Uncertainty On Subdivision Methods.pdf

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Uncertaintyy evaluation on weighing designs M. Sc. Luis Omar Becerra

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Weighing designs Same nominal value, Example:

1 kg, 1 kg (*), 1 kg (**), 1 kg (***)

Subdivision for submultiples of the kilogram Example:

1 kg, 500 g, 200 g, 200 g (*), 100 g, 100 g (*)

S bdi i i for Subdivision f multiples l i l off the h kilogram kil Example:

1 kg, 1 kg (*), 2 kg, 2 kg (*), 5 kg, 10 kg

Luis O. Becerra2

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Example: 1 kg, 1 kg (*), 1 kg (**), 1 kg (***) 1 kg

1 kg  (*)

‐ ‐ ‐

+

1 kg  (**)

1 kg  (***)

= y1

+

= y2

+ ‐ ‐

Observations

+ ‐

= y3 = y4

+ +

= y5 = y6

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Example: 1 kg Standard

‐ ‐

1 kg, 500 g, 200 g, 200 g (*), 100 g, 100 g (*) 500 g 200 g

+ + ‐ ‐

+ + + + ‐ ‐

200 g  (*)

100 g

+ + + + +

+

‐

100 g  (*)

Observations

= y1

+ +

= y2 = y3

+

= y4 = y5

+ + ‐

+ + +

= y6 = y7 = y8

4

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Example: 1 kg Standard

1 kg, 1 kg (*), 2 kg, 2 kg (*), 5 kg, 10 kg 1 kg  (*)

‐ ‐ ‐ ‐ ‐ ‐ ‐

‐ ‐ +

2 kg

2 kg (*)

5 kg

10 kg 

Observations

‐ ‐ ‐ ‐ ‐

‐ ‐ ‐ ‐ + +

‐ ‐ + +

+ +

= y1

+

= y2 = y3 = y4 = y5 = y6 = y7 = y8

5

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Subdivision methods •Lagrange operator

•Gauss Markov (Weighing Least Squares)

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Evaluation of the uncertainty for Lagrange O Operator t approach h Basic mass equation

⎛ ⎞ −1 ⎟ ⎜ − = − − m m ρ V V + Δ ISb −εd ∑j j ∑k k ∑ a⎜∑ k j⎟ j ⎝ k ⎠

Law of propagation of uncertainty for a univariate model

uc =

n

∑[c ⋅ u( x )]

2

i

i

i

=

⎡ ∂Y ⎤ ∑i ⎢ ∂X ⋅ u( xi )⎥ ⎣ i ⎦ n

2

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Lagrange operator Example: 1 kg, 1 kg (*), 1 kg (**), 1 kg (***) N Normal l equations ti

X T Xβ = X T Y

⎡ 3 − 1 − 1 − 1 1⎤ ⎡ β R ⎤ ⎡− 1 − 1 − 1 0 ⎢− 1 3 − 1 − 1 0⎥ ⎢ β ⎥ ⎢ 1 0 0 −1 ⎢ ⎥ ⎢ 1⎥ ⎢ ⎢− 1 − 1 3 − 1 0⎥ ⋅ ⎢ β 2 ⎥ = ⎢ 0 1 0 1 ⎢ ⎥ ⎢ ⎥ ⎢ 0 1 0 ⎢− 1 − 1 − 1 3 0⎥ ⎢ β 2 ⎥ ⎢ 0 0 0 0 0 ⎦⎥ ⎢⎣ λ ⎥⎦ ⎢⎣ 0 0 0 0 ⎣⎢ 1

Solution

(

⎡ y1 ⎤ ⎢ ⎥ 0 0 0⎤ ⎢ y2 ⎥ − 1 0 0 ⎥⎥ ⎢ y3 ⎥ ⎢ ⎥ y ⎥ ⋅ 0 −1 0 ⎢ 4 ⎥ ⎥ ⎢ 1 1 0 ⎥ y5 ⎥ ⎢ ⎥ ⎥ y 0 0 1⎦ ⎢ 6 ⎥ ⎢m ⎥ ⎣ 1kg ⎦ −1 T

)

βˆ = X'T X' X' Y'

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of uncertainty, Ratio of mass between the unknown and the standard

M Mass standard t d d

Air density

Ciu(xi)

hj =

mj mk

, j = 1...k

h j u (mk ) u ( ρ a )(V j − h jVk ) Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of uncertainty

Volume of the standard

Volume of j

ρ a h j u (Vk ),

j = 1...k

ρ a u (V j )

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of uncertainty Difference of Indications

