CN110426059B - An error correction method in measurement considering degrees of freedom - Google Patents

Abstract

The invention discloses a method for correcting errors in measurement considering freedom degrees, belongs to the field of surveying and mapping, and provides a method for calculating a medium error correction coefficient by giving a corresponding tolerance of a high confidence probability under the condition of considering the freedom degrees based on the rule that the confidence probability changes along with the freedom degrees under the condition of giving a confidence interval by t distribution, so that correction of errors corresponding to different freedom degrees is realized, and the reasonability and reliability of precision measurement under the condition of low freedom degrees are ensured. The method comprises the following steps: (1) calculating a mean error of the measured values according to the degrees of freedom of the measured values; (2) determining a medium error correction coefficient according to the degree of freedom; (3) and multiplying the medium error by the correction coefficient to obtain the corrected medium error.

Description

An error correction method in measurement considering degrees of freedom

technical field

The invention belongs to the field of surveying and mapping, and relates to the problem of confidence probability of measurement results, in particular to the problem of confidence probability of the error corresponding to the tolerance in the case of low degree of freedom.

Background technique

In measurement, the occurrence of accidental errors is unavoidable, and it is generally considered that accidental errors are random variables that obey a normal distribution. In order to reflect the degree of dispersion of the error, the medium error is often used as an index to measure the accuracy to reflect the accuracy of the observed value. It is generally believed that the smaller the medium error, the higher the accuracy of the measurement results. Generally, 3 times the medium error is used as the limit value of the accidental error, and the probability that the true error appears in this interval is 99.7%. This is because the accidental error obeys a normal distribution. Generally, the probability of a normal distribution in a given interval is calculated by dividing the chance by the standard deviation into a standard normal distribution for calculation. Due to the limited number of actual observations, only the estimated median error of the standard deviation can be obtained, so the median error is used instead of the standard deviation in the measurement, and the default accidental error divided by the median error obeys the standard normal distribution.

In fact, since the calculation of the median error is related to the degree of freedom n, the accidental error divided by the median error does not obey the standard normal distribution, but obeys the t distribution. In a given interval, the value of the confidence probability of the t distribution increases with the increase of the degree of freedom. When the degree of freedom is low, the confidence probability of the t distribution is much lower than that of the standard normal distribution. Therefore, the method of increasing the number of observations, that is, increasing the degree of freedom, is usually used in the measurement to increase the confidence probability. However, due to the limited number of actual observations, when the number of observations is small, the confidence probabilities of the corresponding tolerances of the medium error of different degrees of freedom are different, and all are smaller than the theoretical value of the confidence probability, that is, the confidence probability of the medium error of 2 times is less than 95.4% , the 3-fold error confidence probability is less than 99.7%. If the medium error calculated in the case of low degrees of freedom is used as the accuracy measure, there will be the following disadvantages:

(1) The confidence probability of the error corresponding to the tolerance is not at the same level, and the measurement of accuracy lacks rationality.

(2) It will reduce the indicators of precision measurement, resulting in the lack of reliability and safety of engineering construction, which is not conducive to the acceptance and quality assessment of the project.

Therefore, the mid-range error should be corrected in consideration of the degree of freedom to ensure that the confidence probability of its corresponding tolerance is at the same level, especially in the case of low degrees of freedom. However, there is currently no correction method for the medium error in the case of low degrees of freedom.

The evaluation of the accuracy of the measurement results has important strategic significance in the measurement work of all walks of life in my country's modernization construction. At present, the reliability of the measurement accuracy is increasingly required. Reasonable accuracy evaluation will help to improve the reliability of measurement results and ensure the safety and economy of engineering construction.

SUMMARY OF THE INVENTION

Technical problem to be solved: Aiming at the deficiencies of the prior art, the present invention proposes an error correction method in measurement that takes into account the degree of freedom, which is used to solve the problem that the confidence probability of the error corresponding limit in the accuracy index of the measurement result under low degree of freedom cannot reach the theoretical level. value issue.

