CN110457748B - Test design method for two equal-level covering arrays - Google Patents
Test design method for two equal-level covering arrays Download PDFInfo
- Publication number
- CN110457748B CN110457748B CN201910601409.1A CN201910601409A CN110457748B CN 110457748 B CN110457748 B CN 110457748B CN 201910601409 A CN201910601409 A CN 201910601409A CN 110457748 B CN110457748 B CN 110457748B
- Authority
- CN
- China
- Prior art keywords
- array
- column
- covering
- arrays
- matrix
- Prior art date
- Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
- Expired - Fee Related
Links
- 238000012360 testing method Methods 0.000 title claims abstract description 96
- 238000013461 design Methods 0.000 title claims abstract description 84
- 238000000034 method Methods 0.000 title claims abstract description 45
- 238000003491 array Methods 0.000 title claims description 71
- 230000003993 interaction Effects 0.000 claims abstract description 78
- 239000011159 matrix material Substances 0.000 claims description 74
- 238000004422 calculation algorithm Methods 0.000 claims description 37
- 238000005457 optimization Methods 0.000 claims description 15
- 230000002068 genetic effect Effects 0.000 claims description 14
- 238000000605 extraction Methods 0.000 claims description 12
- 230000006870 function Effects 0.000 claims description 12
- 239000013598 vector Substances 0.000 claims description 12
- 238000004364 calculation method Methods 0.000 claims description 11
- 238000013401 experimental design Methods 0.000 claims description 10
- 230000002452 interceptive effect Effects 0.000 claims description 8
- 239000000203 mixture Substances 0.000 claims description 7
- 238000010276 construction Methods 0.000 claims description 6
- 238000007405 data analysis Methods 0.000 claims description 5
- 238000002474 experimental method Methods 0.000 claims description 5
- 230000008569 process Effects 0.000 claims description 5
- 230000001131 transforming effect Effects 0.000 claims description 5
- 230000004044 response Effects 0.000 claims description 4
- 238000006243 chemical reaction Methods 0.000 claims description 3
- 238000004880 explosion Methods 0.000 claims description 3
- 230000035772 mutation Effects 0.000 claims description 3
- 239000000758 substrate Substances 0.000 claims description 3
- 241000764238 Isis Species 0.000 claims description 2
- 230000008859 change Effects 0.000 claims description 2
- 238000011156 evaluation Methods 0.000 description 2
- 238000012545 processing Methods 0.000 description 2
- 238000011160 research Methods 0.000 description 2
- 238000013522 software testing Methods 0.000 description 2
- 235000008331 Pinus X rigitaeda Nutrition 0.000 description 1
- 235000011613 Pinus brutia Nutrition 0.000 description 1
- 241000018646 Pinus brutia Species 0.000 description 1
- 230000003042 antagnostic effect Effects 0.000 description 1
- 239000000470 constituent Substances 0.000 description 1
- 230000007547 defect Effects 0.000 description 1
- 238000013400 design of experiment Methods 0.000 description 1
- 238000011161 development Methods 0.000 description 1
- 238000012850 discrimination method Methods 0.000 description 1
- 230000000694 effects Effects 0.000 description 1
- 238000011835 investigation Methods 0.000 description 1
- 230000003252 repetitive effect Effects 0.000 description 1
- 238000007619 statistical method Methods 0.000 description 1
Images
Classifications
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06N—COMPUTING ARRANGEMENTS BASED ON SPECIFIC COMPUTATIONAL MODELS
- G06N3/00—Computing arrangements based on biological models
- G06N3/12—Computing arrangements based on biological models using genetic models
- G06N3/126—Evolutionary algorithms, e.g. genetic algorithms or genetic programming
Landscapes
- Engineering & Computer Science (AREA)
- Physics & Mathematics (AREA)
- Health & Medical Sciences (AREA)
- Life Sciences & Earth Sciences (AREA)
- Biophysics (AREA)
- Bioinformatics & Cheminformatics (AREA)
- Bioinformatics & Computational Biology (AREA)
- Evolutionary Biology (AREA)
- Theoretical Computer Science (AREA)
- Computational Linguistics (AREA)
- Molecular Biology (AREA)
- Biomedical Technology (AREA)
- Genetics & Genomics (AREA)
- Data Mining & Analysis (AREA)
- Evolutionary Computation (AREA)
- General Health & Medical Sciences (AREA)
- Artificial Intelligence (AREA)
- Computing Systems (AREA)
- General Engineering & Computer Science (AREA)
- General Physics & Mathematics (AREA)
- Mathematical Physics (AREA)
- Software Systems (AREA)
- Physiology (AREA)
- Complex Calculations (AREA)
- Management, Administration, Business Operations System, And Electronic Commerce (AREA)
Abstract
The invention belongs to the technical field of combined design, and discloses a test design method of an equal-level two-part covering array, which aims at two-part countermeasure of radar countermeasureClass test, fully covering the factor level interaction problem of both sides, giving two test factors, each with the same level number and total number n, and interaction strength t inside the two factor sets 1 And t 2 Total interaction strength of t = t 1 +t 2 . The invention solves the technical problem of two-party countermeasure tests such as radar countermeasure and the like, and effectively solves the test design problem that factor level interaction of two parties needs to be fully covered in the radar countermeasure test.
Description
Technical Field
The invention belongs to the technical field of combined design, and particularly relates to a test design method of an equal-level two-coverage array in the field of electronic information equipment performance evaluation test design.
Background
Experimental design, also known as experimental design, is a mathematical principle and method of implementation on how to formulate an appropriate experimental plan to a predetermined objective to facilitate effective statistical analysis of experimental results. The design of a test, i.e. the arrangement of the test, requires selection of appropriate factors and levels to allow for the specific procedures and data analysis framework for the implementation of the test, taking into account the type of problem to be solved by the test, the degree of generality to the discussion, how much efficacy is desired to be tested, the homogeneity of the test units, the time and expense of each test, etc.
