Piotr Bilinski
email: piotr.bilinski@inria.fr

Michal Koperski
email: michal.koperski@inria.fr

Slawomir Bak
email: slawomir.bak@inria.fr

Francois Bremond
email: francois.bremond@inria.fr

Representing Visual Appearance by Video Brownian Covariance Descriptor for Human Action Recognition

This paper addresses a problem of recognizing human actions in video sequences. Recent studies have shown that methods which use bag-of-features and space-time features achieve high recognition accuracy. Such methods extract both appearance-based and motion-based features. This paper focuses only on appearance features. We propose to model relationships between different pixel-level appearance features such as intensity and gradient using Brownian covariance, which is a natural extension of classical covariance measure. While classical covariance can model only linear relationships, Brownian covariance models all kinds of possible relationships. We propose a method to compute Brownian covariance on space-time volume of a video sequence. We show that proposed Video Brownian Covariance (VBC) descriptor carries complementary information to the Histogram of Oriented Gradients (HOG) descriptor. The fusion of these two descriptors gives a significant improvement in performance on three challenging action recognition datasets.

Introduction

The automatic recognition of human actions in videos has become one of the most active research topic in computer vision. It plays a key role in many important applications, such as smart surveillance systems, intelligent houses, robots, human-computer interfaces, video data indexing and retrieval, sport event analysis, and virtual reality. Action recognition is a very challenging research problem, mainly due to: intra-class variations, occlusions, background clutter, and viewpoint changes.

Over the last decade, many different action recognition methods have been proposed [START_REF] Dollar | Behavior recognition via sparse spatio-temporal features[END_REF][START_REF] Laptev | Space-time interest points[END_REF][START_REF] Laptev | Learning realistic human actions from movies[END_REF][START_REF] Wang | Action recognition by dense trajectories[END_REF], most of which are based on local spatio-temporal features and bag-of-features approach. The local spatio-temporal features have shown to achieve good accuracy in action recognition over various datasets [START_REF] Bilinski | Evaluation of local descriptors for action recognition in videos[END_REF][START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF]. They are able to capture appearance and motion. They are robust to viewpoint and scale changes. Moreover, they are easy to implement and quick to calculate.

The most popular local spatio-temporal descriptors for action recognition are: HOG, HOF, and MBH descriptor. The Histogram of Oriented Gradients (HOG) descriptor [START_REF] Laptev | Learning realistic human actions from movies[END_REF] encodes visual appearance and shape information, whereas the Histogram of Optical Flow (HOF) descriptor [START_REF] Laptev | Learning realistic human actions from movies[END_REF] and the Motion Boundary Histograms (MBH) descriptor [START_REF] Wang | Action recognition by dense trajectories[END_REF] encode motion information.

Wang et al. [START_REF] Wang | Action recognition by dense trajectories[END_REF] have shown that motion information from HOF descriptor is enough to achieve satisfactory classification performance, but it is not enough to fully describe an action. For example, motion information might distinguish eating a banana from peeling a banana, but motion information might have difficulty distinguishing eating snack chips from eating a banana; conversely, appearance information might be more useful in the second than the first task [START_REF] Messing | Activity recognition using the velocity histories of tracked keypoints[END_REF]. Therefore, it is very important to encode also appearance, which can describe an object involved in a particular action. Recent research [START_REF] Wang | Evaluation of local spatio-temporal features for action recognition[END_REF] also have shown that action recognition model can benefit from complementary appearance information, i.e. from the HOG descriptor.

All the above descriptors, i.e. HOG, HOF, and MBH descriptors, are based on a 1-dimensional histogram representation of individual features, and they directly model values of given features. The joint statistics between individual features are ignored, whereas such information may be informative. Therefore, such descriptors might not be discriminative enough.

In image processing, a novel trend has emerged that ignores explicit values of given features, focusing instead on their pairwise relations. The most known example of such an approach is covariance descriptor [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF].

