3. Uncertainty Analysis

Consider the calculation of electric power from

P = E․I

where E and I are measured as

E = 100 V ± 2 V ,     I =   10 A ± 0.2 A

By taking the worst possible variations,

Pmax = (100 + 2) (10 + 0.2) = 1040.4 W

Pmin = (100 - 2) (10 - 0.2) =  960.4 W

Thus, the uncertainty in the power is 4.04 % and -3.96 %

☜ too severe and quite unlikely

Consider a variable, Xi , which has a known uncertainty, [eqn: rm delta X_i]

[eqn: rm X _{i} `=~X _{i} (measured)`±` delta X _{i}]   (20 : 1)

☞ ∙The best estimate of [eqn: rm X_i ~is ~ X_i] (measured)

∙There is an uncertainty in[eqn: rm X_i] that may be as large as [eqn: rm +- delta X_i]

∙The odds are 20 to 1 (95% confidence) against the uncertainty of [eqn: X_i] being             larger than [eqn: ± delta X _{i}]

	single sample analysis	multiple sample analysis
[eqn: X_i] (measured)	observations	the mean of a set of N observation
[eqn: +- sigma X_i]	[eqn: 2sigma]	∙ [eqn: tS_(N) over SQRT { N}] for random error component ∙ bias limit for fixed errors ∙[eqn: U_0.95] , overall uncertainty in [eqn: X_i]

The result R of the experiment is

[eqn: rm R ~=~ R(X_1 ,~ X_2 ,~ X_3. ~ … ~, X_N )]

Let [eqn: rm w_R] be the uncertainty in the result and [eqn: rm w_1, ~w_2, ~...... ~w_N] be the

uncertainties in the independent variables

[eqn: rm w_R ~=~ left { SUM_ i=1^N left ( deltaR over deltaX_i w_i right )^2 right }]

[eqn: rm =~ left [ left ( deltaR over deltaX_1 delta X_1 right )^2 + left ( deltaR over deltaX_2 delta X_2 right )^2 +..... + left ( deltaR over deltaX_N delta X_N right )^2 right ]]

➡ Basic Equation of Uncertainty Analysis

where  [eqn: rm deltaR over deltaX_i] ; sensitivity coefficient for the result R w.r.t the measurement, Xi

∙ Each of the measurements was independent

∙ Repeated observations of each measurement, if made, would display Gaussian            distributions.

∙ The uncertainty in each measurement was initially expressed at the same odds.

For the result having 'product form' as,

[eqn: R ~=~ X_1^a ~X_2^b ~X_3^c ~… ~X_M^m]

Then,

[eqn: deltaR over R ~=~ left { left(a deltaX_i over X_i right)^2 ~+~ left(b deltaX_2 over X_2 right)^2 ~+~ … ~+~ left(m deltaX_m over X_m right )^2 right }^{1 over 2}]

Here, the exponent of Xi becomes its sensitivity coefficient.

[Ex] resistance of copper

[eqn: rm R ~=~ R_o [1+alpha (T-20)]]

where [eqn: rm R_o = 6`` OMEGA +- 0.3%]  at 20 oC ,  [eqn: rm alpha = 0.004``^o C^-1 +- 1 %] ,  T=30±1oC

The nominal resistance   R = 6.24 Ω

[eqn: rm partialR over partialR_o ~=~ 1+alpha(T-20) ~=~ 1.04]     [eqn: rm ,```````w _{R _{o}} `=~6 TIMES 0.003`=~0.018``` OMEGA]

[eqn: rm partialR over partialalpha ~=~ R_o (T-20) ~=~ 60]          [eqn: rm ,```````w _{alpha } `=~4 TIMES 10 ^{-5} `` ^{o} C ^{-1}]

[eqn: rm partialR over partialT ~=~ R_o ``alpha]                      [eqn: rm ,``````w _{T} `=~1`` ^{o} C]

Thus, the uncertainty in the resistance is

[eqn: rm w_R ~=~ [(1.04)^2 (0.018)^2 + (60)^2 (4 times 10^-5 )^2 + (0.024)^2 (1)^2 ]^1/2 ~=~ 0.0305 `` OMEGA ~o r~ 0.49``` %]

■ Evaluation of uncertainty for complicated data reduction

√ when R is calculated using a large-scale computer program

√ difficult to differentiate

Perturb the variables by [eqn: TRIANGLE x_1 ~, TRIANGLE x_2 ~ ...] and calculate new results

[eqn: rm R(x_i + TRIANGLE x_i ) ~=~ R (x_1 , x_2 ,.., x_i + TRIANGLE x_i , ...)]

