CN1912861A - Method of analog computing synthesis indeterminacy using Monte carlo acounting - Google Patents

Abstract

This invention relates to a method for simulating and computing the uncertainty with Monte Carlo statistic simulation including the following steps: a, introducing a math model of the functional relation of measured input and output volumes Y=f(x1, x2,...xn), in which, Y is the output volume, x1, x2...xn are input volumes, b, introducing the probability distribution and parameters of the uncertainty of input volumes x1, x2,...xn, c, selecting analog volume xi1, xi2,...xin based on the distribution and parameter of the uncertainty, d, selecting the check method for the analog volumes, e, simulating on the computer to compute the output, the standard deviation and synthesized uncertainty accurately associated with the believing level.

Description

A kind of method of calculating synthetic uncertainty with Monte Carlo statistical simulation

[technical field]

The present invention relates to the metrological testing technology field, especially relate to a kind of method of calculating synthetic uncertainty with Monte Carlo statistical simulation.

[background technology]

The notion of uncertainty is seized of consequence in the metrological testing technology field.Arbitrary effective measurement must have the evaluation of uncertainty, so that people can assess the quality of this measuring process, compare and reappear this measuring process.It becomes indispensable part at sciemtifec and technical sphere, has issued " uncertainty of measurement is represented guide " (Guide to the Expression of Uncertainty in Measurement is called for short GUM) so united by the name of 7 international organizations such as International Organization for Standardization in 1993 in the world.China has also issued JJF1059-1999 " evaluation of uncertainty in measurement and expression " in 1999, so that synchronous with the world, can carry out unified assessment to important measuring process, also guaranteed the feasibility and the unitarity of the transmission of quantity value of metering field.

But uncertainty is accompanied by arbitrary measuring process.Many fields, as fields such as industry, commerce and health and safety, the uncertainty that need be accompanied by this measured value has a definite fiducial interval, with expectation with it as foundation, prediction term purpose feasibility.The neither exaggerative predictability of the data (uncertainty) that people's expectation obtains promptly under the situation that probability demands is determined, is dwindled uncertainty; Also do not dwindle foreseeability (if opposite situation takes place), and expectation obtains correct prediction.Because the probability distribution of most of measurement results is difficult to draw with mathematical method, so the combined standard uncertainty u that above-mentioned standard derives according to the uncertainty spreading rate _cMultiply by topped factor k _pThe expanded uncertainty U that obtains _p, not definitely with the probability correlation of confidence level connection (as p=95%,, expression approximately should have 95% measurement result to fall into this interval).

[summary of the invention]

Technical matters to be solved by this invention provides a kind of accurately related method with the synthetic uncertainty of Monte Carlo statistical simulation calculating of final synthetic uncertainty and confidence level that makes, and it comprises the steps: a, introduces the mathematical model Y=f (x that measures input quantity and output quantity funtcional relationship ₁, x ₂... x _n), Y is output quantity, x in the following formula ₁, x ₂... x _nBe input quantity; B, introducing input quantity x ₁, x ₂... x _nThe probability distribution of uncertainty and parameter; C, according to input quantity x ₁, x ₂... x _nThe probability distribution of uncertainty and parameter are selected analog quantity ξ ₁, ξ ₂... ξ _nD, selection are to analog quantity ξ ₁, ξ ₂... ξ _nThe method of inspection; E, carry out analog computation output quantity Y, output quantity Y on computers standard deviation with the accurate relevant synthetic uncertainty of confidence level.Task proposed by the invention also can further be realized by following technical solution: carrying out analog computation on computers is with each random number ξ ₁, ξ ₂... ξ _nSubstitution Y=f (x ₁, x ₂... x _n) obtain y _i=f (ξ _1i, ξ _2i... ξ _Ni), the gamut that the span of each random number distributes at each uncertainty component, the corresponding y that obtains _iBe the probable value of output quantity, i=1,2 ... n produces n analogue value y on computers ₁, y ₂... y _n, output quantity $Y=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i,$ The standard deviation of output quantity Y $s={[\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(y_i-Y)}^2/(n-1)]}^{1/2},$ And the Probability p of the synthetic uncertainty fiducial interval of evaluation as required, making p=m/n, n is the simulation total degree in the formula, m is | y _iThe simulation number of times of-Y|＜δ, δ be need final assessment output quantity Y with the accurate relevant synthetic uncertainty of confidence level.