Sensitivity of the balance

Resolution of the balance

s (ΔI ) Sb u (ΔI ) = Sb n −1

⎛ u (ms ) u (I s ) ⎞ ⎟⎟ ΔIu (Sb ) = ΔI ⎜⎜ + ΔI s ⎠ ⎝ m2

d u (ε d ) = 2 12 Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of uncertainty Fitting of the Least squares

[

u j ( fit ) = s 2 ( X T X) j , j

−1

]

1

2

[ Y − Xβˆ ] [Y − Xβˆ ] = T

with

s

2

m−n

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Standard uncertainty

[

]

u 2 (m j ) = (h j u (mk ) ) + (u ( ρ a )(V j − h jVk )) + ρ a (u (V j ) + h j u (Vk )) 2

2

2

2 ⎡ ⎤ ⎛ ⎞ u (ms ) u (I s ) d ⎞ ⎛ s (ΔI ) ⎞ ⎛ 2 ⎟ ⎜ + + Δ + u +⎜ I 2 + ⎟ ⎢ ⎜ ⎟ ⎥ ⎜ j ( fit ) ⎟ ΔI s ⎠⎦ ⎝ 12 ⎠ ⎝ Sb n ⎠ ⎣ ⎝ m2 2

2

u(m j ) =

n

2 [ ] c ⋅ u ( x ) ∑ i i i

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Degrees of freedom

ν eff (m j ) =

u 4 (m j )

∑

[ciu (xi )]

4

νi

Expanded uncertainty

U (m j ) = t(ν ,α ) × u(m j ) Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Evaluation of the uncertainty for Gauss Markov approach h Basic mass equation

⎛ ⎞ −1 ⎟ ⎜ − = − − m m ρ V V + Δ ISb −εd ∑j j ∑k k ∑ a⎜∑ k j⎟ j ⎝ k ⎠ Law of propagation of uncertainty for a univariate model

uc =

n

∑[c ⋅ u( x )]

2

i

i

i

=

⎡ ∂Y ⎤ ∑i ⎢ ∂X ⋅ u( xi )⎥ ⎣ i ⎦ n

2

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Gauss Markov approach Example: 1 kg, 1 kg (*), 1 kg (**), 1 kg (***) 0 ⎡− 1 1 ⎢− 1 0 1 ⎢ ⎢− 1 0 0 ⎢ ⎢ 0 −1 1 ⎢ 0 −1 0 ⎢ 0 −1 ⎢0 ⎢1 0 0 ⎣

Solution

0⎤ ⎡ y1 ⎤ ⎡ e1 ⎤ ⎢ y ⎥ ⎢e ⎥ 0⎥⎥ 2 2 ⎡β R ⎤ ⎢ ⎥ ⎢ ⎥ 1 ⎥ ⎢ ⎥ ⎢ y3 ⎥ ⎢ e3 ⎥ ⎥ ⎢ β1 ⎥ ⎢ ⎥ ⎢ ⎥ = y − e 0⎥ ⋅ ⎢β2 ⎥ ⎢ 4 ⎥ ⎢ 4 ⎥ 1 ⎥ ⎢ ⎥ ⎢ y5 ⎥ ⎢ e5 ⎥ ⎥ ⎣ β3 ⎦ ⎢ ⎥ ⎢ ⎥ 1⎥ ⎢ y6 ⎥ ⎢e6 ⎥ ⎢m ⎥ ⎢ 0 ⎥ 0⎥⎦ ⎣ R⎦ ⎣ ⎦

(

−1 Y

βˆ = X Ψ X T

Xβ = Y − e

)

−1

X T Ψ −Y1 Y Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Variance – Covariance Matrix

[

Ψ Y = E (y − E(y ))(y − E(y ))

( )

E eeT

⎡ σ y21 E (e1e2 ) ⋅ ⎢ 2 σ y2 ⎢ ⎢ ⋅ = ΨY = ⎢ ⎢ ⎢ ⎢ ⎢⎣ symm

T

⋅

⋅

⋅ ⋅

]

E (e1en )⎤ ⎥ ⎥ ⋅ ⎥ ⎥ ⋅ ⎥ ⋅ ⎥ ⎥ 2 σ yn ⎥⎦ Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Construction of complete variance - covariance matrix of Y

n

Ψ Y = ∑ J xi Ψ xi J

T xi

i

Law of propagation of uncertainty for a multivariate model

analogous to

u ( y) = 2

⎡ ∂Y ⎤ ∑i ⎢ ∂X ⋅ u( xi )⎥ ⎣ i ⎦ n

2

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

With ⎡ ∂y1 ⎤ ⎢ ∂x ⎥ ⎢ i⎥ ⎢ ∂y2 ⎥ ⎢ ∂xi ⎥ ⎢ ⎥ ⎥ J xi = ⎢ ⎢ ⎥ ⎢ . ⎥ ⎢ ⎥ ⎢ . ⎥ ⎢ ∂yn ⎥ ⎢ ∂x ⎥ ⎣ i⎦

⎡σ 2 ( xi ) . . ⎢ 2 (xi ) . σ ⎢ ⎢ . ⎢ Ψ xi = ⎢ ⎢ ⎢ ⎢ ⎢ symm ⎣

.