In this paper we focus on pixel level features such as an intensity, image gradient and image second derivative. We propose to find relationships between mentioned features using Brownian covariance, which is an extension of covariance measure. As covariance measures only linear relationship, it might capture insufficient information in a complex environment such as an action recognition.

The described Brownian covariance can catch all kind of relations between two image patches. We propose a new descriptor called Video Brownian Covariance (VBC), which is an extension of Brownian covariance for spacetime video volumes. Using the proposed descriptor we represent relations between appearance features extracted from a video sequence.

The main contributions of this paper are:

Related Work

Over the last decade, methods based on local spatiotemporal features and bag-of-features approach have become very popular.

There are several popular techniques proposed for extraction of local spatio-temporal video volumes. Laptev and Lindeberg [START_REF] Laptev | Space-time interest points[END_REF] have proposed Harris3D point detector. Dollar et al. [START_REF] Dollar | Behavior recognition via sparse spatio-temporal features[END_REF] have proposed Cuboid detector. Willems et al. [START_REF] Willems | An efficient dense and scale-invariant spatio-temporal interest point detector[END_REF] have proposed Hessian detector. Messing et al. [START_REF] Messing | Activity recognition using the velocity histories of tracked keypoints[END_REF] have proposed to track Harris feature points using KLT tracker. Wang et al. [START_REF] Wang | Action recognition by dense trajectories[END_REF] have proposed to use dense sampling and to track such detected points using optical flow and median filtering.

The most popular local spatio-temporal descriptors are: HOG [START_REF] Dalal | Histograms of oriented gradients for human detection[END_REF][START_REF] Laptev | Learning realistic human actions from movies[END_REF], HOF [START_REF] Laptev | Learning realistic human actions from movies[END_REF], and MBH [START_REF] Wang | Action recognition by dense trajectories[END_REF] descriptor.

Covariance based features have been introduced by Tuzel et al. for object detection and texture classification [START_REF] Tuzel | Region covariance: A fast descriptor for detection and classification[END_REF]. They have been successfully applied for object tracking [START_REF] Porikli | Covariance tracking using model update based means on riemannian manifolds[END_REF], shape modeling [START_REF] Wang | Shape and appearance context modeling[END_REF], and face recognition [START_REF] Pang | Gabor-based region covariance matrices for face recognition[END_REF]. Moreover, covariance based features have been also applied for action recognition [START_REF] Guo | Action recognition from video using feature covariance matrices[END_REF][START_REF] Yuan | Human action recognition under log-euclidean riemannian metric[END_REF]. However, covariance measures only linear relationship between features. Second disadvantage is that covariance matrices do not lie on the Euclidean space, and therefore, we cannot directly use them with state-of-the-art machine learning algorithms. Bak et al. [START_REF] Bak | Brownian descriptor: a Rich Meta-Feature for Appearance Matching[END_REF] have proposed Brownian descriptor for person reidentification. The authors look for relationships between pixel locations and intensity gradients. However, their approach is not suitable for action recognition, as it focuses on exact appearance matching.

In contrast, our novel Video Brownian Covariance (VBC) descriptor is able to capture appearance information based on pixel color, 1st and 2nd intensity derivatives. In addition, it handles a dynamic nature of a video as Brownian covariance is calculated on each patch in space-time video volume.

Brownian Covariance

The classical covariance descriptor measures only information on linear dependence between features. This might not be enough to capture the complex structure of many objects. Covariance descriptor may produce a diagonal matrix, which is not a sufficient condition for statistical independence; actually, non-monotone dependence may exists. This indicates information loss when using the covariance descriptor.

To solve the above issues we propose a novel descriptor for video description based on Brownian covariance [START_REF] Bak | Brownian descriptor: a Rich Meta-Feature for Appearance Matching[END_REF][START_REF] Szekely | Brownian distance covariance[END_REF]. The classical covariance measures only the degree of linear relationship between features, whereas Brownian covariance measures the degree of all kinds of possible relationships between features.