Prior art related to the present invention, conventional test design methods, such as orthogonal design, uniform design, are designed to cover the test design area to the maximum extent with the most balanceThe domain principle is used to design the experimental scheme, so when dealing with the combination of experimental factors and levels, it is desirable to have exactly λ times in the experimental design for all factors to interact, which brings some application disadvantages, such as: the orthogonal matrix is too much relied upon. The construction of orthogonal matrices, especially hybrid orthogonal matrices, has many unsolved problems, and no universal construction method exists, which brings great difficulty to the application of orthogonal design and uniform design [2] . In the field of software testing, people put forward the concept of a coverage array according to the requirements of test combinations, and the greatest difference from the traditional orthogonal design and the like is that when the combination of test factors and levels is processed, the coverage array is at least (not exactly) lambda times in an interactive test design scheme among factors of which the assigned strength (not all factors) is expected, so that the design difficulty is effectively reduced in construction and application practices, and the test design is constructed more flexibly and more efficiently.
However, the conventional orthogonal design, uniform design, new coverage array and other methods are integrated with the factor processing, that is, the importance of the involved factors is considered to be equal during the design, and the combination of the factors is also integrated with each other. In fact, in tests such as electronic warfare, factors participating in the test and their levels are divided into two parts for both warfare, and the test and data analysis also focus on warfare between the two parts, in other words, on the interaction between the factors of the two parts, not the interaction of all factors. The full utilization of the two antagonistic characteristics can greatly reduce the total amount of samples required by the test, thereby more effectively generating a test design scheme, and the new requirement and characteristics provide new challenges for the traditional test design method. Meanwhile, research provides a test design method which is suitable for the special requirement, and the method is a premise and a basis for subsequent work of organizing implementation, data processing and even equipment capability evaluation of the resistance test of the equipment in the complex electromagnetic environment. Abbreviations and Key term definitions
Factor (c): the variable to be considered in the test is called a factor (or a factor). Factor is influence test nodeFruit test parameters [1] 。
Level: the state in which the factor is selectable is referred to as the level of the factor [1] 。
Responding: the result of the test is called the response (or output).
And (3) experimental design: overall arrangement of the centimetre factor and post-levelness tests [1] 。
Covering arrays: x is a set of v-symbols, denoted A = (a) ij )(a ij E.x) is an N X k matrix, each N X t sub-matrix from A contains X t At least λ times, then the matrix A is a covering matrix, denoted (CA) λ (N; t, k, v)). Where N is the coverage size (number of rows in the matrix), t is the coverage intensity, k is the number of factors (degrees), and v is the number of states per factor, i.e., the number of levels.
Two covering arrays: x is a set of v-symbols, denoted A = (a) ij )(a ij E.x) is an N X k matrix, from the preceding k 1 Column and following k 2 The column consists of two parts. If for any subset
Each from A of N × (t) 1 +t 2 ) Subarrays
Comprising X t At least 1 time, the matrix A is then a two-part coverage array, denoted (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)). Where N is the overlay size (number of rows), t = t 1 +t 2 Is the coverage intensity, k is the number of factors (degrees), v is the number of states per factor, i.e., the number of levels. Reference documents: [1]Anchor poem pine, zhou ji xiang, chen glu, experimental design, chinese statistics publisher, 9 months 2012; [2]Beth,T., Jungnickel,D.,Lenz,H.,1999.Design theory.Cambridge University Press,Cambridge。
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides a test design method of an equal-level two-part covering array. The technical scheme of the invention is suitable for the test design problem under the situation of various two-party confrontations.
In order to achieve the purpose, the invention adopts the following technical scheme:
a test design method of an equal-level two-part coverage array aims at the problem of factor level interaction of two sides of radar countermeasure tests and fully covers the countermeasure, two test factors are given, namely a set with the same level number of each factor and a total test number n, and the interaction strength t inside the two factor sets 1 And t 2 Total interaction strength of t = t 1 +t 2 ；
Firstly, calculating the minimum sample capacity required by the two covering arrays, if the capacity is smaller than the specified total number n of tests, reminding a user that the two covering arrays under the specified interaction intensity do not exist, and if not, continuing; transforming the design matrix into an information matrix;
then, transforming the value of the information matrix into an optimization function, and searching an integer solution of the optimization problem by using a genetic algorithm, wherein the integer solution is transformed back to form two covering arrays meeting the requirements; the specific contents are as follows:
a. two-part coverage array proposing and defining
In the electronic countermeasure test, factors participating in the test and their levels are divided into two parts for both of the countermeasures, and the test and data analysis also focus on the countermeasures between the two parts, in other words, the test is intended to focus on the interaction between the factors of the two parts in the test, rather than on the interaction of all the factors; defining two covering arrays, namely two covering arrays; i is v Defined as the set of all positive integers starting from 1 to v,
Defined as the set of u positive integers starting from k +1; note a = (a) ij )(i∈I N ,j∈I k ) Is an N x k matrix, k starting from 1 And k after 2 Column composition in which the j-th column element is taken from the group consisting of the symbols v j Set of compositions V j Where k is 1 +k 2 K (= k); at this time, willA is rewritten as
thus, the concept and definition of the two part locator array is as follows:
definition 1: let X be a set of v-symbols, let A = (a) ij )(a ij E.x) is an N X k matrix, from the previous k 1 Column and following k 2 The column consists of two parts; if for any subset
Each from A of N × (t) 1 +t 2 ) Subarrays
Comprising X t At least 1 time, the matrix A is then a two-part overlay matrix, denoted as (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)); where N is the overlay size (number of rows), t = t 1 +t 2 Is the coverage intensity, k is the number of factors (degrees), v is the number of states per factor, i.e. the number of levels;
note 1: (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)) represents factors that affect the response; the values in the columns represent the setting or value of the factor, i.e., the level of the factor; each row represents a set of tests to be run, with a value specified for each factor; then, in all tests, taken from
All v within the factor combination set t A combination of horizontal values at (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)) occurs at least 1 time in a row;
note 2: if (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)) there is a minimum dimension N, at which time the two coverage arrays are (t) 1 ,t 2 )-BCAN(k 1 ,k 2 V); the minimum size of the two-part coverage array reduces the time and cost of the experiment when other parameters are fixed; the construction and optimization of the two covering arrays are given parameters (t) 1 ,t 2 ,k 1 ,k 2 V) finding two covering arrays (t) satisfying the minimum size 1 ,t 2 )-BCAN(k 1 ,k 2 ,v)；
