Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.gao.spb.ru/english/personal/malkin/publ/zm11f.pdf Äàòà èçìåíåíèÿ: Fri Jan 13 10:39:36 2012 Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 03:18:45 2012 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: m 31

O n c o m p u ta ti o n o f a c o m m o n m e a n
Zinovy Malkin

arXiv:1110.6639v1 [physics.data-an] 30 Oct 2011

Pulkovo Observatory, Pulkovskoe Ch. 65, St. Petersburg 196140, Russia e-mail: malkin@gao.spb.ru
Abstract Combining several indep endent measurements of the same physical quantity is one of the most imp ortant tasks in metrology. Small samples, biased input estimates, not always adequate rep orted uncertainties, and unknown error distribution make a rigorous solution very difficult, if not imp ossible. For this reason, many methods to compute a common mean and its uncertainty were prop osed, each with own advantages and shortcomings. Most of them are variants of the weighted average (WA) approach with different strategies to compute WA and its standard deviation. Median estimate b ecame also increasingly p opular during recent years. In this pap er, these two methods in most widely used modifications are compared using simulated and real data. To overcome some problems of known approaches to compute the WA uncertainty, a new combined estimate has b een prop osed. It has b een shown that the prop osed method can help to obtain more robust and realistic estimate suitable for b oth consistent and discrepant measurements.

1

Intro duction

Computation of a common mean (CM) of several independent measurements of a physical quantity is a common pro cedure in scientific analysis. Typical task include combining results obtained by different analysts or results based on different measurement metho ds or results of several series of measurements, etc. One of the most important applications required computation of a CM is derivation of the best estimate of physical constants. Both accurate CM value and its realistic uncertainty are equally important in such computations. Input information consists of measured values xi , i = 1, . . . , n with asso ciated uncertainties si , usually standard deviations (STD), and correlations between xi are not available. From statistical point of view, it is a classical case of direct measurements of unequal precision. Due to lack of needed information we have to treat them as uncorrelated. Combination pro cedure aimed to get an estimate x of the CM x with asso ciated uncertainty , which ad¯ equately reflects the scatter and uncertainties of input measurements. There is no unambiguous solution of this problem. It is well known the the classical weighted average (WA) is unbiased CM estimate with minimum variance, provided xi are independent unbiased estimators, and s2 are true variances. A solution of the problem i is not straightforward if true variances are unknown. Numerous papers are devoted to 1


computation of a CM and its uncertainty; see e.g. Graybill and Deal (1959); Sinha (1985); Witkovsky and Wimmer (2001); Zhang (2006) and references therein. Unfortunately, results ´ are obtained under rather strong assumptions: the xi are unbiased estimates of x, si are ¯ normally distributed, and number of measurements ni formed each xi is known. Evidently, these assumptions are hardly met in scientific data analysis. The situation becomes even more complicated because small samples are are the rule rather than the exception in such tasks, which makes it difficult to efficiently apply most of statistical metho ds generally used. For these reasons many alternative approaches, not always strongly justified, are mostly used, see e.g. MacMahon et al (2004), Chen et al (2011), and Dataplot software do cumentation1 . We do not aim this paper at investigation of all the existing metho ds. Such a task seems to be impractical because some metho ds cannot be used in our case because we lack needed information, some are developed for specific applications, and some approaches may be to o complicated for routine use, e.g. bo otstrapping (Helene, 2007) or total median (Figueiredo and Gomes, 2004; Cox and Harris, 2004). We consider two basic metho ds most commonly used: WA and median. As to the former, several approaches to compute its uncertainty (STD) are proposed in literature. This paper is devoted to comparison of this metho ds based on simulated and real data. Besides, a new combined approach is proposed to compute the WA uncertainty. This approach was proposed for the first time in Malkin (2001b) and has shown to be useful in practical applications such as computation of Earth orientation parameter (EOP) combined solution (Malkin, 2001a), radio source position catalogues combination (Sokolova and Malkin, 2007), analysis of radio source position time series (Malkin, 2008), and mo deling the galactic aberration (Malkin, 2011). Special attention is given to analysis of small statistical samples.

