CN107145627A - A kind of method for building belt restraining least square maximum entropy tantile function model - Google Patents

A kind of method for building belt restraining least square maximum entropy tantile function model Download PDF

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CN107145627A
CN107145627A CN201710206271.6A CN201710206271A CN107145627A CN 107145627 A CN107145627 A CN 107145627A CN 201710206271 A CN201710206271 A CN 201710206271A CN 107145627 A CN107145627 A CN 107145627A
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温卫东
吴福仙
翁晶萌
陈波
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Nanjing University of Aeronautics and Astronautics
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Abstract

The present invention proposes a kind of method for building belt restraining least square maximum entropy tantile function model, including step 1) set up without constraint least square maximum entropy tantile function model；2) constrain based on Weibull distribution model afterbody and set up belt restraining maximum entropy tantile function model and method without constraint least square maximum entropy tantile function model.Belt restraining least square maximum entropy tantile function model proposed by the present invention can be applied successfully to aerospace component fail-safe analysis tantile Function Estimation problem under Small Sample Size with method, compared to conventional probability statistical method, computational efficiency of the present invention is high, can handle Complicated Random Variables；Compared to classic maximum entropy tantile function model, computational accuracy of the present invention is high and stably.

Description

A kind of method for building belt restraining least square maximum entropy tantile function model
Technical field：
The present invention relates to aerospace component reliability assessment field, and in particular to aerospace component is reliable under Small Sample Size Property analysis in a kind of structure belt restraining least square maximum entropy tantile function model method.
Background technology：
External drive based on aerospace component not only with operating mode about also being influenceed by enchancement factor, the inequality of material structure The influence of dimensional tolerance dispersiveness, aerospace component intensive analysis during the random distributions such as even property, internal flaw and processing and manufacturing During fail-safe analysis the problem of be one common：In structural fatigue design, the fatigue life of given reliability is asked；Structure In rigidity Design, the displacement structure of given reliability is asked to respond etc..
Integrity problem is solved at present typically to use conventional probability statistical method, i.e. probability distribution to assume, verify and be distributed Parameter Estimation, but there are the following problems：(1) calculating process is troublesome, efficiency is low；(2) because probability distribution is assumed, checking link is held Human error is easily introduced, the probability distribution and actual distribution for making estimation have error；(3) unimodal stochastic variable, nothing can only be handled Method is applied in Complicated Random Variables (such as multimodal).
Principle of maximum entropy is a kind of powerful of processing probability statistics of RECENT DEVELOPMENTS.Using principle of maximum entropy fitting point Have during place value function formula and do not assume that probability distribution, minimum artificial disturbance factor, formula unification, strong applicability etc. are excellent in advance Put and can overcome the shortcoming that common probability Distribution Model can not be applied in the complicated probability distribution of fitting.In the case of large sample (sample number is more than 100), maximum entropy tantile function model can accurately estimate sample fractiles value function；But in engineering especially It is that aeronautical product test number (TN) is less (generally below 10), now maximum entropy tantile function model estimation precision is poor.This Maximum entropy tantile function model is referred to as classic maximum entropy tantile function model in text.
The content of the invention：
Goal of the invention：For conventional probability statistical method computational efficiency under Small Sample Size is high and classic maximum entropy point The situation of place value function model computational accuracy difference, the present invention proposes a kind of structure belt restraining least square maximum entropy tantile function The method of model, to improve the computational efficiency and estimation precision of tantile function in the case of small capital sample.