b. Lower bound of sample capacity for an equal horizontal two-coverage array, two-coverage array (t) 1 ,t 2 )-BCAN(k 1 ,k 2 V) a lower bound on size N sample capacity that can help determine whether the two coverage arrays are optimal;
let k 1 ≥t 1 ≥1，k 2 ≥k 2 1 and v 2 are all integers, then
Let A be (t) 1 ,t 2 )-BCAN(k 1 ,k 2 V), note k = k 1 +k 2 ，t＝t 1 +t 2 ；
For any 1 ≦ j 1 ＜j 2 ＜…＜j t ≤k 1 And
let T = ((j) 1 ,x 1 ),…,(j t ,x t ) Then T is one (T) 1 ,t 2 ) Step interaction; note the book
for satisfying 1 ≦ j 1 ＜j 2 ＜…＜j t ≤k 1 And
any t column vectors, since A is a two-part covering array, if A is different (t) 1 ,t 2 ) The order interaction occurs at least once, and there will be a total of v t Then (N-v) remains t ) The rows are repeated (t) 1 ,t 2 ) Step interaction; this indicates that at least v t -(N-v t ) A (t) 1 ,t 2 ) The order interaction occurs exactly once, that is,
in addition, since the array is a two-part array, two (t) are arbitrary 1 ,t 2 ) Order interaction T 1 And T 2 Not in the same row, i.e. not satisfying | ρ (A, T) simultaneously 1 ) L =1 and | ρ (a, T) 2 ) | =1; i.e. each row contains at most one (t) 1 ,t 2 ) The order interaction T is such that | ρ (a, T) | =1; thus, the following inequality is obtained
That is to say that the first and second electrodes,
is solved to obtain
c. Method for discriminating two covering arrays, design matrix A N×k A sufficient condition for being a definition of a coverage array is that all t-order interactions of all factors occur at least once; from a given design matrix A N×k Extracting all interaction information, and judging whether all t-order interactions of all factors occur at least once, but the judgment times required by the judgment is extremely large due to combined explosion, and particularly when k, t and v are large, the calculation amount of the method increases in a factorial manner; therefore, a high-efficiency discrimination algorithm with polynomial level of operation quantity is adopted for the two covering arrays;
design matrix A N×k Extracting t 1 +t 2 Columns, i.e. k from the front, respectively 1 Column extraction t 1 Column, k from the back 2 Column extraction t 1 Column, replacing all the remaining column elements with 0, and finally forming oneA mutual subarray
The total number of sub-arrays can be extracted as
Is made of
Each interactive subarray
Conversion into information vectors
Recording the factor interaction information of each line in the current subarray;
get
To
The transformed vector is F N ＝[1,v 1 ,v 2 ,…,v k ] T
Wherein k is the number of factors and v is the number of levels of each factor; when v is less than or equal to 10, F N Is designed as F N ＝[1,10,10 2 ,…,10 k ] T
Namely, it is
If A is N×k Is a two-part covering array with
Then, the alternative expression is if for A N×k T in (1) 1 +t 2 Column, with non-repeating number of interactions up to v t1 This t 1 +t 2 The column meets the requirement of the second covering;
if for A N×k All of t in 1 +t 2 Column, number of non-repeated interactions can reach v t Then, A can be explained N×k Is one (t) 1 ,t 2 ) A second-order covering array;
at the same time, according to the above
The calculation of (a) is carried out,
the mutual information of each line is stored in the memory, and the repeated mutual lines are reflected as
The same number in (1); therefore, the following method is adopted for statistics
The number of the different numbers in the two covering arrays is compared with the lower bound of the two covering arrays, thereby judging A N×k Whether it is a two-part coverage array;
for is to
Sorting from small to large to obtain
Note book
Is an integer sequence
The difference of the latter term minus the former term, then
Is different from that of
The number of different numbers in the same table is
Number M of non-zero (i) (ii) a When all M (i) All satisfy M (i) ≥v t Then, A can be determined N×k Is a two-part positioning array;
d. method for generating two coverage arrays from an initial design array, if A N×k Whether it is not a two-part covering array, how it is in A N×k Further optimizing on the basis of the obtained data to finally obtain two covering arrays meeting the requirements; converting the problem of searching the two covering arrays into an optimization problem, namely an integer programming problem; solving this integer programming problem by means of a genetic algorithm;
how to then combine A N×k The process of obtaining two coverage arrays by optimizing is designed into an integer programming problem, namely the definition of an objective function of the integer programming, and the definition of the objective function is as follows:
if it is not
Covering two parts (t) 1 ,t 2 ) The order interaction then has M (i) ≥v t If A is N×k Is a two-part array, then there are all M (i) ≥v t Thus, an objective function for integer programming is defined as
Wherein, A N×k ＝(a ij ) N×k ，1≤a ij V is not more than v, an integer
Test design method of equal-level two-part covering array, wherein two parts areThe experimental design of the coverage array was implemented as follows: given two sets of test factors K 1 And K 2 The number of factors of each set is k 1 And k 2 ；k＝k 1 +k 2 The number of levels of each factor is v, the total number of trials N, and the interaction strength t within the two factor sets 1 And t 2 (ii) a Total interaction intensity t = t 1 +t 2 Structure K of 1 And K 2 Upper arbitrary t 1 +t 2 The interactive two-part coverage array comprises several groups of algorithms as follows:
algorithm 1: generating step of initial design matrix
Step 1: for a given number of trials N, a vector h = (h) is generated 1 ,h 2 ,…h k ) Wherein 1= h 1 ＜h 2 ＜…＜h k Are all positive integers from 1 with N, k is these h j The number of (2) is also a factor number;
step 2: generating an Nxk matrix A from h N×k ＝(a ij ) N×k Wherein a is ij ＝ih j (modN) +1; matrix A N×k The method is an initial design with N points, wherein each line corresponds to a design point; it is clear that A N×k Is one permutation of {1,2, \8230;, N };
and 3, step 3: remember the initial design matrix as
And 2, algorithm: a step of changing the matrix array into an information array and judging whether the matrix array meets the requirements of two covering arrays
Step 1: is _ BCA = true
And 2, step: for is to
Repeating the step 2 to the step 8;
and step 3: from design matrix A N×k Front k 1 Column extraction t 1 Column, k from the back 2 Column extraction t 2 Column, replacing all remaining column elements with 0,obtaining a subarray
And 4, step 4:
and 5: for is to
Sorting from small to large to obtain
Step 6: computing
Difference of (2)
Namely, it is
And 7:
and 8: if M is (i) ≥v t Returning to the step 2; otherwise, is _ BCA = false, end;
algorithm 3: optimizing step for two covering arrays by genetic algorithm
Step 1: f (A) N×k )＝0，min F ＝v t ，
As used in the following calculations
Step 2: for is to
Repeating the following steps 3 to 6;
and 3, step 3: according to Algorithm 2, M is calculated (i) ；
And 4, step 4: calculating out
And 5: f (A) N×k )＝F(A N×k )+f i
And 6: if it is not
Turning to step 7; otherwise, turning to the step 2;
and 7: if F (A) N×k ) =0, the optimal solution has been obtained, i.e. the current design matrix is the two covering matrices, and the process ends; otherwise, turning to step 8;
and 8: if F (A) N×k )≥min F ，
Otherwise
min F ＝F(A N×k )；
And step 9: if min F The set m times are not changed, the algorithm is ended, and the current value is taken
Designing a matrix for the optimal;
otherwise, using genetic algorithm to A N×k Performing mutation adjustment to obtain new A N×k Returning to the step 2;
the above procedure is applicable to experimental design under two-party confrontation conditions, and enables the creation of a cover between the two parts (t) 1 ,t 2 ) A design matrix of order interactions.
