CN110426059A - Error correcting method in a kind of measurement that taking freedom degree into account - Google Patents
Error correcting method in a kind of measurement that taking freedom degree into account Download PDFInfo
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- CN110426059A CN110426059A CN201910461833.0A CN201910461833A CN110426059A CN 110426059 A CN110426059 A CN 110426059A CN 201910461833 A CN201910461833 A CN 201910461833A CN 110426059 A CN110426059 A CN 110426059A
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Abstract
The invention discloses error correcting methods in a kind of measurement for taking freedom degree into account, belong to survey field, the rule that fiducial probability changes with freedom degree in the case of determining confidence interval is distributed to based on t, propose the method for accordingly limiting Ratio for error modification in difference calculating by giving high fiducial probability in the case where taking freedom degree into account, the amendment for realizing error in different freedom degrees correspondences, ensure that the reasonability and reliability that precision is measured under low degree-of-freedom.The following steps are included: the middle error of (1) according to the freedom calculation measured value of measured value；(2) according to Ratio for error modification in the determination of freedom degree size；(3) by middle error multiplied by correction factor, middle error after being corrected.
Description
Technical field
The invention belongs to survey fields, are related to the fiducial probability problem of measurement result, especially in the case of low degree-of-freedom in
The fiducial probability problem of the corresponding limit difference of error.
Background technique
In measurement, accidental error is inevitable, it is generally recognized that accidental error be Normal Distribution with
Machine variable.In order to reflect the dispersion degree of error, middle error is commonly used as the index for measuring precision, the precision of Lai Fanying observation
Just, it is generally recognized that middle error is smaller, and the precision of Measurement results is higher.Generally using error in 3 times as the limit of accidental error
Value, it is believed that it is 99.7% that the probability in this section, which occurs, in true error, in practice, also has and is missed using error in 2 times as the limit
Difference, it is believed that its fiducial probability is 95.4%.This is because accidental error Normal Distribution, generally calculates normal distribution and is giving
Determine the probability on section, is that accidentally will be converted into standardized normal distribution divided by standard deviation to calculate.Due to actual observation number
It is limited, error in the valuation of standard deviation can only be found out, therefore standard deviation is replaced with middle error in the measurements, and default accidental error
Standardized normal distribution is obeyed divided by middle error.
In fact, the calculating due to middle error is related with freedom degree n, accidental error is divided by middle error and disobeys standard just
State distribution, but obey t distribution.On given section, the value of t distribution fiducial probability increases with the increase of freedom degree,
When freedom degree is lower, the fiducial probability of t distribution is well below the fiducial probability under standardized normal distribution.Therefore in measurement usually
Using observation frequency is increased, that is, increase the method for freedom degree to increase fiducial probability.However, since actual observation number is limited,
When observation frequency is less, the fiducial probability of the corresponding limit difference of the middle errors of different freedom degrees is different, and respectively less than fiducial probability
Theoretical value, the i.e. fiducial probability of 2 times of middle error less than 95.4%, 3 times in error fiducial probability less than 99.7%.If with low
Following disadvantage will be present as precision measurement index in the middle error calculated in the case of freedom degree:
(1) for the fiducial probability of the corresponding limit difference of error not under same level, the measurement of precision lacks reasonability in.
(2) index that can reduce precision measurement causes engineering construction to lack reliability and safety, is unfavorable for testing for engineering
Receipts and performance rating.
Therefore, it should which in the case where taking freedom degree into account, the error of centralization is modified, and it is general to guarantee that it corresponds to the confidence of limit difference
Rate is under same level, especially in the case where low degree-of-freedom.But there is presently no about mistake middle in the case of low degree-of-freedom
The modification method of difference.
The accuracy assessment of measurement result has important strategy in the measurement work of China's modernization construction all trades and professions
Meaning, it is higher and higher to the reliability requirement of measurement accuracy at present.Reasonable accuracy assessment is beneficial to improve measurement result
Reliability guarantees the safety and economy of engineering construction.
Summary of the invention
Technical problems to be solved: in view of the deficiencies of the prior art, the present invention proposes in a kind of measurement for taking freedom degree into account
Error correcting method, the fiducial probability for solving the corresponding limit difference of error in Measurement results precision index under low degree-of-freedom cannot reach
The problem of to theoretical value.