.

.

.

.

0

.

.

.

.

.

.

.

.

σ 2 ( xi )

⎤ ⎥ . ⎥ . ⎥ ⎥ . ⎥ . ⎥ ⎥ . ⎥ σ 2 ( xi )⎥⎦ .

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Construction of complete variance - covariance matrix of Y,

ΨY

⎛ ⎞ −1 ⎜ ⎟ yi = ρ a i ⋅ ⎜ ∑ V j − ∑ Vk ⎟ + ΔI i Sbi − 2ε d k ⎝ j ⎠

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of matrix Ψ Mass standard

J ms Ψ ms J

Air density y

J ρa Ψ ρa J ρa

Volume of the Standard i

T ms

T

J Vi ΨVi J

T Vi

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of matrix Ψ Difference of Indications

S Sensitivity iti it off the th balance b l

Resolution of the balance

J ΔI i Ψ ΔI i J

T ΔI i

J Sbi Ψ Sbi J

T Sbi

Jε d Ψε d Jε d T

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Contributions of matrix Ψ Fitting of the

J fit Ψ fit J

Least squares

T fit

Solution of Gauss Markov Approach

(

−1 Y

βˆ = X Ψ X T

n

)

−1

X T Ψ −Y1Y

Ψ Y = ∑ J xi Ψ xi J

T xi

i

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

βˆ

Variance – Covariance Matrix of

(

−1 Y

cov = X Ψ X T

)

−1

element j,j of cov = u (m j ) 2

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Evaluation of the uncertainty by numerical simulation (Monte Carlo method)

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

JCGM 102 2009-06-11 Draft Evaluation of measurement data Supplement 2 to the “Guide to the expression of uncertainty in measurement" - Models with any number of output quantities

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Input quantities 3.25 coverage probability probability that the true quantity value is contained within a specified coverage interval or coverage region. 4.13 4 13 In the vector case, case the PDF for X is denoted by gX(ξ), where ξ = (ξ1,….., ξN)T is a variable describing the possible values of the quantity X. This X is considered as a random variable with expectation E(X) and covariance matrix V(X). Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Input quantities 4.4 Measurement function When the Yi are expressed directly as formulae in X, the measurementt model d l is i represented t d by b the th measurement function

Y = f (X ) Or equivalent

Y1 = f1 ( X ) . . Ym = f m ( X )

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Input quantities 7.2 Number of Monte Carlo trials 7.2.2 Because there is no guarantee that any specific pre-assigned number will suffice, a procedure that selects M adaptively, i.e. as the trials progress, can be used. used

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Propagation 7.3 Making draws from probability distributions 7.3.1.1 In an implementation of MCM, M vectors xr, r = ,1….,M, are drawn d f from th PDFs the PDF gXi (ξi) for f the th input quantities X1,….,XN. Draws would be made from the jjoint ((multivariate)) PDF gX(ξ) if appropriate. pp p Recommendations concerning the manner in which these draws can be obtained are given in GUM Supplement 1 for the commonest univariate distributions.

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Propagation 7.4 Evaluation of the model 7.4.1 The multivariate model is evaluated for each of the M draws from the PDFs for the N input quantities. Specially, denote the M draws by x1,….,xM, each of dimension N X 1, 1 where the rth th draw xr contains x1,r …. xNr, with xi,r a draw from the PDF for Xi. Then, the model values are

yr = f ( xr ),

r = 1,....,M Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Primary MCM Output: distribution function for the output quantity 7.5 Discrete representation of the distribution function for the output quantity A discrete representation p of the distribution function for the vector output quantity is formed from the M values of the vector output quantity obtained in 7.4. In general, this representation is a matrix G of dimension mxM whose rth column is the rth value of the vector output quantity. tit For F a univariate i i t model, d l G is i a row vector. t Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Summarizing 7.6 Estimate of the output quantity and the associated covariance matrix

1 ~ ( y1 + .... + yM ) y= M

[

1 ( y1 − ~y1 )( y1 − ~y1 )T + ... + ( yM − ~y1 )( yM − ~y1 )T U ~y = M −1

]

are taken, respectively, as an estimate y of Y and the covariance matrix Uy associated with y.

