CN106021097A - Software reliability index interval estimation method based on test characteristics - Google Patents
Software reliability index interval estimation method based on test characteristics Download PDFInfo
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Abstract
The invention relates to a software reliability index interval estimation method based on test characteristics. The software reliability index interval estimation method comprises the following steps: analyzing the test characteristics from multiple perspectives, designing a corresponding criterion to measure the test characteristics to obtain a test characteristic measurement result; according to a relationship between the test characteristic measurement result and a reliability index, constructing a regression model of the reliability index and test characteristic measurement; and according to the regression model of the reliability index and test characteristic measurement, carrying out interval estimation on the reliability index. The method for constructing the regression model of the reliability index and test characteristic measurement is used for searching the relationship between the reliability index and the test characteristics, has a mathematical theoretical basis, and is scientific and strict. Test characteristic data generated in a practical test is used for assisting in reliability index interval estimation, and the credibility of the reliability index interval estimation is improved.
Description
Technical field
The present invention relates to software reliability evaluation technical field, particularly relate to a kind of software reliability based on test feature
Index method of interval estimation.
Background technology
Software is ubiquitous in our life, and in industry-by-industry every field, software all plays very important
Role.In Safety-Critical System, software failure can cause serious, fatal consequence.The imprevision of software defect
Property makes us not know when software can lose efficacy and how to lose efficacy.Though the reliability of hardware system is in recent years
Develop, but the reliability of software system the most still can not reach our expectation.Exactly because software reliability
Importance and this field there is a lot of an open question, software reliability is of increased attention.
Software reliability, as a software metrics index, occurs the probability lost efficacy to summarize the reliable of software by software
Operation degree.The software reliability recommended practice promulgated according to IEEE, software reliability refers to the " condition in regulation
Descend and in the time of regulation, software does not cause ability or the probability of thrashing ".Thrashing can cause all multi-risk Systems,
Such as economic loss, casualties, so Safety-Critical System requires that reliability reaches certain level, potential to guarantee
Harm software use in will not occur.In industrial circle, various software exploitation standard is all to Software failure probability
It is strict with, particularly with security critical software.
That in software reliability field, achievement is most, of greatest concern is software reliability model (Software
Reliability Model, SRM) research.The software that software reliability model collects when being intended to utilize software test
Fail data, by the method simulation softward failure procedure of modeling, thus provides software reliability assurance value.Software can
It is current software reliability analysis and evaluate the strongest instrument by property model, provides foundation for improving software quality.
The main target of Software Reliability Modeling is one fail data of the matching theoretical distribution about the time, is distributed according to this
The reliability of assessment software and design a rule that can determine software test dwell time.At existing numerous moulds
In type, NHPP model is widely used by software reliability research person.Owing to average failure number function is by NHPP model
Directly give, so the calculating at the average failure number of certain feature time is the simplest.Unknown parameter in model is permissible
Maximum-likelihood estimation or least-squares estimation is used to obtain.
Software test feature is some characterization about test process and test result, and e.g., reliability is based on survey
Test result analyzes the secure status of software, generally uses multiple reliability indexs to be analyzed.Due to test process
Often contain many features, in software reliability evaluation, how to comprehensively utilize these information in test process
Become main contents of the present invention.
Statistics is thought, interval estimation can portray the precision of point estimation, is a kind of important Statistical Inference.Cause
This needs software reliability model unknown parameter is constructed confidence interval, to more accurately descriptive model parameter estimation
Extent of deviation between value and model parameter actual value.
At present both at home and abroad the research method to the interval estimation of reliability model parameter be all directly from reliability model and
Fail data itself considers, does not the most consider test feature factor.
Summary of the invention
In view of above-mentioned analysis, it is desirable to provide a kind of software reliability growth model interval estimation based on test feature
Method, in order to solve the problem that existing method of interval estimation exists.
The purpose of the present invention is mainly achieved through the following technical solutions:
The invention provides a kind of software reliability growth model method of interval estimation based on test feature, including:
From multiple angles, test feature is analyzed, designs corresponding criterion and test feature is measured, obtain
Test feature measurement results；
According to the relation of test feature measurement results Yu reliability index, build reliability index and test feature degree
The regression model of amount；
According to the regression model of reliability index with test feature tolerance, reliability index is carried out interval estimation.
