CN106777928A - Towards the Bayes prior distribation building methods of normal distribution data sample - Google Patents

Abstract

The invention provides a kind of Bayes prior distribation building methods towards normal distribution data sample, assume initially that the joint prior distribation of Parameters of Normal Distribution (average and variance) is distributed for conjugation --- normal state Inv-Gamma distribution, then data sample is divided into N groups, every group of m (N, m is positive integer) data, calculate each group of average and variance of data, respectively obtain N number of average and N number of variance, count the average and variance of the N number of average of gained and N number of variance respectively again, the distributed constant of normal state Inv-Gamma distribution is fitted according to four estimates for obtaining, so as to obtain the conjugation prior distribation of normal distribution data.Technique effect of the invention：1. the point estimation of average and variance is the unbiased esti-mator for testing preceding sample in gained prior distribation；2. equivalent sample size parameter is unrelated with preceding total sample number amount is tested in gained prior distribation, only relevant with every group of number of samples, in actual configuration prior distribation, can flexibly select grouped data amount m according to field samples number.

Description

Towards the Bayes prior distribation building methods of normal distribution data sample

Technical field

The present invention relates to applied statistics technical field, it is related to the Bayes statistical inference sides in aircraft statistics for experiment field Method, is specifically related to a kind of Bayes prior distribation building methods towards normal distribution data sample.

Background technology

In Bayes statistical theories, the construction of prior distribation is key issue therein, before being tested using the theoretical fusions of Bayes , it is necessary to preceding data configuration into specific distribution form will be tested when test data is with field test data, and it is false to be conjugated prior distribation It is provided with and is calculated beneficial to the statistical inference of Bayes posterior distributions, average and variance combines that to be conjugated distribution be normal state-inverse in normal distribution Gamma is distributed.Traditional prior distribation building method is likelihood function method, first assumes to test preceding data distribution form, Ran Hougen Prior distribation parameter is calculated according to likelihood function.

In the case of normal state-Inv-Gamma distribution, the equivalent sample of prior distribation that this likelihood function building method is obtained holds Amount parameter is equal to tests preceding test data number, test data quantity before testing is very big and during little field test data, after testing point Average mainly determines that the weight very little of field test data can be caused with the statistics of variance by testing preceding test data in cloth Test the problem that preceding test data " floods " field test data.

The content of the invention

The present invention is for the existing technical problem of prior distribation construction in above-mentioned existing Bayes statistical theories, it is proposed that A kind of Bayes prior distribation building methods towards normal distribution data sample.

For ease of understanding, provide technical scheme to the present invention and be summarized as follows：Obtain a certain particular problem tests preceding experiment number According to.The preceding data of testing that will be obtained first are divided into N groups, every group of m data, then to its average of the data statistics in every group and side Difference, can obtain N class means and estimate of variance, and the average value and variance of N number of Estimation of Mean value, and N number of side are then calculated again The average value and variance of difference estimate, finally calculate four distributed constants of normal state-Inv-Gamma distribution.The particular problem can be with It is that shell or guided missile are simulated the test data practiced shooting.

Referring to Fig. 1, the Bayes prior distribations building method towards normal distribution data sample that the present invention is provided include with Lower step：

Step S100：The preceding data sample of testing of overall Normal Distribution is divided into N groups, each group includes m data；

Preceding data sample { x is tested according to given_i, i=1 ... n₀, wherein, data sample x_iOverall obey normal state point Cloth, number of samples is n₀.Sample is divided into N number of packet, each packet includes m data so that N × m=n₀, sample can be again It is expressed as { x_ji, j=1 ... N, i=1 ... m, then x_jiMean that i-th data of j-th packet.

Conjugation distribution theory in Bayes statistical theories, it is believed that the mean μ of normal distribution is tested with combining for variance V Before be distributed as its conjugation distribution --- normal state-Inv-Gamma distribution.

Step S200：Calculate every group of Estimation of Mean value for testing preceding dataAnd estimate of varianceAnd it is equal according to gained Value estimateAnd estimate of varianceCalculate the average of Estimation of Mean valueAnd varianceAnd variance evaluation The average of valueAnd varianceEvery variance and average routinely formula can be calculated in the step.

