CN103106139B - Based on the software failure time forecasting methods that relevance vector regression is estimated - Google Patents

Abstract

The invention discloses a kind of software failure time forecasting methods estimated based on relevance vector regression, software failure moment and m inefficacy time data before it are learnt, thus catching between the inefficacy moment in dependence, thus to build based on Method Using Relevance Vector Machine software reliability prediction method by it. Owing to having taken into full account the small sample characteristic of software reliability prediction, adopt kernel function technology that observational variable can be overcome more than the multicollinearity existed between situation and the variable of observation sample number, thus without model " over-fitting " situation produced by the modeling methods such as neutral net occurs. In new Forecasting Methodology, along with software failure constantly occurs, model parameter constantly will be adapted to the dynamic change of failure procedure automatically, thus realizing the adaptive prediction of software reliability, be effectively improved the adaptive capacity of software faults prediction model.

Description

Based on the software failure time forecasting methods that relevance vector regression is estimated

[technical field]

The present invention relates in software reliability test and evaluation process software failure time data Forecasting Methodology next time or in the following long period.

[background technology]

Software reliability refers under prescribed conditions, and at the appointed time, the probability lost efficacy does not occur software. It is that Traditional solutions reflects the statistical philosophy of large sample solving reliability prediction problem, it is easy to study and the problem such as poor for applicability occurred.

Statistical Learning Theory is built upon on a set of more solid theoretical basis, provides a unified framework for solving finite sample problem concerning study. It can, by included for a lot of existing methods, be expected to help to solve many original insoluble problems, such as neural network structure select permeability, local minimum point's problem etc. Method Using Relevance Vector Machine (relevancevectormachine, RVM) it is Tipping in calendar year 2001 proposed a kind of management loading model, very good application is achieved in a lot, tracking such as object, 3D Attitude estimation, 3D model recovery etc., load forecast, channel equalization prediction etc.

[summary of the invention]

The technical problem to be solved is to provide a kind of software failure time forecasting methods estimated based on relevance vector regression, can realize the adaptive prediction of software reliability. For this, by the following technical solutions, it comprises the steps of this utility model:

(1), first observe and record successive software failure dates set, and all of inputoutput data normalization;

(2), by abstraction and it is assumed that software failure time prediction problem is converted into a function regression problem;

(3), the kernel function for predicting the initialization value of given parameters are selected;

(4) the fail data number for learning, is selected;

(5), adopt relevance vector regression algorithm for estimating to carry out study for different failure dates sets to optimize;

(6), finally select the parameter after optimizing that the new out-of-service time is predicted.

Further, described in step (2), software failure time prediction problem is converted into a function regression problem, adopts with the following method:

Assume that the software failure time occurred is t₁,t₂,…,t_n, make t_l=f (t_l-m,t_l-m+1,…,t_l-1), then t_lObey and fix but the conditional distribution function F (t of the unknown_l|t_l-m,t_l-m+1,…,t_l-1), at t₁,t₂,…,t_kTo t under known conditions_k+1It is predicted becoming: known k-m observation (T₁,t_m+1),(T₂,t_m+2),…,(T_k-m,t_k) and kth-m+1 input T_k-m+1When, estimate kth-m+1 output valveWherein, T_iRepresent m dimensional vector [t_i,t_i+1,…,t_m+i];

The kernel function used in step (3) is gaussian kernel function, κ (x, y)=e^{-g < x-y, x-y > 2}, its initial parameter value g=1.

Fail data number in step (4) is the integer between 5-8.

Further, the employing relevance vector regression algorithm for estimating described in step (5) carries out study for different failure dates sets and optimizes, including following process:

(5.1), given a group vectorWith corresponding desired valueAs input, it is assumed that the corresponding relation of x and t meets following function:

p(t_i)=N (t_i|y(x_i; W), σ²)

(5.2) probability distribution, making t is:

$p\left(t\right|w,{\mathrm{\&sigma;}}^2)=\underset{i=1}{\overset{n}{\&Pi;}}N\left(t_i\right|y\left(x_i;w\right),{\mathrm{\&sigma;}}^2)={\left(2{\mathrm{\&pi;\&sigma;}}^2\right)}^{-\frac{N}{2}}\exp (-\frac{\left|\right|t-\mathrm{\&Phi;}w|{|}^2}{2{\mathrm{\&sigma;}}^2})$