Further, described test feature measurement results at least includes:
The code coverage of code coverage, demand coverage rate, complexity weighting, the test of complexity density weighting are used
Number of cases, test case number of variations, newly-increased failure number.
Further, one or more during described reliability index at least includes following index:
Crash rate, reliability, mean time to failure, mean time between failures, crash rate rate of change.
Further, crash rate: crash rate pays close attention to software system failure probability within the unit interval, is defined asWherein R (t) is reliability function, and i.e. before t, system runs the probability occurring without inefficacy, f (t)
For failure dense function,
Reliability: the attention location system degree of reliability within a period of time, i.e. R (t)；
Mean time to failure: software starts first average time lost efficacy occur, if R (t) is continuous from operation
Type function, then
Mean time between failures: pay close attention to the Mean Time Between Replacement that software system is in operation between twice inefficacy of appearance；
Crash rate rate of change: focus on the situation over time that lost efficacy in software running process, pass through
Calculate, λ (t1) and λ (t2) it is respectively t1And t2The crash rate in moment.
Further, the process building the regression model that reliability index is measured with test feature specifically includes:
Calculate reliability index；
Determine the primitive form of regression model；
Inspection independent variable is the most notable on the impact of reliability index, retains and affects significant independent variable, described independent variable
It is test feature measurement results.
Further, the impact the most significantly method of reliability index is included by described inspection independent variable:
If former regression equation is y=β0+β1x1+…+βi-1xi-1+βixi+βi+1xi+1+…+βpxp, remove variable xiAfter
The new regression model obtained is y=β0+β1x1+…+βi-1xi-1+βi+1xi+1+…+βpxp, new regression model is called former
Regression model subtract model, former regression model is called full model, wherein, βiIt is the coefficient of regression model independent variable, x table
Show that independent variable, y represent that dependent variable, p represent independent variable number；
Calculate the regression sum of square U of full model respectively and subtract the regression sum of square U of modeli', thus obtain
Ui=U-Ui′；After obtaining the sum of squares of partial regression that each independent variable is corresponding, relatively each factor is to whole regression effect
The size of contribution, will contribute little person to reject；
If the multiple correlation coefficient of full model square is R2, subtract model multiple correlation coefficient square beDefinition
Null hypothesisAs null hypothesis H0For true time, statistic of test isFor given level of significance α, sample value calculate Fi's
Value, if Fi≥F1-α(1, n-m-1), then refuse H0, i.e.Significantly being not zero, this illustrates xiY is had a significant impact,
X should be added in subtracting modeliMake full model；If Fi＜ F1-α(1, n-m-1), then accept H0, i.e.Aobvious
Work is zero, and this illustrates xiNot notable on y impact, x should be rejected in full modeli, make and subtract model；N represents
Sample size number, m represents independent variable number.
