called data taper (typically bell-shaped curve) Ŝ(d) ( ) called direct spectral estimator critique: loses information at end of series (sample size N effectively shortened) Thomson (1982): multitapering recovers lost info use set of K orthonormal data tapers {a n,t }: N 1 t=0 a n,t a l,t = 1, if n = l; 0, if n l. 0 n, l K 1 22

24 Multitaper Spectral Estimation: II use {a n,t } to form kth direct spectral estimator: Ŝ (mt) k (f) N 1 t=0 a n,t X t e i2πft 2, n =0,...,K 1 simplest form of multitaper SDF estimator: Ŝ (mt) (f) 1 K K 1 n=0 Ŝ n (mt) (f) sinusoidal tapers are one family of multitapers: 1/2 2 a n,t = sin (N +1) (Riedel & Sidorenko, 1995) (n +1)π(t +1) N +1, Figs. 5 and 6: example of multitapering if S( ) slowly varying around S(f) &ifn large, Ŝ (mt) (f) d = S(f)χ2 2K 2K approximately for 0 <f<1/2, impling var {Ŝ(mt) (f)} S2 (f) 4K 2 var {χ2 2K} = S2 (f) K t =0,...,N 1 23

25 Multitaper Spectral Estimation: III define Y (mt) (f j ) log (Ŝ(mt) (f j )) ψ(k) + log (K) can model Y (mt) (f j )as Y (mt) (f j ) log (S(f j )) + η(f j ) log (σɛ 2 ) 2δ log (2πf j )+η(f j ) over low frequencies indexed by 0 <j<j error η(f j ) in linear regression model such that E{η(f j )} =0 var {η(f j )} = ψ (K), a known constant! approximately Gaussian if K 5 correlated, but with simple structure: cov{η(f j ),η(f j+ν )} ψ (K) ( 1 ν ) K+1, if ν K +1; 0, otherwise. generalized least squares procedure yields estimates ˆδ and ˆσ 2 ɛ for δ and σ 2 ɛ estimates of variability in ˆδ and ˆσ 2 ɛ multitaper approach superior to periodogram approach 24

26 Discrete Wavelet Transform (DWT) let X =[X 0,X 1,...,X N 1 ] T be observed time series (for convenience, assume N integer multiple of 2 J 0) let W be N N orthonormal DWT matrix W = WX is vector of DWT coefficients orthonormality says X = W T W,soX W can partition W as follows: W = W 1. W J0 V J0 W j contains N j = N/2 j wavelet coefficients related to changes of averages at scale τ j =2 j 1 (τ j is jth dyadic scale) related to times spaced 2 j units apart V J0 contains N J0 = N/2 J 0 scaling coefficients related to averages at scale λ J0 =2 J 0 related to times spaced 2 J 0 units apart 25

27 Example: Haar DWT Fig. 7: W for Haar DWT with N =16 first 8 rows yield W 1 changes on scale 1 next 4 rows yield W 2 changes on scale 2 next 2 rows yield W 3 changes on scale 4 next to last row yields W 4 change on scale 8 last row yields V 4 average on scale 16 Fig. 8: Haar DWT coefficients for clock

28 DWT in Terms of Filters filter X 0,X 1,...,X N 1 to obtain 2 j/2 W j,t L j 1 l=0 h j,l X t l mod N, t =0, 1,...,N 1; h j,l is jth level wavelet filter (note: circular filtering) subsample to obtain wavelet coefficients: W j,t =2 j/2 W j,2 j (t+1) 1, t =0, 1,...,N j 1, where W j,t is tth element of W j Figs. 9 & 10: four sets of wavelet filters jth wavelet filter is band-pass with pass-band [ 1 2 j+1, 1 2 j ] (i.e., scale related to interval of frequencies) similarly, scaling filters yield V J0 Figs. 11 & 12: four sets of scaling filters J 0 th scaling filter is low-pass with pass-band [0, 1 2 J 0 +1 ] as width L of 1st level filters increases, band-pass & low-pass approximations improve # of embedded differencing operations increases (related to # of vanishing moments ) 27

29 DWT-Based Analysis of Variance consider energy in time series: X 2 = X T X = N 1 t=0 X 2 t energy preserved in DWT coefficients: W 2 = WX 2 = X T W T WX = X T X = X 2 since W 1,...,W J0, V J0 partitions W, have W 2 = J 0 j=1 W j 2 + V J0 2, leading to analysis of sample variance: ˆσ 2 1 N N 1 t=0 X 2 t = 1 N J 0 j=1 W j 2 + V J0 2 scale-based decomposition (cf. frequency-based) 28

30 Variation: Maximal Overlap DWT can eliminate downsampling and use W j,t 1 2 j/2 L j 1 l=0 h j,l X t l mod N, t =0, 1,...,N 1 to define MODWT coefficients W j (& also Ṽj) unlike DWT, MODWT is not orthonormal (in fact MODWT is highly redundant) like DWT, can do analysis of variance because X 2 = J 0 j=1 W j 2 + ṼJ 0 2 unlike DWT, MODWT works for all samples sizes N (i.e., power of 2 assumption is not required) Fig. 13: Haar MODWT coefficients for clock 571 (cf. Fig. 8 with DWT coefficients) can use to track time-varying FD process 29

