Minimum chi-square estimation method

Abstract

The invention relates to a minimum chi-square estimation method. According to the method, a function form of a fitted curve is constructed on the basis of a linearly independent basis function set, a method of iterative calculation is adopted to give each coefficient in a function of the fitted curve, and theoretical values obtained after substituting the coefficients into the fitting function have a characteristic of the minimum chi-square. The minimum chi-square estimation method of the invention avoids the problem of existence of multiple solutions of a fminsearch function, has the characteristics of a unique solution and ability to ensure the chi-square to be minimum, ensures the accuracy of solving, avoids the problem of information missing existing in a solving process, and is high in logicality, easy to program, simple and easy to realize.

Description

Technical field

[0001] The invention relates to a minimal chi-square estimation method, belonging to the technical field of numerical analysis and reliability. Background technique

[0002] There are many functions in scientific experiments and engineering applications, and their analytical expressions are unknown. For example, with the extension of storage time, the performance and parameters of inertial devices will change, but the change process can only be measured by a series of nodes x by experimental observation. i Value on f i . The problem now is to seek the approximation function y(x) of the test point sequence, or to use geometric language to find a curve y(x) to fit (smooth) these n points.

[0003] In the related theory of fitness test, χ 2 Statistics are often used to test whether the common distribution of a group of independent samples belongs to a distribution family with specific properties. Minimal chi-square estimation refers to minimizing χ 2 The parameters o...

Examples

Embodiment 1

[0099] Let the given argument be (x i }Corresponding measurement sequence{f i }, i=1,2,...,n; where n=10, such as figure 2 Shown is the measurement sequence given in the embodiment of the present invention; the specific values ​​of the data are shown in Table 1. Fit a linear curve and select two basis functions with The fitting function constructed by this is y(x)=a 1 x-a 2 .

[0100] Table 1

[0101]

[0102]

[0103] Using the solution method of the present invention, the initial value is a 1 = 1.857130151514106, a 2 =6.939083333409633, after 8 iterations of calculation, a is finally obtained 1 = 1.854996089750836, a 2 =6.975801527797882. That is, the a calculated from the 7th iteration 1 , A 2 The first 10 digits after the decimal point correspond to the same value, which is the optimal value. The iterative process is like image 3 Shown, where image 3 a and image 3 b respectively represents a 1 , A 2 Iterative process. Put a 1 And a 2 Substituting into the fitting functi...

Technical field

[0001] The invention relates to a minimal chi-square estimation method, belonging to the technical field of numerical analysis and reliability. Background technique

[0002] There are many functions in scientific experiments and engineering applications, and their analytical expressions are unknown. For example, with the extension of storage time, the performance and parameters of inertial devices will change, but the change process can only be measured by a series of nodes x by experimental observation. i Value on f i . The problem now is to seek the approximation function y(x) of the test point sequence, or to use geometric language to find a curve y(x) to fit (smooth) these n points.

[0003] In the related theory of fitness test, χ 2 Statistics are often used to test whether the common distribution of a group of independent samples belongs to a distribution family with specific properties. Minimal chi-square estimation refers to minimizing χ 2 The parameters o...

Examples

Embodiment 1

[0099] Let the given argument be (x i }Corresponding measurement sequence{f i }, i=1,2,...,n; where n=10, such as figure 2 Shown is the measurement sequence given in the embodiment of the present invention; the specific values ​​of the data are shown in Table 1. Fit a linear curve and select two basis functions with The fitting function constructed by this is y(x)=a 1 x-a 2 .

[0100] Table 1

[0101]

[0102]

[0103] Using the solution method of the present invention, the initial value is a 1 = 1.857130151514106, a 2 =6.939083333409633, after 8 iterations of calculation, a is finally obtained 1 = 1.854996089750836, a 2 =6.975801527797882. That is, the a calculated from the 7th iteration 1 , A 2 The first 10 digits after the decimal point correspond to the same value, which is the optimal value. The iterative process is like image 3 Shown, where image 3 a and image 3 b respectively represents a 1 , A 2 Iterative process. Put a 1 And a 2 Substituting into the fitting functi...

Images