2
2.1

Computation of CM
Basic WA estimators

The WA estimator is most widely used in various scientific and practical applications. Let we have n values xi with asso ciated standard deviations si , i = 1 . . . n. Then we can compute the following statistics (e.g. Brandt (1999); Bevington and Robinson (2003))
n

1 pi = 2 , si

n

pi xi pi , xw = ¯
i= 1

p=
i= 1

p

.

(1 )

In result, we have the classical WA estimate xw with weights inversely proportional to ¯ 2 variances of input measurements si . We also can compute the following statistics also used as a measure of go o dness of fit
n n

H=
i= 1

pi (xi - xw )2 = ¯

i= 1

(xi - xw ) ¯ si

2

,

(2 )

where H has a 2 distribution with n - 1 degree of freedom (dof ) if s2 are theoretical i variances. In practice, s2 are, as a rule, sample variances, but this fact is usually ignored. i
1

http://www.itl.nist.gov/div898/software/dataplot/

2


is close to unity. Otherwise one can assume that the input measurements xi have systematic errors or si are underestimated. The question is how to estimate the standard error of the mean? Two main approaches can be applied to compute . The classical WA estimate is 1 1 = . p

The H statistics is an indicator of consistency of input measurements. If the measurements are consistent, the value H 2 = (3 ) d of n-1

(4 )

Least squares approach leads to alternative estimate of the WA uncertainty. The solution of the least square problem xi = x + i with weights pi gives the same WA estimate x, but ¯ ¯ another estimate of its uncertainty:
n i= 1

2 =

pi (xi - xw ) ¯ p (n - 1 )

2

.

(5 )

This estimate of the WA uncertainty is, in fact, 1 estimate scaled in such a way to make /dof close to unity: H , (6 ) 2 = 1 n-1
2

which is equivalent to scaling of input si by the factor H/(n - 1) as recommended by Rosenfeld et al (1967) and Brandt (1999). However, such a scaling makes resulting estimate independent of the "scale" of input variances s2 , and dependent only on their ratio. i So, both approaches give the same estimate for the WA, but different estimates for the WA uncertainty. The first value 1 depends on si and do es not depend on the scatter of the input values xi . On the other hand, 2 depends on relative values of input variances s2 and i the scatter of xi . Difference between the two estimates may be attributed to systematic errors in xi or underestimated si . Theoretically, choice between 1 and 2 depends on whether the scatter of xi is a result of random error or there are systematic differences between estimates xi . Obviously, both effects are present in most of practical applications. This is a well recognized problem in data analysis, and its rigorous solution is hardly possible due to generally biased input estimates, not always adequate reported uncertainties, and unknown error distribution. In practice, if xi are close each other and si are greater than the scatter of xi , it seems reasonable to use 1 . Otherwise, if si are much less than the scatter of the input measurements, 2 estimate seems to be more adequate to the data. Indeed such a way to cho ose the best estimate for the WA uncertainty cannot be considered satisfactory. A possible practical, statistically based approach has been proposed by Rosenfeld et al. (1967) and Brandt (1999). According to this approach, 2 criteria is used to decide whether

3


the scatter of xi is a result of random errors. First, both uncertainty estimates 1 and 2 are computed. Then the final WA uncertainty is taken as 3 =


1 , 2 ,

if H 2 (Q, n - 1) , if H > 2 (Q, n - 1) ,

(7 )

where Q is a significance level. One can see that to a first approximation 3 = 1 for consistent measurements and 3 = 2 for discrepant ones, and given Q value is used to distinguish between them. As a consequence, substantially different estimates can be obtained for the same input measurements (xi , si ) but specifying different Q. Similar approach is discussed by Bich et al (2002). It recommends accept 1 as the estimate of the CM mean uncertainty if consistency check H 2 (Q, n - 1) passed at a significance level Q=5%; otherwise supplement studies should be performed, such check of outliers, investigation of input data, etc. Unfortunately, the latter is generally not feasible.

2.2

Combined WA uncertainty estimator

As pointed out in the previous section, the literature recommends using either 1 or 2 depending on some criteria, which can lead to ambiguous results, keeping in mind that these two estimates may differ by several times. So, a more robust estimate is desirable for practical use, which would account of both the scatter of xi and their uncertainties si . After investigation of behavior of all three estimates using simulated and real data, and supplement tests we decided in favor of combined estimate computed by simple formula c =
2 2 1 + 2 .