31 Definition of Wavelet Variance let X t, t Z, be a stochastic process run X t through jth level wavelet filter: W j,t L j 1 l=0 h j,l X t l, t Z definition of time dependent wavelet variance (also called wavelet spectrum): ν 2 X,t(τ j ) var {W j,t }, assuming var {W j,t } exists and is finite ν 2 X,t(τ j ) depends on τ j and t will consider time independent wavelet variance: ν 2 X(τ j ) var {W j,t } (can be easily adapted to time varying situation) rationale for wavelet variance decomposes variance on scale by scale basis useful substitute/complement for SDF 30

32 Variance Decomposition suppose X t has SDF S X (f): 1/2 1/2 S X(f) df =var{x t }; i.e., decomposes var {X t } across frequencies f involves uncountably infinite number of f s S X (f) f contribution to var {X t } due to f s in interval of length f centered at f note: var {X t } taken to be for nonstationary processes with stationary backward differences wavelet variance analog to fundamental result: j=1 ν 2 X(τ j )=var{x t } i.e., decomposes var {X t } across scales τ j recall DWT/MODWT and sample variance involves countably infinite number of τ j s νx(τ 2 j ) contribution to var {X t } due to scale τ j ν X (τ j ) has same units as X t (easier to interpret) 31

33 Spectrum Substitute/Complement because h j,l bandpass over [1/2 j+1, 1/2 j ], ν 2 X(τ j ) 2 1/2 j 1/2 j+1 S X (f) df ( ) if S X (f) featureless, info in ν 2 X(τ j ) info in S X (f) ν 2 X(τ j ) more succinct: only 1 value per octave band recall SDF for FD process: S X (f) = ( ) implies ν 2 X(τ j ) τ 2δ 1 j σ 2 ɛ 2 sin(πf) 2δ σ2 ɛ 2πf 2δ approximately can deduce δ from slope of log (ν 2 X(τ j )) vs. log (τ j ) can estimate δ & σ 2 ɛ by applying regression analysis to log of estimates of ν 2 X(τ j ) 32

34 Estimation of Wavelet Variance: I can base estimator on MODWT of X 0,X 1,...,X N 1 : W j,t L j 1 l=0 h j,l X t l mod N, t =0, 1,...,N 1 (DWT-based estimator possible, but less efficient) recall that W j,t L j 1 l=0 h j,l X t l, t =0, ±1, ±2,... so W j,t = W j,t if mod not needed: L j 1 t<n if N L j 0, unbiased estimator of ν 2 X(τ j )is ˆν 2 X(τ j ) 1 N L j +1 where M j N L j +1 N 1 t=l j 1 Wj,t 2 = 1 N 1 M j can also construct biased estimator of ν 2 X(τ j ): ν 2 X(τ j ) 1 N N 1 t=0 Wj,t 2 = 1 (L j 2 N t=0 Wj,t 2 + N 1 W 2 j,t, t=l j 1 W 2 j,t t=l j 1 1st sum in parentheses influenced by circularity ) 33

35 Estimation of Wavelet Variance: II biased estimator unbiased if {X t } white noise biased estimator offers exact analysis of ˆσ 2 ; unbiased estimator need not biased estimator can have better mean square error (Greenhall et al., 1999; need to reflect X t ) 34

36 Statistical Properties of ˆν 2 X(τ j ) suppose {W j,t } Gaussian, mean 0&SDFS j (f) suppose square integrability condition holds: A j 1/2 1/2 S2 j (f) df < & S j (f) > 0 (holds for FD process if L large enough) can show ˆν 2 X(τ j ) asymptotically normal with mean ν 2 X(τ j ) & large sample variance 2A j /M j can estimate A j and use with ˆν 2 X(τ j ) to construct confidence interval for ν 2 X(τ j ) example Fig. 14: clock errors X t X t (0) along with differences X t (i) X t (i 1) X (i 1) t 1 for i =1, 2 Fig. 15: ˆν X(τ 2 j ) for clock errors Fig. 16: ˆν Y 2 (τ j) for Y t X t (1) Haar ˆν 2 Y (τ j) related to Allan variance σ 2 Y (2,τ j): ν 2 Y (τ j)= 1 2 σ2 Y (2,τ j) 35

37 Summary fractionally differenced processes are able to cover all power laws easy to work with (SDF, ACVS & PACS simply expressed) extensible to composite, ARFIMA & time-varying processes spectral and wavelet analysis can provide estimates of parameters of FD processes decomposition of sample variance across frequencies (in case of spectral analysis) scales (in case of wavelet analysis) complementary analyses wavelet analysis has some advantages for clock noise estimates δ & σɛ 2 somewhat better useful with time-varying noise process can deal with polynomial trends (not covered here) results expressed in same units as Xt 2 a big thank you to conference organizers! 36