CN107423260A - A kind of minimum card side's method of estimation - Google Patents
A kind of minimum card side's method of estimation Download PDFInfo
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- CN107423260A CN107423260A CN201710203816.8A CN201710203816A CN107423260A CN 107423260 A CN107423260 A CN 107423260A CN 201710203816 A CN201710203816 A CN 201710203816A CN 107423260 A CN107423260 A CN 107423260A
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Abstract
The present invention relates to a kind of minimum card side's method of estimation, this method constructs the functional form of matched curve on the basis of the basic function collection of linear independence, each term coefficient in the function of matched curve is provided using the method for iterative calculation, coefficient is substituted into after fitting function obtained theoretical value has the characteristics of card side is minimum, the minimum card side's method of estimation of the present invention, avoid fminsearch functions and more solutions be present, with unique solution and it can guarantee that the characteristics of card side is minimum, it ensure that the accuracy of solution, avoid loss of learning problem present in solution procedure, and the minimum card side's method of estimation of the present invention, logicality is strong, it is easily programmed, method is simple, it is easily achieved.
Description
Technical Field
The invention relates to a minimum chi-square estimation method, and belongs to the technical field of numerical analysis and reliability.
Background
There are many functions in scientific experimental and engineering applications whose analytical expressions are unknown. For example, the performance and parameters of the inertial device change with the storage time, but the change process can only be measured by a series of nodes x through experimental observationiValue f ofi. The problem is now to find an approximation function y (x) for the sequence of test points or, in geometrical terms, a curve y (x) to fit (smooth) the n points.
In the theory relating to the fitness test, χ2Statistics are often used to test whether a common distribution of a set of independent samples belongs to a family of distributions with specific properties. The least chi-square estimation means to minimize chi2The parameters obtained from the statistics are written as the best estimates of the true values
Wherein,
then y (x) is denoted as f (x) with respect to the weight coefficient { omega }iAnd (4) chi-square approximation, and a method for solving an approximation function y (x) by the criterion is called a chi-square estimation method. In the above formula, { ωj(x) Is a linearly independent function set, { a }jIs the corresponding coefficient set.
Defining an m + 1-ary function H (a)0,a1,…,am) Is composed of
The minimum value point should satisfy the requirement of the extreme value of the multivariate function
The above formula is equivalent to
However, since the above formula is a multi-element nonlinear equation system, the coefficient a is directly solved0,a1,…,amIt is difficult.
In the reference "assessment of the storage life of accelerometers under temperature stress" ("Proc. Armour Engineers' Proc. engineering, Vol.28No.3,2014, P28), a method is proposed to solve the minimum by means of the fmisearch function in Matlab software. As introduced herein, "in Matlab, the function fminsearch is used to solve the unconstrained multidimensional extremum problem, which uses a simplex search method without the need to compute gradients".
Through the application of the function in practical application, the following disadvantages are found to exist:
(1) the solution is not unique, which means that the method cannot determine the coefficient value that minimizes the chi-squared;
(2) the mechanism of the method is unclear and the algorithm is not optimal.
(3) Information loss is easily caused.
Therefore, there is a need to provide an improved explicit minimum chi-squared estimation method for the above situation.
Disclosure of Invention
The invention aims to overcome the defects in the prior art and provide a minimum chi-square estimation method, which can provide each coefficient of a fitting function in a precise iterative calculation mode and meet the requirement of chi-square minimization.