In order to solve the above-mentioned technical problems, the present invention adopts the following technical solutions:

An error correction method in measurement considering degrees of freedom, which specifically includes the following steps:

Step 1, calculate the degrees of freedom, and calculate the median error of the measured value;

Step 2: Determine the medium error correction coefficient according to the degree of freedom;

Step 3: Multiply the median error by the correction coefficient to obtain the corrected median error.

其中，V表示改正数，P表示权阵，n为自由度，等于观测次数减去未知数个数。As a further preferred solution of the method for correcting errors in the measurement in consideration of the degree of freedom of the present invention, in step 1, the calculation formula of the intermediate error is:

Among them, V represents the correction number, P represents the weight matrix, and n represents the degree of freedom, which is equal to the number of observations minus the number of unknowns.

As a further preferred solution of the method for correcting errors in measurement in consideration of degrees of freedom of the present invention, in step 2, the determining of the correction coefficients for medium errors according to degrees of freedom includes the following steps:

Step 2.1, given the confidence probability p;

Step 2.2, according to the confidence probability p, use formula (1) to calculate the upper confidence limit m under the standard normal distribution;

According to the confidence probability P, use formula (2) to calculate the upper confidence limit θ of the t distribution under the degree of freedom n;

Among them, Г(x) is the gamma function;

Step 2.3, calculate the correction factor:

k=θ/m (3).

As a further preferred solution of the method for correcting errors in measurement in consideration of degrees of freedom of the present invention, in step 3, the errors after correction are:

As a further preferred solution of the method for correcting errors in measurement in consideration of degrees of freedom of the present invention, in step 2.1, the confidence probability p is 95.4% or 99.7%.

As a further preferred solution of the method for error correction in measurement considering the degree of freedom of the present invention, in step 2.2, the upper confidence limits m and θ can be obtained through table look-up or computer calculation.

Beneficial effects:

The invention corrects the error in the measurement under the condition of taking into account the degree of freedom, and by adding a correction coefficient to the error in the centering, it is ensured that the confidence probability of the corresponding limit error reaches the theoretical value. On the one hand, it ensures that the confidence probability of the error corresponding to the tolerance is the same under different degrees of freedom, so that the accuracy measurement is at the same confidence level, which ensures the rationality of the accuracy measurement. On the other hand, it improves the confidence probability of the limit error in the case of low degrees of freedom, which ensures the reliability of the measurement results, which in turn helps to ensure the safety and economy of the project construction, and facilitates the acceptance and quality assessment of the project.

Description of drawings

Fig. 1 is the method flow chart of the present invention;

Figure 2 is a schematic diagram of the leveling route of the present invention.

Detailed ways

The present invention will be specifically described below with reference to the embodiments.

An error correction method in measurement considering the degree of freedom, as shown in Figure 1, specifically includes the following steps:

Step 1, calculate the degrees of freedom, and use the middle error calculation formula to calculate the middle error of the measured value;

Step 2: Determine the medium error correction coefficient according to the degree of freedom;

Step 3: Multiply the median error by the correction coefficient to obtain the corrected median error.

其中，V表示改正数，P表示权阵，n为自由度，等于观测次数减去未知数个数。Preferably, in step 1, the calculation formula of the middle error is

Among them, V represents the correction number, P represents the weight matrix, and n represents the degree of freedom, which is equal to the number of observations minus the number of unknowns.

Preferably, in step 2, the determining of the middle error correction coefficient according to the degree of freedom includes the following steps:

Step 2.1, given the confidence probability p, calculate the correction coefficient for the medium error with degrees of freedom not greater than 30;

Step 2.2, according to the confidence probability p, use formula (1) to calculate the upper confidence limit m under the standard normal distribution;

According to the confidence probability p, use formula (2) to calculate the upper confidence limit θ of the t distribution under the degree of freedom n;

Among them, Γ(x) is the gamma function.

Step 2.3, calculate the correction factor:

k=θ/m (3)

Preferably, in step 3, the correction value of the medium error

Preferably, in step 2.1, the confidence probability p is generally 95.4% or 99.7%.