In order to solve the above technical problems, the invention adopts the following technical scheme:
Error correcting method in a kind of measurement that taking freedom degree into account, specifically includes the following steps:
Step 1 calculates freedom degree, and calculates the middle error of measured value；
Step 2, according to Ratio for error modification in the determination of freedom degree size；
Step 3, by middle error multiplied by correction factor, middle error after being corrected.
As the further preferred scheme of error correcting method in a kind of measurement for taking freedom degree into account of the present invention, in step 1
In, the calculation formula of middle error isWherein, V indicates that correction, P indicate power battle array, and n is freedom degree, is equal to observation
Number subtracts unknown number number.
As the further preferred scheme of error correcting method in a kind of measurement for taking freedom degree into account of the present invention, in step 2
In, it is described according to freedom degree size determine in Ratio for error modification the following steps are included:
Step 2.1, fiducial probability p is given；
Step 2.2, according to fiducial probability p, the confidence upper limit m under standardized normal distribution is calculated using formula (1)；
It according to fiducial probability P, is calculated at freedom degree n using formula (2), the confidence upper limit θ of t distribution；
Wherein, Г (x) is gamma function；
Step 2.3, correction factor is calculated:
K=θ/m (3).
As the further preferred scheme of error correcting method in a kind of measurement for taking freedom degree into account of the present invention, in step 3
In, middle error after amendment:
As the further preferred scheme of error correcting method in a kind of measurement for taking freedom degree into account of the present invention, in step
In 2.1, fiducial probability p takes 95.4% or 99.7%.
As the further preferred scheme of error correcting method in a kind of measurement for taking freedom degree into account of the present invention, in step
In 2.2, confidence upper limit m and θ can be by tabling look-up or computer is calculated.
The utility model has the advantages that
The present invention is corrected error in measurement in the case where taking freedom degree into account, is added and is corrected by the error of centralization
Coefficient, it is ensured that its fiducial probability for corresponding to limit difference reaches theoretical value.On the one hand, it ensure that the corresponding limit of middle error under different freedom degrees
The fiducial probability of difference is identical, so that the measurement of precision under same confidence level, ensure that the reasonability that precision is measured.Another party
Face improves the fiducial probability of low degree-of-freedom situation lower limit difference, ensure that the reliability of measurement result, and then advantageously ensures that work
The safety and economy of Cheng Jianshe facilitates the examination and performance rating of engineering.
Detailed description of the invention
Fig. 1 is flow chart of the method for the present invention；
Fig. 2 is leveling line schematic diagram of the present invention.
Specific embodiment
The present invention is illustrated with reference to embodiments.
Error correcting method in a kind of measurement that taking freedom degree into account, as shown in Figure 1, specifically includes the following steps:
Step 1 calculates freedom degree, and error calculation formula calculates the middle error of measured value in utilization；
Step 2, according to Ratio for error modification in the determination of freedom degree size；
Step 3, by middle error multiplied by correction factor, middle error after being corrected.
Preferably, in step 1, the calculation formula of middle error isWherein, V indicates that correction, P indicate
Quan Zhen, n are freedom degree, subtract unknown number number equal to observation frequency.
Preferably, in step 2, it is described according to freedom degree size determine in Ratio for error modification the following steps are included:
Step 2.1, fiducial probability p is given, the middle error for freedom degree no more than 30 is modified the calculating of coefficient；
Step 2.2, according to fiducial probability p, the confidence upper limit m under standardized normal distribution is calculated using formula (1)；
It according to fiducial probability p, is calculated at freedom degree n using formula (2), the confidence upper limit θ of t distribution；
Wherein, Γ (x) is gamma function.
Step 2.3, correction factor is calculated:
K=θ/m (3)
Preferably, in step 3, the corrected value of middle error
Preferably, in step 2.1, fiducial probability p generally takes 95.4% or 99.7%.
Preferably, in step 2.2, confidence upper limit m and θ can be by tabling look-up or computer is calculated.
In levelling network, the elevation as known to Fig. 2, A and B are the bench marks of known elevation and set these points is error free.In figure
C, D and E point is point to be located.A and B point height, observation height difference and corresponding leveling line are shown in Table 1.In 95.4% fiducial probability
Under, the middle error of point to be located C, D vertical adjustment value is sought in examination.
