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Monotonic vector space (MVS), as a novel model in which there exist some monotonic mappings, is proposed. MVS model is an abstract of many practical problems (such as image processing, system capability engineering, etc.) and includes many useful important operations. This paper, as the first one of series papers, discusses the MVS framework, relative important concepts and important operations including partition, synthesis, screening, sampling, etc. And algorithms for these operations are the focus of this paper. The application of these operations in system capability engineering will be dealt with in the second part of this series of papers.

MVS is a special vector space where there exist some monotonic mappings. This model actually represents a lot of practical problems. Several examples are given as follows:

Example 1 (

Example 2 (

Example 3. A monotonic vector space P for a ground-to-air missile system may include the searching radar detecting capability metric dimension

Many practical problems can be solved with the operations defined in MVS. For example, for a vector space composed of system capability metrics [1,2], the requirement analysis of capability can be done with partitioning operation in MVS [

Definition 1. Monotonic Vector Space (MVS)

According to the different type combination of dimension and mapping, MVS can be classified into four types, i.e. discrete deterministic type, continuous deterministic type, discrete stochastic type, and continuous stochastic type. The MVS researching framework is shown in

• Partitioning. This operation is used to attain the effective area, which covers all the points satisfying certain requirement, through segmenting MVS according to certain requirement criteria. For example, the effective detecting range of a jammed radar, the effective capability indicators fields, and so on, can be gotten through carrying out partitioning operation in MVS.

• Synthesizing. This operation is used to get the synthesized area of several effective areas, each of which contains points meeting different requirement targets. For example, acquiring the intersection of several effective areas in MVS is a typical synthesizing operation.

• Screening. This operation is used to find the essential elements such as key dimensions, key subarea, etc, in the MVS. With this operation, some less important dimensions or subarea of MVS can be overlooked in the tackling process of the practical problems. And thus the computation burden of other operations in MVS can be decreased remarkably.

• Sampling. If the mapping in MVS is of stochastic type, sampling the vector points in a large-scale MVS with high efficiency and quality have significance on stochastic simulation experiment (might be considered as stochastic mapping) in MVS.

• Searching. This operation is to find high sensitivity region for some mappings in MVS.

• Other operations. There are some other operations except for the above mentioned, such as slicing, projecting, rotating, etc. These operations, similar to OLAP or data mining, are useful for getting interesting information in the high dimensional MVS.

To solve practical problems, some new operations (except for those mentioned above) in MVS may need to be studied. In the following sections of this paper, the algorithms for partitioning operation in continuous/discrete-type MVS, synthesizing operation in continuous-deterministic-type MVS, sampling operation in stochastic-type MVS, screening operation and Searching operation in corresponding MVS, and so on, will be dealt with in detail as the following sections.

Partitioning is a very valuable operation in MVS and can be used to solve many practical problems such as capability requirement analysis, imaging processing, multi-objective optimization, etc. The requirement-based partitioning operation in MVS of continuous deterministic type is to be discussed in this section.

In the mapping

For simplicity, all the dimensions of MVS are assumed to be monotonically increasing (monotonically decreasing dimension can also be easily transformed into monotonically increasing dimension). MVRL is the set of points meeting

According to

Definition 2. The minimal envelop hyperbox (MEH). MEH refers to the cross product of each dimension’s interval [lower bound, upper bound] and it may also be represented as such simplified form

For example, assuming that x, y are the indexes of a system and that the MVRL is the point set meeting the inequation

Definition 3. The maximal included hyperbox (MIH) and its opposite hyperbox. The MIH is wholly contained in the MVRL and its diagonal is in the superposition state with the diagonal of the MEH. We can attain the MIH through dividing the interval of the MEH’s each dimension with the same proportion. We give the rigid mathematics definition of the MIH as follow. Set a coefficient

and

For the example mentioned above,

is the maximal included hyperbox (MIH) and

Theorem 1. Any point in the opposite hyperbox of the maximal included hyperbox does not meet the system requirement, i.e., is not in the MVRL.

Let

Assume

Let

Let

and

(according to the monotonic property of

In addition,

So

This algorithm recursively approximates MVRL with MIHs, and the algorithm steps are as follows.