Preferably, every group of Estimation of Mean value for testing preceding data is calculated by formula (1)And estimate of varianceAnd according to Gained Estimation of Mean valueAnd estimate of varianceThe average of Estimation of Mean value is calculated by formula (2)~(3)And varianceAnd the average of estimate of varianceAnd variance

By this calculating, gained estimate closer to and actual value.

For the data after packet, the estimate of every group of average of data and variance is calculatedJ=1,2 ..., N, computational methods are

The estimate of N number of average and variance can be so obtained respectively.

According to the N number of average and the estimate of variance that obtainJ=1,2 ..., N, further calculate N number of average EstimateThe average and variance of j=1,2 ..., N, and N number of estimate of varianceJ=1,2 ..., N estimates Average and variance, i.e.,

Step S300：Every group for being obtained in step S200 tests the Estimation of Mean value of preceding dataEstimate of varianceThe average of Estimation of Mean valueAnd varianceAnd the average of estimate of varianceAnd varianceSubstitute into In formula (6), (7) and (9), normal state-inverse Gamma prior distribation parameter μs are obtained₀,η₀,α₀,β₀Estimate, gained is distributed The estimate of parameter is substituted into the probability density function as shown in formula (4), and preceding data are tested so as to be met normal distribution The conjugation prior distribation of sample；

According in Bayes statistical theories on conjugation distribution principle, the average for testing preceding data sample of normal distribution with Variance also meets conjugation prior distribation for the distribution of normal state-Inv-Gamma distribution, i.e. normally distributed variable μ and V meets such as lower probability Density function：

Wherein, μ, V are respectively the average and variance of normal distribution, μ₀,η₀,α₀,β₀It is distributed constant, 1/ η₀For normal state-inverse Equivalent sample size parameter in Gamma distributions, f (μ, V) is probability density function, Γ (β₀) it is Gamma functions.

According to above-mentioned distribution character, orderCan obtain

AndWherein V estimatesInstead of can obtain

According to Inv-Gamma distribution property, have

Its sample estimate is respectivelyOrderIt is available

Four parameter μs in normal state-Inv-Gamma distribution are thus obtained₀,η₀,α₀,β₀Estimation, that is, obtain normal state- Inverse Gamma prior distribation parameters.

Compared with the prior art, technique effect of the invention：

1st, the Bayes prior distribation building methods towards normal distribution data sample that the present invention is provided, for Bayes systems Normal distribution sample prior distribation is constructed during meter is inferred, assumes initially that combining for Parameters of Normal Distribution (average and variance) tests preceding point Cloth is distributed for conjugation --- normal state-Inv-Gamma distribution, data sample is then divided into N groups, every group m (N, m are positive integer) Data, calculate each group of average and variance of data, respectively obtain N number of average and N number of variance, then statistics gained is N number of respectively The average and variance of value and N number of variance, four estimates according to gained are fitted the distributed constant of normal state-Inv-Gamma distribution, from And obtain the conjugation prior distribation of normal distribution data.So that the point estimation of average and variance is to test in the N number of prior distribation of gained The unbiased esti-mator of preceding sample.So that equivalent sample size parameter is unrelated with preceding total sample number amount is tested in gained prior distribation, only with Every group of number of samples is relevant, so as to realize in actual configuration prior distribation, can flexibly select packet according to field samples number Data volume m.

2nd, the Bayes prior distribation building methods towards normal distribution data sample that the present invention is provided, its equivalent sample Capacity (1/ η₀) only relevant (the two is equal in mathematic expectaion) with every group of number of samples m, test preceding number of samples n with total₀Nothing Close.In the middle of actual, m can be selected according to field samples number so that the two is substantially suitable, and preceding test data is tested so as to avoid The problem of " flooding " field test data.

The Bayes prior distribations building method towards normal distribution data sample of the invention is specifically refer to propose Various embodiments it is described below, will cause that above and other of the invention aspect is apparent.

Brief description of the drawings

Fig. 1 is that the Bayes prior distribation building methods flow towards normal distribution data sample that the present invention is provided is illustrated Figure；

Fig. 2 is that guided missile carries out 300 simulations and practices shooting the longitudinal offset landings statistics Nogatas of experiment in the preferred embodiment of the present invention Figure；

Fig. 3 is the probability density function figure of normal state-Inv-Gamma distribution in the preferred embodiment of the present invention, wherein a) be on The 2D plans of mean μ；B) it is the 2D plans on variance V.

