CN111784193A - Product performance consistency inspection method based on normal distribution - Google Patents
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Abstract
The invention provides a product performance consistency testing method based on normal distribution, which comprises the following steps: acquiring performance parameters of a plurality of products; selecting two groups of measurement data samples from the performance parameters, wherein the two groups of samples are subjected to normal distribution; respectively estimating normal distribution parameters of the two groups of measurement data samples according to the two groups of samples; further obtaining the intersection point of the normal distribution probability density function curves of the two groups of samples; then obtaining the area of the overlapping part of the probability density function of the two groups of samples and the area surrounded by the abscissa axis as consistency measurement; finally, the two groups of samples are compared with a given significance level or confidence level to determine whether the two groups of samples have consistency at the given confidence level. The consistency test method is different from a classical consistency test method, does not need to construct statistics, defines consistency measurement by starting from the concept and the mathematical meaning of a probability density function, and avoids the difficulty of constructing sufficient statistics.
Description
Technical Field
The invention relates to the technical field of product performance consistency inspection methods, in particular to a normal distribution-based product performance consistency inspection method.
Background
The normal distribution is a distribution form most commonly used in the engineering field, and occurs in many fields, such as thermal noise current in semiconductor devices, weight of bagged cement, filling amount of liquid rocket propellant, temperature in constant-temperature storage containers in industrial production, and the like, and measurement data thereof are subject to the normal distribution. When the physical quantities are measured in multiple batches, two or more groups of different data are obtained, and whether the two or more groups of data are subjected to the same normal distribution needs to be judged so as to analyze the consistency of the measured data and judge the stability of the production process and technical means, such as the stability of temperature control in a constant-temperature storage container, the stability of a bagged cement production line and the like.
In engineering practice, two different sets of measurement data which are subject to normal distribution are obtained, such as the magnitude of thermal noise current in two batches of semiconductor devices, the weight of two batches of bagged cement produced in different time periods, the temperature in a constant-temperature storage container under two power-on starts and the like, theoretical analysis and actual measurement data show that the two sets of measurement data are both subject to normal distribution, but whether the two sets of data are subject to the same normal distribution, namely whether normal distribution parameters are the same or not, further analysis is needed, and an analysis conclusion can be used for judging the stability of the production process. For two sets of samples that obey a normal distribution (referred to as measured data in the aforementioned practice), a consistency test is performed on the two sets of samples to determine whether the two sets of samples obey the same distribution at a given confidence level or significance level. The normal distribution consistency test method adopts a classical mean-variance test method, firstly, the F distribution is used for carrying out consistency test on the variances of two groups of data, if the variance of two groups of data passes the test, the t distribution is continuously used for carrying out the consistency test on the means of two groups of data, if the variance of two groups of data passes the test, the two groups of data are considered to be obeyed by the same normal distribution, and if any parameter of the variance and the mean does not pass the consistency test, the two groups of normal distribution data are considered to be not obeyed by the same normal distribution. This method requires the construction of test statistics that are subject to F-distribution and t-distribution, respectively, to be implemented.
Disclosure of Invention
The invention aims to provide a method for testing the consistency of product performance based on normal distribution, which solves the problems.
The invention provides a product performance consistency testing method based on normal distribution, which comprises the following steps:
acquiring performance parameters of a plurality of products; the performance parameters are extracted from different batches of products or extracted from different moments of the products;
selecting two groups of measurement data samples from the performance parameters, wherein the two groups of measurement data samples are subjected to normal distribution;
respectively estimating the mean value and the mean square error of normal distribution of two groups of measurement data samples by utilizing a maximum likelihood estimation method according to the normal distribution probability density function;
obtaining the intersection point of the normal distribution probability density function curves of the two groups of measurement data samples according to the mean value and the mean square error of the normal distribution of the two groups of measurement data samples;
according to the intersection point of the probability density function curves of the two groups of measurement data samples, the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis is obtained;
and judging whether the two groups of measurement data samples have consistency or not according to the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the area surrounded by the abscissa axis and a preset significance level or confidence coefficient.