Due to the adoption of the technical scheme, the invention has the advantages of:
a test design method of an equal-level two-part covering array is characterized in that a method for transforming a test design matrix into an information matrix is adopted; judging whether the original test design matrix is a method of two covering matrixes or not through the information matrix; namely, the test design matrix is a lower bound calculation method of the total amount of samples of the two covering arrays. Can be adapted to the problem of experimental design under two-sided confrontation conditions, can produce coverage between two parts (t) 1 ,t 2 ) A design matrix of order interactions.
The invention solves the technical problem of two-party countermeasure tests such as radar countermeasure and the like, and effectively solves the test design problem that the factor level interaction of the two parties needs to be fully covered in the radar countermeasure test.
Drawings
FIG. 1 is a flow chart of an experiment of an equal level two-coverage array.
Detailed Description
The present invention will be described in further detail below with reference to specific embodiments and drawings.
As shown in fig. 1, a method for designing an equal-level two-coverage-array test is to provide a two-coverage-array test design method for a two-party countermeasure test such as radar countermeasure, aiming at the factor level interaction problem of two parties of sufficient coverage countermeasure, so as to effectively solve the test design problem that the factor level interaction of two parties of sufficient coverage countermeasure is required in the radar countermeasure test.
The invention researches two-party countermeasure tests such as radar countermeasure and provides a two-part coverage array test design method aiming at the problem of factor level interaction of two parties of sufficient coverage countermeasure. The technical scheme is as follows: given two trial factors, the number of levels of each factor is the same, the total number of sets and trials n, and the strength of interaction t within the two factor sets 1 And t 2 (ii) a Total interaction intensity t = t 1 +t 2 。
Firstly, calculating the minimum sample capacity required by the two covering arrays, if the capacity is smaller than the specified total number n of tests, reminding a user that the two covering arrays under the specified interaction strength do not exist, and if not, continuing; converting the design matrix into an information matrix, then converting the value of the information matrix into an optimization function, and searching an integer solution of the optimization problem by using a genetic algorithm, wherein the integer solution is converted back to be the two covering arrays meeting the requirements; the specific contents are as follows:
1.1 two-part coverage array development and definition
In some cases, one does not need to cover all factor interactions in the corresponding system. In tests such as electronic countermeasure, factors participating in the test and their levels are divided into two parts for both countermeasures, and the test and data analysis also focus on the countermeasures between the two parts, in other words, the test is intended to focus on the interaction between the factors of the two parts under investigation, rather than on the interaction of all the factors. In this sense, a two-part coverage array is more suitable for this type of test, for which the following definition of two-part coverage array is proposed.
First, some symbols used in the present invention are agreed:
I v defined as the set of all positive integers starting from 1 to v.
Note a = (a) ij )(i∈I N ,j∈I k ) Is an N x k matrix, k starting from 1 And k after 2 Column, where the jth column element is taken from the group consisting of the symbols v j Set of constituents V j Where k is 1 +k 2 K (= k). At this time, we can A rewrite as
Thus, we propose the concept and definition of the two arrays as follows.
Definition 1: let X be a set of v-symbols, let A = (a) ij )(a ij E.x) is an N X k matrix, from the previous k 1 Column and following k 2 The column consists of two parts. If for any subset
Each from A of N × (t) 1 +t 2 ) Subarrays
Comprising X t At least 1 time, the matrix A is then a two-part coverage array, denoted (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)). Where N is the overlay size (number of rows), t = t 1 +t 2 Is the coverage intensity, k is the number of factors (degrees), v is the number of states per factor, i.e., the number of levels.
Remarks 1: similar to the widespread use of coverage arrays in software testing and design of experiments, a two-part coverage array would also be a very useful tool, especially for two-part challenge-type experiments. (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)) represents factors that affect the response; the values in the column represent the setting or value of the factor (i.e., the level of the factor); each row represents a set of tests to be run, with a value assigned to each factor. Then, in all tests, taken from
Within the factor combination set, all v t A combination of horizontal values at (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)) occurs at least 1 time in a row.
Remarks 2: if (BCA (N; t) 1 ,t 2 ,k 1 ,k 2 V)) there is a minimum dimension N, at which time the two coverage arrays are (t) 1 ,t 2 )-BCAN(k 1 ,k 2 V). The minimum size of the two-part coverage array is of particular interest when other parameters are fixed, as it reduces the time and cost of the experiment. The construction and optimization of the two covering arrays are given parameters (t) 1 ,t 2 ,k 1 ,k 2 V) finding two covering arrays (t) satisfying the minimum size 1 ,t 2 )-BCAN(k 1 ,k 2 ,v)。
1.2 lower bound on sample Capacity of equal-level two-covering array (t) 1 ,t 2 )-BCAN(k 1 ,k 2 V) size N, lower bound of sample volume, which canWe are helped in the following algorithm to determine if the two coverage arrays are optimal.