Further, the process that reliability index carries out interval estimation specifically includes:
Assume to represent reliability index, x with y1～xpRepresent test feature measurement results respectively, it is assumed that the recurrence obtained
Model is y=β0+β1x1+β2x2+β3x3+...+βpxp+ ε, if it is assumed that different phase reliability index estimated value phase
The most independent, then deviation ε Normal Distribution；
In order to utilize regression equation to forecast, providing x1,x2,…,xpA class value x01,x02,…,x0pTime, note
x0=(1, x01,x02,…,x0p) ', obtains y0=x '0β+ε0, E (ε0)=0, Var (ε0)=σ2And y0Predictive valueE (X) represents the average of X, and Var (X) represents the variance of X, σ2For mark
The variance of quasi normal distribution, x '0Represent x0Transposition；
Can be derived by further by above characterThen y0Confidence level be 1-α
Confidence interval beT represents t-distribution, n-p-1
For the degree of freedom of t-distribution, X for organize independent variable (x more1,x2,...xp) matrix that forms, X ' represents the transposition of X；
In order to simplify mathematical model, it is assumed herein that each stage reliability index estimated value and reliability index previous stage
The degree of association of estimated value is ρ, if identifying bias vector (ε with Σ1,ε2,...,εn), wherein n represents test phase number,
Then have
If the above different phase separate hypothesis of reliability index estimated value meets, then ρ=0 in matrix, now
Represent with WUnderstanding W is positive definite matrix, there is matrix Z and makes Z2=W；
On equation Y=XB+ Σ both sides with taking advantage of Z-1Obtain Z-1Y=Z-1XB+Z-1Σ, wherein Y represents reliability index
Vector (y1,y2,...,yn) ', X representsB represents β vector, and Σ represents bias vector,
P represents independent variable number；After converting, Var (Z-1Σ)=Z-2Var (Σ)=W-1σ2W=σ2I, if therefore
Make Y*=Z-1Y, X*=Z-1X, Σ*=Z-1Σ, available Y*=X*B+Σ*, Y*,Σ*Normal Distribution,
Available Y*Confidence interval, and then according to Y*=Z-1Y can get the confidence interval of Y.
The present invention has the beneficial effect that:
Reliability index and test is found with the method that test feature measures regression model by building reliability index
Relation between feature, possesses pure mathematics basis, more scientific and precise.
The test feature data produced when utilizing reality test assist the interval estimation carrying out reliability index, improve
The credibility of Interval Estimator of The Reliability Indexes.
Other features and advantages of the present invention will illustrate in the following description, and, becoming from description of part
Obtain it is clear that or understand by implementing the present invention.The purpose of the present invention and other advantages can be by being write
Structure specifically noted in description, claims and accompanying drawing realizes and obtains.
Accompanying drawing explanation
Fig. 1 is the schematic flow sheet of method described in the embodiment of the present invention.
Detailed description of the invention
Specifically describing the preferred embodiments of the present invention below in conjunction with the accompanying drawings, wherein, accompanying drawing constitutes the application part,
And together with embodiments of the present invention for explaining the principle of the present invention.
As it is shown in figure 1, Fig. 1 is the schematic flow sheet of method described in the embodiment of the present invention, specifically can include walking as follows
Rapid:
Step 101: be analyzed test feature from multiple angles, measure test feature, obtains including all
Measurement results such as the following aspects:
Code coverage, demand coverage rate, the code coverage of consideration complexity density, the survey of consideration complexity density
Try out and between number of cases mesh, test phase, between test case number of variations and test phase, increase failure number newly.
Step 102: calculate reliability index, and according to reliability index and test feature measurement results, structure can
Regression model by property index with test feature tolerance.That is, intend using Maximum-likelihood estimation or least-squares estimation to soft
The unknown parameter of part reliability model (such as Schneidewind) carries out point estimation, thus by software reliability model meter
Calculate the estimation of corresponding reliability index, then use the method for recurrence to measure with test feature to describe reliability index
Between relation, set up corresponding regression model.
The reliability index that the present invention pays close attention to includes: crash rate, reliability, mean time to failure (MTTF),
Mean time between failures (MTBF) and 5 reliability indexs of crash rate rate of change.Crash rate reflection software is in unit
Failure probability in time, reliability reflect that software failure probability within a period of time, reliability growth reflection are reliable
Property at the growth of a period of time, the reflection crash rate change within a period of time of crash rate rate of change, MTTF and MTBF
The time interval attribute that reflection was lost efficacy.Each reliability index has clear and definite computing formula, and its core is to estimate model
Parameter, once obtain corresponding model parameter estimation, it is possible to direct basis formula estimates corresponding software can
By property index.
5 reliability indexs be specifically defined and computational methods are as follows:
1) crash rate: crash rate pays close attention to software system failure probability within the unit interval, generally represents with λ.
Crash rate is the most relevant with time t, and crash rate changes in time, the change of reflection reliability.General definable
ForWherein R (t) is reliability function, and i.e. before t, system runs the probability occurring without inefficacy,
F (t) is failure dense function,
2) reliability: the attention location system degree of reliability within a period of time, i.e. R (t).If t is beyond given
The time range of multistage test data, the most referred to as predicting reliability, the most referred to as test phase reliability.This project
Because paying close attention to the reliability status analysis not carrying out section plane test, the reliability in Main Analysis test phase.