(8 )

As a variant, 2 computed with unit weights can be used, which provides clear separation of impacts at combined estimate c from the uncertainties (1 ) and scatter (2 ) of the input measurements. However, in this case, the result is generally more sensitive to measurements suspected to be outliers. Unfortunately, we cannot suggest a rigorous theoretical ground of this approach, which is common for other practical recommendations to o. Our considerations are as follows. Suppose we can represent each input value as xi = x + i + 0i , where x is the true value of the CM, i is a random error distributed as N (0, s2 ) and 0i is a systematic error of the xi i 2 measurement distributed as N (0, 0 ). Here 0 is considered as a measure of the scatter of the set of systematic errors in input measurements. Evidently, 0i is unknown, otherwise it would be accounted for in the reported value of xi . We can suppose that 0i biases xi but do es not bias si . Thus the mathematical expectation of each xi is E (xi ) = x + 0i . (9 )

Now we can use the set of n equations (9) for i = 1, . . . , n as an equations of condition to be solved by the least squares metho d. As a result of this solution, we obtain an estimate of x and its uncertainty 0 , which can be expected close to 2 . Then we can consider 0 2 as an additive error in the WA uncertainty. Combining this error with 1 computed under the assumption of absence of systematic errors in xi we get c as defined by Eq 8. 4


Finally, let us notice that (8) can be rewritten as c = 1 H 1+ p n-1 . (1 0 )

So, to obtain c estimate, there is no need to compute separately both 1 and 2 and then use Eq. (8).

2.3

Median

Another approach routinely used to get the estimate of a CM is computation of a median xm . ¯ The median is known as a robust statistics less influenced by outliers. However its standard definition do es not provide an estimate of error of a median value (it makes it immune to unreliable uncertainties though). A possible approach to compute a median uncertainty was proposed by Muller (1995, ¨ 2000a). Let xm be the median of xi , i.e. xm = med{xi }. Now we can compute the median ¯ ¯ of the absolute deviations (MAD) as M AD = med{|xi - xm |} . ¯ The uncertainty of m is then taken as ¯ 1.8582 M AD . m = n-1 (1 2 ) (1 1 )

One can see that this estimate of the median uncertainty depends only on the data scatter and not on input uncertainties. Later Muller (2000b) proposed a metho d to take account ¨ of the uncertainty in input data and thus compute weighted median and its uncertainty. However its practical realization, as pointed out by the author, is more cumbersome, and the testing results and discussion given therein do not show clear advantage of using weighted median.

3

Tests with artificial data

In this section, results of two tests with simulated data are presented. These tests were constructed to investigate in more details the behavior of the estimates intro duced in the previous section. Indeed, many of features discussed here can be seen directly from corresponding equations, but not so demonstrative. Table 1 shows some numerical examples of computation of WA for two measurements, and its standard deviation. To compute 3 we used Q=99%, which corresponds to 2 (0.99,1)=6.635. Using Q=95%, 2 (0.95,1)=3.841 do es not change main conclusions. Note that, unlike general practice, we keep several significant digits in uncertainty just to better show the difference between various estimates. Classical example 1 shows that 2 cannot provide reasonable estimate for whatever how large input uncertainty are given. Examples 2­7 and 8­15 show how estimates change with grown si for the same xi . Examples 16­23 show how estimates change with grown x2 5


