CN111812080A - Nonlinear calibration precision analysis method based on iterative optimal power - Google Patents

Nonlinear calibration precision analysis method based on iterative optimal power Download PDF

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CN111812080A
CN111812080A CN202010589024.0A CN202010589024A CN111812080A CN 111812080 A CN111812080 A CN 111812080A CN 202010589024 A CN202010589024 A CN 202010589024A CN 111812080 A CN111812080 A CN 111812080A
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CN111812080B (en
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何章鸣
马正芳
王大轶
王炯琦
侯博文
吕东辉
魏居辉
周萱影
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Abstract

The embodiment of the invention provides a nonlinear scaling precision analysis method based on an iterative optimal power, which comprises the following steps: acquiring a set number of reagent light values and a corresponding reagent concentration data set; constructing a nonlinear function from the reagent light value to the reagent concentration according to the data set by a nonlinear scaling mode based on the iterative optimal power; and carrying out precision analysis on the nonlinear function. By adopting the method, the problems of poor calibration accuracy and imperfect calibration precision evaluation method in the prior art can be solved, so that the reagent concentration can be more accurately determined through the reagent light value when the chemiluminescence immunoassay reagent is applied.

Description

Nonlinear calibration precision analysis method based on iterative optimal power
Technical Field
The invention relates to the technical field of analytical reagent calibration, in particular to a nonlinear calibration precision analysis method based on iterative optimal power.
Background
The light values of chemiluminescent immunoassay reagents are mapped to reagent concentrations in a one-to-one manner, as represented by a nonlinear function, and the scaling is the identification of a nonlinear function from concentration to light value using a finite number of data sets. This non-linear function is a strictly monotonically increasing function and depends on four unknown parameters. Given the functional structure, knowing the light value allows the concentration to be calculated by a non-linear inverse function. The reagent concentration range is 0 to 1000 (a common reagent concentration range is 0.1 to 3.5), and the corresponding reagent light value measurement range is 100 to 10000000. At present, many calibration methods exist, such as steepest descent method, newton method, gauss-newton method, hybrid method, calibration method based on shrinkage-expansion variation step length and weighted nonlinear parameter estimation, and so on.
In the process of implementing the invention, the inventor finds that at least the following problems exist in the prior art:
(1) the conventional methods usually use the sum of squares of residuals as an objective function, and solve parameters to minimize the objective function, and these methods cannot meet the relative residual error requirement of low concentration, which will result in reduced calibration accuracy at low concentration. A decrease in the calibration accuracy will lead to an inaccuracy of said non-linear function and finally to an inaccuracy of the result when determining the reagent concentration from the light values.
(2) The existing method does not consider the buffer parameter range between absolute least squares and relative least squares, and the range may have more optimal calibration parameters, so that the nonlinear function of calibration is not accurate enough, and the result when determining the concentration of the reagent through an optical value is not accurate enough.
(3) The estimated parameters cannot represent the completion of the calibration work, and the calibration precision analysis is still an important work, so that a good precision analysis method is favorable for accurately evaluating the calibration accuracy. For a single variable system, a stationary system and a linear system, the precision evaluation is mature, and a 3 sigma criterion, a precision factor and the like exist. However, for the non-stationary nonlinear system corresponding to the chemiluminescence immunoassay reagent, the precision evaluation is not perfect, and the conventional precision evaluation factors, such as the sum of squares of residuals, cannot meet the calibration requirement. The precision evaluation needs to give not only the sum of squares of the residuals, but also the sum of squares of the relative residuals, the uncertainty of the parameters, the uncertainty of the fitted values, etc.
Disclosure of Invention
The embodiment of the invention provides a nonlinear calibration precision analysis method based on iterative optimal power. In order to determine the nonlinear function, a method of performing mild nonlinear calibration based on the optimal power and relative weight is adopted, so that the requirement of low-concentration relative residual error can be met, the determination result of the nonlinear function is more accurate, and finally the result of determining the concentration of the reagent through an optical value is more accurate; and simultaneously, parameters such as a residual sum of squares, a relative residual sum of squares, a parameter precision pipeline, a fitting value precision pipeline and the like are given so as to solve the problem that the calibration precision evaluation method in the prior art is incomplete.