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

MCM Summarizing 7.7 Coverage region for a vector output quantity 7.7.1 The determination from G of coverage regions for vector output quantities is not straightforward, straightforward be-cause be cause the operation of sorting vector data is generally not welldefined. Some approaches have been proposed, including the sorting of such data using the metric

( yr − a )T Σ −1 ( yr − a ) where a is a location statistic, such as the average, for the set y1,…,yyM and Σ is a dispersion statistic, statistic such as the covariance matrix Uy, for the set. Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico Distribution for 5 kg/F89 X <=0 2.275% 2 275%

1.2

Valu ues in 10^ 3

1

Statistic

X <=0 97.725% 97 725% M ean = 3.051933E-03

0.8 0.6 04 0.4 0.2 0 -5

-4

-3

-2

-1

Values in 10^-3

Distribution for 5 kg/F89 X <=0 2.275%

Values in 10^ 3

1

X <=0 97.725% M ean = 3.051933E-03

0.8 0.6 0.4

0 -5

-4

-3

Values in 10^-3

-2

-1

Value

Minimum

-0.00473

5%

-0.00366

Maximum

-0.00116

10%

-0.00353

Mean

-0.00305

15%

-0.00343

Std Dev

0.00037

20%

-0.00336

Variance

1.36E-07

25%

-0.00330

Skewness

6.09E-03

30%

-0.00325

Kurtosis

3.00E+00

35%

-0.00319

Median

-0.00305

40%

-0.00314

Mode

-0.00339

45%

-0.00310

Left X

-0.00379

50%

-0.00305

Left P

2.28%

55%

-0.00301

Right X

-0.00231

60%

-0.00296

Right P

97 72% 97.72%

65%

-0.00291 0 00291

Diff X

0.00148

70%

-0.00286

Diff P

95.45%

75%

-0.00280

0

80%

-0.00274

Filter Min

85%

-0 00267 -0.00267

Filter Max

90%

-0.00258

95%

-0.00244

#Errors 0.2

Summary Statistics Value %tile

#Filtered

0

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Examples Weighing Desing (1, 1*, 1**, 1***) GM x u(x) 1 kg 0.430 mg 0.013 g g (*) () 0.563 mg 0.021 1 kg 1 kg (**) 0.542 mg 0.017 1 kg (***) 0.011 mg 0.016 0.0 χ2 evaluated 7.8 χ2 critical value

Weighing Desing (1, 1*, 1**, 1***) GM x u(x) g g 0.430 0 30 mg 0.013 0 0 3 1 kg 1 kg (*) 0.591 mg 0.032 1 kg (**) 0.556 mg 0.030 1 kg (***) 0.027 mg 0.030 12.1 χ2 evaluated critical t ca value a ue 7.8 8 χ2 c χ

mg g mg mg mg

GM-S II x 0.430 0.563 0.542 0.011

mg g mg mg mg

GM-S II x 0.430 0 30 0.591 0.556 0.027

LO mg g mg mg mg

u(x) 0.013 0.027 0.015 0.013

x mg 0.430 g mg 0.563 mg 0.542 mg 0.011 F evaluated F critical value

mg g mg mg mg

u(x) 0.013 0.027 0.015 0.013 0.3 3.9

mg g mg mg mg

LO-S II x 0.430 0.563 0.542 0.011

mg g mg mg mg

OL-S II x 0.430 0 30 0.588 0.554 0.023

OL mg g mg mg mg

u(x) 0.013 0 0 3 0.037 0.029 0.028

x mg g 0.430 0 30 mg 0.588 mg 0.554 mg 0.023 F evaluated Fc critical t ca value a ue

mg g mg mg mg

u(x) 0.013 0 0 3 0.041 0.034 0.033 1428.5 3.9 3 9

mg g mg mg mg

u(x) 0.013 0.027 0.015 0.013

mg g mg mg mg

mg g mg mg mg

u(x) 0.013 0 0 3 0.041 0.034 0.033

mg g mg mg mg

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 10 kg -8.43 mg 0.37 5 kg -3.03 mg 0.19 2 kg -0.96 mg 0.08 2k kg (*) -0.55 0 55 mg 0 08 0.08 1 kg 0.563 mg 0.027 1 kg (*) 0.016 mg 0.044 0.0 χ2 evaluated 8.0 χ2 critical value Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 10 kg -8.39 mg 0.60 5k kg -3.05 3 05 mg 0 31 0.31 2 kg -0.89 mg 0.13 2 kg (*) -0.54 mg 0.13 1 kg 0.563 mg 0.027 1 kg (*) -0.029 mg 0.084 2 evaluated l t d 23 7 23.7 χ2 8.0 χ2 critical value