In formula, Φ=[φ (x₁),φ(x₂),…φ(x_N)]^T, φ (x_n)=[1, k (x_n,x₁),k(x_n,x₂),…,k(x_n,x_N)]^T;

W=[w₀,w₁,…w_N]^T,

(5.3), to each weights ω_iDefinition prior probability distribution:

$p\left(w\right|\mathrm{\&alpha;})=\underset{i=0}{\overset{N}{\&Pi;}}\frac{{\mathrm{\&alpha;}}_i}{\sqrt{2\mathrm{\&pi;}}}\exp (-\frac{{\mathrm{\&alpha;}}_i{w_i}^2}{2}),$ In formula, α_iIt is determine w_iThe hyper parameter of prior distribution,

α=(α₁,α_i,…α_N)。

(5.4) Posterior distrbutionp of unknown quantity, is calculated: $p(w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2|t)=\frac{p\left(t\right|w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)p(w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)}{p\left(t\right)}$

(5.5), after integration, abbreviation obtains:

$p\left(w\right|t,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}\right)}^{-\frac{N+1}{2}}\left|\mathrm{\&Sigma;}{|}^{-\frac{1}{2}}\exp \right\{-\frac{{(w-\mathrm{\&mu;})}^T{\mathrm{\&Sigma;}}^{-1}(w-\mathrm{\&mu;})}{2}\},$

$p\left(t\right|\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}\right)}^{-\frac{N}{2}}\left|\mathrm{\&Omega;}{|}^{-\frac{1}{2}}\exp \right\{-\frac{t^T{\mathrm{\&Omega;}}^{-1}t}{2}\}$

μ=σ^-2ΣΦ^TT, Σ=(A+ σ^-2Φ^TΦ)^-1, A=diag (α₀,α₁,…α_N), Ω=σ²I+ΦA^-1Φ^T,

(5.6), p (t is calculated_*| approximate solution t):

$p({\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2)=\arg \underset{\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2}{\max }p(\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2|t),p\left(t_{*}\right|t)\&ap;\&Integral;p\left(t_{*}\right|w,{\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2)p(w,{\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2|t)dw$

(5.7), following formula iterative α is used_MP,

${\mathrm{\&alpha;}}_i^{new}=\frac{{\mathrm{\&gamma;}}_i}{{\mathrm{\&mu;}}_i^2},{\left({\mathrm{\&sigma;}}^2\right)}^{new}=\frac{\left|\right|t-\mathrm{\&Phi;}\mathrm{\&mu;}|{|}^2}{N-{\mathrm{\&Sigma;}}_{i=0}^N{\mathrm{\&gamma;}}_i},{\mathrm{\&gamma;}}_i=1-{\mathrm{\&alpha;}}_i{\mathrm{\&Sigma;}}_{ii}.$

Owing to adopting technical scheme, the present invention uses RVM that software failure moment and m inefficacy time data before it are learnt thus catching between the inefficacy moment in dependence, thus builds based on Method Using Relevance Vector Machine software reliability prediction method. By the application of kernel function technology, software reliability prediction problem is converted into a regression estimation problem, and applies relevance vector regression algorithm for estimating to solve this problem. In new Forecasting Methodology, along with software failure constantly occurs, model parameter constantly will be adapted to the dynamic change of failure procedure automatically, thus realizing the adaptive prediction of software reliability.

[accompanying drawing explanation]

Fig. 1 is the flow chart of invention software out-of-service time Forecasting Methodology.