To achieve the above object, an embodiment of the present invention provides a nonlinear scaling precision analysis method based on an iterative optimal power, where the method includes:
acquiring a set number of reagent light values and a corresponding reagent concentration data set;
constructing a nonlinear function from the reagent light value to the reagent concentration according to the data set by a nonlinear scaling mode based on the iterative optimal power;
and carrying out precision analysis on the nonlinear function.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of a nonlinear scaling precision analysis method based on iterative optimal power according to the present invention;
FIG. 2 is a flow chart of an embodiment of the present invention;
FIG. 3 is a parametric pipeline diagram in accordance with an embodiment of the present invention;
FIG. 4 is a fitting value pipeline diagram according to an embodiment of the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
As shown in FIG. 1, the present invention provides a nonlinear scaling precision analysis method based on iterative optimal power, which comprises:
s101, acquiring a set number of reagent light values and a corresponding reagent concentration data set;
s102, constructing a nonlinear function from the reagent light value to the reagent concentration through a nonlinear scaling mode based on the iterative optimal power according to the data set;
and S103, carrying out precision analysis on the nonlinear function.
Further, the constructing a nonlinear function from the reagent light value to the reagent concentration by a nonlinear scaling based on an iterative optimal power according to the data set includes:
determining the nonlinear function model and setting parameters to be estimated;
determining an initial value of a parameter to be estimated and an initial value of a relative error of the nonlinear function model according to the data group;
iteratively calculating and updating the parameter to be estimated and the iterative correction relative error, and determining a primary optimal weight matrix of the parameter to be estimated;
through the initially selected optimal weight matrix, the parameters to be estimated and the iterative correction relative error are calculated and updated again in an iterative mode, and the optimal values of the parameters to be estimated under the optimal power and the optimal weight matrix are determined;
and constructing a nonlinear function from the reagent light value to the reagent concentration through the optimal value of the parameter to be estimated.
Further, the determining the nonlinear function model and setting the parameter to be estimated includes:
the nonlinear function is modeled as
Figure BDA0002554781380000031
Wherein the parameter to be estimated of the nonlinear function is β ═ a, b, c, d]T。
Further, the determining an initial value of a parameter to be estimated and an initial value of a relative error of the nonlinear function model according to the data group includes:
measurement data from m sets of concentration and light values
Figure BDA0002554781380000032
Wherein m is more than or equal to 4;
determining an initial value beta of an unknown parameter beta0＝[a0,b0,c0,d0]T(2)
Wherein the content of the first and second substances,
Figure BDA0002554781380000033
Figure BDA0002554781380000034
determining an initial value r of a relative error0：
Figure BDA0002554781380000035
Further, the iteratively calculating and updating the parameter to be estimated and the iteratively correcting relative error, and determining the initial optimal weight matrix of the parameter to be estimated includes:
determining an initial value k of the iteration number k to be 0;
determining iteration coefficients
Figure BDA0002554781380000041
Wherein the division number k of the power1,max＝50，ImIs an identity matrix;
updating the weight matrix ∑
Figure BDA0002554781380000042
Wherein the weight matrix Σ is | y1|p,…,|ym|pA diagonal matrix being a diagonal element;
the given threshold value is 0.000001, and when | | | delta β | | < then, the parameter β to be estimated is updated
Figure BDA0002554781380000043
Wherein x ═ x1,…,xm]T,y＝[y1,…,ym]T,f(x,β)＝[f(x1,β),…,f(xm,β)]TJ is Jacobian matrix
Figure BDA0002554781380000044
Updating relative error
Figure BDA0002554781380000045
Wherein the content of the first and second substances,
Figure BDA0002554781380000046
when r < rbestWhen, specify
Figure BDA0002554781380000047
Updating the iteration number k ═ k +1 (9);
when k is less than or equal to k1,maxWhen the error is +1, the initial step of iterative calculation and updating the parameter to be estimated and iterative correction relative error is transferred, namely updating the weight matrix sigma;
to this end, the sigma obtained after iterative computation may be determinedbestAnd initially selecting an optimal weight matrix for the parameter to be estimated.