3) mean time to failure: software starts first average time lost efficacy occur from operation, is abbreviated as MTTF,
Also may be interpreted as the expected time that software system no-failure runs.If R (t) is continuous function, then
4) mean time between failures: although many times losing efficacy occurs in running software, but still can keep certain
Duty.Mean time between failures pays close attention to the equispaced that software system is in operation between twice inefficacy of appearance
Time, it is expressed as MTBF.In practice, crash rate, MTTF and MTBF both can be directly by fail data degree of carrying out
Amount, it is also possible to estimated by selected reliability model.
5) crash rate rate of change: focus on the situation over time that lost efficacy in software running process, mathematically come
Say the derivative corresponding to λ (t).Practice can be passed throughCalculate, λ (t1) and λ (t2) respectively
For t1And t2The crash rate in moment..
From definition above it can be seen that there is certain derivation relation between reliability index, if obtaining reliable
Degree, then mean time to failure can directly be tried to achieve by integration.Thus obtain one of them reliability index is put
After letter interval estimation, the confidence interval of other indexs also can be obtained by corresponding derivation relation.
Reliability index, x is represented with y1～xpRepresent that such as code coverage, demand coverage rate, complexity add respectively
The test case number of the code coverage of power, complexity density weighting, test case number of variations, newly-increased failure number etc.
Test feature factor.Firstly the need of determining which kind of curve to describe y and x withi(i=1 ..., relation p).If assume y with
xiBetween be linear relationship, then the regression model obtained is multiple linear regression model, and primitive form is
Y=β0+β1x1+β2x2+β3x3+...+βpxp+ε；If using exponential curve to describe y and x1Relation, and use
Straight line describes y and other xi(i=2 ..., relation p), then the regression model primitive form obtained isThis shows with different curves describe y with
xi(i=1 ..., relation p), different regression models will be obtained.In order to solve this problem, we to investigate certainly
Variable xiWith the domain knowledge background of y, if according to domain background still cannot determine use which kind of curve x is describedi
With y, then can be according to y about xiScatterplot carry out the curve type that selected shape is close.
After determining the primitive form of regression model, need to check independent variable the most notable on the impact of y, retain impact
Significantly independent variable, the method solving this problem can be sum of squares of partial regression or partial F test.
1) sum of squares of partial regression
Regression sum of square U is the contribution to the total variance of y of all independent variables.If rejecting a variable, then return flat
Side and will reduce, the contribution of numerical value this variable of the biggest explanation of minimizing is the biggest.We are eliminating independent variable xiRear time
Return the numerical value U that quadratic sum U is reducediIt is referred to as variable xiSum of squares of partial regression.
If former regression equation is y=β0+β1x1+…+βi-1xi-1+βixi+βi+1xi+1+…+βpxp, remove variable xiAfter
The new regression model obtained is y=β0+β1x1+…+βi-1xi-1+βi+1xi+1+…+βpxp, new regression model is called former
Regression model subtract model, former regression model is called full model.Wherein, βiIt is the coefficient of regression model independent variable, i=1
To 6, x represents that independent variable, y represent dependent variable.
Calculate the regression sum of square U of full model respectively and subtract the regression sum of square U of modeli', thus obtain
Ui=U-Ui′.After obtaining the sum of squares of partial regression that each independent variable is corresponding, each factor can be compared to whole recurrence
The size of effect contribution, in order to little person will be contributed to reject.
2) partial F test
If the multiple correlation coefficient of full model square is R2, subtract model multiple correlation coefficient square beDefinitionDue to independent variable x many in full modeliIf, soAlmost nil explanation xiY is not had
Have a significant impact.Therefore, here it is null hypothesisWork as H0For true time, inspection system
Metering isFor given level of significance α, sample value calculate
Go out FiValue, if Fi≥F1-α(1, n-m-1), then refuse H0, i.e.Significantly being not zero, this illustrates xiY is had aobvious
Write impact, x should be added in subtracting modeliIt is allowed to referred to as full model.If Fi＜ F1-α(1, n-m-1), then accept H0,
I.e.Being significantly zero, this illustrates xiNot notable on y impact, x should be rejected in full modeli, it is allowed to be referred to as subtracting model.