Further, the estimating the mean and mean square error of the normal distribution of the two groups of measurement data samples by using a maximum likelihood estimation method according to the normal distribution probability density function includes:
the normal distribution probability density function is:
wherein mu is the mean value of normal distribution, and sigma is the mean square error of normal distribution;
according to the maximum likelihood estimation method, the estimated values of two groups of data normal distribution parameters are respectively
Wherein n is the number of samples in the sample set X, and m is the number of samples in the sample set Y;
then the normal distribution probability density function obeyed by the two groups of data is respectively:
wherein the content of the first and second substances,
is the mean of a normal distribution under the first set of samples,
is the mean square error of a normal distribution under the first set of samples,
is the mean of a normal distribution for the second set of samples,
is the mean square error of the normal distribution for the second set of samples.
Further, the obtaining an intersection point of the normal distribution probability density function curves of the two groups of measurement data samples according to the normal distribution probability density functions of the two groups of measurement data samples includes:
making the two probability density functions equal, i.e. fx(r)＝fy(r) then
Wherein the first set of measurement data samples is X ═ { X1,x2,…,xnY ═ Y in the second set of measurement data samples1,y2,…,ymN is the number of the first group of samples, and m is the number of the second group of samples; f. ofx(r) probability density function corresponding to the first set of samples, fy(r) represents a probability density function corresponding to the second set of samples, r representing a variable;
namely, it is
An equation can be obtained
When in use
When there is a crossing point
When in use
When, there are two intersections, respectively
Wherein
Further, according to the intersection point of the probability density function curves of the two groups of measurement data samples,obtaining the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis, wherein the area comprises the following steps: when in use
When there is a crossing point
If it is not
The calculation formula of the area of the overlapped part is
If it is not
The calculation formula of the area of the overlapped part is
If it is not
The two probability density function curves are completely overlapped, and the area of the overlapped part
cr＝1
Wherein c isrIndicating the size of the consistency.
Further, the obtaining of the area of the overlapping portion of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis according to the intersection point of the probability density function curves of the two groups of measurement data samples further includes:
when in use
When, there are two intersections, respectively
Order to
The calculation of the area of the overlapped part can be divided into three parts
When in use
When, let f1(r)＝min(fx(r),fy(r)) to obtain
When in use
When, let f2(r)＝min(fx(r),fy(r)) to obtain
When in use
When, let f3(r)＝min(fx(r),fy(r)) to obtain
The area of the overlapped part of the two normally distributed probability density function curves and the area surrounded by the abscissa axis is
cr＝c3+c2+c1。
Further, the determining whether the two groups of measurement data samples have consistency according to the area of the overlapping portion of the normal distribution probability density function curves of the two groups of measurement data samples and the area surrounded by the abscissa axis and a preset significance level or confidence degree includes:
if the preset significance level is alpha, the judgment rule of whether the two groups of samples are consistent is
If c isr1- α, the two groups of samples were considered consistent at a significance level of α;
if c isr< 1- α, the two groups of samples were considered to be inconsistent at a significance level of α.
The invention has the technical effects that:
(1) the invention provides a product performance consistency test method based on normal distribution, which utilizes the area of the overlapping part of two groups of sample experience probability density functions and the area surrounded by the abscissa axis as the consistency measurement of two samples and provides a corresponding calculation method and a judgment criterion. Since the probability density function and the cumulative distribution function are two basic concepts in statistics, the probability density function has an integral of 1 throughout the domain of the random variable. Under the condition of a given random variable interval, the area enclosed by the probability density function and the interval is the cumulative distribution function under the interval, namely the probability. According to the concept and the meaning of the probability density function, if the overlapping area of the two normal distribution probability density function curves and the area surrounded by the abscissa axis is larger, the approximation degree of the two probability density functions is larger, and the consistency of the two groups of corresponding samples is better. The innovation point of the method is that the area of the overlapping part of the two groups of normal distribution sample empirical probability density function curves and the area surrounded by the abscissa axis is defined as the measure of the consistency of the two groups of normal distribution samples, a new method for judging the consistency of the two groups of normal distribution samples is provided, the calculation process of the consistency is provided, the basic problem in the consistency test of the two groups of normal distribution samples is solved, and a feasible method is provided for the consistency test of the normal distribution.
(2) The product performance consistency testing method based on normal distribution provided by the invention has the advantages of clear mathematical concept, clear calculation steps, reasonability and feasibility. Moreover, the product performance consistency test method based on normal distribution provided by the invention does not need to construct sample-based statistics and even know the distribution form of the statistics, but starts with the concept and mathematical meaning of the probability density function, defines consistency measurement, and provides a feasible calculation method for solving the consistency test problem of two groups of normal distribution samples.