Further, through the initially selected optimal weight matrix, the parameters to be estimated and the iterative correction relative error are re-iteratively calculated and updated, and the optimal values of the parameters to be estimated under the optimal power and the optimal weight matrix are determined, including:
determining the number of iterations k2Initial value k of2＝0；
Determining iteration coefficients
Figure BDA0002554781380000051
Wherein the division number k of the power2,max＝50；
Updating the weight matrix ∑2
Figure BDA0002554781380000052
Wherein the weight matrix Σ2So as to make
Figure BDA0002554781380000053
A diagonal matrix being a diagonal element;
the given threshold value is 0.000001, and when | | | delta β | | < then, the parameter β to be estimated is updated
Figure BDA0002554781380000054
Updating the relative error r;
when r < rbesAt time t, specify
Figure BDA0002554781380000055
Wherein, betabestObtaining the optimal value of the parameter to be estimated under the optimal power and the optimal weight matrix;
updating iteration number k2＝k2+1 (14)；
When k is2＜k2,maxThen, the method shifts to the initial step of re-iterative calculation and updating the parameters to be estimated and iterative correction of relative errors, namely updating the weight matrix sigma2；
To this end, the beta obtained after the iterative calculation can be determinedbestAnd the optimal value of the parameter to be estimated is obtained.
Further, the constructing a non-linear function from the reagent light value to the reagent concentration by the optimal value of the parameter to be estimated includes:
determining the nonlinear function as f (x, beta) according to the optimal value of the parameter to be estimatedbest) And finishing the calibration work.
Further, the performing precision analysis on the nonlinear function specifically includes:
determining fitting values
f(x,βbest)＝[f(x1,βbest),…,f(xm,βbest)]T(15)；
Determining variance
Figure BDA0002554781380000061
Determining a covariance matrix
S＝(JTJ)-1(17)；
Determining a projection matrix
H＝JSJT(18)；
Given a level of significance a, a quantile t is determined from the level of significance a1-α/2
Figure BDA0002554781380000062
Wherein f (x, m-4) is a density function of a t-distribution with a degree of freedom of m-4;
estimating parameter accuracy radius
Figure BDA0002554781380000063
Wherein s isiiIs the ith diagonal of S;
estimating the radius of precision of the fit value
Figure BDA0002554781380000064
Wherein h isiiIs the ith diagonal of H.
Further, the given significance level α satisfies α ∈ [0.01,0.05 ].
FIG. 2 is a flow chart of an embodiment of the present invention, which is now demonstrated by an example of a calibration and accuracy evaluation method using the flow chart of FIG. 2 and a set of calibration measurement data, wherein the calibration measurement data comprises 6 sets of reagent concentrations and light values, as shown in Table 1.
Concentration (pg/m) xi 0 3.04 14.13 100.76 302.28 600.47
Light value (without unit) yi 398.5 14141.5 65256 468031.5 1389821 2696539
TABLE 1 calibration of measurement data
The calibration accuracy analysis of this example can be roughly divided into five steps:
step 1: and (5) rough calibration.
After the model is determined, m is 6 concentration-light value data
Figure BDA0002554781380000071
Wherein m is more than or equal to 4, initializing unknown parameters based on monotonicity and a linear model, and specifically comprising the following steps of: determining a model, determining an initial value, synthesizing parameters, calculating relative errors and the like.
1.1, determining a model. The model is
Figure BDA0002554781380000072
And 1.2, determining an initial value. Order to
Figure BDA0002554781380000073
Note the book
Figure BDA0002554781380000074
Then
Figure BDA0002554781380000075
1.3, initial values of synthesis parameters.
β0＝[a0,b0,c0,d0]T＝[2696539,-2696140.5,0.0013,1.1443]T
And 1.4, calculating a relative error.
Figure BDA0002554781380000076
Wherein
Figure BDA0002554781380000077
Step 2: and (5) fine calibration. Based on the parameters in the coarse scaling, the parameters at the optimal power are searched. The method specifically comprises the following steps: determining iteration related parameters, updating a weight matrix, iteratively calculating unknown parameters, calculating relative errors, recording optimal parameters and the like as follows.