Wherein, n represents sample size number, and m represents independent variable number.
Step 103: according to the regression model of reliability index with test feature tolerance, reliability index is carried out interval
Estimate.Obtaining after the regression model of test feature tolerance and reliability index, on the basis of regression analysis
Reliability index is estimated confidence interval by the actual features of contact multistage test.
Owing to test the defect finding and repairing is a part for software all defect, the fail data that test obtains
It it is an observation sample of out-of-service sequence all to software.Based on the parameter estimation result obtained by this sample not necessarily with
Software truth is consistent, and therefore the present invention is by the basis of the reliability index estimated, utilizes test feature to measure
Result carries out Estimating Confidence Interval to reliability index, and target is to obtain following reliability index result:WhereinFor the estimated result of certain reliability index (such as MTTF), r is that software is true
Real reliability index result (although unknown, but exist), 1-α is confidence level, and w is confidence interval length.
The present invention is concerned with how to obtain the confidence interval of the confidence level for regulation.
Software test and Reliability Practice experience have shown that, test feature concentrated expression can test effect, such as test feature
In coverage rate factor can reflect and test for the coverage condition of tested software, coverage rate is the highest, then in testing
Later observations to invalid cost and trend just use the inefficacy being likely to occur to be more nearly in the future with software, thus based on
The reliability index estimated result that corresponding test data obtain just with achieved reliability index closer to.
This shows that test feature can assist in length w of the confidence interval of reliability index to a certain extent.This
Invention plan first passes through regression model and builds the relation between test feature tolerance and reliability index, then passes through back
Model is returned to provide the confidence interval of reliability index estimated value.
Assume to represent reliability index, x with y1～xpRepresent that code coverage, demand coverage rate, complexity add respectively
The test case number of the code coverage of power, complexity density weighting, test case number of variations, newly-increased failure number etc.
Test feature factor.Assume that the regression model obtained is y=β0+β1x1+β2x2+β3x3+...+βpxp+ ε, if false
If different phase reliability index estimated value is separate, then deviation ε Normal Distribution.
In order to utilize regression equation to forecast, providing x1,x2,…,xpA class value x01,x02,…,x0pTime, note
x0=(1, x01,x02,…,x0p) ', obtains y0=x '0β+ε0, E (ε0)=0, Var (ε0)=σ2And y0Predictive valueWherein, E (X) represents the average of X, and Var (X) represents the side of X
Difference, σ2For the variance of standard normal distribution, x '0Represent x0Transposition, E () represents and takes average, and Var () represents
Take variance；
There is Several Properties:
(1)It is the Unbiased estimtion of y, i.e.
(2) at y0All linear unbiased forecast in,Variance minimum；
(3) ε～N (0, σ2In),AndWithThe most solely
Vertical, whereinFor residual sum of squares (RSS).
Can be derived by further by above characterThen y0Confidence level be 1-α
Confidence interval be
Wherein, t represents t
Distribution, n-p-1 is the degree of freedom of t-distribution, and X for organize independent variable (x more1,x2,...xp) matrix that forms, X ' represents the transposition of X；
But, test experience according to the actual multistage, after-stage test is usually on the basis of testing previous stage
The adjustment made, so being false above with respect to the hypothesis that different phase reliability index estimated value is separate, rear one
Stage reliability index estimated value is relevant to reliability index estimated value previous stage, in order to simplify mathematical model,
It is assumed herein that the degree of association of each stage reliability index estimated value and reliability index estimated value previous stage is ρ.
If identifying bias vector (ε with Σ1,ε2,...,εn), wherein n represents test phase number, then have
If the above different phase separate hypothesis of reliability index estimated value meets, then ρ=0 in matrix, now
Represent with WUnderstanding W is positive definite matrix, there is matrix Z and makes Z2=W.
