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The D numbers theory is a novel theory to express uncertain information. It successfully overcomes some shortcomings of Dempster-Shafer theory, such as the conditions of exclusiveness hypothesis and completeness constraint. However, the combination rule of D numbers does not satisfy the associative property, which leads to limitations in practical application for D numbers. In this paper, the improved D numbers theory is proposed to overcome the weakness based on the analysis of D numbers’ combination rule. A new algorithm is constructed with the strict proof to simplify the combination rule. The similarities and differences among DS theory, D numbers, and the improved D numbers are introduced with the numerical analysis. An illustrative example of the radiation source identification is presented to demonstrate the effectiveness of the improved method.

In real applications, uncertainty reasoning is widely applied to information fusion, risk assessment, pattern recognition, artificial intelligence, decision-making, and so forth. Many methods such as fuzzy sets theory, Dempster-Shafer theory, and possibility theory, have been proposed to model uncertain information.

Dempster [

The DS evidence theory combination rule cannot effectively deal with high-conflict information [

On the application side, Zuo [

Though D numbers theory overcomes some defects of DS theory and is widely used in different fields, the combination rule of D numbers does not satisfy the associative property. In order that multiple D numbers can be combined correctly and efficiently, a combination operation for multiple D numbers is developed in [

The remainder of this paper is organized as follows. A brief introduction about the DS theory and D numbers is given in Section

Let

A mass function is a mapping

In real applications, for the same problem, there may be many different sources that acquire various evidences.

Considering two pieces of evidence from different and independent information sources, denoted by two BPAs

D numbers theory is more suitable to the framework and is defined as follows.

Let

Let

Like the mass function, D numbers theory also has the combination rule to combine two D numbers.

Let

Let

In the meanwhile, for the special D number, an aggregation operator is defined as follows.

Let

As a novel theory to express uncertain information, D numbers’ combination rule does not preserve the associative property. For n D numbers, there are

The combination rule of D numbers and improved D numbers.

In order for D numbers’ combination rule to satisfy the associative property, in this paper, the improved D numbers’ combination rule with parallel structure is proposed.

Before that, a novel algorithm

The parameter

When

The parameter

When

The improved D numbers’ combination rule is defined as follows.

Let

When

Similarly, the parameter

In the case of obtaining n D numbers at the same time, the proposed method is able to satisfy the associative law. Unfortunately, the more common situation in practical engineering applications is that we first get a part of D numbers and then obtain the others and calculate their fusion results step by step. The following is a study of the D number combination rule for time series evidence.

In order to facilitate the explanation of the problem, it is worthwhile to record the parameter

Flowchart of evidence D combination rule flow chart.

Though the D numbers derive from DS theory, the two theories solve different problems. The specific similarities and differences of the two theories are summarized as shown in Table

Similarities and differences of D numbers and DS theory.

DS theory | D numbers | ||
---|---|---|---|

Differences | The elements in the frame of discernment | Mutually exclusive | |

Basic probability assignment | Complete | | |

Major application fields | An extension of probability theory and Bayesian theory | Multiattribute decision-making | |

Combination rule | Satisfies the associative property | Doesn’t satisfy the associative property | |

| |||

Similarities | A theory to express uncertain information |

To make a comprehensive assessment of a student, we can get the following basic probability assignment based on DS theory.

If we apply the D numbers theory to analyze the problem, we can get the expression as follows.

As we can see from formulas (

Let

Then, we use the combination rules of D numbers and the improved D numbers separately to calculate the combined D number’s integration representation

Combined D number’s integration representation.

| D numbers | Improved D numbers |
---|---|---|

| 0.1719 | 0.4756 |

| 0.2267 | 0.4756 |

| 0.2302 | 0.4756 |

It can be seen from Table

In this section, in order to illustrate the application of the improved D numbers, a simple example about radiation source identification is given.

Radiation source identification mainly includes the analysis and decision-making of the detected signals. The detected signals of radiation source are radio frequency information, which have added up the noise and been modulated by the atmosphere. As a result, the radiation source identification can be regarded as an uncertainty reasoning process.