And 2.1, determining iteration related parameters. k is 0, the number of divisions of a given power k1,max＝50，ImExpress identity matrix
Figure BDA0002554781380000078
And 2.2, updating the weight matrix. The weight matrix Σ is y1|p,…,|ym|pA diagonal matrix sigma of diagonal bins, as follows
Figure BDA0002554781380000081
And 2.3, iteratively calculating unknown parameters. Given a threshold of 0.000001, iteratively update β ═ a, b, c, d]TUntil | |. Δ β | <, as follows
Figure BDA0002554781380000082
Wherein x ═ x1,…,xm]T,y＝[y1,…,ym]T,f(x,β)＝[f(x1,β),…,f(xm,β)]TAnd J is the Jacobian matrix, as follows
Figure BDA0002554781380000083
And 2.4, calculating a relative error. Relative error r, as follows
Figure BDA0002554781380000084
Wherein
Figure BDA0002554781380000085
And 2.5, recording the optimal parameters. 1.2766% < r%best＝16.4330％
Figure BDA0002554781380000086
And 2.6, continuing searching. k is k +1, k is less than or equal to k1,max+1, return to step 2.2.
Finally, the
Figure BDA0002554781380000087
And step 3: and (5) final calibration. And continuously searching the parameters under the optimal relative weight based on the parameters in the fine calibration and the predicted relative error.
And 3.1, determining iteration related parameters. k is a radical of20, give the highest number of searches k2,max＝50。
Figure BDA0002554781380000091
And 3.2, updating the weight matrix. As follows
Figure BDA0002554781380000092
And 3.3, iteratively calculating the unknown parameters by adopting the formula in the step 2.3. Given a threshold of 0.000001, β is iteratively updated until | | | Δ β | <.
And 3.4, calculating the relative error r by adopting the formula in the step 2.4.
3.5, if r < rbest，
Figure BDA0002554781380000093
3.6 if k < k2,maxThen returning to the step 3.2; otherwise
Recording the optimal value beta of the unknown parameter beta under the optimal relative weight at the momentbest。
Finally, the parameter to be estimated, which generates a nonlinear function between the light value of the chemiluminescent immunoassay reagent and the reagent concentration, is,
βbest＝[44601102,-44600703,0.00010025,1.01041577]T，
and obtaining the optimal relative error of the final calibration stage as follows:
rbest＝0.0214％
the optimal relative errors of the rough calibration, the fine calibration and the final calibration of the present embodiment are 16.4330%, 0.0229% and 0.0214%, respectively, as shown in table 2:
coarse calibration Fine calibration Final scaling
rbest 16.4330％ 0.0229％ 0.0214％
TABLE 2 Performance index in three stages
And 4, step 4: and calculating a fitting result. The method comprises the following specific steps: calculating fit values, estimating variances, calculating covariance matrices, calculating projection matrices, and the like.
4.1, calculating fitting value
f(x,βbest)＝[f(x1,βbest),…,f(xm,βbest)]T＝[398,14144,65247,468120,1390064,2696662]T
4.2, estimating the variance.
Figure BDA0002554781380000101
And 4.3, calculating a covariance matrix.
Figure BDA0002554781380000102
And 4.4, calculating a projection matrix.
Figure BDA0002554781380000103
And 5: and calculating the calibration precision. The method comprises the following specific steps: calculating quantile, calculating parameter precision radius and calculating fitting value precision radius as follows.
5.1, given significance level α ═ 0.05, quantile t1-α/2＝4.3027
t1-α/2＝4.3027
5.2, calculating the precision radius of the parameter as follows
Figure BDA0002554781380000104
Wherein s isiiIs the ith diagonal of S.
A parametric pipeline diagram of a specific embodiment of the present invention is shown in fig. 3.
5.3, calculate the radius of accuracy of the fit value, as follows
Figure BDA0002554781380000105
Wherein h isiiIs the ith diagonal of H.
A pipeline plot of fit values for a specific embodiment of the present invention is shown in fig. 4.
By way of example, it can be seen that:
(1) the calibration performance of the mild nonlinear calibration method based on the optimal power and the relative weight is gradually optimized along with the progress of the steps;
(2) the mild nonlinear calibration precision evaluation method based on the optimal power and the relative weight can provide a parameter pipeline and a fitting pipeline, and lays a foundation for realizing online calibration precision evaluation of a user.