for the same x1 and si . One can see that no one of 1 , 2 , 3 provides a satisfactory estimate of for all the examples. Several observations from Table 1 are as follows. The 1 estimate sometimes it is clearly underestimated (examples 8­12). Examples 2­7 and 8­15 illustrate that 2 (2) cannot provide satisfactory estimate, especially in the cases 7, 14, 15, where it seems to be underestimated. Estimate 3 gives more realistic result, but not in all the cases, e.g. 2­3 and 8­11. Moreover, 3 value depends not only on data sample {xi , si } but also on sub jective choice of Q. Besides, as can be seen from this test and Eq. 7 that 3 may show significant jumps caused by small changes in input data or confidence level. For these reasons, it was decided not to use 3 in further work. In contrast to 1 , 2 , and 3 , one can see that c approach can provide stable and realistic estimate of the standard deviation of the WA. In the second test, we use the same set of five measurements with different errors (see Fig. 1). In the upper row the data have minimal uncertainties; in the middle row, all the uncertainties are increased by factor of 3; and in the bottom row, all the uncertainties are increased again by factor of 3. From this Figure, one can see that 1 became greater as si grow, as expected. However it lo oks underestimated in case a. The 2 estimate remains the same for all three examples because it do es not depend on absolute values of si , but only on the dispersion ratio, which is the same in all the cases. One would expect however that should be greater in two last examples as compared to the previous ones. Median estimate is close to 2 as expected because both of them depend on the xi scatter only, and hence m shows the same problems as 2 . The c estimate appears to be optimal because it shows a steady increase from case a to case c accounting both for input si value and xi scatter. In case a with small si values c is determined mainly by the data scatter, and is close to 2 . In case b with equa ontribution lc of input data uncertainties and scatter (1 = 2 ), c is just greater by factor of 2. In case c, c is defined mainly by si , which are much greater than the data scatter, and it is close to 1 . We can say that the c estimate "automatically" takes account of both input measurements scatter and uncertainties without any need in supplement assumptions or parameters like significance level.

4

Application to real data

In this section, the tests with real data are presented. In the first test, the height differences are analyzed between marks 107 and 109 of the lo cal geo detic network of the Svetlo e Radio Astronomical Observatory of the Institute of Applied Astronomy, St. Petersburg, Russia (Kazarinov and Malkin, 1997; Finkelstein et al, 2006). This analysis includes four levelling surveys performed in 1998­2003. Distance between the marks is about 135 m. Results of computations are shown in Fig 2. In this case, the scatter of the measurements is rather large as compared to measurement uncertainties. For this reason, 1 seems to be underestimated. Combined estimate 3 is close to 2 , but may be preferable because accounts also for input uncertainties. Median uncertainty is close to 2 . In this example, it is difficult to decide which of the two latter estimates should be preferred. Both of them lo oks equally realistic. 6


Table 1: Examples of computation of WA of two measurements x1 , x2 with asso ciated uncertainties s1 , s2 . Results of computations are mean x and four estimates of its uncertainty ¯ 1 , 2 , 3 , and c computed by (4), (5), (7), and (8) respectively. H is computed by (2) and used to compute 3 No. 1 2 3 4 5 6 7 x1 1.0 1.0 x2 1.0 2.0 s1 , s
2

x ¯ 1.0 1.5

H 0.00 50.0 12.5 5.5 2.0 0.5 0.1 0 0 6 0 0 2

1

2

3 0.354 0 0 0 0 0 1 5 5 5 5 2 3 7 4 0 0 0 0 2 2 3 3 .5 .5 .2 .3 .7 .4 .0 .0 .0 .0 .1 .5 .0 .1 .7 .7 .7 .7 .0 .5 .0 .5 0 0 1 5 0 1 0 0 0 0 2 3 7 4 0 0 0 0 0 0 0 0 0 0 2 4 7 4 0 0 0 0 1 6 1 21 7 7 7 7 0 0 0 0

c 0.354 0 0 0 0 0 1 5 5 5 5 5 6 8 5 0 0 1 1 2 2 3 3 .5 .5 .5 .6 .8 .5 .0 .0 .0 .1 .4 .1 .6 .0 .7 .8 .2 .6 .1 .5 .0 .5 0 2 4 1 6 0 0 1 5 9 3 2 6 0 0 6 2 5 2 9 8 7 5 0 3 2 6 0 0 2 0 6 1 4 0 0 7 6 5 8 1 8 2 1

