Uncertanty and auto-correlaton n arxv:1707.03276v2 [physcs.data-an] 30 Dec 2017 Measurement Markus Schebl Federal Offce of Metrology and Surveyng (BEV), 1160 Venna, Austra E-mal: markus.schebl@bev.gv.at Abstract Although a system s descrbed by a well-known set of equatons leadng to a determnstc behavor, n the real world the value of a measurand obtaned by an experment wll mostly scatter. Accordngly, an uncertanty s assocated wth that value of the measurand due to apparently random fluctuaton. Ths papers deals wth the queston why ths dscrepancy exst. Furthermore t wll be shown how the uncertanty of one ndvdual observaton s calculated and consequently how the best estmate and ts correspondng uncertanty consderng auto-correlatons s determned. Introducton A measurand s determned from other quanttes trough a functonal relatonshp f by 1, y = f (x 1, x 2,..., x n ) (0.1) where x 1, x 2,..., x n are nput parameters. These quanttes are often n turn nfluenced by other quanttes trough a functonal relatonshp g by, 1
x = g (ε 1, ε 2,.., ε k ) (0.2) where the parameters ε 1, ε 2,.., ε k are fundamental snce they determne the characterstcs of the measurand va Eq. (0.1,0.2). Unfortunately, not all fundamental parameters may be known, e.g., convecton of ar causng dynamc pressure durng a weghng measurement. Ths leads to the generaton of addtonal chaotc forces dependng on the velocty pattern of the ar flow n the close vcnty of the pan of the balance. If the velocty pattern s not determned as well as the mpact of t on the balance, ts nfluence on the measurand s not known. Or, vbratons cause forces actng on the balance due to acceleratons. But f acceleratons are not measured and addtonally, ther mpact are not known, the nfluence of them on the measurand s not known ether. Therefore, the fundamental parameters can be dstngushed between known and unknown quanttes such as, x = g (ε 1, ε 2,.., ε l, h 1, h 2,.., h j ) (0.3) where h j are hdden parameters. Thus the measurand s gven by, y = f (g 1 (ε 1, ε 2,.., ε l, h 1, h 2,.., h j ),..., g n (ε 1, ε 2,.., ε l, h 1, h 2,.., h j )) (0.4) Consequently, the measurand becomes a functon of fundamental known and hdden parameters, y = (ε 1, ε 2,.., ε l, h 1, h 2,.., h j ) (0.5) 2
Orgn of apparent random fluctuatons of the Measurand Let us frst assume that hdden fundamental parameters do not exst and consequently the system s fully descrbed by a well-known set of equatons. That means, that all fundamental parameters and ther mpact on the behavor of the measurand are known. Hence the measurand s gven by, y = (ε 1, ε 2,.., ε l ) (0.6) Thus, at tme t = t 0, the measurand s gven by, y (t 0 ) = ( ε 1 t0, ε 2 t0,.., ε l t0 ) (0.7) at the tme t = t 0 + t, the measurand s gven by, y (t 0 + t) = ( ε 1 t0 + t, ε 2 t0 + t,.., ε l t0 + t ) (0.8) If the values of all known fundamental parameter at tme t = t 0 + t, are equal to the values at tme t = t 0, the value of the measurand would be the same, ε 1 t0 + t = ε 1 t0 ε 2 t0 + t = ε 2 t0. ε l t0 + t = ε l t0 y (t 0 + t) = y (t 0 ) (0.9) Consequently the system s fully determnstc n that case. Now let us assume that hdden fundamental parameters exst. The measurand n that case s gven by, y = (ε 1, ε 2,.., ε l, h 1, h 2,.., h j ) (0.10) 3
Agan, at tme t = t 0, the measurand s gven by, y (t 0 ) = ( ε 1 t0, ε 2 t0,.., ε l t0, h 1 t0, h 2 t0,.., h j t0 ) (0.11) and at tme t = t 0 + t, the measurand s gven by, y (t 0 ) = ( ε 1 t0 + t 0, ε 2 t0 + t 0,.., ε l t0 + t 0, h 1 t0 + t 0, h 2 t0 + t 0,.., h j t0 + t 0 ) (0.12) However, snce the mpact of hdden parameters can not be evaluated leads to the fact, that although n the case that the values of the known parameters at both tmes are equal, the values of the measurand at both tmes are not necessarly equal, h 1 t0 + t = h 1 t0 ε 1 t0 + t = ε 1 t0 ε 2 t0 + t = ε 2 t0. ε l t0 + t = ε l t0 y (t 0 + t) = y (t 0 ) h 2 t0 + t = h 2 t0. h l t0 + t = h l t0 (0.13) y (t 0 + t) y (t 0 ) h t0 + t h t0 Strctly speakng, only n the case that the values of all nput parameters (known and hdden) are exactly the same, the value of the measurand at both tmes would be equal. If only the value of one hdden parameter s dfferent at dfferent tmes, the value of the measurand would be dfferent too. Ths leads to the fact, that for equal sets of known nput parameters the measurand can reach dfferent values. Thus due to lack of nformaton of the system, t apparently behaves not necessarly determnstc but rather reveals a stochastc behavor. Ths phenomenon s depcted n Fg. (0.1) Nota bene, the stochastc behavor of the system s a consequence of the exstence of hdden parameters. 4