It should be understood that the specific order or hierarchy of steps in the processes disclosed is an example of exemplary approaches. Based upon design preferences, it is understood that the specific order or hierarchy of steps in the processes may be rearranged without departing from the scope of the present disclosure. The accompanying method claims present elements of the various steps in a sample order, and are not intended to be limited to the specific order or hierarchy presented.
In the foregoing detailed description, various features are grouped together in a single embodiment for the purpose of streamlining the disclosure. This method of disclosure is not to be interpreted as reflecting an intention that the claimed embodiments of the subject matter require more features than are expressly recited in each claim. Rather, as the following claims reflect, invention lies in less than all features of a single disclosed embodiment. Thus, the following claims are hereby expressly incorporated into the detailed description, with each claim standing on its own as a separate preferred embodiment of the invention.
The previous description of the disclosed embodiments is provided to enable any person skilled in the art to make or use the present invention. To those skilled in the art; various modifications to these embodiments will be readily apparent, and the generic principles defined herein may be applied to other embodiments without departing from the spirit or scope of the disclosure. Thus, the present disclosure is not intended to be limited to the embodiments shown herein but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.
What has been described above includes examples of one or more embodiments. It is, of course, not possible to describe every conceivable combination of components or methodologies for purposes of describing the aforementioned embodiments, but one of ordinary skill in the art may recognize that many further combinations and permutations of various embodiments are possible. Accordingly, the embodiments described herein are intended to embrace all such alterations, modifications and variations that fall within the scope of the appended claims. Furthermore, to the extent that the term "includes" is used in either the detailed description or the claims, such term is intended to be inclusive in a manner similar to the term "comprising" as "comprising" is interpreted when employed as a transitional word in a claim. Furthermore, any use of the term "or" in the specification of the claims is intended to mean a "non-exclusive or".
The above-mentioned embodiments are intended to illustrate the objects, technical solutions and advantages of the present invention in further detail, and it should be understood that the above-mentioned embodiments are merely exemplary embodiments of the present invention, and are not intended to limit the scope of the present invention, and any modifications, equivalent substitutions, improvements and the like made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims (9)

1. A nonlinear scaling precision analysis method based on iterative optimal power is characterized by comprising the following steps:
acquiring a set number of reagent light values and a corresponding reagent concentration data set;
constructing a nonlinear function from the reagent light value to the reagent concentration according to the data set by a nonlinear scaling mode based on the iterative optimal power;
and carrying out precision analysis on the nonlinear function.
2. The method for non-linear scaled precision analysis based on iterative optimal powers of claim 1, wherein said constructing a non-linear function from reagent light value to reagent concentration by a non-linear scaling based on iterative optimal powers according to said data set comprises:
determining the nonlinear function model and setting parameters to be estimated;
determining an initial value of a parameter to be estimated and an initial value of a relative error of the nonlinear function model according to the data group;
iteratively calculating and updating the parameter to be estimated and the iterative correction relative error, and determining a primary optimal weight matrix of the parameter to be estimated;
through the initially selected optimal weight matrix, the parameters to be estimated and the iterative correction relative error are calculated and updated again in an iterative mode, and the optimal values of the parameters to be estimated under the optimal power and the optimal weight matrix are determined;
and constructing a nonlinear function from the reagent light value to the reagent concentration through the optimal value of the parameter to be estimated.