[detailed description of the invention]

1) data normalization

When using regression estimation algorithm to carry out study prediction, it is necessary first to all of inputoutput data is normalized to interval [0.1,0.9], the concrete formula that converts is: $y=\frac{0.8}{\mathrm{\&Delta;}}x+(0.9-0.8\&times;\frac{x_{max}}{\mathrm{\&Delta;}}),$ Wherein, y is the value after normalization, and x is actual value, x_maxIt is the maximum in data set, x_minIt is minima, Δ=x_max-x_min, it was predicted that after terminating, adopt following mapping that data are mapped back to actual value:

2) problem converts

Assume that the software failure time occurred is t₁,t₂,…,t_n, make t_l=f (t_l-m,t_l-m+1,…,t_l-1), then t_lObey and fix but the conditional distribution function F (t of the unknown_l|t_l-m,t_l-m+1,…,t_l-1), use RVM that software failure time data is learnt, it is possible to catch the dependence of out-of-service time inherence. The input of RVM is m dimensional vector [t_l-m,t_l-m+1,…,t_l-1], it is output as t_l, then total for RVM list entries is t₁,t₂,…,t_n...;

Output sequence is: t_m+1,t_m+2,…,t_n,t_n+1,…。

If being t for the RVM inefficacy moment sequence carrying out learning₁,t₂,…,t_k(k > m), then at t₁,t₂,…,t_kTo t under known conditions_k+1It is predicted becoming: known k-m observation (T₁,t_m+1),(T₂,t_m+2),…,(T_k-m,t_k) and kth-m+1 input T_k-m+1When, estimate kth-m+1 output valveWherein, T_iRepresent m dimensional vector [t_i,t_i+1,…,t_m+i].?As input, then can predictIn like manner can obtain

The mean value function of predictive value is given by:

$\widehat{t_{k+1}}=\underset{i=1}{\overset{m}{\&Sigma;}}{w}_{i}K(T_{i+1},T_i)+w_0$

Probabilistic forecasting distribution function is:

$p\left(t_{k+1}\right|T)\&ap;N\left(t_{k+1}\right|y_{k+1},{\mathrm{\&sigma;}}_{k+1}^2)$

3) kernel function for predicting the initialization value of given parameters are selected

4) value of kernel functional parameter is determined

Kernel functional parameter select permeability, its essence is exactly an optimization problem, adopts grid data service to carry out kernel functional parameter selection, such as when predicting with SVM, adopt gaussian kernel function, it is thus necessary to determine that two parameters and penalty factor and kernel functional parameter g, based on gridding method by C ∈ [C₁,C₂], change step is C_s, and g ∈ [g₁,g₂], change step is g_t, for every pair of parameter, (C, g) is trained, and chooses a pair best parameter of effect as model parameter

5) Relevance vector machine for regression algorithm for estimating

Solve regression problem with RVM can be described as: given a group vectorWith corresponding desired valueAs input, it is desirable to find out x_iWith t_iBetween corresponding relation so that running into a new vector x_*Time, it is possible to dope the desired value t that it is corresponding_*, t_iIt it is any real number. The corresponding relation of x and t meets following function:

p(t_i)=N (t_i|y(x_i; W), σ²)

It may be reasonably assumed thatIt is random variable independent of each other,

KnownWith σ²Under condition, the probability distribution of t is

$p\left(t\right|w,{\mathrm{\&sigma;}}^2)=\underset{i=1}{\overset{n}{\&Pi;}}N\left(t_i\right|y\left(x_i;w\right),{\mathrm{\&sigma;}}^2)={\left(2{\mathrm{\&pi;\&sigma;}}^2\right)}^{-\frac{N}{2}}\exp (-\frac{\left|\right|t-\mathrm{\&Phi;}w|{|}^2}{2{\mathrm{\&sigma;}}^2})$

In formula, Φ=[φ (x₁),φ(x₂),…φ(x_N)]^T, φ (x_n)=[1, k (x_n,x₁),k(x_n,x₂),…,k(x_n,x_N)]^T;

W=[w₀,w₁,…w_N]^T, to each weights ω_iDefinition prior probability distribution:

$p\left(w\right|\mathrm{\&alpha;})=\underset{i=0}{\overset{N}{\&Pi;}}\frac{{\mathrm{\&alpha;}}_i}{\sqrt{2\mathrm{\&pi;}}}\exp (-\frac{{\mathrm{\&alpha;}}_i{w_i}^2}{2})$

In formula, α_iIt is determine w_iThe hyper parameter of prior distribution, α=(α₁,α_i,…α_N)。

