Statistical Aalysis o Ucertaity for Autocorrelated Measuremets ad its Applicatios to Key Comparisos Nie Fa Zhag Natioal Istitute of Stadards ad Techology Gaithersburg, MD 0899, USA
Outlies. Itroductio. Statioary processes 3. Ucertaity of a sample mea for statioary measuremets 4. Cofidece itervals for the mea value 5. Applicatios to key Comparisos 6. Discussios ad coclusios
. Itroductio I metrology, for give repeated measuremets, of u = S {,..., } of size Here we assume that {,..., } have same mea ad variace ad are ucorrelated. However the coditio of ucorrelatedess does ot always hold. I may cases measuremets are autocorrelated or selfcorrelated. Here are two examples.
The figure shows 400 liewidth measuremets for a omially 500 m lie by a SEM.
This shows the ACF plot of the liewidth measuremets with a 95% cofidece bad for WN.
The ACF plot is a plot for the sample autocorrelatios, which measures the autocorrelatio of the data. The cofidece bad is cetered at zero ad with limits of ±.96 with = 400 The cofidece bad is for the autocorrelatios of a ucorrelated sequece or white oise. Namely, if the data are statistically idepedet, the its sample autocorrelatio has a mea of zero ad a approximate stadard deviatio of Obviously, the liewidth measuremets are autocorrelated ad thus the ucertaity give i the above may ot be appropriate to use.
High precisio weight measuremets with differeces from the kg check stadards.
The ACF plot shows the autocorrelatios ad a 95% cofidece bad for white oise. Obviously, the measuremets are autocorrelated ad thus the traditioal approach to calculate the ucertaity of the average of measuremets is ot appropriate. If the mea is estimated by a weighted mea, the whe the ucertaity of the weighted mea is calculated, the traditioal approach u = w i S w i= w i i i= is also ot appropriate. Thus, appropriate approaches are eeded to calculate the correspodig ucertaities. For autocorrelated processes, a importat class is the oe of statioary processes. We propose a practical approach to compute the ucertaity of the average of the measuremets from a statioary process. = w
. Statioary processes A discrete time series {, t =,,...} is (weakly) statioary if () () (3) E [ t ] = µ Va r[ ] = σ < t R( τ ) is the autocovariace of { } at lag of τ. The autocorrelatio is defied as Whe the measuremets are from a statioary process, they have the same mea ad variace for all t. Cov[, ] R( τ ) t t+ τ = t t ρ( τ) = R( τ) R(0) R(0) = σ
Examples of statioary processes () White oise ---- mea=0 ad ρ( τ ) = 0 for all τ 0 () First order autoregressive process (AR()) ( µ ) = φ ( µ ) + a t t t Whe φ < the process is statioary. ρ( k) k = φ whe k 0 (3) Movig average process (MA(q)) µ = a θa... θ a t t t q t q ρ ( k) = 0 whe k > q
3. Ucertaity of a sample mea for statioary measuremets For the sample mea +... + = whe ( i) ρ( i)] i= σ Var[ ] = + is statioary where σ is the variace of { t }. Whe the measuremets are ucorrelated, σ Var[ ] = Whe { } is a AR() process t { t } Var[ ] = φ φ + φ ( φ) + σ
The ucertaity of the sample mea is give by where u = + i= ( i) ˆ ρ( i)] S = ( k ) k= S ˆ( ρ i) = i k= ( )( ) k k= k+ i ( ) k
Two issues i usig the above estimator: () Whe i is close to the umber of product i the umerator is small ad the estimate ˆ( ρ i) will ot be good. () Sice ˆ( ρ i) 0whe i we eed to cosider if ay ˆ( ρ i) is statistically sigificat from zero. For (), a rule of thumb: ˆ( ρ i) ca oly be used whe 50 i 4 Thus, we oly use these ˆ( ρ i) with i 4 For (), by Wold Decompositio Theorem, a realizatio of a statioary process ca be approximated by a fiite order MA process µ = a θ a t t k t k k= q ad
Based o that, for a set of time series data, the order of the correspodig MA model, q eeds to be foud to have all ˆ( ρ i) the ucertaity formula sigificatly differet from zero. For a MA(q) process ad a sufficietly large, ˆ( ρ i) for i > q approximately ormally distributed with a mea of zero ad a variace of q + ρ ( k) k= σ ˆ ρ () i i is I particular, whe { i } is a ucorrelated sequece with same mea ad variace, σ ˆ ρ () i which was used to build a cofidece bad of ACF for white oise.