3. The method for analyzing nonlinear scaling precision based on iterative optimal power as claimed in claim 2, wherein said determining the nonlinear function model and setting the parameter to be estimated comprises:
the nonlinear function is modeled as
Figure FDA0002554781370000011
Wherein the parameter to be estimated is beta ═ a, b, c, d]T。
4. The method for analyzing nonlinear scaling precision based on iterative optimal power as claimed in claim 3, wherein said determining initial values of parameters to be estimated and initial values of relative errors of said nonlinear function model according to said data set comprises:
measurement data from m sets of concentration and light values
Figure FDA0002554781370000012
Wherein m is more than or equal to 4;
determining an initial value beta of an unknown parameter beta0＝[a0,b0,c0,d0]T(2)
Wherein the content of the first and second substances,
Figure FDA0002554781370000013
Figure FDA0002554781370000021
determining an initial value r of a relative error0：
Figure FDA0002554781370000022
。
5. The method for analyzing nonlinear scaling precision based on iterative optimal power as claimed in claim 4, wherein said iteratively calculating and updating the parameter to be estimated and iteratively correcting the relative error to determine the initial optimal weight matrix of the parameter to be estimated comprises:
determining an initial value k of the iteration number k to be 0;
determining iteration coefficients
Figure FDA0002554781370000023
Wherein the division number k of the power1,max＝50，ImIs a unit matrix, sigmabestInitially selecting an optimal weight matrix;
updating the weight matrix ∑
Figure FDA0002554781370000024
Wherein the weight matrix Σ is | y1|p,…,|ym|pA diagonal matrix being a diagonal element;
the given threshold value is 0.000001, and when | | | delta β | | < then, the parameter β to be estimated is updated
Figure FDA0002554781370000025
Wherein x ═ x1,…,xm]T,y＝[y1,…,ym]T,f(x,β)＝[f(x1,β),…,f(xm,β)]TJ is Jacobian matrix
Figure FDA0002554781370000031
Updating relative error
Figure FDA0002554781370000032
Wherein the content of the first and second substances,
Figure FDA0002554781370000033
when r < rbestWhen, specify
Figure FDA0002554781370000034
Updating the iteration number k ═ k +1 (9);
when k is less than or equal to k1,maxWhen +1, move to iterative computation and updateAnd starting the parameters to be estimated and iteratively correcting the relative error.
6. The method for analyzing nonlinear scaling precision based on iterative optimal power as claimed in claim 5, wherein the determining the optimal value of the parameter to be estimated under the optimal power and the optimal weight matrix by the initially selected optimal weight matrix, re-iterative computing and updating the parameter to be estimated and iterative correction of relative errors comprises:
determining the number of iterations k2Initial value k of2＝0；
Determining iteration coefficients
Figure FDA0002554781370000035
Wherein the division number k of the power2,max＝50；
Updating the weight matrix ∑2
Figure FDA0002554781370000036
Wherein the weight matrix Σ2So as to make
Figure FDA0002554781370000037
A diagonal matrix being a diagonal element;
the given threshold value is 0.000001, and when | | | delta β | | < then, the parameter β to be estimated is updated
Figure FDA0002554781370000041
Updating the relative error r;
when r < rbestWhen, specify
Figure FDA0002554781370000042
Wherein, betabestObtaining the optimal value of the parameter to be estimated under the optimal power and the optimal weight matrix;
updating iteration number k2＝k2+1 (14)；
When k is2＜k2,maxAnd then, transferring to the initial step of re-iterative calculation and updating the parameters to be estimated and iterative correction of relative errors.
7. The method according to claim 6, wherein the constructing a nonlinear function from the reagent light value to the reagent concentration by the optimal value of the parameter to be estimated comprises:
determining the nonlinear function as f (x, beta) according to the optimal value of the parameter to be estimatedbest)。
8. The method according to claim 7, wherein the performing precision analysis on the nonlinear function specifically includes:
determining fitting values
f(x,βbest)＝[f(x1,βbest),…,f(xm,βbest)]T(15)；
Determining variance
Figure FDA0002554781370000043
Determining a covariance matrix
S＝(JTJ)-1(17)；
Determining a projection matrix
H＝JSJT(18)；
Given a level of significance a, a quantile t is determined from the level of significance a1-α/2
Figure FDA0002554781370000044
Wherein f (x, m-4) is a density function of a t-distribution with a degree of freedom of m-4;
estimating parameter accuracy radius
Figure FDA0002554781370000051
Wherein s isiiIs the ith diagonal of S;
estimating the radius of precision of the fit value
Figure FDA0002554781370000052
Wherein h isiiIs the ith diagonal of H.
9. The method for nonlinear scaled precision analysis based on iterative optimal powers of claim 8 wherein the given significance level α satisfies α e [0.01,0.05 ].
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