The above and other aspects of the present invention will become apparent from the following description, which refers in particular to various embodiments of the method for checking the consistency of product properties based on normal distribution according to the present invention.
Drawings
FIG. 1 is a schematic flow diagram of a preferred embodiment method of the present invention;
fig. 2 is a schematic area diagram of an overlapping portion of two normal distribution probability density function curves and a region surrounded by an abscissa axis, and an intersection point of the two probability density function curves is given, and a diagonally shaded portion in the diagram is the overlapping portion.
Detailed Description
The accompanying drawings, which are incorporated in and constitute a part of this application, illustrate embodiments of the invention and, together with the description, serve to explain the invention and not to limit the invention.
Referring to fig. 1, the method for checking the consistency of product performance based on normal distribution includes the following steps:
step one, acquiring performance parameters of a plurality of products; the performance parameters are extracted from different batches of products or extracted from different moments of the products; selecting two groups of measurement data samples from the performance parameters, wherein the two groups of measurement data samples are subjected to normal distribution;
secondly, respectively estimating the mean value and the mean square error of normal distribution of two groups of measurement data samples by using a maximum likelihood estimation method according to the probability density function of the normal distribution;
step three, obtaining the intersection point of the normal distribution probability density function curves of the two groups of measurement data samples according to the mean value and the mean square error of the normal distribution of the two groups of measurement data samples;
obtaining the area (shown in figure 2) of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis according to the intersection point of the probability density function curves of the two groups of measurement data samples, and taking the area as the measure of the consistency of the two groups of samples;
and fifthly, judging whether the two groups of measurement data samples have consistency or not according to the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the area surrounded by the abscissa axis and a preset significance level or confidence coefficient.
Since the probability density function and the cumulative distribution function are two basic concepts in statistics, the probability density function has an integral of 1 throughout the domain of the random variable. Under the condition of a given random variable interval, the area enclosed by the probability density function and the interval is the cumulative distribution function under the interval, namely the probability. According to the concept and the meaning of the probability density function, if the overlapping area of the two normal distribution probability density function curves and the area surrounded by the abscissa axis is larger, the approximation degree of the two probability density functions is larger, and the consistency of the two groups of corresponding samples is better. The innovation point of the method is that the area of the overlapping part of the two groups of normal distribution sample empirical probability density function curves and the area surrounded by the abscissa axis is defined as the measure of the consistency of the two groups of normal distribution samples, a new method for judging the consistency of the two groups of normal distribution samples is provided, the calculation process of the consistency is provided, the basic problem in the consistency test of the two groups of normal distribution samples is solved, and a feasible method is provided for the consistency test of the normal distribution.
Specifically, the second step further comprises:
the normal distribution probability density function is:
wherein mu is the mean value of normal distribution, and sigma is the mean square error of normal distribution;
according to the maximum likelihood estimation method, the estimated values of two groups of data normal distribution parameters are respectively
Wherein n is the number of samples in the sample set X, and m is the number of samples in the sample set Y;
then the normal distribution probability density function obeyed by the two groups of data is respectively:
wherein the content of the first and second substances,
is the mean of a normal distribution under the first set of samples
Is the mean square error of a normal distribution under the first set of samples,
is the mean of a normal distribution under the first set of samples
Is the mean square error of the normal distribution under the first set of samples.
Specifically, the third step further comprises:
making the two probability density functions equal, i.e. fx(r)＝fy(r) then
Wherein the first set of measurement data samples is X ═ { X1,x2,…,xnY ═ Y in the second set of measurement data samples1,y2,…,ymN is the number of the first group of samples, and m is the number of the second group of samples; f. ofx(r) probability density function corresponding to the first set of samples, fy(r) represents a probability density function corresponding to the second set of samples, r representing a variable;
namely, it is
An equation can be obtained
When in use
When there is a crossing point
When in use
When, there are two intersections, respectively
Wherein
Specifically, the fourth step further comprises:
when in use
When there is a crossing point
If it is not
The calculation formula of the area of the overlapped part is
If it is not
The calculation formula of the area of the overlapped part is
If it is not
The two probability density function curves are completely overlapped, and the area of the overlapped part
cr＝1
Wherein c isrIndicating the size of the consistency.