Let k 1 ≥t 1 ≥1，k 2 ≥k 2 1 and v.gtoreq.2 are integers. Then
And (3) proving that: let A be (t) 1 ,t 2 )-BCAN(k 1 ,k 2 V). Note k = k 1 +k 2 ，t＝t 1 +t 2 。
For any 1 ≦ j 1 ＜j 2 ＜…＜j t ≤k 1 And
let T = ((j) 1 ,x 1 ),…,(j t ,x t ) Then T is one (T) 1 ,t 2 ) And (4) step interaction. Note book
for satisfying 1 ≦ j 1 ＜j 2 ＜…＜j t ≤k 1 And
any t column vectors, since A is a two-part covering array, if A is different (t) 1 ,t 2 ) The order interaction occurs at least once, and there will be a total of v t Then (N-v) remains t ) The rows are repeated (t) 1 ,t 2 ) And (4) step interaction. This indicates that at least v t -(N-v t ) A (t) 1 ,t 2 ) The order interaction occurs exactly once, that is,
in addition, since the array is a two-part array, two (t) are arbitrary 1 ,t 2 ) Order interaction T 1 And T 2 Not in the same row, i.e. not satisfying | ρ (A, T) simultaneously 1 ) L =1 and | ρ (a, T) 2 ) L =1. In other words, each row contains at most one (t) 1 ,t 2 ) The order interaction T is such that | ρ (a, T) | =1. From this we can get the following inequality
That is to say that the first and second electrodes,
is solved to obtain
1.3 discrimination method of two covering arrays, design matrix A N×k It is sufficient condition for the definition of the coverage array that all t-order interactions of all factors occur at least once. Naturally, we wish to derive a given design matrix A from N×k And extracting all interaction information, and judging whether all t-order interactions of all factors occur at least once, but the judgment times required by the judgment is extremely large due to combined explosion, and particularly when k, t and v are large, the calculation amount of the judgment is increased upwards in terms of factorial numbers. Therefore, an efficient discrimination algorithm with polynomial level operation quantity is creatively designed, so that the design and discrimination of the two covering arrays in practice have realizability.
We derive the design matrix A from N×k Extracting t 1 +t 2 Columns (i.e. k from the front, respectively) 1 Column extraction t 1 Column, k from the back 2 Column extraction t 1 Columns), all the remaining column elements are replaced by 0, and finally an interactive subarray is formed
In total, the number of sub-arrays can be extractedNumber is
Is that
We will each interact with the subarray
Conversion into information vectors
And recording the information of the factor interaction of each row in the current subarray.
We get
To
The transformed vector is F N ＝[1,v 1 ,v 2 ,…,v k ] T
Where k is the number of factors and v is the number of levels of each factor.
When v is less than or equal to 10, F can also be counted for simple calculation and understanding N Is designed as F N ＝[1,10,10 2 ,…,10 k ] T
Namely, it is
From the conclusion in the above section, it is known that if A N×k Is a two-part covering array with
Then, the alternative expression is if for A N×k T in (1) 1 +t 2 Column, the number of its non-repetitive interactions reaches v t2 This t 1 +t 2 The column meets the two-part coverage requirement. If for A N×k All of t in 1 +t 2 Column, number of non-repeated interactions can reach v t Then, A can be explained N×k Is one (t) 1 ,t 2 ) A second order covering array.
At the same time, according to the above
As can be seen from the calculation of (a),
the mutual information of each line is stored in the memory, and the repeated mutual lines are reflected as
The same number in (1). Therefore we use the following method to make statistics
The number of the different numbers in the two covering arrays is compared with the lower boundary of the two covering arrays, thereby judging A N×k Whether it is a two-part coverage array.
For is to
Sorting from small to large to obtain
Note book
Is an integer sequence
The difference of the latter term minus the former term, then
Is different from that of
The number of different numbers in (B) is
Number M of non-zero (i) . When all M (i) All satisfy M (i) ≥v t Then, it can be judged that A is N×k Is a two-part positioning array.
1.4 method of generating two coverage arrays from an initial design matrix
In the previous section, how to determine a given design matrix A N×k Whether it is a two-part coverage array, this section is that if A N×k Whether or not it is not a two-part covering matrix, how we are at A N×k Further optimization is carried out on the basis, and finally two covering arrays meeting the requirements are obtained. Thus, we transform the problem of finding two coverage arrays into an optimization problem, specifically, an integer programming problem. Since the genetic algorithm has a good application effect in the optimization problem, especially in the solution of the integer programming problem, we use the genetic algorithm to solve the integer programming problem, and refer to relevant documents for the principle and method of the genetic algorithm, which are not described herein again.
The algorithm problem is solved, and the only problem existing now is how to apply A N×k The process of obtaining two coverage arrays by optimization is described as an integer programming problem, namely how to define an objective function of the integer programming. The definition of the objective function is described below.
From the previous section, if
Covering two parts (t) 1 ,t 2 ) The order interaction then has M (i) ≥v t If A is N×k Is a two-part array, then there are all M (i) ≥v t . Thus, we can define an integer-programming objective function as
Wherein A is N×k ＝(a ij ) N×k ，1≤a ij V is not more than v, an integer
The design of the test of the two equal-level covering arrays is implemented, and two test factor sets K are given 1 And K 2 (ii) a The number of factors in each set is k 1 And k 2 。k＝k 1 +k 2 The number of levels of each factor is v; total number of trials N, and interaction strength t within two factor sets 1 And t 2 (ii) a Total interaction intensity t = t 1 +t 2 Structure K of 1 And K 2 Upper arbitrary t 1 +t 2 The interactive two-part coverage array is shown as the following sets of algorithms.
Algorithm 1: generating step of initial design matrix
Step 1: for a given number of trials N, a vector h = (h) is generated 1 ,h 2 ,…h k ) Wherein 1= h 1 ＜h 2 ＜…＜h k Are all positive integers from 1 with N, k being these h j The number of (2) is also a factor number.