Specific embodiment

The accompanying drawing for constituting the part of the application is used for providing a further understanding of the present invention, schematic reality of the invention Apply example and its illustrate, for explaining the present invention, not constitute inappropriate limitation of the present invention.

The Bayes prior distribation building methods towards normal distribution data sample that the present invention is provided, comprise the following steps that：

Step one：Preceding data sample { x is tested according to given_i, i=1 ... n₀, wherein, data sample x_iOverall obedience Normal distribution, number of samples is n₀.Sample is divided into N number of packet, each packet includes m data so that N × m=n₀, sample { x can be again expressed as_ji, j=1 ... N, i=1 ... m, then x_jiMean that i-th data of j-th packet.According to Bayes Conjugation distribution theory in statistical theory, it is believed that the mean μ of normal distribution is its conjugation point with the prior distribation of combining of variance V Cloth --- normal state-Inv-Gamma distribution.

Step 2：For the data after packet, the estimate of every group of average of data and variance is calculated respectively J=1,2 ..., N, computational methods are

So can be obtained by the estimate of N class means and variance.

Step 3：According to the N class means and the estimate of variance that obtainJ=1,2 ..., N, further calculate N Individual Estimation of Mean valueThe average and variance of j=1,2 ..., N, and N number of estimate of varianceJ=1,2 ..., N The average and variance of estimate, i.e.,

Step 4：According to the principle on conjugation distribution in Bayes statistical theories, the average of normal distribution and being total to for variance Yoke prior distribation is normal state-Inv-Gamma distribution, i.e. the distribution of normally distributed variable μ and V meets following probability density function：

Wherein μ, V are respectively the average and variance of normal distribution, μ₀,η₀,α₀,β₀It is distributed constant, 1/ η₀For normal state-inverse Equivalent sample size parameter in Gamma distributions, f (μ, V) is probability density function, Γ (β₀) it is Gamma functions.

According to above-mentioned distribution character, orderCan obtain

AndV estimatesInstead of can obtain

According to Inv-Gamma distribution property, have

Its sample estimate is respectivelyOrderIt is available

Four parameter μs in normal state-Inv-Gamma distribution are thus obtained₀,η₀,α₀,β₀Estimation, that is, obtain normal state- Inverse Gamma prior distribation parameters.

On the Bayes prior distribation building methods towards normal distribution data sample that the present invention is provided, have Technique effect proves as follows：

Property 1：For the constructed in groups method, just like drawing a conclusion：

Prove：Sample data Normal Distribution N (μ, V), from Normal Distribution Theory, every group of sample variance of dataIt is the chi square distribution of m-1 to obey the free degree.

By the average and variance of chi square distribution, can obtain

Then

Again according to the building method

Make formula (11) and formula (13), formula (12) equal respectively with formula (14), simultaneously because what mathematic expectaion obtained is to estimate Progressive nature, so

Card is finished.

Property 2：The point estimation of average is unbiased esti-mator in the building method.

Prove：For the building method,

The point estimation of average is the arithmetic average of sample i.e. in the building method, and it is the unbiased esti-mator of average.

Card is finished.

Property 3：The point estimation of variance is the unbiased esti-mator of true variance in the building method.

Prove：For the building method, every group of sample variance of data obeys chi square distribution, by the property of chi square distribution, Order

It is the chi square distribution of N (m-1) that then Y obeys the free degree, according to chi square distribution property, is had

First formula in (3) formula is substituted into (20) formula simultaneously, then Y can be expressed as

Then can obtain

The variance evaluation for being obtained using constructed in groups method is the unbiased esti-mator of true variance.

Card is finished.

Property 4：For above-mentioned building method, have

Prove：Data sample Normal Distribution N (μ, V), in classical statistical theory, each group of data under normal distribution Estimation of Mean value obey lower column distribution

WhereinRepresent normpdf.

I.e.

According to formula (7), can obtain

Or

Card is finished.

Constructed in groups method proposed by the present invention is mainly characterized by its equivalent sample size (1 η₀) only with every group of sample Number m is relevant (equal in mathematic expectaion), and preceding number of samples n is tested with total₀It is unrelated.