Brownian Covariance

Brownian descriptor is based the theory in mathematical statistics related to the Brownian motion [START_REF] Szekely | Brownian distance covariance[END_REF]. The descriptor is based on the distance covariance statistics that measures the dependence between random vectors in the arbitrary dimension. The mathematical notations and formulas provided in this section are in accordance with [START_REF] Szekely | Brownian distance covariance[END_REF].

3.1.1 Distance Covariance V 2
Let X ∈ R p and Y ∈ R q be random vectors, where p and q are natural numbers. f X and f Y denote the characteristic functions of X and Y , respectively, and their joint characteristic function is denoted as f X,Y . In terms of characteristic functions, X and Y are independent if and only if f X,Y = f X f Y . Thus, a natural way of measuring the dependence between X and Y is to find a suitable norm to measure the distance between f X,Y and f X f Y . Distance covariance V 2 [START_REF] Szekely | Brownian distance covariance[END_REF] is a new measure of dependence between random vectors and can be defined as:

V 2 (X, Y ) = ||f X,Y (t, s) -f X (t)f Y (s)|| 2 (1) = 1 cpcq R p+q |f X,Y (t,s)-f X (t)f Y (s)| 2 |t| 1+p p |s| 1+q q dtds, (2)
where c p and c q are constants determining norm function in

R p × R q , t ∈ X, s ∈ Y .
This measure is analogous to classical covariance, but with the important property that V 2 (X, Y ) = 0 if and only if X and Y are independent.

The paper [START_REF] Szekely | Brownian distance covariance[END_REF] provides us the following definition of a sample distance covariance V 2 n . For a random sample

(X, Y) = {(X k , Y k ) : k = 1 . . . n} of n i.i.d random vectors (X, Y ) from their joint distribution, compute the Euclidean distance matrices (a kl ) = (|X k -X l | p ) and (b kl ) = (|Y k -Y l | q ). Define: A kl = a kl -āk• -ā•l + ā•• , k, l = 1, . . . , n, (3)
where:

āk• = 1 n n l=1 a kl , ā•l = 1 n n k=1 a kl , ā•• = 1 n 2 n k,l=1
a kl .

(4) Similarly, we define

B kl = b kl -bk• -b•l + b•• .
Having these simple linear functions of the pairwise distances between sample elements of X and Y distributions, we define distance covariance as:

V 2 n (X, Y ) = 1 n 2 n k,l=1 A kl B kl . (5)
Although, the relation of equations ( 1) and ( 5) is not straightforward, THEOREM 2 from [START_REF] Szekely | Brownian distance covariance[END_REF] justifies it:

If E|X| p < ∞ and E|Y | q < ∞, then almost surely lim n→∞ V n (X, Y ) = V(X, Y ). (6)
Standardization. Similarly to covariance, which has its standardized counterpart ρ, V 2 n has its standardized version referred to as distance correlation R 2 n , defined as:

R 2 n (X, Y ) =    V 2 n (X,Y ) √ V 2 n (X)V 2 n (Y ) , V 2 n (X)V 2 n (Y ) > 0; 0, V 2 n (X)V 2 n (Y ) = 0, (7)
where:

V 2 n (X) = V 2 n (X, X) = 1 n 2 n k,l=1 A 2 kl . (8)

Video Brownian Covariance Descriptor

In this section, we present our Video Brownian Covariance (VBC) descriptor. Our descriptor encodes a spacetime video volume of size S × S pixels and t frames.

Image Low-Level Appearance Features

For each frame of a video volume, we compute seven low-level appearance features. For every pixel we extract intensities in red, green, and blue channels, first and second order derivatives of grey scale intensity image along x and y axis. Thus, every pixel at time t can be expressed in the following vector form:

f t = R, G, B, ∂I ∂x , ∂I ∂y , ∂ 2 I ∂x 2 , ∂ 2 I ∂y 2 , ( 9
)
where I is a gray scale intensity image. The examples of the extracted low-level appearance features are presented in Figure 1.