On equation Y=XB+ Σ both sides with taking advantage of Z-1Obtain Z-1Y=Z-1XB+Z-1Σ, wherein Y represents reliability index vector
(y1,y2,...,yn) ', X representsB represents β vector, and Σ represents bias vector.Warp
After crossing conversion,If therefore making Y*=Z-1Y,
X*=Z-1X, Σ*=Z-1Σ, available Y*=X*B+Σ*, Y*,Σ*Normal Distribution, available Y*Put
Letter interval, and then according to Y*=Z-1Y can get the confidence interval of Y.
In sum, a kind of software reliability growth model interval estimation side based on test feature is embodiments provided
Method, has the advantages that
(1) method by building reliability index and test feature tolerance regression model find reliability index with
Relation between test feature, possesses pure mathematics basis, more scientific and precise.
(2) compare traditional existing method to ignore test feature and directly according to fail data, reliability index is carried out district
Between estimate, when the present invention utilizes actual test, the test feature data of generation assist the interval carrying out reliability index to estimate
Meter, improves the credibility of Interval Estimator of The Reliability Indexes.
It will be understood by those skilled in the art that all or part of flow process realizing above-described embodiment method, can be by meter
Calculation machine program instructs relevant hardware and completes, and described program can be stored in computer-readable recording medium.Its
In, described computer-readable recording medium is disk, CD, read-only store-memory body or random store-memory body etc..
Although the present invention of being described in detail and advantage thereof it should be appreciated that without departing from by appended claim
Various change can be carried out in the case of the spirit and scope of the present invention limited, substitute and convert.And, this Shen
Scope please is not limited only to the specific embodiment of the process described by description, equipment, means, method and steps.This
Those of ordinary skill in field will readily appreciate that from the disclosure, according to the present invention can use execution with
Function that corresponding embodiment described herein is essentially identical or obtain the result essentially identical with it, existing and future
Process, equipment, means, method or step to be developed.Therefore, appended claim is intended to their model
Such process, equipment, means, method or step is included in enclosing.
The above, the only present invention preferably detailed description of the invention, but protection scope of the present invention is not limited thereto,
Any those familiar with the art in the technical scope that the invention discloses, the change that can readily occur in or replace
Change, all should contain within protection scope of the present invention.
Claims (7)
1. a software reliability growth model method of interval estimation based on test feature, it is characterised in that including:
From multiple angles, test feature is analyzed, designs corresponding criterion and test feature is measured, obtain test feature measurement results；
According to the relation of test feature measurement results Yu reliability index, build the regression model of reliability index and test feature tolerance；
According to the regression model of reliability index with test feature tolerance, reliability index is carried out interval estimation.
Method the most according to claim 1, it is characterised in that described test feature measurement results includes such as descending aspect:
The code coverage of code coverage, demand coverage rate, complexity weighting, the test case number of complexity density weighting, test case number of variations, newly-increased failure number.
Method the most according to claim 1, it is characterised in that it is one or more that described reliability index at least includes in following index:
Crash rate, reliability, mean time to failure, mean time between failures, crash rate rate of change.
Method the most according to claim 3, it is characterised in that
Crash rate: crash rate pays close attention to software system failure probability within the unit interval, is defined asWherein R (t) is reliability function, and i.e. before t, system runs the probability occurring without inefficacy, and f (t) is failure dense function,
Reliability: the attention location system degree of reliability within a period of time, i.e. R (t)；
Mean time to failure: software starts first average time lost efficacy occur, if R (t) is continuous function, then from operation
Mean time between failures: pay close attention to the Mean Time Between Replacement that software system is in operation between twice inefficacy of appearance；
Crash rate rate of change: focus on the situation over time that lost efficacy in software running process, pass throughCalculate, λ (t1) and λ (t2) it is respectively t1And t2The crash rate in moment.
Method the most according to claim 1, it is characterised in that the process building the regression model that reliability index is measured with test feature specifically includes:
Calculate reliability index；
Determine the primitive form of regression model；
Inspection independent variable is the most notable on the impact of reliability index, retains and affects significant independent variable, and described independent variable is test feature measurement results.