Embodiment

Embodiment 1

With reference to a specific embodiment, the prior distribation constructed in groups method to normal distribution data sample of the present invention is done Further describe, concrete application background is ignored in the embodiment, wherein every initial parameters are set for by this, with Proof the method provided by the present invention can apply in all kinds of engineering problems.

Step one：Given sample { x_i, i=1 ... 3000, x_iNormal Distribution, average is 0.1, and variance is 1, sample Quantity is 3000, and sample is divided into 500 groups, and every group of data amount check is 6, i.e. N=500, m=6.

Step 2：Every group of average and variance of sample are counted, 500 sample average estimates are obtainedEstimate with variance Evaluation(j=1,2 ..., 500).

Step 3：According to 500 sample averages and the estimate of variance that are obtainedJ=1,2 ..., 500, The average and variance of further average statistical estimate, and the average of estimate of variance is with variance

Step 4：According to the principle on conjugation distribution in Bayes statistics, before normal distribution average is tested with the conjugation of variance It is distributed as normal state-Inv-Gamma distribution.

According to the property of above-mentioned distribution, according to formula (6), formula (7), can obtain

μ₀=0.0996, η₀=0.167 (32)

According to formula (9), can obtain

α₀=3.498, β₀=4.54 (33)

Thus four distributed constant μ in normal state-Inv-Gamma distribution have been obtained₀,η₀,α₀,β₀, gained normal state-inverse Gamma distributions are the prior distribation that the method for the invention is obtained.

Embodiment 2

With reference to the specific embodiment of certain type guided missile simulation Targeting data, to normal distribution data sample of the present invention Prior distribation building method be described in further details, it is comprised the following steps that：

In Targeting is simulated, the point of impact and deviation of the impact point in range direction (longitudinal direction) are a stochastic variables, The present carries out 300 simulation target practice experiments to same type of missile, and the numerical value for measuring guided missile longitudinal direction offset landings is as shown in table 1.Table 1 In data shown in each cell be the longitudinal offset landings amount of gained in once testing.

1 guided missile of table, 300 simulations are practiced shooting and test longitudinal offset landings tables of data (unit：Rice)

4.1	18.9	0.3	37.9	36.7	-33.4	6.4	1.8	28.5	-48.7	36.2	0.2	0.1	-9.9	29.8
															14.3	-0.4	29.7	20.0	42.7	2.4	8.1	12.6	-37.3	36.0	2.4	-8.9	-24.7	-2.5	23.0
-3.4	44.8	0.1	2.7	-9.6	-10.4	1.4	2.9	14.2	-13.5	-11.4	-8.7	-2.1	13.5	-18.0
															6.4	-15.3	-7.7	16.5	-1.9	52.8	19.6	9.3	-44.2	-9.4	10.8	23.7	12.1	7.3	-0.3
-29.3	-21.8	-19.8	8.7	14.2	6.5	-2.0	14.4	-1.1	21.7	-17.6	-19.2	41.7	-5.2	-27.5
															25.0	37.7	26.4	16.5	13.1	12.5	9.7	-27.3	13.0	-6.3	22.6	-2.2	23.0	25.4	16.6
-14.8	-13.5	-20.9	7.5	32.7	12.9	-12.4	5.3	-29.1	-15.0	-13.9	-18.6	10.0	-23.4	43.0
															-0.6	1.2	-1.5	5.2	3.6	2.8	-14.7	4.2	20.2	17.8	-7.4	17.9	24.7	55.9	36.6
8.4	-14.8	-10.6	36.8	36.9	-31.6	-0.9	13.1	9.1	-0.1	-34.8	-16.8	-5.3	-2.2	23.4
															22.9	-18.5	-10.5	7.5	-11.7	46.8	21.5	11.1	24.2	-17.4	-14.0	-29.0	-21.2	13.0	-13.8
-11.0	-10.5	-0.7	17.8	6.6	12.2	8.9	-19.9	14.7	38.0	10.2	-20.8	-8.7	-9.1	-9.7
															5.5	1.7	7.9	20.6	11.5	-10.9	11.0	7.6	-3.9	-1.1	29.0	15.9	-18.7	-15.4	8.6
-6.8	-4.0	9.7	31.5	-14.3	0.5	-9.2	-15.0	11.5	-10.5	27.2	47.7	24.6	27.7	2.4
															-16.7	22.3	-11.9	1.9	6.2	36.9	-8.8	-2.1	4.6	-22.1	-27.3	-6.5	2.3	5.8	22.4
4.3	-3.4	11.8	17.6	-0.3	8.1	16.5	7.1	-32.8	-1.5	-2.1	21.5	-18.8	9.6	31.3
															1.6	30.4	15.3	6.3	22.2	8.6	13.9	24.1	11.8	36.1	4.8	37.3	0.9	7.5	0.0
4.2	-11.4	2.2	-4.7	15.9	-1.8	-21.5	14.3	11.8	17.6	4.7	29.3	-11.9	18.9	3.6
															-15.6	-26.2	0.5	3.0	42.1	-25.5	-25.0	10.3	13.5	-15.6	9.3	37.3	-1.3	39.8	26.9
-5.3	10.9	-19.2	17.1	24.9	-9.2	25.3	-23.5	7.3	1.0	8.6	-7.0	33.5	18.3	14.8
															11.7	31.4	11.2	-9.9	7.1	-12.3	4.4	7.2	2.6	22.2	-5.4	25.8	34.8	18.8	-28.6