The present invention is adopted the following technical scheme that：
A kind of method for building belt restraining least square maximum entropy tantile function model, this method comprises the following steps：
(1) set up any one stochastic variable X without constraint least square maximum entropy tantile function model：
X (u) is stochastic variable X without constraint least square maximum entropy tantile functional value；U (x) is X cumulative distribution letter Numerical value, u (x)=P (X≤x) and 0≤u of satisfaction (x)≤1；λLs-qf, jFor Lagrange multiplier, i.e. undetermined coefficient, j=0, 1 ..., m, Lagrange multiplier number is m+1；
(2) stochastic variable X one group of sample point is chosen as Median rank point, and calculates the empirical cumulative minute of each Median rank point Cloth functional value, definitionFor Median rank point xiEmpirical cumulative distribution function value, x1≤x2≤xi…≤xn, n is sample Number；With without constraint least square maximum entropy tantile function model fitting Median rank point, and ensure fitting after without constraint most A young waiter in a wineshop or an inn multiplies each point on maximum entropy tantile function curve and is more than 0 on the slope of tantile, obtains without constraint least square most Big entropy tantile function curve x (u)Slope ＞ 0, the expression x of subscript slope ＞ 0 (u) derivative is more than 0；
(3) definition is without constraint least square maximum entropy tantile function curve x (u)Slope ＞ 0On point with (cumulative distribution letter Numerical value, tantile functional value) form represents, chooses x (u)Slope ＞ 0Stage casing curve point formation point set C1, C1=(ur, x (ur)Slope ＞ 0), r=1 ..., M, whereinurFor from curve x (u)Slope ＞ 0On the accumulation uniformly chosen Distribution function value, ur∈[umin, umax], [umin, umax] it is default threshold interval；M is the cumulative distribution function value of selection Sum；
(4) sample selected using Weibull distribution model fit procedure (2), obtains Weibull tantile function curve x (u)weibull；Choose Weibull tantile function curve both sides afterbody curve point formation point set C2And C3：
Wherein, usFor from curve x (u)weibullThe cumulative distribution function value uniformly chosen, s=1 ..., M；
(5) by point set C1, C2, C3Constitute big point set C, i.e. C=C1∪C2∪C3；With without constraint least square maximum entropy point Place value function model is fitted point set C, obtains belt restraining least square maximum entropy tantile function curve.
Further, it is described using without constraint least square maximum entropy tantile function model any one group of sample of fitting Step is：
(2-1) defines sample to be fitted for d1, d2..., dP, dpCumulative distribution function value be up, p=1,2 ..., P；
(2-2) builds majorized functionIn formula：
(2-3) solves Min Q by least square method, obtains Lagrange multiplier
(2-4) is by the Lagrange multiplier obtainedSubstitute into without constraint least square maximum entropy tantile function Model, just can determine that the least square maximum entropy tantile function of correspondence sample.
Further, the Median rank point xiEmpirical cumulative distribution function value calculation formula be：
Further, umin=0.001, umax=0.999.
The present invention has the advantages that：
The present invention can be applied successfully to aerospace component fail-safe analysis tantile Function Estimation under Small Sample Size Problem, compared to conventional probability statistical method, computational efficiency of the present invention is high, can handle Complicated Random Variables；Compared to classic maximum Entropy tantile function model, computational accuracy of the present invention is high and stably.
Brief description of the drawings：
Fig. 1 is maximum by Weibull distribution model, without constraint least square respectively for 8 random samples of Normal Distribution Entropy tantile function model is fitted obtained tantile function curve with belt restraining least square maximum entropy tantile function model Computational accuracy comparison diagram；
Fig. 2 is the tantile function curve fit procedure based on belt restraining least square maximum entropy tantile function model Figure；
Fig. 3 is 8 random samples of Normal Distribution respectively by classic least square maximum entropy tantile Function Modules Type is fitted obtained tantile function curve computational accuracy with belt restraining least square maximum entropy tantile function model and contrasted.
Embodiment：
