Error theory and adjustment basic study notes (Ⅰ)

1. Preliminary knowledge

1. Error

1 Source of error

Errors are inevitably included in the observations, and the observation conditions (human, instrument, ring) determine the quality of the observations.

• Measuring instrument (instrument): the accuracy of the instrument, the error of the instrument itself
• Observer: sensory organ identification ability, technical proficiency, work attitude
• Outside condition (outside condition): air pressure, humidity, wind, sunlight, atmospheric refraction

2 Error types

Error is divided into gross error, systematic error, random error

• Gross error (also called error): Refers to the error greater than the tolerance, and refers to the large error caused by the observer's carelessness or other interfering factors, that is, the error.
  Processing method: make redundant observations, use gross error detection to identify gross errors, and eliminate its influence through data processing.
• Systematic error (regular error): A series of observations are made under the same observation conditions. If the error has a certain law, this type of error is called systematic error.
  Processing method: establish a function model (semi-parametric model, adjustment model with additional system parameters), and adopt corresponding observation methods
• Random error (random error): A series of observations are made under the same observation conditions. If the error shows contingency in both size and sign, that is, from the point of view of a single error, the size and sign of the error in the column are not Regularity; but in terms of a large number of errors as a whole, there is a certain statistical law.
  Processing method: the best estimate of redundant observation adjustment method

Statistical characteristics of accidental errors:

• The finiteness of the absolute value of the error
• An error with a small absolute value is more likely to occur than a large error
• The probability of positive and negative errors with equal absolute value is equal
• The theoretical mean of accidental error is 0

2. Random variables

Concept: As far as a single error is concerned, the size of the symbol of its value is accidental, random, and irregular, but for a large number of accidental errors, it has certain regularity, so it is called statistical regularity. In mathematical statistics, variables with the above characteristics are called random variables.

1 Digital features

• Mathematical expectation: (Mathematical expectation) reflects the numerical characteristics of the concentrated position of random variables
  ​- Variance: (variance) reflects the degree of dispersion of random variables from the concentrated position
• Covariance & correlation coefficient: (covariance & correlation coefficient) a numerical feature reflecting the degree of correlation between two random variables x and y

E(x)=∑xi pi
E(C)=C
E(CX)=CE(X)
E(X+Y)=E(X)+E(Y)
E(XY)=E(X)E(Y)
-------------------
D(x)=E{
    
    [x-E(x)]²}
D(C)=0
D(CX)=C²D(X)
D(X)=E(X²)-E²(X)
D(X+Y)=D(X)+D(Y)
-------------------
Dxy=E{
    
    [x-E(x)]*[y-E(y)]}
ρxy=Dxy/DxDy

2 Metrics to measure accuracy

name	English	Calculation method
Variance and median error	variance and mean square error	σ² & σ
average error	average error	θ = sqrt (2 / π) σ (0.7979σ)
Contingent error	probable error	ρ≈⅔ σ (0.6745σ)
Limit error	limit error	Δ = 3σ

Δ> σ> θ> ρ

3 Indicators to measure the quality of observations

name	meaning	Calculation method
Precision	precision describes the accidental error, that is, the degree of deviation of the observed value from the mathematical expectation; expressed by the variance or the median error	σ² & σ
Accuracy	accuracy describes the system error, which can be described by the difference between the true value of the observation and the mathematical expectation of the observation	ε=Ã-E(A)
Accuracy	Describe the combined effects of accidental errors and systematic errors. The accuracy can be described by the mean square error of the observations.	MSE(A)=E[(A-Ã)²]=σ²+(E(A)-Ã)²=σ²+ε²

As shown in the above figure:
Figure 1 shows the high precision of shooting, which is like high precision of the measurement data, but the accuracy is poor;
Figure 2 shows the high accuracy of shooting, which is like the high accuracy of the measurement data, but the precision is poor;
Figure 3 shows The precision and accuracy are both good, just like the precision and accuracy of the measurement data are good, that is, the accuracy is high.

2. Types of adjustment

For more information, please refer to: https://blog.csdn.net/Gou_Hailong/article/details/108216533