The minimum longitudinal direction offset landings that measurement is obtained are -48.0m, and maximum longitudinal direction offset landings are 65.1m, accordingly will longitudinal direction Offset landings data are divided into 15 intervals, count the frequency and frequency of the sample value in each minizone, obtain table 2.According to It is longitudinal offset landings statistic histogram that table 2 obtains Fig. 2.

2 guided missile of table, 300 simulations are practiced shooting and test longitudinal offset landings interval frequency and frequency table

Step one：Using existing 300 experimental data samples as sample { x_i, i=1 ... 300 can from Fig. 2 Go out x_iApproximate Normal Distribution, sample size is 300, and sample is divided into 30 groups, and every group contains 10 data, i.e. N=30, m =10.

Step 2：Every group of average and variance of sample are counted, 30 sample average estimates are obtainedAnd variance evaluation Value(j=1,2 ..., 30), it is listed below

Step 3：According to 30 sample averages and the estimate of variance that obtainJ=1,2 ..., 30, enter one Step statistics its average and variance, i.e.,

Step 4：According to the principle on conjugation distribution in Bayes statistics, before normal distribution average is tested with the conjugation of variance It is distributed as normal state-Inv-Gamma distribution.

According to the property of above-mentioned distribution, according to formula (6), formula (7), can obtain

μ₀=4.948, η₀=0.1021 (36)

According to formula (9), can obtain

α₀=1707.6, β₀=5.849 (37)

Thus four distributed constant μ in normal state-Inv-Gamma distribution have been obtained₀,η₀,α₀,β₀, will 4 distributions above Parameter is substituted into formula (4), you can obtain obtaining probability density function by 4 distributed constant constructions of gained.And by gained probability Density function is the prior distribation that the method for the invention is obtained as gained normal state-Inv-Gamma distribution.Wherein on average The marginal probability density function of μ is

Marginal probability density function on variance V is

The probability density function of normal state-Inv-Gamma distribution is as shown in Figure 3 in the present embodiment.Gained normal state of the invention-inverse Average in Gamma distributions is equal to sample average and sample variance with variance, and its equivalent sample size is 9.79, much smaller than original Preceding sample size (300) is tested, and is had no truck with, this has a significant impact to Bayes posteriori estimations.

Those skilled in the art will be clear that the scope of the present invention is not restricted to example discussed above, it is possible to which it is carried out Some changes and modification, without deviating from the scope of the present invention that appended claims are limited.Although oneself is through in accompanying drawing and explanation The present invention is illustrated and described in book in detail, but such explanation and description are only explanations or schematical, and it is nonrestrictive. The present invention is not limited to the disclosed embodiments.

By to accompanying drawing, the research of specification and claims, when the present invention is implemented, those skilled in the art can be with Understand and realize the deformation of the disclosed embodiments.In detail in the claims, term " including " be not excluded for other steps or element, And indefinite article " one " or " one kind " are not excluded for multiple.Some measures quoted in mutually different dependent claims The fact does not mean that the combination of these measures can not be advantageously used.It is right that any reference marker in claims is not constituted The limitation of the scope of the present invention.