Prior distribution according to weights and sample set likelihood function, the Posterior distrbutionp of unknown quantity can be calculated by Bayesian formula and obtain:

$p(w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2|t)=\frac{p\left(t\right|w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)p(w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)}{p\left(t\right)}$

Therefore, a given new vector x_*Time, t_*Probability distribution prediction be:

p(t_*| t)=∫ p (t_*|w,α,σ²)p(w,α,σ²|t)dwdαdσ²,

p(w,α,σ²| t)=p (w | t, α, σ²)p(α,σ²|t)

Thus, have

$p\left(w\right|t,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)=\frac{p(w,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2|t)}{p(\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2|t)}=\frac{p\left(t\right|w,{\mathrm{\&sigma;}}^2)p\left(w\right|\mathrm{\&alpha;})}{p\left(t\right|\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)}=\frac{p\left(t\right|w,{\mathrm{\&sigma;}}^2)p\left(w\right|\mathrm{\&alpha;})}{\&Integral;p\left(t\right|w,{\mathrm{\&sigma;}}^2)p\left(w\right|\mathrm{\&alpha;})dw}$

P in above formula (t | w, σ²) it is all the product of Gaussian function with p (w | α), after integration, abbreviation obtains:

$p\left(w\right|t,\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}\right)}^{-\frac{N+1}{2}}\left|\mathrm{\&Sigma;}{|}^{-\frac{1}{2}}\exp \right\{-\frac{{(w-\mathrm{\&mu;})}^T{\mathrm{\&Sigma;}}^{-1}(w-\mathrm{\&mu;})}{2}\},$

$p\left(t\right|\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2)={\left(2\mathrm{\&pi;}\right)}^{-\frac{N}{2}}\left|\mathrm{\&Omega;}{|}^{-\frac{1}{2}}\exp \right\{-\frac{t^T{\mathrm{\&Omega;}}^{-1}t}{2}\}$

Wherein, μ=σ^-2ΣΦ^TT, Σ=(A+ σ^-2Φ^TΦ)^-1, A=diag (α₀,α₁,…α_N), Ω=σ²I+ΦA^-1Φ^T, such that it is able to find p (t_*| approximate solution t):

$p({\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2)=\arg \underset{\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2}{\max }p(\mathrm{\&alpha;},{\mathrm{\&sigma;}}^2|t),p\left(t_{*}\right|t)\&ap;\&Integral;p\left(t_{*}\right|w,{\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2)p(w,{\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2|t)dw$

Two products being all Gaussian function in integration type. So, after definite integral, result is:

$p\left(t_{*}\right|t)\&ap;N\left(t_{*}\right|y_{*},{\mathrm{\&sigma;}}_{*}^2),y_{*}={\mathrm{\&mu;}}^T\mathrm{\&phi;}\left(x_{*}\right),{\mathrm{\&sigma;}}_{*}^2={\mathrm{\&sigma;}}_{MP}^2+\mathrm{\&phi;}{\left(x_{*}\right)}^T\mathrm{\&Sigma;}\mathrm{\&phi;}\left(x_{*}\right),$

φ(x_*)=[1, k (x_*,x₁),k(x_*,x₂),…,k(x_*,x_N)]^T

Finally, remaining issues is to solve for ${\mathrm{\&alpha;}}_{MP},{\mathrm{\&sigma;}}_{MP}^2:{\mathrm{\&alpha;}}_i^{new}=\frac{{\mathrm{\&gamma;}}_i}{{\mathrm{\&mu;}}_i^2},{\left({\mathrm{\&sigma;}}^2\right)}^{new}=\frac{\left|\right|t-\mathrm{\&Phi;}\mathrm{\&mu;}|{|}^2}{N-{\mathrm{\&Sigma;}}_{i=0}^N{\mathrm{\&gamma;}}_i},{\mathrm{\&gamma;}}_i=1-{\mathrm{\&alpha;}}_i{\mathrm{\&Sigma;}}_{ii},$

Wherein Σ_iiIt is i-th element on the diagonal in Σ, first provides α, σ²Conjecture value, then constantly updated by above formula, just can approach α_MP,

In order to provide rational comparison and analysis to the model set up, adopt 10 and carried out experimental analysis from the model that the true fail data set pair of dissimilar software is proposed, as shown in table 2. These data sets describe the failure procedure of each software system, and each data point comprises two kinds of observation statistics set: accumulative execution time and accumulative Failure count. In an experiment, training set includes starting rear complete thrashing process from test, in order to allow kernel function learn fully, in experimentation, take all data sets first three point one as learning data, compare with truthful data after 2/3rds data below are predicted.