Fgure 0.1: Measurand as a functon of hdden and known parameter. a) The measurand value s gven for nstance by Y = 2T 2 4T P +1000. The parameter T denotes the known temperature and P s the unknown pressure. The red balls ndcate values of the measurand gven by specfc values of the known and unknown varable. Here the mpact of P on the measurand value s consdered to be known. b) Due to the fact, that the hdden parameter s not accessble, only the projecton (plane TY) of Y s vsble. Thus, the measurand values apparently scatter. Uncertanty of ndvdual value of measurand Bascally, the true value of a quantty s often not known. For nstance, consderng hydrostatc weghng for the determnaton of lqud densty. Usually a sold body s mmersed nto the lqud and the apparent loss of ts mass (due to a lft force) s measured by usng a balance. The lft depends on the volume of the body whch n turn depends on temperature. But, the temperature of the body s not measured drectly snce one avods any generaton of contact forces actng on the body. Solely, the temperature of the flud s measured n the vcnty of the body. Thus, one can only estmate the true value of the body temperature. To derve a relaton for the uncertanty n that case, one calculates the change of the value of the measurand for a small change of the nput parameters. Ths s gven by, dy = l (ε ε 0 ) + ε ε10,...,ε l0 j k (h k h k0 ) (0.14) h k h10,...h j0 5
Now t s assumed that the true value of ε and h k les between [ε 0 ε, ε 0 + ε ] respectvely [h k0 h k, h k0 + h k ] so that ε and h k defnes the range n whch we beleve the true value les wth a specfed lkelhood. Hence t s reasonable to chose ε = ε 0 + ε, and h k = h k0 + h k thus Eq. (0.14) becomes, dy = l ε + ε ε10,...,ε l0 j k h k (0.15) h k h10,...h j0 The value (dy) 2 s a measure for the measurand uncertanty ( (dy) 2 nstead of dy snce the uncertanty should be postve). Thus, u 2 (y) = (dy) 2 = l 2 l 1 l ε ε + 2 ε ε ε10,...,ε l0 ε k ε k k=+1 ε10,...,ε l0 ε10,...,ε l0 j + 2 j 1 j h + 2 h h k h k h h10,...h j0 h k k=+1 h10,...h j0 h10,...h j0 l j +2 ε h k ε h k k ε10,...,ε l0 h10,...h j0 (0.16) Obvously, snce the mpact of hdden parameters can not be quantfed, the uncertanty of the value of the measurand can not be determned ether. However, a reasonable procedure to determne the uncertanty s to consder the varance of the measurand at constant known fundamental parameters. Ths s clear f we look on equaton (0.9). For constant known fundamental parameters the value of the measurand s also constant. Hence, f any fluctuaton (scatter) of the measurand s observed at constant known parameters one can readly conclude that ths fluctuatons must be caused by hdden parameters (Fg. 0.1). Thus all values of the measurand must be transformed to the same set of known parameters (Fg. 0.3). Ths can be acheved by calculatng a ft functon of the measurand values. Thus the transformed measurand values, γ, are gven by, 6
γ = f : y (ε 1, ε 2,..., ε n ) y(ε 1 = A 1, ε 2 = A 2,..., ε n = A n ) (0.17) where the parameters A n are constants. Nota bene, ths fluctuaton of the measurand value does not really exst. They are quas exstng due to the lack of full nformaton of the system. Fgure 0.2: Scatter of the measurand. (a) At constant known parameter T the measurand exhbts a characterstc behavor dependent on the hdden parameter P. (b) The measurand value apparently shows an stochastc behavor n the accessble projecton plane TY. Thus, the measurand values scatters due to lack of nformaton of the system. As t s depcted, t s obvous that the scatter nterval (A,B,C) may depent on the known parameter T. Hence, the ndvdual uncertanty of the measurand value becomes, u 2 (y) = (dy) 2 = TYPE B { }}{ l 2 l 1 l ε + 2 ε ε k ε ε ε k ε10,...,ε l0 ε10,...,ε l0 k=+1 ε10,...,ε l0 +V ar (γ) }{{} TYPE A (0.18) where the frst term on the rght hand sde of Eq. (0.18) determnes the Type B contrbuton and the second term determnes the Type A contrbuton to the overall uncertanty. Type B uncertantes are calculated by deducton from an gven probablty densty functon, 7