The above purpose of the invention is mainly realized by the following technical scheme:
a method of least chi-squared estimation, comprising:
given a sequence of measurements fiAnd a sequence of weight coefficients ωi}，i＝1,2,…,n；
Determining a set of linearly independent basis functions
Construct a measurement sequence { fiCurve fitting function psi (x) ofWherein { ajIs the corresponding set of unknown coefficients;
determining the set of coefficients { a ] according to an optimization criterionjThe optimum value of { a }jI.e. thatSolving the optimal value { a } according to the following formulaj}：
The optimal value { a }jSubstituting into the curve fitting function ψ (x) to obtain a function expression y (x) of the best-fit curve estimated by the minimum chi-square:
in the above estimation method, the optimal value { a } is solvedjThe concrete method is as follows:
(1) solving the coefficient a according to the following formula0,a1,…,amAn initial value of (d);
(2) a coefficient of friction0,a1,…,amSubstituting the initial value of (A) into the following formula to solve gi：
(3) G is prepared byiSubstituting the following formula (5), and solving the coefficient a according to the following formula (5)0,a1,…,amA set of new values of:
(4) returning to the step (2) and enabling the coefficient a to be0,a1,…,amSubstituting a group of new values as initial values into the formula (4), circularly calculating,up to the calculated coefficient a0,a1,…,amAnd the coefficient a obtained by the last calculation0,a1,…,amIf the deviation between the new values of the L-1 th group meets the requirement, the new value of the L-th group is taken as the optimal value { a }j}; wherein: l is a positive integer.
In the above estimation method, the coefficient a in the step (4)0,a1,…,amAnd the coefficient a obtained by the last calculation0,a1,…,amThe deviation between the new values of the L-1 th group meets the requirement that: coefficient a0,a1,…,amIs compared with the coefficient a0,a1,…,amThe corresponding p bits of each coefficient in the L-1 th new group of values are the same after the decimal point, namely a in the L-th new group of values0With a in the new value of the L-1 th group0The values of p bits before and after decimal point are the same, and a in the Lth group of new values1With a in the new value of the L-1 th group1The values of the p bits before and after the decimal point are the same, … …, a in the Lth set of new valuesmWith a in the new value of the L-1 th groupmThe values of the front p bits after the decimal point are the same, wherein: p is a positive integer.
In the estimating method, the coefficient a in the step (1)0,a1,…,amThe initial value of (a) is obtained by solving the following formula:
in the above estimation method, the coefficient a in the step (3)0,a1,…,amIs solved by the following formula:
compared with the prior art, the invention has the following beneficial effects:
(1) the minimum chi-square estimation method provided by the invention constructs a function form of a fitting curve on the basis of a linearly independent basis function set, adopts an iterative calculation method to give each coefficient in the function of the fitting curve, and substitutes the coefficient into the fitting function to obtain a theoretical value with the minimum chi-square characteristic.
(2) The minimum chi-square estimation method provided by the invention provides a method different from the method for applying the fmisearch function in Matlab software, avoids the problem of multiple solutions of the fmisearch function, and has the characteristics of unique solution and capability of ensuring the minimum chi-square.
(3) The coefficient of the fitting function is solved by adopting iterative computation, so that the problem that a nonlinear equation set cannot be solved is solved; moreover, the accuracy of the solution is ensured, and the problem of information loss in the solution process is avoided.
(4) The minimum chi-square estimation curve fitting method provided by the invention has the advantages of strong logic, easiness in programming, simplicity and easiness in implementation.
Drawings
FIG. 1 is a flow chart of curve fitting based on relative minimization of error according to the present invention;
FIG. 2 is a measurement sequence given in an embodiment of the present invention;
FIG. 3 is an iterative process of an embodiment of the present invention;
FIG. 4 is a graph of a fit obtained by an embodiment of the present invention;
FIG. 5 is a graph comparing a fitted curve obtained by an example of the present invention with the least square method.
Detailed Description
The invention is described in further detail below with reference to the following figures and specific examples:
the invention provides a minimum chi-square estimation method, which is characterized in that a function form of a fitting curve is constructed on the basis of a linearly independent basis function set, each coefficient in the function of the fitting curve is given by adopting an iterative calculation method, and a theoretical value obtained by substituting the coefficient into the fitting function has the characteristic of minimum chi-square.