Table 1
Observation frequency is 7 in the problem, and unknown number number is 3, then freedom degree
N=7-3=4
By measurement adjustment knowledge it is found that correction
V=[- 0.2 2.9-4.2-0.1-3.9-0.6-1.2]T
Power battle array P is unit battle array；
Middle error
Given fiducial probability p=95.4%；
It is calculated by computer, the confidence upper limit m=2 under standardized normal distribution, the confidence under the t distribution that freedom degree is 4
Upper limit θ=2.859；
Correction factor k=θ/2=1.429；
The correction value of middle error under 95.4% fiducial probability
Then under 95.4% fiducial probability, middle error is 3.1mm after amendment, further improves the index of precision, is protected
The reliability of measurement is demonstrate,proved.
Further, the correction factor that the freedom degree in step 2 is 1~30 is provided.
Given 95.4% fiducial probability；
The specific value that given 95.4% fiducial probability t is distributed lower confidence upper limit θ is found out by computer, such as table 2, wherein
Freedom degree n=1,2 ..., 30, taking the effective digital of θ is 3 after decimal point.Under 95.4% fiducial probability shown in table 2 it is different from
By the θ value spent
Table 2
It finds out under standardized normal distribution, the confidence upper limit m=2 of 95.4% fiducial probability；
It is θ/2 correction factor k=corresponding to 1~30 that freedom degree n, which can be calculated, by table 2, and such as table 3, table 3 is indicated
The value of the corresponding correction factor k of difference freedom degree under 95.4% fiducial probability, taking the effective digital of k is 3 after decimal point.
Table 3
The present invention gives error correcting methods in the measurement for taking freedom degree under low degree-of-freedom into account, guarantee middle error after amendment
The fiducial probability of corresponding limit difference can reach theoretical value, while give specific embodiment, and with 95.4% fiducial probability
For give freedom degree be 1~30 middle error correction factor.
The foregoing is merely presently preferred embodiments of the present invention, is not intended to limit the invention, it is all in spirit of the invention and
Within principle, any modification, equivalent replacement, improvement and so on be should all be included in the protection scope of the present invention.
Claims (6)
1. error correcting method in a kind of measurement for taking freedom degree into account, which is characterized in that specifically includes the following steps:
Step 1 calculates freedom degree, and calculates the middle error of measured value；
Step 2, according to Ratio for error modification in the determination of freedom degree size；
Step 3, by middle error multiplied by correction factor, middle error after being corrected.
2. error correcting method in a kind of measurement for taking freedom degree into account according to claim 1, it is characterised in that: in step
In one, the calculation formula of middle error isWherein, V indicates that correction, P indicate power battle array, and n is freedom degree, is equal to and sees
It surveys number and subtracts unknown number number.
3. error correcting method in a kind of measurement for taking freedom degree into account according to claim 1, which is characterized in that in step
In two, it is described correction factor determined according to freedom degree size the following steps are included:
Step 2.1, fiducial probability p is given；
Step 2.2, according to fiducial probability p, the confidence upper limit m under standardized normal distribution is calculated using formula (1)；
It according to fiducial probability p, is calculated at freedom degree n using formula (2), the confidence upper limit θ of t distribution；
Wherein, Γ (x) is gamma function；
Step 2.3, correction factor is calculated:
K=θ/m (3).
4. error correcting method in a kind of measurement for taking freedom degree into account according to claim 1, which is characterized in that in step
In three, middle error after amendment:
5. error correcting method in a kind of measurement for taking freedom degree into account according to claim 3, which is characterized in that wherein,
In step 2.1, fiducial probability p takes 95.4% or 99.7%.
6. error correcting method in a kind of measurement for taking freedom degree into account according to claim 3, which is characterized in that wherein,
In step 2.2, confidence upper limit m and θ can be by tabling look-up or computer is calculated.
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| Publication number | Priority date | Publication date | Assignee | Title |
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| CN111895953A (en) * | 2020-09-03 | 2020-11-06 | 长沙理工大学 | Method for detecting and correcting buried depth of guardrail upright column |
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