Claims (3)

1. a kind of Bayes prior distribation building methods towards normal distribution data sample, it is characterised in that including following step Suddenly：

Step S100：The preceding data sample of testing of overall Normal Distribution is divided into N groups, each group includes m data；

Step S200：Calculate every group of Estimation of Mean value for testing preceding dataAnd estimate of varianceAnd estimated according to gained average EvaluationAnd estimate of varianceCalculate the average of Estimation of Mean valueAnd varianceAnd estimate of variance AverageAnd variance

Step S300：Every group for being obtained in step S200 tests the Estimation of Mean value of preceding dataEstimate of varianceAverage The average of estimateAnd varianceAnd the average of estimate of varianceAnd varianceSubstitution formula (6), (7) and in (9), normal state-inverse Gamma prior distribation parameter μs are obtained₀,η₀,α₀,β₀Estimate, by estimating for gained distributed constant Evaluation is substituted into probability density function as shown in formula (4), and being total to for preceding data sample is tested so as to be met normal distribution Yoke prior distribation

${\mathrm{\μ}}_0=\frac{1}{N}\underset{i=1}{\overset{N}{\Σ}}{\widehat{\mathrm{\μ}}}^{\left(j\right)}---\left(6\right)$

${\mathrm{\η}}_0=\frac{Var\left({\widehat{\mathrm{\μ}}}^{\left(j\right)}\right)}{\hat{V}}---\left(7\right)$

${\mathrm{\β}}_0=\frac{{\left(\hat{V}\right)}^2}{Var\left({\hat{V}}^{\left(j\right)}\right)}+2,{\mathrm{\α}}_0=\hat{V}({\mathrm{\β}}_0-1)---\left(9\right)$

$f(\mathrm{\μ},V)=\frac{1}{\sqrt{2{\mathrm{\π\η}}_{0}V}}\exp (-\frac{{(\mathrm{\μ}-{\mathrm{\μ}}_0)}^2}{2{\mathrm{\η}}_{0}V})\·\frac{{{\mathrm{\α}}_0}^{{\mathrm{\β}}_0}}{\mathrm{\Γ}\left({\mathrm{\β}}_0\right)}{V}^{-({\mathrm{\β}}_0+1)}{e}^{-\frac{{\mathrm{\α}}_0}{V}}---\left(4\right)$

Wherein, 1/ η₀Equivalent sample size parameter in for normal state-Inv-Gamma distribution, Γ (β₀) it is Gamma functions.

2. Bayes prior distribation building methods towards normal distribution data sample according to claim 1, its feature exists In calculating every group of Estimation of Mean value for testing preceding data by formula (1)And estimate of variance

${\widehat{\mathrm{\μ}}}^{\left(j\right)}=\frac{1}{m}\underset{i=1}{\overset{m}{\Σ}}{x}_{ji},{\hat{V}}^{\left(j\right)}=\frac{1}{m-1}\underset{i=1}{\overset{m}{\Σ}}{(x_{ji}-{\widehat{\mathrm{\μ}}}^{\left(j\right)})}^2---\left(1\right).$

3. Bayes prior distribation building methods towards normal distribution data sample according to claim 2, its feature exists In by the average of formula (2)~(3) calculating Estimation of Mean valueAnd varianceAnd the average of estimate of varianceWith Variance

$,\widehat{\mathrm{\μ}}=\frac{1}{N}\underset{j=1}{\overset{N}{\Σ}}{\widehat{\mathrm{\μ}}}^{\left(j\right)},Var\left({\widehat{\mathrm{\μ}}}^{\left(j\right)}\right)=\frac{1}{N-1}\underset{j=1}{\overset{N}{\Σ}}{({\widehat{\mathrm{\μ}}}^{\left(j\right)}-\widehat{\mathrm{\μ}})}^2---\left(2\right)$

$\hat{V}=\frac{1}{N}\underset{j=1}{\overset{N}{\Σ}}{\hat{V}}^{\left(j\right)},Var\left({\hat{V}}^{\left(j\right)}\right)=\frac{1}{N-1}\underset{j=1}{\overset{N}{\Σ}}{({\hat{V}}^{\left(j\right)}-\hat{V})}^2---\left(3\right).$