The present invention is further described with reference to the accompanying drawings and examples.
Fig. 2 show the tantile function curve based on belt restraining least square maximum entropy tantile function model and was fitted Cheng Tu, following steps：
1) set up without constraint least square maximum entropy tantile function model
If stochastic variable X, u (x)=P (X≤x) are X cumulative distribution function value, and meet 0≤u (x)≤1, then at random Variable without constraint least square maximum entropy tantile function x (u) be：
In formula, λLs-qf, j(j=0,1 ..., m) be Lagrange multiplier, that is, undetermined coefficient；Lagrange multiplier Number is m+1.Cumulative distribution function u (x) with without constraint least square maximum entropy tantile function x (u) inverse function each other.
The determination step of Lagrange multiplier is as follows：
11) formula (1) is derived again to obtain：
The random sample point of stochastic variable X one group of a certain probability distribution of obedience is obtained by numerical simulation or number is truly tested According to random sample point Normal Distribution in this example, if this group of random sample point is：x1, x2..., xnAnd x1≤x2≤…≤ xn。
12) u is definediIt is xiCumulative distribution function value, wherein xiIt is stochastic variable X i-th of order statistic, i ∈ [1,2 ..., n]；uiObtained by appropriate middle position rank technique：
13) by least square method solving-optimizing function Min Q, Lagrange multiplier is obtained
In formula：
By the Lagrange multiplier obtainedSubstitute into formula (1), just can determine that stochastic variable X without constraint most A young waiter in a wineshop or an inn multiplies maximum entropy tantile function (LSMEQFM).But in some cases, no constraint least square maximum entropy tantile function It is not the monotonic function of cumulative distribution function value and tantile function curve afterbody error is larger, therefore when calculating Lagrange multiplier Constraints is introduced, it is further proposed that belt restraining least square maximum entropy tantile function model (LSMEQFMCC).
2) belt restraining least square maximum entropy tantile function model is set up, step is：
21) with (x1, x2..., xn) it is Median rank point, calculate the empirical cumulative distribution function value of sampleWherein：
22) the derivative x ' without constraint least square maximum entropy tantile function x (u) on cumulative distribution function value u is calculated (u) it is：
To meet without constraint least square maximum entropy tantile function x (u) be cumulative distribution function value u dull letter Number, then need to meet x ' (u) ＞ 0.As x ' (u) ＞ 0, becauseMore than 0, soMore than 0.Need to increase about on the basis of without constraint least square maximum entropy score value function model Beam condition：
Wherein, r is ukIn interval [umin umax] uniform value number.
Using without constraint least square maximum entropy score value function model fitting sample (x1, x2..., xn), and ensure to intend It is more than 0 on the slope of tantile without each point on constraint least square maximum entropy tantile function curve after conjunction, obtains nothing Constrain least square maximum entropy tantile function curve x (u)Slope ＞ 0。
23) definition is without constraint least square maximum entropy tantile function curve x (u)Slope ＞ 0On point with (cumulative distribution letter Numerical value, tantile functional value) form represents, chooses x (u)Slope ＞ 0Stage casing curve point formation point set C1, C1=(ur, x (ur)Slope ＞ 0), r=1 ..., M, whereinurFor from curve x (u)Slope ＞ 0On the accumulation uniformly chosen Distribution function value, ur∈[umin, umax], [umin, umax] it is default threshold interval；M is the cumulative distribution function value of selection Sum；
24) fine fitting character is had based on Weibull distribution curve tail, sample is fitted using Weibull distribution model (x1, x2..., xn), obtain Weibull tantile function curve x (u)weibull；Choose Weibull tantile function curve both sides Afterbody curve point formation point set C2And C3：
Wherein, usFor from curve x (u)weibullThe cumulative distribution function value uniformly chosen, s=1 ..., M；M is selection The sum of cumulative distribution function value.
25) by point set C1, C2, C3Big point set C is constituted, using without constraint least square maximum entropy tantile Function Modules Type fitting point set C obtains belt restraining least square maximum entropy tantile function curve.
Technical solution of the present invention is described further below by specific embodiment.