The present invention can make final synthetic uncertainty accurately related with confidence level, particularly the synthetic uncertainty of high confidence level (confidence level 95% or 99% etc.).If combined standard uncertainty u according to the derivation of uncertainty spreading rate _cMultiply by topped factor k _pThe expanded uncertainty U that obtains _p, because the distribution of output quantity is often unknowable, and imprecisely with the probability correlation connection of confidence level.The present invention will be in a lot of fields, as the very big effect of fields such as industry, commerce, medical and health performance.

[embodiment]

Table 1 is the computer program flow process of analog computation.

Table 2 is analog computation Y=f (D ₀, M _x, D _x) distribution.

Table 3 is 10 double counting values (different simulation number of times) of analog computation.

Embodiment: present embodiment is the example of micro scale on the calibration scan electron microscope document image, wherein measures mathematical model and is the computing formula of calculating calibration value:

M＝D ₀×M _x/D _x

In the formula:

The calibration value of M---micro scale;

D ₀---the length value of standard substance;

M _x---the length measurements of micro scale on the document image;

D _x---the length measurements of standard substance.

The method of each input quantity Determination of Uncertainty and generation random number

Mathematical model f (the D that is measuring ₀, M _x, D _x)=D ₀* M _x/ D _xUnder the situation about having set up, each input quantity D wherein ₀, M _x, D _xUncertainty hypothesis determine as follows:

The length value D of standard substance ₀=4.6 μ m ± 0.05 μ m, and uncertainty Normal Distribution N (a, σ), a=4.6 μ m here; σ=0.05 μ m.The length value of standard substance is with simulating number ξ _1iExpression, select the normal distribution N that common Monte Carlo method introduces (a, the method that random number σ) produces can obtain:

${\mathrm{\ξ}}_{1i}=\mathrm{\σ}(\underset{k=1}{\overset{48}{\mathrm{\Σ}}}{r}_k-24)/2+a$

$=0.05(\underset{k=1}{\overset{48}{\mathrm{\Σ}}}{r}_k-24)/2+4.6$ (r wherein _kFor (0,1) goes up equally distributed random number)

The length measurements M of micro scale on the document image _x=15mm ± 0.5mm, uncertainty is obeyed (a, b) Qu Jian even distribution, a=15mm-0.5mm=14.5mm here; B=15mm+0.5mm=15.5mm.The length measurements of micro scale is with simulating number ξ on the document image _2iExpression, select that common Monte Carlo method introduces to arbitrary region (a b) goes up the method that equally distributed random number produces and produces, and can obtain:

ξ _2i=a+ (b-a) r _2i=14.5+r _2i(r wherein _2iFor (0,1) goes up equally distributed random number)

The length measurements D of standard substance _x=36mm ± 0.5mm, uncertainty is obeyed at (a, b) Qu Jian even distribution, a=36mm-0.5mm=35.5mm here; B=36mm+0.5mm=36.5mm.The standard substance length measurements is with simulating number ξ _3iExpression, select that common Monte Carlo method introduces to arbitrary region (a b) goes up the method that equally distributed random number produces and produces, and can obtain:

ξ _3i=a+ (b-a) r _3i=35.5+r _3i(r wherein _3iBe that (0,1) go up equally distributed random number) check of simulation number

Because this example is to adopt the computer program of product Visual Basic 6.0 software programmings of U.S. Microsoft company to carry out analog computation, the generation of random number is that the appended randomizer of this software produces.Can think such acquisition (0,1) goes up equally distributed random number series by parameters check, uniformity testing and independence test, so in calculation procedure, we only to structure on this random number basis get up at arbitrary region (a, b) go up equally distributed random number and normal distribution N (a, the mean value of random number σ) $\stackrel{\‾}{y}=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i$ Carry out parametric test, its method is as follows:

For arbitrary region (a, b) mathematical expectation and the variance that goes up uniform random number is expressed as:

E(y)＝(a+b)/2

σ ²＝E(y ²)-[E(y)] ²＝(b ³-a ³)/3-(a+b) ²/4

Know statistic by central limit theorem:

$u=[\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i-\mathrm{nE}\left(y\right)]/(\mathrm{\σ}\×\sqrt{n})=\sqrt{n}[\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i-\frac{a+b}{2}]/{[\frac{b^3-a^3}{3}-\frac{{(a+b)}^2}{4}]}^{1/2}$

When N is fully big, progressively obeys N (0,1) and distribute.

For normal distribution N (a, the mathematical expectation of random number σ) and variance directly are exactly a and σ, so obtain statistic by central limit theorem:

$u=[\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i-\mathrm{nE}\left(y\right)]/(\mathrm{\σ}\×\sqrt{n})=\frac{\sqrt{n}}{\mathrm{\σ}}(\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i-a)$

When N is fully big, progressively obeys N (0,1) and distribute.

Get level of significance α=0.1, then when | u|＞1.645, computer program will produce random number again, until the random number that produces is proceeded analog computation then by parametric test.

The computer program flow process of analog computation is referring to table 1.

The analog computation result:

Analog computation Y=f (D ₀, M _x, D _x) distribution referring to table 2.

The analog computation mean value of Y: E (y)=1.91670410155901 μ m; Analog computation number of times: n=100000; The analog computation standard deviation of Y: S=4.62415096429435E-02 μ m; The fiducial probability of simulation value is 95% synthetic uncertainty: U ₉₅=0.086317873244385 μ m.

The present invention can also adopt repeatedly double counting to obtain mean value, so that obtain higher precision.10 double counting values of analog computation (different simulation number of times) are referring to table 3.

If getting 10 double counting mean values of n=100000 is net result, can gets calibration value and uncertainty and be:

Micro scale calibration value M=1.917 μ m on the image, standard deviation s=0.046 μ m, U ₉₅(M)=0.085 μ m.

Spreading rate according to the uncertainty of regulation among " uncertainty of measurement is represented guide " (Guide to the Expression of Uncertainty inMeasurement is called for short GUM) and the CNS JJF1059-1999 " evaluation of uncertainty in measurement and expression " $u_c^2\left(y\right)=\underset{i=1}{\overset{N}{\mathrm{\Σ}}}{\left[\frac{\∂f}{{\∂x}_i}\right]}^2{u}^2\left(x_i\right)$ Calculating can get calibration value and uncertainty is: micro scale calibration value M=1.917 μ m on the image, combined standard uncertainty u _c(M)=0.045 μ m, expanded uncertainty U (M)=0.090 μ m; K=2.

Table 1

Table 2

The simulation frequency n	Analog computation mean value E (y)	Standard deviation S	Synthetic uncertainty (95%)
				1000	1.91999960561	0.04343572944	0.07719564026
1000	1.91431399520	0.04693235069	0.08898753965
				1000	1.91650754874	0.04502600662	0.08558322496
1000	1.91461375245	0.04853157404	0.09428388535
				1000	1.92130136760	0.04419618587	0.08387538353
1000	1.92078692475	0.04687640923	0.08692830841
				1000	1.91278062577	0.04706553280	0.08933426860
1000	1.92021422396	0.04675727676	0.08659899333
				1000	1.92007606678	0.04442139537	0.08172215466
1000	1.91660246479	0.04489642036	0.08437755711
				Mean value	1.91771965757	0.04581388812	0.08588869559
Standard deviation	0.00311924152	0.00162946694	0.00462960147
10000	1.91327348144	0.04485437348	0.08474359067
				10000	1.91471483038	0.04687660217	0.08655682969
10000	1.91336180236	0.04237230110	0.08064296981
				10000	1.91254033916	0.04510784008	0.08615734388
10000	1.91822420335	0.04547925248	0.08737239349
				10000	1.91401236881	0.04731933820	0.08725464655
10000	1.91526978118	0.04781703640	0.09169709697
				10000	1.91719845107	0.04706752749	0.08899017738
10000	1.92087012419	0.04491498246	0.08518137817
				10000	1.92042430766	0.04588298945	0.08492132702
Mean value	1.91598896896	0.04576922433	0.08635177536
				Standard deviation	0.00302165441	0.00160376252	0.00290713773