There are many theories to express uncertain information. In this paper, we will use the improved theory to identify the source of radiation. The problem is described in detail below.

There are 3 sources of information to investigate the target separately, indicated as

The decision matrix of multiple source.

| | | |||||||
---|---|---|---|---|---|---|---|---|---|

Attributes | | | | | | | | | |

| 0.5 | 0.3 | 0.4 | 0.2 | 0.5 | 0.6 | 0.2 | 0.7 | 0.2 |

| 0.4 | 0.5 | 0.2 | 0.4 | 0.4 | 0.1 | 0.1 | 0.0 | 0.7 |

| 0.1 | 0.1 | 0.1 | 0.0 | 0.1 | 0.1 | 0.5 | 0.1 | 0.1 |

| 0.0 | 0.1 | 0.3 | 0.4 | 0.0 | 0.2 | 0.2 | 0.2 | 0.0 |

The accuracy and detection range of the source can vary with the battlefield environment and climate conditions. Table

The weights of different sources.

| | | |
---|---|---|---|

| 0.4 | 0.6 | 0.5 |

| 0.3 | 0.1 | 0.4 |

| 0.3 | 0.3 | 0.1 |

In this paper, we use the improved D numbers to solve this problem. The application process is presented as follows.

Express these assessment results of Table

Take alternative

The expression of D numbers for

| D numbers |
---|---|

| |

| |

| |

Apply the improved D numbers’ combination rule to combine the D numbers of Table

The results of

| | | | | |
---|---|---|---|---|---|

0.3000 | 0.0556 | 0.2333 | 0.0519 | 0.2667 | 0.0519 |

0.4667 | 0.0519 | 0.4000 | 0.0481 | 0.4333 | 0.0481 |

0.3000 | 0.0407 | 0.2333 | 0.0370 | 0.2667 | 0.0370 |

0.4000 | 0.0370 | 0.3333 | 0.0333 | 0.3667 | 0.0333 |

0.5667 | 0.0333 | 0.5000 | 0.0296 | 0.5333 | 0.0296 |

0.4000 | 0.0222 | 0.3333 | 0.0185 | 0.3667 | 0.0185 |

0.4333 | 0.0444 | 0.3667 | 0.0407 | 0.4000 | 0.0407 |

0.6000 | 0.0407 | 0.5333 | 0.0370 | 0.5667 | 0.0370 |

0.4333 | 0.0296 | 0.3667 | 0.0259 | 0.4000 | 0.0259 |

Apply formula (

Calculate and rank the integration representations of

The ranking of decisions.

D number theory | Ranking | Proposed algorithm | Ranking | |
---|---|---|---|---|

| 0.2314 | 1 | 0.3956 | 1 |

| 0.1210 | 3 | 0.2963 | 2 |

| 0.2012 | 2~1 | 0.1378 | 4 |

| 0.0984 | 4 | 0.1704 | 3 |

Analyze the data of Table

From Table

In this paper, the improved D numbers theory is proposed for radiation source identification. In the improved theory, a novel combination rule of D numbers theory is represented to satisfy the associative property. Meanwhile, a new algorithm is constructed with the strict proof to simplify the combination rule. For the application of radiation source identification, the novel theory is simpler and more effective. An illustrative example has shown the improved theory’s effectiveness.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

Xin Guan and Haiqiao Liu conceived the concept and performed the research. Haiqiao Liu conducted the experiments to evaluate the performance of the proposed information fusion algorithm based on the improved D numbers method. Xiao Yi and Jing Zhao reviewed the manuscript. All authors have read and approved the final manuscript.

The work is partially supported by the special fund for Taishan scholar project: 201712072, National Natural Science Foundation of China: 61671463, National Natural Science Foundation of China: 91538201, Excellent Youth Scholar of the National Defense Science and Technology Foundation of China. Professor Xiao Yi provided many suggestions about the arithmetic operations. The Ph.D. students of the authors Jing Zhao and Haiqiao Liu in Naval Aviation University gave some valuable discussion about the definitions.