Fgure 0.3: Transformaton of the measurand values. (a) In order to calculate the varance, the measurand values has to be transformed accordng to the fundamental relatonshp Y = 2T 2 4T P C +1000 where P C s equal to a gven pressure value. (b) However, the parameter P s hdden. Thus n turn t s necessary to approxmate the temperature characterstcs of the measurand Y. Thus the measurand values have to be transformed accordng to a ft functon to a specfc value of the known parameter T (usually the mean value) (dashed black lne). Otherwse the varance would be overestmated due to an over szed scatter nterval (A). The red bold lne shows the ft functon (n that case a lnear ft was chosen). The red balls ndcates the transformed measurand values. It s also evdent, that a lnear transformaton (lnear ft) s just a approxmaton. In fact, the black sold lnes depcts the functonal relatonshp between the measurand and the temperature at gven preassures. It s clear, that ths functonal relatonshp would be the best ft functon to transform every specfc data pont. But unfurtunatelly, P s hdden, and one only observes the stuaton depcted on the rght fgure wthout any nformaton of the true functonal relatonshp between T and Y 8
p (ε) by 3, ε 2 = ˆ + dε (ε ε ) 2 p (ε) = ε 2 ε 2 (0.19) whereas Type A contrbutons are calculated by nducton va the varance whch s gven by, V ar (γ) = N (γ γ) 2 N 1 (0.20) The mean value of the measurand s gven by, γ = N N γ (0.21) It s mportant to emphass that, V ar (γ) j 2 j 1 j h h + 2 h h h10,...h j0 h k h k k=+1 h10,...h j0 h10,...h j0 l j +2 ε h k ε h k k ε10,...,ε l0 h10,...h j0 (0.22) Dependng on the magntude of parameters the varance could be, h k h k and on the stablty (varaton) of the hdden V ar (γ) j 2 j 1 j h + 2 h h k h h h10,...h j0 h k k=+1 h10,...h j0 h10,...h j0 l j +2 ε ε h k h k k ε10,...,ε l0 h10,...h j0 (0.23) 9
or, V ar (γ) j 2 j 1 j h + 2 h h k h h h10,...h j0 h k k=+1 h10,...h j0 h10,...h j0 l j +2 ε h k ε h k k ε10,...,ε l0 h10,...h j0 (0.24) Generally, n most cases (see Fg. 0.2) due to lack of nformaton (exstence of hdden parameters) the total uncertanty for a sngle observaton of the measurand value wll be overrated by applyng statstcal methods (Fg. 0.4). It s just a tool to account for uncertantes related to hdden parameters. Nota bene, for a non-lnear relatonshp between the fundamental parameters and the measurand, y, the Type B uncertanty accordng to Eq. (0.18) would gve a wrong contrbuton to the overall uncertanty of the measurand. In such a case a sutable procedure s gven by the Monte Carlo (MC) method to calculate the Type B uncertanty contrbuton. Accordng to ths method, one calculates the measurand several tmes where the fundamental parameters are pcked from a probablty densty dstrbuton, P εn (ε n, ε n ) wth expectaton value, ε n, and varance ε n, It s gven by, ŷ j = ( ε 1j, ε 2j,..., ε nj ) (0.25) wth, ε nj = P εn (ε n, ε n ) j (0.26) The Type B uncertanty for an ndvdual measurand value by applyng the MC method would then be gven by, 10
M j u 2 B (y ) = V ar (ŷ ) = (ŷ j ŷ ) 2 M 1 (0.27) wth, ŷ = M j M ŷ j (0.28) where M s the number of trals. Fgure 0.4: Uncertanty of a sngle observaton. (a) For a determnstc system (no hdden parameters) the uncertanty s gven by Eq. (0.16). (b) Due to lack of nformaton the uncertanty for a sngle measurand value may be overrated dependng on the scatter nterval (see Fg. (0.3b). The mean value of the measurand and ts uncertanty Accordng to the statements n sectons (1), (2), and (3) t s nevtably clear that the mean value of the measurand has to be evaluated for a constant set of known fundamental parameters. Thus wth Eq. (0.17)t s gven by, y = N y (ε 1 = A 1, ε 2 = A 2,..., ε n = A n ) N = N γ N = γ (0.29) 11