Table lists the AE value of each model on ten data sets, wherein model 1-6 represents SRGMWithLogisticTEF, SRGMWithRayleighTEF, DelayedS-ShapedModelWithLogisticTEF, DelayedS-ShapedModelWithRayleighTEF, G-Omodel, YamadaDelayedS-Shaped respectively; Model 7 represents the method that the present invention adopts, and a, b, c, d represent kernel function respectively GaussianFunction, LinearFunction, PolynomialFunction, SymmetricTriangleFunction of adopting.

The AE value of each model prediction on 1:10 data set of table

Conclusion: on different pieces of information collection, when adopting different kernel functions and adopt different regression estimation methods, model prediction performance is all variant, adopts the prediction model of software reliability based on relevance vector regression algorithm for estimating can be effectively improved estimated performance and the suitability of model.

Claims (3)

1. the software failure time forecasting methods estimated based on relevance vector regression, is characterized in that, it comprises the steps of:

(1), first observe and record successive software failure dates set, and all of inputoutput data normalization;

(2), by abstraction and it is assumed that software failure time prediction problem is converted into a function regression problem;

(3), the kernel function for predicting the initialization value of given parameters are selected;

(4) the fail data number for learning, is selected;

(5), adopt relevance vector regression algorithm for estimating to carry out study for different failure dates sets to optimize;

(6), finally select the parameter after optimizing that the new out-of-service time is predicted;

Employing relevance vector regression algorithm for estimating described in step (5) carries out study for different failure dates sets and optimizes, including following process:

(5.1), given a group vectorWith corresponding desired valueAs input, it is assumed that the corresponding relation of x and t meets following function:

p(t_i)=N (t_i|y(x_i; W), σ²)

(5.2) probability distribution, making t is:

In formula, Φ=[φ (x₁),φ(x₂),…φ(x_N)]^T, φ (x_n)=[1, k (x_n,x₁),k(x_n,x₂),…,k(x_n,x_N)]^T;

W=[w₀,w₁,…w_N]^T,

(5.3), to each weights ω_iDefinition prior probability distribution:

In formula, α_iIt is determine w_iThe hyper parameter of prior distribution,

α=(α₁,α_i,…α_N),

(5.4) Posterior distrbutionp of unknown quantity, is calculated:

(5.5), after integration, abbreviation obtains:

μ=σ^-2ΣΦ^TT, Σ=(A+ σ^-2Φ^TΦ)^-1, A=diag (α₀,α₁,…α_N), Ω=σ²I+ΦA^-1Φ^T,

(5.6), p (t is calculated_*| approximate solution t):

(5.7), following formula iterative α is used_MP,

2. the as claimed in claim 1 software failure time forecasting methods estimated based on relevance vector regression, is characterized in that, described in step (2), software failure time prediction problem are converted into a function regression problem, adopt with the following method:

Assume that the software failure time occurred is t₁,t₂,…,t_n, make t_l=f (t_l-m,t_l-m+1,…,t_l-1), then t_lObey and fix but the conditional distribution function F (t of the unknown_l|t_l-m,t_l-m+1,…,t_l-1), at t₁,t₂,…,t_kTo t under known conditions_k+1It is predicted becoming: known k-m observation (T₁,t_m+1),(T₂,t_m+2),…,(T_k-m,t_k) and kth-m+1 input T_k-m+1When, estimate kth-m+1 output valveWherein, T_iRepresent m dimensional vector [t_i,t_i+1,…,t_m+i]。

3. the software failure time forecasting methods estimated based on relevance vector regression as claimed in claim 1, is characterized in that, the kernel function used in step (3) is gaussian kernel function,Its initial parameter value g=1; Fail data number in step (4) is the integer between 5-8.