0.5 0 0 0 0 1 2 0 0 1 2 3 5 10 20 .1 .2 .3 .5 .0 .0

0.354 0.000 0 0 0 0 0 1 0 0 0 1 2 3 7 4 0 0 0 0 0 0 0 0 .0 .1 .2 .3 .7 .4 .0 .3 .7 .4 .1 .5 .0 .1 .7 .7 .7 .7 .7 .7 .7 .7 7 4 1 5 0 1 7 5 0 1 2 3 7 4 0 0 0 0 0 0 0 0 1 1 2 4 7 4 1 4 7 4 1 6 1 2 7 7 7 7 7 7 7 7 0 0 0 0 0 0 5 5 5 5 5 5 5 5 0 0 1 1 2 2 3 3 .5 .5 .5 .5 .5 .5 .0 .0 .0 .0 .0 .0 .0 .0 .0 .5 .0 .5 .0 .5 .0 .5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 01 0 0 0 0 0 0 0 0

1 1 1 1 1 1 1 1 1 1 2 2 2 2

8 10.0 20.0 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 .0 .0 .0 .0 .0 .0 .0 .0 1 1 1 1 1 1 1 1 0 1 2 3 4 5 6 7 .0 .0 .0 .0 .0 .0 .0 .0

.1 15.0 5000.00 .5 200.00 .0 50.00 .0 12.50 .0 5.56 .0 2.00 .0 0.50 .0 0.12 1 0 0 1 1 2 2 3 3 .0 .5 .0 .5 .0 .5 .0 .5 0 0 2 4 8 12 18 24 .0 .5 .0 .5 .0 .5 .0 .5 0 0 0 0 0 0 0 0

1.0 1 1 1 1 1 1 1 1

7


a

x1 x2 x3 x4 x5

= = = = =

2 1 2 2 2

0 8 9 1 3

.0 .7 .8 .1 .3

± ± ± ± ±

1 1 1 1 1

.4 .7 .4 .6 .0

40 30 20 10

xw1 ¯ xw2 ¯ xwc ¯ xm ¯ x1 x2 x3 x4 x5

= = = =

23.00 ± 0.60 23.00 ± 1.81 23.00 ± 1.91 21.10 ± 2.04 0 8 9 1 3 .0 .7 .8 .1 .3 ± ± ± ± ± 4 5 4 4 3 .2 .1 .2 .8 .0

1

2 3 4 Measurement number

5 1 2 c m

b

= = = = =

2 1 2 2 2

40 30 20 10

xw1 ¯ xw2 ¯ xwc ¯ xm ¯ x1 x2 x3 x4 x5

= = = =

23.00 ± 1.81 23.00 ± 1.81 23.00 ± 2.56 21.10 ± 2.04 0 8 9 1 3 .0 .7 .8 .1 .3 ± ± ± ± ± 12.6 15.3 12.6 14.4 9.0

1

2 3 4 Measurement number

5 1 2 c m

c

= = = = =

2 1 2 2 2

40 30 20 10

xw1 ¯ xw2 ¯ xwc ¯ xm ¯

= = = =

13.00 ± 5.42 13.00 ± 1.81 13.00 ± 5.71 11.10 ± 2.04

1

2 3 4 Measurement number

5 1 2 c m

Figure 1: Testing four metho ds to compute CM with simulated data. The left group of points (discs) represents measurements, and the right group of points (circles) represents CM estimates. In all three examples, averaged measurements xi are the same, but their uncertainties differ. Uncertainties in example b are three times greater than those in example a, and uncertainties in example c are three times greater than those in example b; xw1 is ¯ the WA computed by Eq. (1) with the uncertainty 1 computed by (4). xw2 is the same ¯ WA with the uncertainty 2 computed by (5), xwc is the same WA with the uncertainty c ¯ computed by (8), xm is the median with the uncertainty m computed by (12) ¯

8


dH-3800, mm

x1 x2 x3 x4

= = = =

3 3 3 3

8 8 8 8

4 4 4 4

7 7 6 8

.5 .9 .6 .1

± ± ± ±

0 0 0 0

.4 .4 .5 .2

49

48

xw1 ¯ xw2 ¯ xwc ¯ xm ¯

= = = =

3847.83 ± 0.16 3847.83 ± 0.26 3847.83 ± 0.31 3847.70 ± 0.32

47

46 1998 2000 Year 2002 2004 1 2 c m

Figure 2: Testing four metho ds to compute a CA with real data: a case of determination of the height difference between two geo detic marks. The designations are the same as in Fig 1