For small enough values of △x,

[eqn: rm partialR over partialx_i ~=~ {R(x_i + TRIANGLE x_i ) -R(x_i ) } over { TRIANGLE x_i }]

☞ increment may be [eqn: delta X_i]

☞ when the result R is likely to be a strongly nonlinear junction of [eqn: X_i]

In the above example,

[eqn: TRIANGLE R_o = 0.01~ ,~~~ TRIANGLE alpha = 1 times 10^-1 ~,~~~ TRIANGLE T = 0.1]

Then,     [eqn: rm partialR over partialR_o APPROX {R(R_o + TRIANGLE R_o ) -R} over {TRIANGLE R_o }= {6.2504-6.24} over 0.01 = 1.04]

Also,     [eqn: rm partialR over partialalpha ~=~ 60~, ~~~ partialR over partialT ~=~ 0.024]

All the derivatives are the same as in the above example so the uncertainty in R

would be the same, or 0.0305 Ω.

그림 1  Uncertainty of Computed Heat Transfer Coefficient

[eqn: rm input ~=~ Q'' , ~T_1 , ~T_2 , ~delta Q'' , ~ delta T_1 , ~ delta T_2]

Base Case        Run 1                  Run 2                   Run 3

[eqn: rm Q'' , ~T_1 , ~T_2]  [eqn: rm ( Q'' + delta Q'' ) , T_1 , T_2]    [eqn: rm Q'' , (T_1 + delta T_1 ), ~T_2]        [eqn: rm Q'' , T_1 ,T_2 + delta T_2]

[eqn: rm h_o = Q'' over { T_1 - T_2 }][eqn: rm h_{TRIANGLE Q } = {Q'' ~+~ deltaQ''} over {T_1 - T_2 }] [eqn: rm h _triangleT_1 = Q over {(T_1 + deltaT_1 ) ~-~ T_2 }] [eqn: rm h_{ TRIANGLE T_2 }= Q over { T_1 - (T_2 + deltaT_2 ) }]

(ho)          [eqn: rm (h_{ TRIANGLE Q''} - h_o )]          [eqn: rm (h_{ TRIANGLE T_1} - h_o )]               [eqn: rm (h_{ TRIANGLE T_2} - h_o )]

ho           [eqn: rm deltah ~=~ left{ left( {partial h} over {partial Q''} deltaQ'' right )^2 ~~+~~ left ( {partial h} over partialT_1 } deltaT_1 right )^2 ~~+~~ left( {partial h} over {partial T_2 } deltaT_2 right)^2 right}^1/2]

(1) Statistical Analysis of Experimental Data

Since precision errors are random errors, it is necessary to consider how we can      determine some quantitative value of the "scatter" in the measurements of a variable      and the degree of confidence to associate with such a value

○ parent distribution or population ; infinite number of readings

○ sample distribution             ; finite number of readings

● arithmathic mean   ;   [eqn: rm x_m ~=~ 1 over N SUM_ i=1^N ```x_i]

● deviation           ;   [eqn: rm d_i ~=~ x_i - x_m ~~~~~(~bar d_i = 0~)]

● population standard deviation  ;    [eqn: rm sigma ~=~ left[ 1 over N SUM_ i=1^N ```(x_i - x_m )^2 right]^1/2]

- desirable to have at least 20 measurements

● sample standard deviation     ;    [eqn: rm sigma ~=~ left[ 1 over N-1 SUM_ i=1^N ```(x_i - x_m )^2 right]^1/2]

- for small sets of data

● median  ;   the value that devides the data points in half.

[Ex]  10, 12, 13, 14, 15 kΩ   ->   mean ; 12.8 kΩ

median ; 13 kΩ

● geometric mean   ;   [eqn: rm x_g ~=~ [x_1 CDOT x_2 cdot x_3 cdot cdot cdot x_N ]^1/N]

- biological process or growth rate in financial resources

● probability distribution

- shows how the probability of success in a

certain event is distributed over the distance x

- The area under the curve is unity

① binomial distribution

- gives a number of sucesses n out of N possible

independent events when each event has a probability of success p

[eqn: rm p(n) ~=~ {N`!} over {(N-n)`! ````n`!} p^n (1-p)^N-n]

② Poisson distribution

- limit of binomial distribution as N → ∞ , such as Np = a ; constant

[eqn: rm p_a (n) ~=~ { a^n e^-a } over {n`!}]   ,      [eqn: rm sigma ~=~ sqrt a]

- applicable to the calculation of the decay of radioactive nuclei

③ Gaussian or Normal Error distribution

; the smooth distribution of an infinite number of readings whose variation                results from a combination of many small errors of equal magnitude with each           of the errors being equally likely to be positive or negative

[eqn: rm f(X) ~=~ 1 over {sigma sqrt 2pi}``e^{-(X-mu )^2 ``/2`` sigma``^2}]                                     (1)

where  f(X)dx ; the probability that a single measurement of X will be lie between                       X and X+dX

μ(distribution mean) = [eqn: rm lim from { N -> inf } ``1 over N SUM from { { i}=0} to N ``X_i]

[eqn: sigma] (distribution standard deviation) =  [eqn: rm lim from { N -> inf } ``left[1 over N SUM from { { i}=0} to N ``(X_i -mu)^2 right]^1/2]

Thus, larger [eqn: sigma] means larger scatter of readings and larger precision error.