As shown in fig. 1, a curve fitting flowchart based on the minimum relative error of the present invention is shown, and the minimum chi-square estimation method of the present invention includes the following steps:
one, given a measurement sequence { fiAnd a sequence of weight coefficients ωi}，i＝1,2,…,n；
Two, determining a set of linearly independent basis functions
Thirdly, constructing a measurement sequence { fiCurve fitting function psi (x) ofWherein { ajIs the corresponding set of unknown coefficients;
fourthly, determining the coefficient set { a) according to an optimal criterionjThe optimum value of { a }j}. To minimize the chi-square, there are
Defining an m + 1-ary function H (a)0,a1,…,am) Is composed of
The minimum value point should satisfy the requirement of the extreme value of the multivariate function
The above formula is equivalent to
However, in the above formula, each coefficient a0,a1,…,amAppearing in the denominator, it is difficult to solve. Iterative solution is to be used. The optimal value { a } can be solved by the following methodjThe method specifically comprises the following steps:
(1) solving the coefficient a for the following formula0,a1,…,amAn initial value of (d);
defining an m + 1-ary function H1(a0,a1,…,am) Is composed of
The minimum value point should satisfy the requirement of the extreme value of the multivariate function
The above formula is equivalent to
The initial value a of the coefficient can be obtained0,a1,…,amA calculation method of
(2) A coefficient of friction0,a1,…,amSubstituting the initial value of (A) into the following formula to solve gi：
(3) G is prepared byiSubstituting the following formula (5), and solving the coefficient a according to the following formula (5)0,a1,…,amA set of new values of:
the derivation process of equation (5) is as follows:
the above formula is equivalent to:
namely:
the coefficient a can be obtained0,a1,…,amOne of the calculation methods of (1) is:
(4) returning to the step (2) to obtain the coefficient a0,a1,…,amSubstituting a set of new values of (a) into the formula (4) as initial values, and calculating g againiG is mixingiSubstituting into formula (5) to obtain coefficient a0,a1,…,amThen returning to the step (2), substituting the second new value as the initial value into the formula (4), repeating the steps, and performing cyclic calculation until the coefficient a is obtained0,a1,…,amAnd the coefficient a obtained by the last calculation0,a1,…,amThe deviation between the new values of the L-1 th group meets the requirement, namely, each error coefficient tends to a fixed value, the new value of the L-1 th group is taken as the optimal value { a }j}; wherein: l is a positive integer.
Coefficient a in the above step (4)0,a1,…,amAnd the coefficient a obtained by the last calculation0,a1,…,amThe deviation between the new values of the L-1 th group meets the requirement that: coefficient a0,a1,…,amIs compared with the coefficient a0,a1,…,amThe corresponding p bits of each coefficient in the L-1 th new group of values are the same after the decimal point, namely a in the L-th new group of values0With a in the new value of the L-1 th group0The values of p bits before and after decimal point are the same, and a in the Lth group of new values1With a in the new value of the L-1 th group1The values of the p bits before and after the decimal point are the same, … …, a in the Lth set of new valuesmWith a in the new value of the L-1 th groupmThe values of the front p bits after the decimal point are the same, wherein: p is a positive integer, for example, in this embodiment, p is 10, that is, the first 10 after m +1 coefficient decimal points are the same, it is considered that the deviation requirement is satisfied.
Fifthly, the optimal value { a }is setjSubstituting into the curve fitting function ψ (x) to obtain a function expression y (x) of the best-fit curve estimated by the minimum chi-square:
the method can be used for performing curve fitting on the tested data in practical application, and is used for performing service life evaluation on the equipment, judging and predicting the validity period and the storage life of the equipment.
Example 1
Let a given argument be { xiCorresponding measurement sequence fi1,2, …, n; where n is 10, the measurement sequence given in the embodiment of the present invention is shown in fig. 2; specific values of the data are shown in table 1. Fitting by linear curve, selecting two basis functionsAndthe fitting function thus constructed is y (x) a1x-a2。
TABLE 1
With the solving method of the invention, the initial value is a1＝1.857130151514106、a26.939083333409633, and finally obtaining a after 8 times of iterative computation1＝1.854996089750836、a26.975801527797882. I.e. a calculated from the 7 th iteration1、a2The first 10 bits after the decimal point are correspondingly equal and are the optimal value. The iterative process is shown in FIG. 3, in which FIG. 3a and FIG. 3b respectively show a1、a2The iterative process of (2). A is a1And a2Substituting the fitting function into y (x) ═ a1x-a2The ideal data obtained in (1) is shown as the "+" dot in fig. 4, and fig. 4 shows the fitted curve obtained by the embodiment of the present invention, in the figure ". "means measured value.