Fig. 1 is maximum by Weibull distribution model, without constraint least square respectively for 8 random samples of Normal Distribution Entropy tantile function model is fitted obtained tantile function curve with belt restraining least square maximum entropy tantile function model Computational accuracy comparison diagram；Belt restraining least square maximum entropy tantile function model computational accuracy is higher, overcomes minimum without constraint Two when multiplying maximum entropy tantile function model fitting tantile function, the poor situation of computational accuracy at curve tail A and C.
S1：If 8 random samples selected are x1, x2..., x8, and meet x1≤x2≤…≤x8, calculate each sample point Empirical cumulative distribution function value, i.e.,
Random sample point is called Median rank point, x1, x2..., x8Corresponding tantile and the tool of cumulative distribution function value Volume data is as shown in table 1：
The Median rank point data of table 1
S2：Using without constraint least square maximum entropy tantile function model fitting Median rank point x1, x2..., x8, and Ensure that each point is more than 0 on the slope of tantile on curve, obtains without constraint least square maximum entropy tantile function curve, As shown in Figure 2.As shown in Figure 2, the calculation error of no constraint least square maximum entropy tantile function curve afterbody is larger, such as A, At C two.
S3：M=1000 is made, [umin, umax] it is [0.001,0.999], if the point on tantile function curve is by (iterated integral Cloth functional value, tantile functional value) represent, choose the stage casing curve without constraint least square maximum entropy tantile function curve Point, is shown at the B in Fig. 2, and is set to point set C1=(ur, x (ur)Slope ＞ 0), whereinur∈ [0.001, 0.999],WithValue can be determined according to actual conditions, in this example
S4：Using three-parameter Weibull distribution models fitting random sample (x1, x2..., x8), obtain Weibull tantile Function curve x (u)weibull, as shown in Figure 2.As can be seen from Figure 2 Weibull tantile function curve and theoretical tantile letter Number curve more coincide in afterbody, illustrates that Weibull Function has preferable afterbody fitting characteristic.
Weibull tantile function both sides afterbody curve point is chosen, sees at the D in Fig. 2 and at E, is set to point set C2And C3：
C2=(us, x (us)weibull), 0.001≤us≤0.1；
C3=(us, x (us)weibull), 0.9≤us≤0.999；
Wherein us(s=1 ..., 1000) is the cumulative distribution function value uniformly chosen from interval [0.001,0.999].
S5：By point set C1, C2, C3Constitute big point set C, i.e. C=C1∪C2∪C3.Using maximum without constraint least square Entropy tantile function model fitting point set C obtains belt restraining least square maximum entropy tantile function curve, as shown in Figure 1. It can be seen that belt restraining least square maximum entropy tantile function curve coincide preferably, and in song with theoretical tantile function curve Line tail calculations precision is high compared with without constraint least square maximum entropy tantile function curve.
In above-mentioned technical proposal, any one group of sample is fitted using without constraint least square maximum entropy tantile function model The step of it is as follows：
A) sample to be fitted is defined for d1, d2..., dP, dpCumulative distribution function value be up, p=1,2 ..., P；
B) majorized function is builtIn formula：
C) Min Q are solved by least square method, obtains Lagrange multiplier
D) Lagrange obtained is multipliedSubstitute into without constraint least square maximum entropy tantile Function Modules Type, just can determine that the least square maximum entropy tantile function of correspondence sample.
Fig. 1, Fig. 3 show the different maximum entropy tantile function models calculating essence for stochastic variable under Small Sample Size Degree contrast, it can be seen that belt restraining least square maximum entropy tantile function model computational accuracy highest.Each maximum entropy tantile The specific data of function model computational accuracy are as shown in table 2：
The different maximum entropy tantile function model computational accuracy contrasts of table 2
RMSE represents root-mean-square error in table 2：
Wherein,For random sample value xiEstimate.
Described above is only the preferred embodiment of the present invention, it is noted that for the ordinary skill people of the art For member, under the premise without departing from the principles of the invention, some improvements and modifications can also be made, these improvements and modifications also should It is considered as protection scope of the present invention.