Oort constant A

x1 x2 x3 x4 x4

= = = = =

1 1 1 1 1

5 4 1 4 4

.0 .4 .3 .8 .5

± ± ± ± ±

0 1 1 0 1

.8 .2 .1 .8 .5

16

14

12

xw1 ¯ xw2 ¯ xwc ¯ xm ¯ x1 x2 x3 x4 x4

= = = =

14.21 ± 0.44 14.21 ± 0.65 14.21 ± 0.79 14.50 ± 0.28 0 2 3 2 2 .0 .0 .9 .4 .0 ± ± ± ± ± 1 2 0 0 3 .2 .8 .9 .6 .0

10

8 1 2 3 4 Measurement number 5 1 2 c m

Oort constant -B

= = = = =

-1 -1 -1 -1 -1

16

14

12

xw1 ¯ xw2 ¯ xwc ¯ xm ¯

= = = =

-12.42 ± 0.45 -12.42 ± 0.59 -12.42 ± 0.74 -12.00 ± 0.37

10

8 1 2 3 4 Measurement number 5 1 2 c m

Figure 3: Testing four metho ds to compute a CA with real data: a case of determination of the Oort constants A (top) and B (bottom). The designations are the same as in Fig 1

9


In the second test, results of determination of average values of the Oort constants in Klaka (2009) are revised (see Fig 3). For comparison, the author's estimates are 14.2 ± 0.5 c for A, and -12.4 ± 0.5 for B, with uncertainties computed as 1 and evidently rounded up. This value of the WA uncertainty is likely underestimated as compared to input data uncertainties and scatter. Median uncertainty also seems underestimated because it accounts only for relatively small scatter, and ignores relatively large measurement uncertainties. In this example, c estimate again lo oks the most realistic and corresponding to input data.

5

C o n clu s io n

Although computation of a CM, in particular WA, is widely used in data analysis, this problem has no definite, unambiguity solution yet. In particular, a very important problem in most of applications is to obtain a "realistic" estimate for the CM uncertainty. Both underestimation and overestimation are equally undesirable. In this study, we mostly investigated several basic approaches to compute a WA. Currently used metho ds for computation of the WA uncertainty do not provide satisfactory result for many practical tasks. The classical WA uncertainty estimate 1 often yields underestimated value because it is rigorously justified for unbiased xi only. Another estimate 2 , which can be derived from a least squares solution or by appropriate scaling of the input uncertainties, do es not take account of input uncertainties, but only of the variance ratio. In this paper, we propose a new approach to compute the WA uncertainty, combined estimate c given by (8) and (10), which is able to account for both uncertainties and scatter of input data. We did not propose a rigorous statistical background for this estimate. However, it can be shown that in the case when input values xi are obtained from a normally distributed populations, and each value xi has a normally distributed systematic error, proposed estimate c can be derived from a least squares solution. It appears probable that this estimate is suitable for practical use until the deviation from normality is very large. Several tests with simulated and real measurements have demonstrated that using c is a simple and effective approach equally suitable for both discrepant and consistent measurements. It is also important that it provides realistic estimate even for very small samples of 2­3 measurements. As to the median approach, it is known as a more robust CM estimate, but computation ¨ of its uncertainty is not so straightforward. A simple approach by Muller (1995, 2000a) seems to be not satisfactory in some cases, as follows from our tests. In this respect, the bo otstrap metho d and extended bo otstrap metho d (MacMahon et al, 2004) deserve consideration; this metho ds may be to o complicated for routine use though. In conclusion, one should never forget the following. Despite the metho d used, the CM uncertainty is only a part of real measurement accuracy, namely a Type A uncertainty according to the standard metrological terminology. By definition, Type A uncertainty is computed from data using statistical pro cedures, while a Type B uncertainty is obtained using supplement information and theoretical considerations (see e.g. Bucher (2004). Sometimes, the type B uncertainty is evaluated from supplement testing data pro cessing, e.g. using subsets of the original data set. It can also be a result not of mathematical computation, but considerations based on the knowledge of the measurement pro cedures, observational history, 10


and previous experience. Such pro cedures were used e.g. to derive the best estimates of some quantities related to Solar System dynamics (Pitjeva EV, 2009), and to compute realistic errors in radio source positions for the second realization of the International Celestial Reference Frame (Ma et al, 2009).

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