And 2, step: generating an Nxk matrix A from h N×k ＝(a ij ) N×k Wherein a is ij ＝ih j (modN) +1. Matrix A N×k Is an initial design with N points, one for each row. It is clear that A N×k Is one permutation of 1,2, \8230;, N.
And step 3: noting the initial design matrix as
And 2, algorithm: a step of changing the matrix array into an information array and judging whether the two covering arrays are satisfied
Step 1: is _ BCA = true;
step 2: to pair
Repeating the following steps 2 to 8;
and step 3: from design matrix A N×k Front k 1 Column extraction t 1 Column, k from the back 2 Column extraction t 2 Column, all the remaining column elements are replaced by 0 to obtain a subarray
And 4, step 4:
and 5: to pair
Sorting from small to large to obtain
Step 6: computing
Difference of (2)
Namely that
And 7:
and step 8: if M is (i) ≥v t Returning to the step 2; otherwise, is _ BCA = false, end.
And (3) algorithm: optimizing step for two covering arrays by using genetic algorithm
Step 1: f (A) N×k )＝0，min F ＝v t ，
As used in the following calculations
Step 2: to pair
Repeating the following steps 3 to 6;
and step 3: according to Algorithm 2, M is calculated (i) ；
And 4, step 4: computing
And 5: f (A) N×k )＝F(A N×k )+f i ；
Step 6: if it is not
Turning to step 7; otherwise, turning to the step 2;
and 7: if F (A) N×k ) =0, the optimal solution has been obtained, i.e. the current design matrix is two covering matrices, and end; otherwise, turning to step 8;
and 8: if F (A) N×k )≥min F ，
Otherwise
min F ＝F(A N×k ) (ii) a And 8: if min F The set m times are reached without change, the algorithm is ended, and the current time is taken
Designing a matrix for the optimum; otherwise, using genetic algorithm to A N×k Performing mutation adjustment to obtain new A N×k And returning to the step 2.
The technical scheme is suitable for the test design problem under the two-party confrontation condition, and can cover the space between two parts (t) 1 ,t 2 ) A design matrix of order interactions. The advantages of the method are briefly described below by comparison with conventional overlay arrays.
TABLE 1 size ratio of two covering arrays to conventional covering array at t-order interaction in factor 4 level of 7
It is obvious from the above table that, under the condition of fully considering the characteristics of the two parts of confrontation factors, the design size required by the two covering arrays in the invention is less than 10% in proportion, even about 5% in half of the proportion compared with the traditional covering array. That is, the test through the design of the two covering arrays is greatly reduced compared with the traditional covering array method, thereby greatly reducing the test consumption and the test time,
in a specific practical application example, the invention has completely implemented a test design of a radar countermeasure test, in the test, a radar side has 5 controllable factors, an interference side has 4 controllable factors, and the settable state of each factor is 2. The two coverage arrays (1, 1) -BCAN (5, 4, 3) generated by the present invention are shown in Table 2 below.
TABLE 2 example of a Radar challenge test design
Claims (2)
1. A test design method of an equal-level two-part covering array is characterized by comprising the following steps: is directed to radar countermeasureTwo-party confrontation test, which fully covers the factor level interaction problem of two confrontation parties, and gives two test factors, namely the same set of level number of each factor and the total number n of tests, and the interaction strength t inside the two factor sets 1 And t 2 Total interaction strength of t = t 1 +t 2 ；
Firstly, calculating the minimum sample capacity required by the two covering arrays, if the capacity is smaller than the specified total number n of tests, reminding a user that the two covering arrays under the specified interaction intensity do not exist, and if not, continuing; transforming the design matrix into an information matrix;
then, transforming the value of the information matrix into an optimization function, and searching an integer solution of the optimization problem by using a genetic algorithm, wherein the integer solution is transformed back to form two covering arrays meeting the requirements; the specific contents are as follows:
a. two-part coverage array proposing and defining
In the electronic countermeasure test, factors participating in the test and their levels are divided into two parts for both of the countermeasures, and the test and data analysis also focus on the countermeasures between the two parts, in other words, the test is intended to focus on the interaction between the factors of the two parts in the test, rather than on the interaction of all the factors; defining two covering arrays, namely two covering arrays; i is v Defined as the set of all positive integers starting from 1 to v,
Defined as the set of u positive integers starting from k +1; note a = (a) ij )(i∈I N ，j∈I k ) Is an N x k matrix, k starting from 1 And k after 2 Column composition in which the j-th column element is taken from the group consisting of the symbols v j Set of compositions V j Where k is 1 +k 2 = k; at this time, a is rewritten as a = (a) 1 ：A 2 )，A 1 ＝(b ij )，A 2 ＝(c lm )，
thus, the concept and definition of the two part locator array is as follows:
definition 1: let X be a set of v-symbols, let A = (a) ij )(a ij E.x) is an N X k matrix, from the previous k 1 Column and following k 2 The column consists of two parts; if for any subset
Each from A of N × (t) 1 +t 2 ) Subarrays
Comprising X t At least 1 time, the matrix A is then a two-part overlay matrix, denoted as (BCA (N; t) 1 ，t 2 ，k 1 ，k 2 V)); where N is the overlay size (number of rows), t = t 1 +t 2 Is the coverage intensity, k is the number of factors (degrees), v is the number of states per factor, i.e. the number of levels;
note 1: (BCA (N; t) 1 ，t 2 ，k 1 ，k 2 V)) represents factors that affect the response; the values in the columns represent the setting or value of the factor, i.e., the level of the factor; each row represents a set of tests to be run, with a value specified for each factor; then, in all tests, taken from
All v within the factor combination set t A combination of horizontal values at (BCA (N; t) 1 ，t 2 ，k 1 ，k 2 V)) occurs at least 1 time in a row;
note 2: if (BCA (N; t) 1 ，t 2 ，k 1 ，k 2 V)) there is a minimum dimension N, at which time the two coverage arrays are (t) 1 ，t 2 )-BCAN(k 1 ，k 2 V); the minimum size of the two-part coverage array reduces the time and cost of the experiment when other parameters are fixed; the construction and optimization of the two covering arrays are given parameters (t) 1 ，t 2 ，k 1 ，k 2 V) finding two covers satisfying the minimum sizeCover array (t) 1 ，t 2 )-BCAN(k 1 ，k 2 ，v)；
b. Lower bound of sample capacity for an equal horizontal two-coverage array, two-coverage array (t) 1 ，t 2 )-BCAN(k 1 ，k 2 V) a lower bound on size N sample capacity that can help determine whether the two coverage arrays are optimal;
let k 1 ≥t 1 ≥1，k 2 ≥k 2 1 and v 2 are all integers, then
Let A be (t) 1 ，t 2 )-BCAN(k 1 ，k 2 V), note k = k 1 +k 2 ，t＝t 1 +t 2 ；
For any 1 ≦ j 1 ＜j 2 ＜…＜j t ≤k 1 And
let T = ((j) 1 ，x 1 )，…，(j t ，x t ) Then T is one (T) 1 ，t 2 ) Step interaction; note the book
for satisfying 1 ≦ j 1 ＜j 2 ＜…＜j t ≤k 1 And