When in use
When, there are two intersections, respectively
Order to
The calculation of the area of the overlapped part can be divided into three parts
When in use
When, let f1(r)＝min(fx(r),fy(r)) to obtain
When in use
When, let f2(r)＝min(fx(r),fy(r)) to obtain
When in use
When, let f3(r)＝min(fx(r),fy(r)) to obtain
The area of the overlapped part of the two normally distributed probability density function curves and the area surrounded by the abscissa axis is
cr＝c3+c2+c1。
Specifically, the fifth step further comprises:
if the preset significance level is alpha, the judgment rule of whether the two groups of samples are consistent is
If c isr1- α, the two groups of samples were considered consistent at a significance level of α;
if c isr< 1- α, the two groups of samples were considered to be inconsistent at a significance level of α.
In order to better explain the technical scheme provided by the invention, the following description is made in conjunction with specific examples.
(1) Assuming that two different batches of semiconductor devices are produced in a factory, the thermal noise current of the two batches of devices is measured, and two sets of measurement data samples are obtained, wherein X is ═ { X ═ respectively1,x2,…,xn}，Y＝{y1,y2,…,ymAnd the two sets of measurement data are subjected to consistency test to judge whether the thermal noise performance of the semiconductor products in two batches is the same or not. Where the data set X has a total of 18 samples, i.e. n 18. The data set Y has 15 samples, i.e., m 15.
X＝{4.61 -4.63 0.19 -0.22 1.46 1.44 -2.09 0.41 0.01 2.38 3.78 3.83 -2.09 0.73 -3.14 -2.84 0.48 5.10}，
Y＝{-2.08 2.49 0.10 5.47 -3.36 1.13 3.21 5.40 7.18 1.34 -4.97 -1.97 -3.25 10.40 -1.46}。
(2) According to the normal distribution probability density function, respectively estimating normal distribution parameter parameters under two groups of data by utilizing a maximum likelihood estimation method
And
. For the first set of data X, the distribution parameter estimate is
For the second set of data Y, the distribution parameter estimate is
Wherein n is the number of samples in the sample set X, and m is the number of samples in the sample set Y.
Then the normal distribution probability density function obeyed by the two groups of data is respectively:
(3) after the normal distribution probability density functions of the two groups of data are obtained, the intersection point of the two probability density function curves is calculated. Let fx(r)＝fy(r) due to
The equation can be obtained
Can obtain
Then
(4) And calculating the area of the overlapped part of the two normal distribution probability density function curves and the region surrounded by the abscissa axis according to the intersection point characteristics of the two probability density function curves, and taking the area as the measure of the consistency of the two groups of samples. Order to
Due to the fact that
The calculation of the area of the overlapped part can be divided into three parts
1) When in use
When, let f1(r)＝min(fx(r),fy(r))＝fx(r) obtaining
2) When in use
When, let f2(r)＝min(fx(r),fy(r))＝fy(r) obtaining
3) When in use
When, let f3(r)＝min(fx(r),fy(r))＝fx(r) obtaining
The area of the overlapped part of the two normally distributed probability density function curves and the area surrounded by the abscissa axis is
cr＝c3+c2+c1＝0.7688
(5) Judging whether the two groups of data samples have consistency or not according to a given significance level or confidence;
assuming a significance level of α of 0.2 and a confidence level of 1- α of 0.8, c is assignedr< 1- α ═ 0.8, i.e., the two sets of samples X and Y were considered to have no agreement at significance level 0.2.
The product performance consistency testing method based on normal distribution provided by the invention has clear mathematical concept, clear calculation steps, and is reasonable and feasible. Moreover, the product performance consistency test method based on normal distribution provided by the invention does not need to construct sample-based statistics and even know the distribution form of the statistics, but starts with the concept and mathematical meaning of the probability density function, defines consistency measurement, and provides a feasible calculation method for solving the consistency test problem of two groups of normal distribution samples.
It will be clear to a person skilled in the art that the scope of the present invention is not limited to the examples discussed in the foregoing, but that several amendments and modifications thereof are possible without deviating from the scope of the present invention as defined in the attached claims. While the invention has been illustrated and described in detail in the drawings and the description, such illustration and description are to be considered illustrative or exemplary and not restrictive. The invention is not limited to the disclosed embodiments.
Variations to the disclosed embodiments can be understood and effected by those skilled in the art in practicing the claimed invention, from a study of the drawings, the disclosure, and the appended claims. In the claims, the term "comprising" does not exclude other steps or elements, and the indefinite article "a" or "an" does not exclude a plurality. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. Any reference signs in the claims shall not be construed as limiting the scope of the invention.