Method the most according to claim 5, it is characterised in that the impact the most significantly method of reliability index is included by described inspection independent variable:
If former regression equation is y=β0+β1x1+…+βi-1xi-1+βixi+βi+1xi+1+…+βpxp, remove variable xiAfter the new regression model that obtains be y=β0+β1x1+…+βi-1xi-1+βi+1xi+1+…+βpxp, new regression model is called the model that subtracts of former regression model, and former regression model is called full model, wherein, and βiBeing the coefficient of regression model independent variable, p represents independent variable number, and x represents that independent variable, y represent dependent variable；
Calculate the regression sum of square U of full model respectively and subtract the regression sum of square U of modeli', thus obtain Ui=U-Ui′；After obtaining the sum of squares of partial regression that each independent variable is corresponding, the size that relatively whole regression effect is contributed by each factor, little person will be contributed to reject；
If the multiple correlation coefficient of full model square is R2, subtract model multiple correlation coefficient square beDefinition
Null hypothesisAs null hypothesis H0For true time, statistic of test isFor given level of significance α, sample value calculate FiValue, if Fi≥F1- α(1, n-m-1), then refuse H0, i.e.Significantly being not zero, this illustrates xiY is had a significant impact, x should be added in subtracting modeliMake full model；If Fi＜ F1- α(1, n-m-1), then accept H0, i.e.Being significantly zero, this illustrates xiNot notable on y impact, x should be rejected in full modeli, make and subtract model；N represents sample size number, and m represents independent variable number.
Method the most according to claim 1, it is characterised in that the process that reliability index carries out interval estimation specifically includes:
Assume to represent reliability index, x with y1～xpRepresent test feature measurement results respectively, it is assumed that the regression model obtained is y=β0+β1x1+β2x2+β3x3+β4x4+...+βpxp+ ε, if it is assumed that different phase reliability index estimated value is separate, then deviation ε Normal Distribution；
In order to utilize regression equation to forecast, providing x1,x2,…,xpA class value x01,x02,…,x0pTime, remember x0=(1, x01,x02,…,x0p) ', obtains y0=x '0β+ε0, E (ε0)=0, Var (ε0)=σ2And y0Predictive valueE (X) represents the average of X, and Var (X) represents the variance of X, σ2For the variance of standard normal distribution, x '0Represent x0Transposition, E () represents and takes average, and Var () represents and takes variance,；
Can be derived by further by above characterThen y0The confidence interval that confidence level is 1-α beT represents t-distribution, and n-p-1 is the degree of freedom of t-distribution, and X for organize independent variable (x more1,x2,...xp) matrix that forms, x ' represents the transposition of X；
In order to simplify mathematical model, it is assumed herein that the degree of association of each stage reliability index estimated value and reliability index estimated value previous stage is ρ, if identifying bias vector (ε with Σ1,ε2,...,εn), wherein n represents test phase number, then have
If the above different phase separate hypothesis of reliability index estimated value meets, then ρ=0 in matrix, now
Represent with WUnderstanding W is positive definite matrix, there is matrix Z and makes Z2=W；
On equation Y=XB+ Σ both sides with taking advantage of Z-1Obtain Z-1Y=Z-1XB+Z-1Σ, wherein Y represents reliability index vector (y1,y2,...,yn) ', X representsB represents β vector, and Σ represents bias vector；After converting, Var (Z-1Σ)=Z-2Var (Σ)=W-1σ2W=σ2Therefore I, if making Y*=Z-1Y, X*=Z-1X, Σ*=Z-1Σ, available Y*=X*B+Σ*, Y*,Σ*Normal Distribution, available Y*Confidence interval, and then according to Y*=Z-1Y can get the confidence interval of Y.
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Application publication date: 20161012 Assignee: Zhengzhou Yunhai Technology Co.,Ltd. Assignor: BEIHANG University Contract record no.: X2021990000107 Denomination of invention: Interval estimation method of software reliability index based on test features Granted publication date: 20190226 License type: Common License Record date: 20210218 |