Preferably, in step 2.2, the upper confidence limits m and θ can be calculated by looking up a table or by a computer.

In the leveling network, as shown in Figure 2, A and B are benchmarks with known elevations, and these points are assumed to have known elevations without error. Points C, D and E in the figure are to be determined. The elevations of points A and B, the observed height difference and the corresponding leveling route are shown in Table 1. Under the confidence probability of 95.4%, try to find the median error of the height adjustment values of the points C and D to be determined.

Table 1

In this problem, the number of observations is 7 and the number of unknowns is 3, then the degree of freedom

n=7-3=4

Through the knowledge of measurement adjustment, it can be known that the correction number

V=[-0.2 2.9 -4.2 -0.1 -3.9 -0.6 -1.2] ^T

The power matrix P is the unit matrix;

medium error

Given a confidence probability p=95.4%;

Calculated by computer, the upper confidence limit m=2 under the standard normal distribution, and the upper confidence limit θ=2.859 under the t distribution with 4 degrees of freedom;

Correction coefficient k=θ/2=1.429;

Corrected value of medium error at 95.4% confidence probability

Under the confidence probability of 95.4%, the error after correction is 3.1mm, which further improves the accuracy index and ensures the reliability of the measurement.

Further, a correction coefficient with degrees of freedom ranging from 1 to 30 in step 2 is given.

Given a 95.4% confidence probability;

Calculate the specific value of the upper confidence limit θ under a given 95.4% confidence probability t distribution by computer, as shown in Table 2, where the degrees of freedom are n=1, 2, ..., 30, and the significant figures of θ are 3 decimal places. . Table 2 shows the θ values of different degrees of freedom under 95.4% confidence probability

Table 2

Find the upper confidence limit m=2 of the 95.4% confidence probability under the standard normal distribution;

According to Table 2, the correction coefficient k=θ/2 corresponding to the degree of freedom n from 1 to 30 can be calculated, as shown in Table 3. Table 3 shows the value of the correction coefficient k corresponding to different degrees of freedom under the 95.4% confidence probability, take k The significant figures are 3 decimal places.

table 3

The invention provides a method for correcting errors in measurement taking into account the degree of freedom under low degrees of freedom, ensuring that the confidence probability of the corresponding limit error of the intermediate errors after the correction can reach the theoretical value, and at the same time, a specific implementation method is given, and the confidence probability is 95.4%. As an example, the correction coefficients for medium errors with degrees of freedom ranging from 1 to 30 are given.

The above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection of the present invention. within the range.

Claims (4)

1. a kind of error correction method in the measurement taking into account the degree of freedom, is characterized in that, specifically comprises the following steps: Step 1, calculate the degrees of freedom, and calculate the median error of the measured value; Step 2: Determine the medium error correction coefficient according to the degree of freedom; Step 3: Multiply the middle error by the correction coefficient to obtain the corrected middle error; In step 1, the calculation formula of the medium error is:

Among them, V represents the correction number, P represents the weight matrix, and n represents the degree of freedom, which is equal to the number of observations minus the number of unknowns; In step 2, the described determination of the correction coefficient according to the degree of freedom includes the following steps: Step 2.1, given the confidence probability p; Step 2.2, according to the confidence probability p, use formula (1) to calculate the upper confidence limit m under the standard normal distribution;

According to the confidence probability p, use formula (2) to calculate the upper confidence limit θ of the t distribution under the degree of freedom n;

Among them, Γ(x) is the gamma function; Step 2.3, calculate the correction factor: k=θ/m (3). 2. a kind of error correction method in consideration of degree of freedom according to claim 1, is characterized in that, in step 3, after correction error:

3 . The method for correcting errors in measurement considering degrees of freedom according to claim 1 , wherein, in step 2.1, the confidence probability p is 95.4% or 99.7%. 4 . 4 . The method for correcting errors in measurement considering degrees of freedom according to claim 1 , wherein, in step 2.2 , the upper confidence limits m and θ can be obtained by look-up table or computer calculation. 5 .

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