Step 1 Determine the requirement

Step 2 Find the minimal envelop hyperbox (MEH):

Step 2.1 Set

Step 2.2 All dimensions except the kth dimension are set to 0 (the minimal value). Due to the monotonic property of

Step 2.3 If

Step 3 Find the MIH:

Step 3.1 Judge whether the volume of MEH is smaller than the value E (which represents the exit condition). If true, make interpolation and exit. If the interval length of one certain dimension in the MEH is smaller than a certain value, this dimension need not be divided anymore and thus will be deleted;

Step 3.2 According to the monotonic property of

So, the binary method is applied to get the

Step 3.3 Each dimension interval of the MEH is divided into two parts. One part is inside the MIH and marked by “0”, and the other is outside the MIH and marked by “1”. So, there are altogether ^{n}−1. Code “0” whose binary digits are all 0 represents the MIH where any point meets the system requirement and will be saved. Code “1” whose binary digits are all 1 represents the opposite hyperbox of the MIH where any point does not meet the system requirement according to Theorem 1 and will be removed. Regard each of the rest combinations as a new MEH and recursively go to Step 3.

There exist monotonic realtions in MVS, thus we can get the special benefit of high efficiency as many famous algorithms which reduce calculation based on monotonicity (i.e. apriori algorithm in data mining [

This algorithm can be applied to solve many problems in which there exists the monotonicity. For example,it can be applied to efficiently draw the radar detecting range when the radar is jammed by jammer and efficiently seek the pareto locus in multi-object programming.

This algorithm is essentially a partition technique for some special figures. If the assumptions is met, it is superior to the classic triangulation method [11,12] by triangular or tetrahedron for the following reasons:

1) As the above mentioned, hyperbox is more suitable for many kinds of operations than triangular or tetrahedron.

2) It is effective and efficient to partition 3-dimensional or more than 3-dimensioal figure for which triangulation method is very difficult to be applied [

An effective algorithm for partitioning in MVS of discrete type is proposed in this section. Based on the assumption that each dimension of MVS is divided into several segments, this algorithm can efficiently find out the combinations which meet the specified requirements. If each dimension of m-dimension MVS is divided into

Step 1 Specify a representative value (RV) for each combination and fill all the RVs into the combination table. RVs will be used to determine which combination is the next one to be judged whether meeting the requirement or not. Definition of RV is based on two other basic conceptions, i.e., the lower-cut set and the uppercut set. Explanations for these conceptions will be given with the following example. For example, if the combination #3#3#3#4 with m = 4 and n = 5 meet the specified requirement (i.e.,

In the above formula,

Step 2 Scan the combination table, which is dimensioned by a heap structure to make the combination with the maximal RV found quickly, and select the combination with the maximal RV by using the greedy tactics. Firstly, assume combination #I1#I2#I3#I4 is the one with the maximal RV at present.

1) If the combination #I1#I2#I3#I4 meets the specified requirement, all the elements in its lower-cut set will be deleted from the combination table, which likely makes the lower-cut set element number of any other combination decreased. The method for adjusting the lower-cut set element number of any other combination (such as #J1#J2#J3#J4) is not so complicated. With setting #Ki =min(#Ii, #Ji)

2) If the combination #I1#I2#I3#I4 does not meet the specified requirement, all the elements in its upper-cut set will be deleted from the combination table, which likely makes the upper-cut set element number of any other combination decreased. The method for adjusting the upper-cut set element number of any other combination (such as #J1#J2#J3#J4) is not so complicated. After setting #Ki = min(#Ii, #Ji)

Step 3 Adjust the combination table. If the selected combination meets the specified requirement, this combination and all the elements in its lower-cut set will be deleted from the combination table, or else this combination and all the elements in its upper-cut set will be deleted from the combination table. For example, if combination #3#3#3#4 meets the specified requirement, all the elements in its lower-cut set such as #1#1#1#1, #1#2#2#2, #3#3#3#3, etc. in the combination table would be deleted.

Step 4 Go to step 2 until the combination table is empty.

To illustrate the above algorithm in detail, it is necessary to discuss an example. In the following example, a discrete MVS with four dimensions (i.e.,

Now the partitioning operation can be carried out under the guidance of the above algorithm.

Step 1 Determine the corresponding RVs for all the combinations, the number of which is 256

Step 2 Single out the combination with the maximal RV from combination table. At the very beginning, combination #3#3#2#2 is found to be the combination with the maximal RV

Step 3 Adjust the combination table. For example, since combination #3#3#2#2 meets the specified requirement, on the one hand, this combination and all the elements in its lower-cut set are to be deleted from the combination table; on the other hand, the lower-cut set element number of any remaining combination in the combination table may also need to be modified. Now take the combination #4#1#1#1 as an example. Combination