The probability that a reading from a Gaussian parent population will fall within a       specified range [eqn: [mu- TRIANGLE X ~,~mu+ TRIANGLE X``]]

[eqn: rm Prob( TRIANGLE X) ~=~ int from {mu- TRIANGLE x} to {mu+ TRIANGLE x} 1 over {sigma sqrt 2pi}``e^{-(X-mu )^2 ``/2`` sigma``^2}``dX]                     (2)

Normalized deviation ;

[eqn: tau ~==~ X-mu over sigma]

Eq. (2) will be

[eqn: rm P ro b( tau_1 ) ~=~ 1 over {sqrt 2pi} int from -tau_1 to tau_1 ``e^{-tau^2 ``/2}``dX]   : "two-tailed" probability

where      [eqn: tau_1 ~=~ TRIANGLE X / sigma]

Note that  50% of readings   ;  within ±0.675[eqn: sigma] of the mean

68.3% of readings  ;  within ±1.0[eqn: sigma] of the mean

95% of readings   ;  within ±1.96[eqn: sigma] of the mean

1) Confidence intervals in a Gaussian distribution

If we were to take one more reading Xi , within what interval could we be           95% confident the reading would fall ?

Then from Table of Gaussian distribution, τ=1.96 of Prob(τ) = 0.95

[eqn: rm P r ob left(-1.96 <= {X_i - mu } over sigma <= 1.96 right) ~=~ 0.95]                              (3)

Or    [eqn: rm P rob (mu - 1.96`` sigma ~<=~ X_i~ <=~ mu + 1.96`` sigma) ~=~ 0.95]

Thus, we can be 95% confident that a single reading will fall within [eqn: +- 1.96 sigma]

Or.

Within what interval about a single reading Xi would we expect the mean value         of the distribution to lie at a confidence level of 95% ?

From Eq. (3),     [eqn: rm P rob (X_i - 1.96`` sigma ~<=~ mu~ <=~ X_i + 1.96`` sigma) ~=~ 0.95]           (4)

2) Gaussian sample population

- impractical to expect anything approaching an infinite number of readings in a            real experiments

Consider a finite number of samples drawn from a parent distribution.

o sample population mean  ; [eqn: rm x_m ~=~ 1 over N SUM_ i=1^N ```x_i]

o precision index of sample population  ;   [eqn: rm sigma_x ~=~ left[ 1 over N-1 SUM_ i=1^N ```(x_i - x_m )^2 right]^1/2]

How good (or precise) is the arithmetic mean which is taken as the best                estimate of the true value of a set of measurements ?

-> necessary to repeat the set of measurements and find a new arithmetic mean

Consider the standard deviation of the mean of a single set of data from the true        value

[eqn: rm sigma_m ~=~ sigma over sqrt N]                                                    (5)

where   [eqn: rm sigma_m]  ; standard deviation of the mean value

[eqn: sigma]  ; standard deviation of the set of measurements

N  ; number of measurements in the set

3) Confidence intervals in sample populations

From Eqs. (4) and (5),

[eqn: rm P rob (x_m - 1.96`` sigma over sqrt N ~<=~ mu~ <=~ x_m + 1.96`` sigma over sqrt N ) ~=~ 0.95]

Thus, we can also say with 95% confident that the mean [eqn: mu] of the parent               distribution is within [eqn: +- 1.96 sigma] of a single reading from Gaussian distribution

Actually we do not know [eqn: sigma] but only know[eqn: sigma_m] . If N is very large, two values

will approach each other.  Thus, in order to determine 95% confidence interval,

we have to find t which satisfy the following, according to Eq. (3).

[eqn: rm P r ob left(-t ~<= {x - mu } over sigma_x <= ~t right) ~=~ 0.95]  and  [eqn: rm P r ob left(-t~ <= {x_m - mu } over {sigma_m / sqrt N} <= ~t right) ~=~ 0.95]

Here t is not 1.96. (例 ; t=2.57 for N=6,  t=2.04 for N=31)

※ For N < 31 , use t in Table.   But use t = 2.0  for N > 31

Precision limit, P  ; 95% confidence limit for a sample of N measurements of x                              drawn from Gaussian distributions

[eqn: rm P_x ~=~ t``sigma_x]      and      [eqn: rm P_x_m ~=~ t``sigma_m ~=~ t``sigma_x / sqrt N]

Thus ±P interval will be the interval within which we expect [eqn: mu] to lie with 95%        confidence.