Claims (4)

1. a kind of method for building belt restraining least square maximum entropy tantile function model, it is characterised in that this method includes Following steps：
(1) set up any one stochastic variable X without constraint least square maximum entropy tantile function model：
<mrow> <mi>x</mi> <mrow> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>exp</mi> <mrow> <mo>(</mo> <mo>-</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>m</mi> </munderover> <mo>(</mo> <mrow> <msub> <mi>&amp;lambda;</mi> <mrow> <mi>l</mi> <mi>s</mi> <mo>-</mo> <mi>q</mi> <mi>f</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>&amp;CenterDot;</mo> <msup> <mi>u</mi> <mi>j</mi> </msup> </mrow> <mo>)</mo> <mo>)</mo> </mrow> </mrow>
X (u) is stochastic variable X without constraint least square maximum entropy tantile functional value；U (x) is X cumulative distribution function Value, u (x)=P (X≤x) and 0≤u of satisfaction (x)≤1；λLs-qf, jFor Lagrange multiplier, i.e. undetermined coefficient, j=0,1 ..., M, Lagrange multiplier number is m+1；
(2) stochastic variable X one group of sample point is chosen as Median rank point, and calculates the empirical cumulative distribution letter of each Median rank point Numerical value, definitionFor Median rank point xiEmpirical cumulative distribution function value, x1≤x2≤xi...≤xn, n is number of samples；With Without constraint least square maximum entropy tantile function model fitting Median rank point, and ensure fitting after without constraint least square Each point is more than 0 on the slope of tantile on maximum entropy tantile function curve, obtains without constraint least square maximum entropy point Place value function curve x (u)Slope ＞ 0, the expression x of subscript slope ＞ 0 (u) derivative is more than 0；
(3) definition is without constraint least square maximum entropy tantile function curve x (u)Slope ＞ 0On point with (cumulative distribution function Value, tantile functional value) form represents, chooses x (u)Slope ＞ 0Stage casing curve point formation point set C1, C1=(ur, x (ur)Slope ＞ 0), r=1 ..., M, whereinurFor from curve x (u)Slope ＞ 0On the accumulation uniformly chosen Distribution function value, ur∈[umin, umax], [umin, umax] it is default threshold interval；M is the cumulative distribution function value of selection Sum；
(4) sample selected using Weibull distribution model fit procedure (2), obtains Weibull tantile function curve x (u)weibull；Choose Weibull tantile function curve both sides afterbody curve point formation point set C2And C3：
C2=(us, x (us)weibull),
C3=(us, x (us)weibull),
Wherein, usFor from curve x (u)weibullThe cumulative distribution function value uniformly chosen, s=1 ..., M；
(5) by point set C1, C2, C3Constitute big point set C, i.e. C=C1∪C2∪C3；With without constraint least square maximum entropy tantile Function model is fitted point set C, obtains belt restraining least square maximum entropy tantile function curve.
2. a kind of method for building belt restraining least square maximum entropy tantile function model according to claim 1, its Be characterised by, it is described use without constraint least square maximum entropy tantile function model be fitted any one group of sample the step of for：
(2-1) defines sample to be fitted for d1, d2..., dP, dpCumulative distribution function value be up, p=1,2 ..., P；
(2-2) builds majorized functionIn formula：
<mfenced open = "{" close = ""> <mtable> <mtr> <mtd> <msub> <mi>s</mi> <mi>p</mi> </msub> <mo>=</mo> <mi>ln</mi> <mi> </mi> <msub> <mi>x</mi> <mi>p</mi> </msub> </mtd> </mtr> <mtr> <mtd> <mrow> <msub> <mi>t</mi> <mi>p</mi> </msub> <mo>=</mo> <munderover> <mo>&amp;Sigma;</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>m</mi> </munderover> <mrow> <mo>(</mo> <msub> <mi>&amp;lambda;</mi> <mrow> <mi>l</mi> <mi>s</mi> <mo>-</mo> <mi>q</mi> <mi>f</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msubsup> <mi>u</mi> <mi>p</mi> <mi>j</mi> </msubsup> <mo>)</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mfenced>
(2-3) solves Min Q by least square method, obtains Lagrange multiplier
(2-4) is by the Lagrange multiplier obtainedSubstitute into without constraint least square maximum entropy tantile Function Modules Type, just can determine that the least square maximum entropy tantile function of correspondence sample.
3. a kind of method for building belt restraining least square maximum entropy tantile function model according to claim 2, its It is characterised by, the Median rank point xiEmpirical cumulative distribution function value calculation formula be：
<mrow> <msub> <mi>cdf</mi> <msub> <mi>x</mi> <mi>i</mi> </msub> </msub> <mo>=</mo> <mfrac> <mrow> <mi>i</mi> <mo>-</mo> <mn>0.35</mn> </mrow> <mi>n</mi> </mfrac> <mo>.</mo> </mrow>
4. a kind of method for building belt restraining least square maximum entropy tantile function model according to claim 3, its It is characterised by, umin=0.001, umax=0.999.
CN201710206271.6A 2017-03-31 2017-03-31 A kind of method for building belt restraining least square maximum entropy tantile function model Pending CN107145627A (en)

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Cited By (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110135002A (en) * 2019-04-16 2019-08-16 上海城市交通设计院有限公司 A method of measuring new energy car battery charge accumulation capacity loss reliability
CN110362879A (en) * 2019-06-25 2019-10-22 中国人民解放军军事科学院国防科技创新研究院 The Prior Fusion and update method and priori compensation process of two layers and multilayered structure

Cited By (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN110135002A (en) * 2019-04-16 2019-08-16 上海城市交通设计院有限公司 A method of measuring new energy car battery charge accumulation capacity loss reliability
CN110135002B (en) * 2019-04-16 2022-10-14 上海城市交通设计院有限公司 Method for measuring reliability of charge accumulated capacity loss of new energy automobile battery
CN110362879A (en) * 2019-06-25 2019-10-22 中国人民解放军军事科学院国防科技创新研究院 The Prior Fusion and update method and priori compensation process of two layers and multilayered structure
CN110362879B (en) * 2019-06-25 2020-09-04 中国人民解放军军事科学院国防科技创新研究院 Priori fusion and updating method and priori supplement method for two-layer and multi-layer structure

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