#K1#K2#K3#K4 = #3#1#1#1 with #I1#I2#I3#I4 = #3#3#2#2 and #J1#J2#J3#J4 = #4#1#1#1 can be easily gotten by using the method mentioned above. Because combination #3#1#1#1 is found to be still in the combination table, the lower-cut set element number and RV of combination #4#1#1#1 must be modified. So the adjusted lower-cut set element number of #4#1#1#1 is equal to its original lower-cut set element number minus the lower-cut set element number of combination #3#1#1#1 (i.e.,

Step 4 Go to step 2 until the combination table is empty. In this example, all the combinations meeting the specified requirement such as #1#2#2#4, #2#1#2#4,···, #3#3#2#2, etc., are acquired after 67 iterations, which means that only about one quarter of all the combinations are calculated and the algorithm elaborated in this section is of high efficiency.

For a practical problem, there are usually several different kinds of requirement targets, each of which obviously corresponds to a different MVRL. Therefore, intersection operation of multiple MVRLs is of great significance to the solution of one practical problem. In this section, two algorithms for one kind of synthesizing operations (i.e., intersection operation of multiple MVRLs) are presented.

The shape of a MVRL approached by many hyperboxes is usually irregular. The intersection of multiple MVRLs is approximately to be the intersection of the hyperboxes included in the MVRLs. Therefore, the former can be acquired through finding out the latter. The method of seeking the total intersection of all the hyperboxes is very simple. However, its time cost is large. For example, if there are two MVRLs whose numbers of hyperboxes are respectively

Actually, all the MEHs and MIHs acquired in the process of producing a MVRL can be naturally organized into an including relation tree (heap) shown in

If all the requirements have the same constraint direction on any system dimension, i.e., achievement degree of all the requirement indices becoming larger or smaller with any system dimension value increasing or decreasing, approaching method can be used to seek the intersection of multiple MVRLs. The algorithm based on approaching method is stated as follows.

Step 1 Get the intersection of the MEHs of all the MVRLs, i.e., the MEH of the intersection of all the MVLRs.

Step 2 Seek all the MIHs, which approximately compose the intersection of all MVRLs, with the related algorithm mentioned above.

The above two steps are illustrated in

If this tree is saved in the memory, the time cost of the direct method may be largely reduced. The reason is that if the hyperbox of one MVRL has null intersection with one certain MEH of another MVRL, it will have no common intersection with all the MIHs included in this MEH, and so, the related intersection operations can be omitted.

Besides intersection operation, which is a relatively simple and typical synthesizing operation, there are still some other complicated synthesizing operations such as union operation, union-intersection mixed operation (i.e., carrying out intersection operation on some dimensions and union operation on the others). And all of these synthesizing operations are very useful in solving practical problems and should be studied thoroughly in the future.

Dimensions screening is very important for all kinds of operations in MVS because it can alleviate calculation burden to a great extent. Pareto principle or 20 - 80 rule implies that only a few factors are really important. The purpose of screening is to find out the really important ones among all the dimensions of MVS. As mentioned above, in the stochastic type MVS, the mapping

Sequential bifurcation (SB) method originally developed by Bettonvil in his doctoral dissertation [

1) The metamodel of

2) The metamodel is a first-order polynomial with

The dimensions

To explain some mathematical details of SB, the following notations are used.

A simple estimate of this group effect based on replication r is

SB starts with calculating the two most extreme scenarios: in scenario 1, all k dimensions are set at their low levels so

so

Likewise it follows that the individual main effect is estimated:

The SB procedure is sequential. Its first step places all dimensions into a single group, and tests whether or not that group of dimensions has an important effect. If the group indeed has an important effect, then the second step splits the group into two subgroups (i.e., bifurcates) and tests each of these subgroups for importance. The next steps continue in a similar way, discarding unimportant subgroups and splitting important subgroups into smaller subgroups. In the Final step, all individual dimensions that are not in subgroups identified as unimportant, are estimated and tested. The steps are showed below:

1) Select the group whose dimensions are from

2) If this group’s

3) If this group’s

This section introduces a simple algorithm (i.e., sequential bifurcation) for screening operation in MVS. SB method based on some assumptions is effective and efficient in the solving of many practical problems(i.e., supply chain analysis [6), And some investigator, such as the authors of [7-9] have successfully extended the simple SB.

Sampling operation is important for experiments in MVS. The purpose of this operation is to effectively and efficiently select the points used in all kinds of experiments. Latin hypercube sampling (LHS) is very fit for MVS. The algorithm for acquiring

The relative advantages of LHS method can be revealed with an example. The estimation value

In the above expressions,

Supported by NSFC, China, PRC, Grant No 70871120.

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