any t column vectors, since A is a two-part covering array, if A is different (t) 1 ，t 2 ) The order interaction occurs at least once, and there will be a total of v t Then (N-v) remains t ) The rows being repetitions(t 1 ，t 2 ) Step interaction; this indicates that at least v t -(N-v t ) A (t) 1 ，t 2 ) The order interaction occurs exactly once, that is,
in addition, since the array is a two-part array, two (t) are arbitrary 1 ，t 2 ) Order interaction T 1 And T 2 Not in the same row, i.e. not satisfying | ρ (A, T) simultaneously 1 ) L =1 and | ρ (a, T) 2 ) | =1; i.e. each row contains at most one (t) 1 ，t 2 ) Order interaction T is such that | ρ (a, T) | =1; thus, the following inequality is obtained
That is to say that the first and second electrodes,
is solved to obtain
c. Method for discriminating two covering arrays, design matrix A N×k A sufficient condition for being a definition of a coverage array is that all t-order interactions of all factors occur at least once; from a given design matrix A N×k Extracting all interaction information, and judging whether all t-order interactions of all factors occur at least once, but the judgment times required by the judgment is extremely large due to combined explosion, and particularly when k, t and v are large, the calculation amount of the method increases in a factorial manner; therefore, a high-efficiency discrimination algorithm with polynomial levels of operation quantity is adopted and used for the two covering arrays;
design matrix A N×k Extracting t 1 +t 2 Columns, i.e. k from the front respectively 1 Column extraction t 1 Column, k from the back 2 Column extraction t 1 Column, all the rest column elements are replaced by 0, and finally an interactive subarray is formed
The total number of the sub-arrays which can be extracted is
Is made of
Each interactive subarray
Conversion into information vectors
Recording the factor interaction information of each line in the current subarray;
get
To
The transformed vector is F N ＝[1，v 1 ，v 2 ，…，v k ] T
Wherein k is the number of factors and v is the number of levels of each factor; when v is less than or equal to 10, F is added N Is designed as F N ＝[1，10，10 2 ，…，10 k ] T
Namely, it is
If A is N×k Is a two-part covering array with
Then, the alternative expression is if for A N×k T in (1) 1 +t 2 Column, with non-repeating number of interactions up to v t1 This t 1 +t 2 The column meets the requirement of two-part coverage;
if for A N×k All of t in 1 +t 2 Column, number of non-repeated interactions can reach v t Then, A can be explained N×k Is one (t) 1 ，t 2 ) A second-order covering array;
at the same time, according to the above
Is calculated by the calculation of (a) and (b),
the mutual information of each line is stored in the memory, and the repeated mutual lines are reflected as
The same number in (1); therefore, the following method is adopted for statistics
The number of the different numbers in the two covering arrays is compared with the lower bound of the two covering arrays, thereby judging A N×k Whether it is a two-part coverage array;
to pair
Sorting from small to large to obtain
Note book
Is an integer sequence
The difference of the latter term minus the former term, then
Is different from that of
The number of different numbers in the same table is
Number M of non-zero (i) (ii) a When all M (i) All satisfy M (i) ≥v t Then, A can be determined N×k Is a two-part positioning array;
d. method for generating two coverage arrays from an initial design array, if A N×k Whether it is not a two-part covering array, how it is in A N×k Further optimizing on the basis of the obtained data to finally obtain two covering arrays meeting the requirements; converting the problem of searching the two coverage arrays into an optimization problem-an integer programming problem; solving this integer programming problem by means of a genetic algorithm;
how to then combine A N×k The process of obtaining two coverage arrays by optimizing is designed into an integer programming problem, namely the definition of an objective function of the integer programming, and the definition of the objective function is as follows:
if it is not
Covering two parts (t) 1 ，t 2 ) The order interaction then has M (i) ≥v t If A is N×k Is a two-part array, then there are all M (i) ≥v t Thus, an objective function for integer programming is defined as
Wherein A is N×k ＝(a ij ) N×k ，1≤a ij V is not more than v, an integer
2. The method of claim 1, wherein the method comprises: the experimental design of the two covering arrays is implemented as follows: given two sets of test factors K 1 And K 2 The number of factors in each set is k 1 And k 2 ；k＝k 1 +k 2 The number of levels of each factor is v, the total number of trials N, and the interaction strength t within the two factor sets 1 And t 2 (ii) a Total interaction intensity t = t 1 +t 2 Structure K of 1 And K 2 Upper arbitrary t 1 +t 2 The interactive two-part coverage array comprises several groups of algorithms as follows:
algorithm 1: generating step of initial design matrix
Step 1: for a given number of trials N, a vector h = (h) is generated 1 ，h 2 ，…h k ) Wherein 1= h 1 ＜h 2 ＜…＜h k Are all positive integers from 1 with N, k is these h j The number of (2) is also a factor number;
and 2, step: generating an Nxk matrix A from h N×k ＝(a ij ) N×k Wherein a is ij ＝ih j (mod N) +1; matrix A N×k The method is an initial design with N points, wherein each line corresponds to a design point; it is clear that A N×k Is one permutation of {1,2, \8230;, N };
and step 3: remember the initial design matrix as
And 2, algorithm: a step of changing the matrix array into an information array and judging whether the two covering arrays are satisfied
Step 1: is _ BCA = true
And 2, step: to pair
Repeating the step 2 to the step 8;
and step 3: from the design matrix A N×k Front k 1 Column extraction t 1 Column, k from the back 2 Column extraction t 2 Column, all the remaining column elements are replaced by 0 to obtain a subarray
And 4, step 4:
and 5: for is to
Sorting from small to large to obtain
And 6: computing
Difference of (2)
Namely that
And 7:
and step 8: if M is (i) ≥v t Returning to the step 2; otherwise, is _ BCA = false, end;
algorithm 3: optimizing step for two covering arrays by using genetic algorithm
Step 1: f (A) N×k )＝0，min F ＝v t ，
As used in the following calculations
Step 2: to pair
Repeating the following steps 3 to 6;
and step 3: according to Algorithm 2, M is calculated (i) ；
And 4, step 4: computing
And 5: f (A) N×k )＝F(A N×k )+f i
Step 6: if it is not
Turning to step 7; otherwise, turning to the step 2;
and 7: if F (A) N×k ) =0, the optimal solution has been obtained, i.e. the current design matrix is the two covering matrices, and the process ends; otherwise, turning to step 8;
and 8: if F (A) N×k )≥min F ，
Otherwise
min F ＝F(A N×k )；
And step 9: if min F The set m times are reached without change, the algorithm is ended, and the current time is taken
Designing a matrix for the optimal;
otherwise, using genetic algorithm to A N×k Performing mutation adjustment to obtain new A N×k Returning to the step 2;
the above procedure is applicable to experimental design under two-party confrontation conditions, and enables the creation of a cover between the two parts (t) 1 ，t 2 ) A design matrix of order interactions.