Claims (6)
1. The method for checking the consistency of product performance based on normal distribution is characterized by comprising the following steps:
acquiring performance parameters of a plurality of products; the performance parameters are extracted from different batches of products or extracted from different moments of the products;
selecting two groups of measurement data samples from the performance parameters, wherein the two groups of measurement data samples are subjected to normal distribution;
respectively estimating the mean value and the mean square error of normal distribution of two groups of measurement data samples by utilizing a maximum likelihood estimation method according to the normal distribution probability density function;
obtaining the intersection point of the normal distribution probability density function curves of the two groups of measurement data samples according to the mean value and the mean square error of the normal distribution of the two groups of measurement data samples;
according to the intersection point of the probability density function curves of the two groups of measurement data samples, the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis is obtained;
and judging whether the two groups of measurement data samples have consistency or not according to the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the area surrounded by the abscissa axis and a preset significance level or confidence coefficient.
2. The method for checking the product performance consistency based on normal distribution according to claim 1, wherein the step of respectively estimating the mean value and the mean square error of the normal distribution of two groups of measurement data samples by using a maximum likelihood estimation method according to the probability density function of the normal distribution comprises the following steps:
the normal distribution probability density function is:
wherein mu is the mean value of normal distribution, and sigma is the mean square error of normal distribution;
according to the maximum likelihood estimation method, the estimated values of two groups of data normal distribution parameters are respectively
Wherein n is the number of samples in the sample set X, and m is the number of samples in the sample set Y;
then the normal distribution probability density function obeyed by the two groups of data is respectively:
wherein the content of the first and second substances,
is the mean of a normal distribution under the first set of samples,
is the mean square error of a normal distribution under the first set of samples,
is the mean of a normal distribution for the second set of samples,
is the mean square error of the normal distribution for the second set of samples.
3. The method for checking the product performance consistency based on normal distribution according to claim 1, wherein obtaining the intersection point of the normal distribution probability density function curves of the two groups of measurement data samples according to the normal distribution probability density functions of the two groups of measurement data samples comprises:
making the two probability density functions equal, i.e. fx(r)＝fy(r) then
Wherein the first set of measurement data samples is X ═ { X1,x2,…,xnY ═ Y in the second set of measurement data samples1,y2,…,ymN is the number of the first group of samples, and m is the number of the second group of samples; f. ofx(r) probability density function corresponding to the first set of samples, fy(r) represents a probability density function corresponding to the second set of samples, r representing a variable;
namely, it is
An equation can be obtained
When in use
When there is a crossing point
When in use
When, there are two intersections, respectively
Wherein
4. The method for checking the product performance consistency based on normal distribution according to claim 3, wherein the step of obtaining the area of the overlapping part of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis according to the intersection point of the probability density function curves of the two groups of measurement data samples comprises the following steps:
when in use
When there is a crossing point
If it is not
The calculation formula of the area of the overlapped part is
If it is not
The calculation formula of the area of the overlapped part is
If it is not
The two probability density function curves are completely overlapped, and the area of the overlapped part
cr＝1
Wherein c isrIndicating the size of the consistency.
5. The method for checking product performance consistency according to claim 3, wherein the area of the overlapping portion of the normal distribution probability density function curves of the two groups of measurement data samples and the region surrounded by the abscissa axis is obtained according to the intersection point of the probability density function curves of the two groups of measurement data samples, and the method further comprises:
when in use
When, there are two intersections, respectively
Order to
The calculation of the area of the overlapped part can be divided into three parts
When in use
When, let f1(r)＝min(fx(r),fy(r)) to obtain
When in use
When, let f2(r)＝min(fx(r),fy(r)) to obtain
When in use
When, let f3(r)＝min(fx(r),fy(r)) to obtain
The area of the overlapped part of the two normally distributed probability density function curves and the area surrounded by the abscissa axis is
cr＝c3+c2+c1。
6. The method for checking the product performance consistency based on normal distribution according to claim 1, wherein the step of judging whether the two groups of measurement data samples have consistency according to the area of the overlapping part of the normal distribution probability density function curve of the two groups of measurement data samples and the area surrounded by the abscissa axis and a preset significance level or confidence level comprises the following steps:
if the preset significance level is alpha, the judgment rule of whether the two groups of samples are consistent is
If c isr1- α, the two groups of samples were considered consistent at a significance level of α;
if c isr< 1- α, the two groups of samples were considered to be inconsistent at a significance level of α.
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