In prncple, the constants A n are arbtrary but t s reasonable to choose the mean values of the known parameters thus A n = ε n. Hence, the mean value of the measurand becomes, y = N y (ε 1 = ε 1, ε 2 = ε 2,..., ε n = ε n ) N (0.30) Nota bene, f no hdden parameters would exst, each measurand y for a constant set of known fundamental parameters would be gven accordng to Eq. (0.6) by, y = y = (ε 1 = A 1, ε 2 = A 2,..., ε n = A n ) (0.31) Wth Eq. (0.30) ths would lead to the fact, that, y = (ε 1 = ε 1, ε 2 = ε 2,..., ε n = ε n ) (0.32) However, n that specfc case, the measurand s totally determnstc and the concept of a mean value and varance loses ther meanng. Furthermore, bear n mnd, that the constants A, B, C,... are totally arbtrary. The uncertanty of the mean s gven wth Eq. (0.18) choosng ε 10 = ε 1,..., ε l0 = ε l by, u 2 (y) = (dy) 2 = TYPE B { }}{ l 2 l 1 l ε + 2 ε ε k ε ε ε1,..,ε l ε k k=+1 ε1,..,ε l ε1,..,ε l [ + 1 N ] N 1 N V ar (γ) + 2 COV (γ, γ j ) N 2 =1 =1 j=+1 } {{ } TYPE A (0.33) where COV (γ, γ j ) accounts the correlaton between the transformed measurand value γ and γ j. The correlaton term can be wrtten as, 12
COV (γ, γ j ) = r γ,γ j V ar (γ ) V ar (γ j ) (0.34) where r γ,γ j s the correlaton coeffcent. Snce the values γ and γ j belongs to the same measurand, r γ,γ j s called auto correlaton coeffcent. Thus, the Type A contrbuton to the overall uncertanty of the mean of the measurand becomes, u 2 A(y) = 1 N [ V ar (γ) + 2 N N 1 N =1 j=+1 ] r y,y j V ar (γ ) V ar (γ j ) (0.35) Case 1: V ar (γ ) = V ar (γ j ) = V ar (γ), r γ,γ j = r 0 If we assume that all varances are equal as well as the auto correlaton coeffcent between two transformed measurand value γ and γ j, Eq. (0.35) becomes 1, ( u 2 A(y) = V ar (γ) r + 1 r ) N (0.36) It s evdent, that for correlated system, the contrbuton of Type A uncertantes to the overall uncertanty of the mean becomes n the lmt of N, lm n u2 A(y) = rv ar (γ) (0.37) In practce usually one encounters the fact, that the auto correlaton of the data s not consdered n the total uncertanty of the mean. Thus t s just often calculated by, u 2 A(y) = V ar (γ) N (0.38) Hence the uncertanty of the mean vanshes n the lmt of N. Equaton (0.38) should be utlzed wth care because t can result n a strong underestmaton of the uncertanty. 1 N 1 N =1 j=+1 = 1 2N (N 1) 13
Case 2: V ar (γ ) V ar (γ j ), r γ,γ j r 0 Usually the correlaton coeffcent between two random varables x, y s gven by 4, r x,y = N N (x x) (y y) (0.39) (x x) 2 N (y y) 2 Fgure 0.5: From correlaton to auto correlaton. Two separated random varables are merged to one data set. Mergng both varables (Fg. (0.5)) the correlaton coeffcent can be calculated as, r x,y = r x,x+5 = N/2 N/2 (x x 5 ) (x +5 x >5 ) (0.40) (x x 5 ) 2 N/2 (y x >5 ) 2 wth the mean values gven by, x 5 = 1 N/2 N/2 =1 x x >5 = 1 N/2 N/2 Hence n general, the auto correlaton coeffcent s gven by, =6 x (0.41) wth, r γ,γ j = r γk,γ k+m = N (k+m) =0 (γ k+ γ k ) (γ k+m+ γ k+m ) N (k+m) =0 (γ k+ γ k ) 2 N (k+m) =0 (γ k+m+ γ k+m ) 2 (0.42) 14
N (k+m) V ar (γ k ) = (γ k+ γ k ) 2 V ar (γ k+m ) = N (k+m) =0 =0 (γ k+m+ γ k+m ) (0.43) Fgure 0.6: Auto correlaton. (a) Auto correlaton coeffcent calculaton wth a small step sze m. (b) Large step sze leads to no overlap zone between the data. For example, Fg. shows the auto correlaton coeffcent of a data set wth N = 150. Fgure 0.7: Auto correlaton coeffcent for a specfc data set. (a) Auto correlaton coeffcent. (b) Contour plot of the auto correlaton coeffcent. 15
References (1) Evaluaton of measurement data - Gude to the expresson of uncertanty n measurement, JCGM 100:2008 (2) I. Hughes, Measurements And Ther Uncertantes: A practcal gude to modern error analyss, Oxford Unversty Press, 2009 (3) B. Pesch, Messunscherhet: Basswssen fuer Ensteger und Anwender, Books on Demand, 2010 (4) J. R. Taylor, An Introducton to Error Analyss: The Study of Uncertantes n Physcal Measurements, Unv Scence Books, 1997 (5) Gude to the Expresson of Uncertanty n Measurement, Internatonal Organzaton for Standardzaton (ISO), 1955 16