(continuous nextpage) (brought forward)

50000	1.91445048858	0.04601576090	0.08670459993
				50000	1.91490416155	0.04514804391	0.08322765646
50000	1.91764740887	0.04391093842	0.08278820786
				50000	1.91899204445	0.04468988012	0.08517379323
50000	1.92114890430	0.04589111150	0.08504693505
				50000	1.91788347790	0.04502908213	0.08452275663
50000	1.91936685187	0.04553810318	0.08506173059
				50000	1.91788666864	0.04758123999	0.08746756429
50000	1.91578996295	0.04386265770	0.08274164036
				50000	1.91312546807	0.04560360155	0.08617017878
Mean value	1.91711954372	0.04532704194	0.08489050632
				Standard deviation	0.00249433394	0.00108890950	0.00162074597
				100000	1.91645610604	0.04647044109	0.08786854215
100000	1.91847265710	0.04554862898	0.08565389052
				100000	1.91560375003	0.04577514149	0.08593928581
100000	1.91709709493	0.04626752522	0.08563484604
				100000	1.91789391940	0.04447459245	0.08307888582
100000	1.91695612417	0.04530520963	0.08518167415
				100000	1.91808268402	0.04508823893	0.08446613380
100000	1.91783382781	0.04513480701	0.08416785468
				100000	1.91522932069	0.04712540374	0.08666412406
100000	1.91642910731	0.04621093718	0.08563994081
				Mean value	1.91700545915	0.04574009257	0.08542951779
Standard deviation	0.00108284077	0.00078684744	0.00133235532

Table 3

Claims (2)

1. the method with the synthetic uncertainty of Monte Carlo statistical simulation calculating is characterized in that it comprises the steps:

Mathematical model Y=f (the x of input quantity and output quantity funtcional relationship is measured in a, introducing ₁, x ₂... x _n), Y is output quantity, x in the following formula ₁, x ₂... x _nBe input quantity;

B, introducing input quantity x ₁, x ₂... x _nThe probability distribution of uncertainty and parameter;

C, according to input quantity x ₁, x ₂... x _nThe probability distribution of uncertainty and parameter are selected analog quantity ξ ₁, ξ ₂... ξ _n

D, selection are to analog quantity ξ ₁, ξ ₂... ξ _nThe method of inspection;

E, carry out analog computation output quantity Y, output quantity Y on computers standard deviation with the accurate relevant synthetic uncertainty of confidence level.

2. a kind of method with the synthetic uncertainty of Monte Carlo statistical simulation calculating according to claim 1, it is characterized in that: carrying out analog computation on computers is with each random number ξ ₁, ξ ₂... ξ _nSubstitution Y=f (x ₁, x ₂... x _n) obtain y _i=f (ξ _1i, ξ _2i... ξ _Ni), the gamut that the span of each random number distributes at each uncertainty component, the corresponding y that obtains _iBe the probable value of output quantity, i=1,2 ... n produces n analogue value y on computers ₁, y ₂... y _n, output quantity $Y=\frac{1}{n}\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{y}_i,$ The standard deviation of output quantity Y is $s={[\underset{i=1}{\overset{n}{\mathrm{\Σ}}}{(y_i-Y)}^2/(n-1)]}^{1/2},$ And the Probability p of the synthetic uncertainty fiducial interval of evaluation as required, making p=m/n, n is the simulation total degree in the formula, m is | y _iThe simulation number of times of-Y|＜δ, δ be need final assessment output quantity Y with the accurate relevant synthetic uncertainty of confidence level.