[eqn: & Prob(X_i ~-~ P_X ~<=~ mu ~<=~ X_i ~+~ P_X ) ~=~ 0.95 ## & Prob(bar X ~-~ P_barX ~<=~ mu ~<=~ bar X~ ~+~ P_barX ) ~=~ 0.95]

Ex) A thermometer immersed in an insulated beaker of fluid was read                        independently by 24 students to the nearest 0.1℉. The mean of the sample              of 24 readings is [eqn: bar T~ ~=~97.22℉] and the precision index of the sample is               [eqn: sigma_T ~=~ 0.082℉]. Find the precision limits for this case and discuss their                  meaning.

For N=24,    t=2.069  from Table

∴ [eqn: rm P_T ~= ~t sigma_T ~=~ 2.069 ~times ~0.082 ~= 0.17℉]

[eqn: P_Tm ~=~ t`` sigma_T / sqrt N~ ~=~ 0.0346℉]

Thus the next reading will lie within the following interval with 95%

confidence.

[eqn: bar T~ ~+-~ P_T ~=~ 97.2℉ ~+- ~0.2℉]

The mean value[eqn: mu_T] of the parent population will fall within the interval of

following with 95% confidence.

[eqn: bar T~ ~+-~ P_Tm ~=~ 97.22℉ ~+-~ 0.03℉]

4) Statistical rejection of "wild" readings from a sample population.

- "outlier" or "wild" points : readings that appear to be significantly out of line             with the other readings.

- If obvious or verifiable cases.  ➡  discarded

- If not, uses statistical criterion to identify points that might be considered for           rejection.

- Chauvenet‘s criterion

: specifies that all points should be retained that fall within a band around the             mean value that corresponds to a probability of 1-1/(2N)

For 6 readings ,  1 - 1/12 = 0.917

Then the nondirectional deviation [eqn: tau] equal 1.73 from Table

∴ [eqn: tau ~=~ {x_i - x_m} over sigma_x ~=~ x_max over sigma_x]

Therefore, all readings that deviate from the mean by more than 1.73[eqn: sigma_x] can be          rejected.

Then a new mean value and a new precision index are calculated from the

readings that remain. No further application of the criterion to the sample is

allowed - Chauvenet criterion can be applied only once fro a given sample of

readings.

TABLE 2.1 Chauvenet's Criterion for Rejecting a Reading
Number of Readings (N)	Ratio of Maximum Acceptable Deviation to Precision Index([eqn: x_max / S_X])
3	1.38
4	1.54
5	1.65
6	1.73
7	1.80
8	1.87
9	1.91
10	1.96
15	2.13
20	2.24
25	2.33
50	2.57
100	2.81
300	3.14
500	3.29
1,000	3.48

5) Effect of Non-Gaussian error distribution

ⅰ) When is it inadvisable to assume normality?

ⅱ) What is magnitude of the effect if errors are from a nonnormal distribution?

➡ low-probability events ; the details of the tails of the error distributions are                                      important.

(Ex) accidents in NPP or natural disaster(100 years flood, etc)

6) Bias error estimation

ⅰ) precision error : tS

ⅱ) bias error ; same in every reading but never know unless we know the true

value

Thus define a bias limit, B, which is an estimated value of the limit of confidence       interval on β with same confidence level of the precision limit.

7) Overall uncertainly in a measurement

How P and B should be combined to obtain the overall unncatainty U in a             measured variable ?

ⅰ) Combination by the toot-sum-square (RSS)

[eqn: rm U_X ~=~ (B^2 ~+~ P_X^2 )^1/2]

where           [eqn: rm P_X ~=~ t``S_X]

Or         [eqn: rm U_barX ~=~ left[B^2 ~+~ (P_barX )^2 right]^1/2]

[eqn: ##]    where            [eqn: rm P_barX ~=~ t``S_barX ~=~ t``S_X / sqrt N]

Thus, [eqn: rm U_RSS] is said to be a 95% coverage estimate when B and P are 95%             confidence values.

ⅱ) Combination by straight addition (ADD)

[eqn: rm U_X_ADD ~=~ B ~+~ P_X]

or

[eqn: rm U_{barX}_ADD ~=~ B ~+~ P_ barX]

[eqn: rm U_ADD] provides approximately 99% coverage when neither B nor P is negligible           relation to the other. However [eqn: rm U_ADD]  provides about 95% coverage when either         B or P is negligible relation to the other.