I practice for i > q a estimator of is q + ˆ ρ ( k) ˆ σ ˆ ρ () i = k= σ ˆ ρ () i Sice ˆ( ρ i) are asymptotically ormally distributed with for i > q q + ˆ ρ ( k) k= ρ ˆ( i) >.96 E[ ˆ ρ( i)] 0 is evidece agaist MA(q) at the α = 0.05 level. Namely, if a MA(q) is to be cosidered, ˆ( ρ i) should remai withi the limits for i > q
Thus, a cut-off lag is give by ˆ ˆ c = max{ i ρ( i) >.96 σ ˆ ρ () i } Thus, the upper limit i the summatio i the ucertaity formula is replaced by = mi{,[ 4]} r c To warraty that oly reasoable good ˆ( ρ i) are icluded. Thus, the ucertaity of the average of measuremets from a statioary process ca be calculated by u + A factor or a ratio is i= r = + r ( i) ˆ ρ( i)] r ( i) ˆ ρ( i)] i= S
4. Cofidece itervals for the mea value Cetral Limit Theorem: Whe {,..., } are statistically idepedet ad idetically distributed with µ ad σ whe is large eough σ N( µ, ) For a statioary process, CLT still holds whe some regular coditios are met. Thus, whe is large eough, the 95% cofidece limits are ±.96u For the first example, = 400 ad r = 3 the u = 0.3776 which is much larger tha 0.0964, the ucertaity based o idepedece assumptio. R, the ratio = 3.9.
ACF plot of liewidth measuremets with a 95 % cofidece bad
Sice = 450.6 a 95 % cofidece bad of the mea is 450.6 ±.96 0.38 = 450.6 ± 0.74 For the secod example of the weights measuremets, =7. r = = 7 c u = 0.0067 which is much larger tha 0.004, the ucertaity based o idepedece assumptio. R, the ratio is.8. A 95 % cofidece bad is give by 9.4645 ±.96 0.0067 = 9.4645 ± 0.03
ACF plot of the weights measuremets with a 95 % cofidece bad
5. Applicatios to Key Comparisos For a Key Compariso, if the measuremets by ay lab are autocorrelated, the the Type A ucertaity for the lab should be determied based o the autocorrelatios assumig the data are from a statioary process. If there are more tha oe travelig stadard ad the measuremets by ay lab are autocorrelated, the the Type A ucertaity for this stadard ad this lab eeds to be determied based o the correspodig autocorrelatios. If the Type A ucertaity of ay lab is obtaied by combiig all travelig stadards, the the calculatio should be based o the autocorrelatio structures of measuremets for all stadards.
6. Discussios ad coclusios Whe repeated measuremets are autocorrelated, it is ot appropriate to use the traditioal approach to calculate the ucertaity of the average of the measuremets. We propose a practical approach to calculate the ucertaity whe the data are from a statioary process. () The result ca be exteded to the case of a weighted mea. w i i i= Whe { } is statioary, it ca be show that t = w i w = σ i + ρ j j+ i i= i= j= Var[ ] { w ( i)[ w w ]} The correspodig ucertaity ca be calculated.
() We assume that time series data were collected at equally spaced (time) iterval. However, i practice a time series dataset ofte cotais more or less missig values. I time series aalysis, various attempts have bee made to deal with such a kid of case. Therefore, i metrology oce the measuremets are determied to be autocorrelated, it is very importat to make sure the data are collected at equal time iterval.
(3) A autocorrelated process or a time series ca be ostatioary such as a rasom walk: = + a t t t with = ad { } white oise. It is obvious that 0 0 a t t t = a i= i Thus, Var[ ] = t σ t a If measuremets are from a o-statioary process usig the average value ad the correspodig variace to characterize the measuremet stadard may be misleadig.