The data in table 1 can also adopt a least square method, and the solved coefficient is a1＝1.696152760852661、a26.574443934342034. FIG. 5 is a graph comparing a least square method with a fitted curve obtained by an embodiment of the present invention, wherein "dotted line" in FIG. 5 is the fitted curve obtained by the least square method, "solid line" is the fitted curve obtained by the method of the present invention, and "dotted line" is the measured sequence { f }iThe fitted curve of. To compare the differences between the two methods, the respective values are determined
(1) Least square method: -0.4517058218079164;
(2) the method comprises the following steps: -0.5265079715177377.
It can be seen that the chi-square given by the invention is minimum, has a unique solution and has no information loss problem.
The above description is only one embodiment of the present invention, but the scope of the present invention is not limited thereto, and any changes or substitutions that can be easily conceived by those skilled in the art within the technical scope of the present invention are included in the scope of the present invention.
The invention has not been described in detail in part of the common general knowledge of those skilled in the art.
Claims (5)
1. A minimum chi-square estimation method is characterized by comprising the following steps: the method comprises the following steps:
given a sequence of measurements fiAnd a sequence of weight coefficients ωi}，i＝1,2,…,n；
Determining a set of linearly independent basis functions
Constructing a curve fitting function psi (x) of the measurement sequence { fi }, and enablingWherein { ajIs the corresponding set of unknown coefficients;
determining the set of coefficients { a ] according to an optimization criterionjThe optimum value of { a }jI.e. solving the optimal value { a } according to the following formulaj}：
The optimal value { a }jSubstituting into the curve fitting function ψ (x) to obtain a function expression y (x) of the best-fit curve estimated by the minimum chi-square:
2. the estimation method according to claim 1, characterized in that: solving for the optimal value { a }jThe concrete method is as follows:
(1) solving the coefficient a according to the following formula0,a1,…,amAn initial value of (d);
(2) a coefficient of friction0,a1,…,amSubstituting the initial value of (A) into the following formula to solve gi：
(3) G is prepared byiSubstituting the following formula (5), and solving the coefficient a according to the following formula (5)0,a1,…,amA set of new values of:
(4) returning to the step (2) and enabling the coefficient a to be0,a1,…,amSubstituting a group of new values as initial values into the formula (4), and circularly calculating until the calculated coefficient a is obtained0,a1,…,amAnd the coefficient a obtained by the last calculation0,a1,…,amIf the deviation between the new values of the L-1 th group meets the requirement, the new value of the L-th group is taken as the optimal value { a }j}; wherein: l is a positive integer.
3. The estimation method according to claim 2, characterized in that: coefficient a in the step (4)0,a1,…,amAnd the coefficient a obtained by the last calculation0,a1,…,amThe deviation between the new values of the L-1 th group meets the requirement that: coefficient a0,a1,…,amIs compared with the coefficient a0,a1,…,amThe corresponding p bits of each coefficient in the L-1 th new group of values are the same after the decimal point, namely a in the L-th new group of values0With a in the new value of the L-1 th group0The values of p bits before and after decimal point are the same, and a in the Lth group of new values1With a in the new value of the L-1 th group1The values of the p bits before and after the decimal point are the same, … …, a in the Lth set of new valuesmWith a in the new value of the L-1 th groupmThe values of the front p bits after the decimal point are the same, wherein: p is a positive integer.
4. The estimation method according to claim 2, characterized in that: coefficient a in the step (1)0,a1,…,amThe initial value of (a) is obtained by solving the following formula:
5. the estimation method according to claim 2, characterized in that: coefficient a in the step (3)0,a1,…,amIs solved by the following formula:
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