Priority Applications (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910601409.1A CN110457748B (en) | 2019-07-04 | 2019-07-04 | Test design method for two equal-level covering arrays |
Applications Claiming Priority (1)
| Application Number | Priority Date | Filing Date | Title |
|---|---|---|---|
| CN201910601409.1A CN110457748B (en) | 2019-07-04 | 2019-07-04 | Test design method for two equal-level covering arrays |
Publications (2)
| Publication Number | Publication Date |
|---|---|
| CN110457748A CN110457748A (en) | 2019-11-15 |
| CN110457748B true CN110457748B (en) | 2022-10-11 |
Family
ID=68482222
Family Applications (1)
| Application Number | Title | Priority Date | Filing Date |
|---|---|---|---|
| CN201910601409.1A Expired - Fee Related CN110457748B (en) | 2019-07-04 | 2019-07-04 | Test design method for two equal-level covering arrays |
Country Status (1)
| Country | Link |
|---|---|
| CN (1) | CN110457748B (en) |
Family Cites Families (3)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN105046079B (en) * | 2015-07-17 | 2018-04-27 | 北京航空航天大学 | A kind of field mouthful test design method based on the optimal interior table designs of D- |
| US10133656B2 (en) * | 2016-02-08 | 2018-11-20 | Tata Consultancy Services Limited | Systems and methods for generating optimized covering arrays |
| CN109407045B (en) * | 2018-10-10 | 2020-05-22 | 苏州大学 | Non-uniform sensor array broadband signal direction-of-arrival estimation method |
-
2019
- 2019-07-04 CN CN201910601409.1A patent/CN110457748B/en not_active Expired - Fee Related
Also Published As
| Publication number | Publication date |
|---|---|
| CN110457748A (en) | 2019-11-15 |
Similar Documents
| Publication | Publication Date | Title |
|---|---|---|
| Song et al. | Efficient residual dense block search for image super-resolution | |
| Satapathy et al. | Weighted teaching-learning-based optimization for global function optimization | |
| Agarwal et al. | Notice of Violation of IEEE Publication Principles: K-means versus k-means++ clustering technique | |
| Wang et al. | An orthogonal design based constrained evolutionary optimization algorithm | |
| CN110383282A (en) | The system and method calculated for mixed signal | |
| US20170357889A1 (en) | Sparse neuromorphic processor | |
| Vignac et al. | Top-n: Equivariant set and graph generation without exchangeability | |
| Qin et al. | Lightweight hierarchical residual feature fusion network for single-image super-resolution | |
| Johner et al. | Efficient evolutionary architecture search for CNN optimization on GTSRB | |
| Rebain et al. | Attention beats concatenation for conditioning neural fields | |
| Liu et al. | An efficient group skip-connecting network for image super-resolution | |
| Vasicek et al. | Evolution of efficient real-time non-linear image filters for fpgas | |
| CN110457748B (en) | Test design method for two equal-level covering arrays | |
| Colbourn et al. | Subspace restrictions and affine composition for covering perfect hash families | |
| Che et al. | Auto-spikformer: Spikformer architecture search | |
| Tumpa et al. | Utilizing genetic evolution to enhance cellular automata for accurate image edge detection | |
| Liao et al. | Parameter distribution balanced CNNs | |
| Zou et al. | LMSN: a lightweight multi-scale network for single image super-resolution | |
| Rikvold et al. | Individual-based predator-prey model for biological coevolution: Fluctuations, stability, and community structure | |
| Wang et al. | Slim-RFFNet: Slim deep convolution random Fourier feature network for image classification | |
| CN114237548A (en) | Method and system for complex dot product operation based on nonvolatile memory array | |
| Ghimire et al. | Loss-aware automatic selection of structured pruning criteria for deep neural network acceleration | |
| Liu et al. | Improving residual block for semantic image segmentation | |
| Morales-Romero et al. | Time-multiplexing cellular neural network in FPGA for image processing | |
| CN107038451B (en) | Network learning method and training method suitable for gray scale picture |
Legal Events
| Date | Code | Title | Description |
|---|---|---|---|
| PB01 | Publication | ||
| PB01 | Publication | ||
| SE01 | Entry into force of request for substantive examination | ||
| SE01 | Entry into force of request for substantive examination | ||
| GR01 | Patent grant | ||
| GR01 | Patent grant | ||
| CF01 | Termination of patent right due to non-payment of annual fee |
Granted publication date: 20221011 |