mg mg mg mg mg mg

GM-S GM S II x -8.43 -3.03 -0.96 -0.55 0 55 0.563 0.016

mg mg mg mg mg mg

GM-S II x -8.40 -3.05 3 05 -0.89 -0.54 0.563 -0.029

LO mg mg mg mg mg mg

u(x) 0.55 0.28 0.11 0 11 0.11 0.027 0.055

x mg -8.43 mg -3.03 mg -0.96 mg -0.55 0 55 mg 0.563 mg 0.016 F evaluated F critical value

mg mg mg mg mg mg

u(x) 0.55 0.28 0.11 0 11 0.11 0.027 0.056 0.3 3.9

mg mg mg mg mg mg

LO-S LO S II x -8.43 -3.03 -0.96 -0.55 0 55 0.563 0.016

mg mg mg mg mg mg

OL-S II x -8.39 -3.05 3 05 -0.89 -0.54 0.563 -0.026

OL mg mg mg mg mg mg

u(x) 0.73 0 37 0.37 0.16 0.16 0.027 0.091

x mg -8.39 mg -3.05 3 05 mg -0.89 mg -0.54 mg 0.563 mg -0.027 F evaluated l t d F critical value

mg mg mg mg mg mg

u(x) 0.73 0 37 0.37 0.16 0.16 0.027 0.091 162 0 162.0 3.9

mg mg mg mg mg mg

u(x) 0.56 0.28 0.11 0 11 0.11 0.027 0.057

mg mg mg mg mg mg

mg mg mg mg mg mg

u(x) 0.73 0 37 0.37 0.16 0.16 0.027 0.092

mg mg mg mg mg mg

Luis O. Becerra

SIM Workshop on Subdivision Methods October 2010, Juriquilla Mexico

Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 1 kg -0.141 mg 0.010 500 g -0.045 mg 0.014 200 g -0.026 mg 0.010 200 g (*) 0 024 mg 0.024 0 010 0.010 100 g -0.036 mg 0.010 100 g (*) -0.163 mg 0.010 3.4 χ2 evaluated 11.1 χ2 critical value Weighing Desing (10, 5, 2, 2*, 1, 1*) GM x u(x) 1 kg -0.141 mg 0.010 500 g -0.061 0 061 mg 0 022 0.022 200 g -0.011 mg 0.016 200 g (*) 0.021 mg 0.016 100 g -0.036 mg 0.017 100 g (*) -0.156 mg 0.017 2 evaluated l t d 16 2 16.2 χ2 11.1 χ2 critical value

mg mg mg mg mg mg

GM-S GM S II x -0.141 -0.045 -0.026 0 024 0.024 -0.036 -0.163

mg mg mg mg mg mg

GM-S II x -0.141 -0.061 0 061 -0.011 0.021 -0.036 -0.156

LO mg mg mg mg mg mg

u(x) 0.009 0.014 0.011 0 011 0.011 0.010 0.010

x mg -0.141 mg -0.045 mg -0.026 mg 0 024 0.024 mg -0.036 mg -0.163 F evaluated F critical value

mg mg mg mg mg mg

u(x) 0.010 0.019 0.014 0 014 0.014 0.014 0.014 1.2 3.9

mg mg mg mg mg mg

LO-S LO S II x -0.141 -0.045 -0.026 0 024 0.024 -0.036 -0.163

mg mg mg mg mg mg

OL-S II x -0.141 -0.060 0 060 -0.011 0.021 -0.036 -0.156

OL mg mg mg mg mg mg

u(x) 0.009 0 022 0.022 0.016 0.016 0.017 0.017

x mg -0.141 mg -0.061 0 061 mg -0.011 mg 0.021 mg -0.036 mg -0.156 F evaluated l t d F critical value

mg mg mg mg mg mg

u(x) 0.010 0 027 0.027 0.020 0.020 0.020 0.020 60 6.0 3.9

mg mg mg mg mg mg

u(x) 0.010 0.015 0.011 0 011 0.011 0.011 0.011

mg mg mg mg mg mg

mg mg mg mg mg mg

u(x) 0.010 0 024 0.024 0.017 0.017 0.018 0.018

mg mg mg mg mg mg

Luis O. Becerra