Video Brownian Covariance Descriptor

For each frame of the video volume we compute Brownian covariance between all mentioned appearance features. We use 7 low level appearance features, thus each frame is represented by a vector of m = 28 unique pairwise relations between features. We define the Video Brownian Covariance (VBC) descriptor D as an average descriptor among all the Brownian covariance descriptors extracted from all video frames:

D = [d 1 , d 2 , ..., d m ], d i = 1 t t j=1 b i,j , (10)
where b i,j is the i-th value of the Brownian covariance descriptor in the j-th video volume frame.

Normalization

Each value of our descriptor encodes a different relationship between two appearance features and it has a different meaning. Therefore, to make the descriptor values more uniform, we apply the min-max normalization to each descriptor value separately, i.e.:

N ormalized(D i ) = D i -Θ min i Θ max i -Θ min i , (11)
where D i is the i-th value of the descriptor D, Θ min i is the minimum value among all the i-th values of the training descriptors, and Θ max i is the maximum value among all the i-th values of the training descriptors.

Approach Overview

In the first step of our approach, we extract local spatiotemporal from a video sequence, and we represent each patch by our Video Brownian Covariance (VBC) and the Histogram of Oriented Gradients (HOG) descriptors (Section 5.1). Then, we apply Fisher vectors on the extracted descriptors to represent videos (Section 5.2). Finally, we apply Support Vector Machines for action classification (Section 5.3).

VBC and HOG Features Extraction

We use dense trajectories [START_REF] Wang | Action recognition by dense trajectories[END_REF] to extract local spatiotemporal patches 1 . By extracting dense trajectories, we provide a good coverage of a video and we ensure extraction of meaningful features. We limit the length of trajectories to t = 15 frames. Short trajectories are more robust than long trajectories, in particular in the presence of fast irregular motions and when the trajectories are drifting. Moreover, short trajectories are necessary for the recognition of short actions like smiling or doing a hive five.

Similarly to [START_REF] Wang | Action recognition by dense trajectories[END_REF], we extract a space-time volume (i.e. a patch) of size S × S pixels and t frames around each trajectory. The volume is subdivided into 3 temporal cells of l = 5 frames. For each cell we compute a descriptor, and we concatenate the descriptor of each cell to create a final trajectory descriptor. As a descriptor we use our Video Brownian Covariance (VBC) descriptor and the HOG descriptor. Therefore, each trajectory is represented by two appearance descriptors.

Action Representation

Once the descriptors are extracted, we use them to create video representations. We encode a video sequence using first and second order statistics of a distribution of a feature set X, based on Fisher vectors [START_REF] Perronnin | Large-Scale Image Retrieval with Compressed Fisher Vectors[END_REF][START_REF] Perronnin | Improving the Fisher Kernel for Large-Scale image classification[END_REF]. We model features with a generative model and compute the gradient of their likelihood with respect to the parameters of the model, i.e. ∆ λ log p(X|λ). We describe how the set of features deviates from an average distribution of features, modeled by a parametric generative model. Firstly, during the preliminary learning stage, we fit a M -centroid Gaussian Mixture Model (GMM) to our training features, which can be regarded as a soft visual vocabulary:

p(x i |λ) = M j=1 w j g(x i |µ j , Σ j ), (12)
s.t. ∀ j : w j ≥ 0, M j=1 w j = 1, (13)
g(x i |µ j , Σ j ) = 1 (2π) D/2 |Σ j | 1/2 e -1 2 (xi-µj ) T Σ -1 j (xi-µj ) , (14)
where x i ∈ X is a D-dimensional feature vector, {g(x i |µ j , Σ j )} M j=1 are the component Gaussian densities and λ = {w j , µ j , Σ j } M j=1 are the parameters of the model, respectively the mixture weights w j ∈ R + , the mean vector µ j ∈ R D and the positive definite covariance matrices Σ j ∈ R D×D of each Gaussian component. We learn the parameters λ using the Expectation Maximization restricting the covariance of the distribution to be diagonal. To estimate the GMM parameters, we randomly sample a subset of 100, 000 features from the training set and we set the number of Gaussians to M = 128. To increase the precision, we initialize GMM ten times and we keep the codebook with the lowest error. We define the soft assignment of descriptor x i to the Gaussian j as a posteriori probability γ(j|x i , λ) for component j:

γ(j|x i , λ) = w j g(x i |µ j , Σ j ) M l=1 w l g(x i |µ l , Σ l ) , (15)
Then, we compute the gradients of the j-th component with respect to µ and σ, using the following derivations:

G X µ,j = 1 N x √ w j Nx l=1 γ(j|x l , λ) x l -µ j σ j , G X σ,j = 1 N x 2w j Nx l=1 γ(j|x l , λ) (x l -µ j ) 2 σ 2 j -1 , (16)
where N x is the cardinality of the set X. Finally, we encode a set of local descriptors X as a concatenation of partial derivatives with respect to the mean G X µ,j and standard deviation G X σ,j parameters for all M components:

V = [G X µ,1 , G X σ,1 , ..., G X µ,M , G X σ,M ] T . (17)
The dimension of the Fisher vector representation is 2DM .

Action Recognition

Linear classifier has shown to be efficient and has shown to provide good results with high dimensional video representations like Fisher vectors. Therefore, we use linear Support Vector Machines for action classification. Given a set of n instance-label pairs (x i , y i ) i=1..n , x i ∈ R k , y i ∈ {-1, +1}, we solve the following unconstrained optimization problem:

min w 1 2 w T w + C n i=1 ξ(w; x i , y i ), (18)
where C is a penalty parameter (C > 0) and ξ(w; x i , y i ) is a loss function max(1-y i w T x i , 0), referred to as L1-SVM.

We set the parameter C to C = 200, which has shown good results on a subset of training samples across various datasets. For multi-class classification, we implement the one-vs-all strategy.

Experiments

In this section, we present the evaluation of our approach using three state-of-the-art challenging datasets: URADL, MSRDailyActivity3D, and HMDB51.

We study the performance of appearance descriptors, i.e. VBC and HOG, separately and together using late fusion technique. Moreover, we study the performance of these descriptors using Principal Component Analysis (PCA). The HOG descriptor is the most popular appearance descriptor for action recognition, and therefore, it provides a good baseline for comparison.

Many authors [START_REF] Laptev | Learning realistic human actions from movies[END_REF][START_REF] Liu | Recognizing realistic actions from videos "in the wild[END_REF][START_REF] Wang | Action recognition by dense trajectories[END_REF] combine appearance descriptors with motion-based descriptors (e.g. HOF, MBH). However, in this paper we focus only on the appearance features and not motion features.

URADL Dataset

The URADL (University of Rochester Activities of Daily Living) dataset [START_REF] Messing | Activity recognition using the velocity histories of tracked keypoints[END_REF] 2 contains 10 types of human activities of daily living, selected to be useful for an assisted cognition task. The full list of activities is: answering a phone, dialing a phone, looking up a phone number in a 2 http://www.cs.rochester.edu/ ˜rmessing/uradl/ telephone directory, writing a phone number on a whiteboard, drinking a glass of water, eating snack chips, peeling a banana, eating a banana, chopping a banana, and eating food with silverware. Each action is performed three times by five different people. In total, the dataset contains 150 video sequences recorded with 30 fps frame rate and 1280×720 pixels spatial resolution. The videos were downsampled to the 640 × 360 pixels spatial resolution. The dataset contains a set of challenges like: different shapes, sizes, genders and ethnicities of people, and difficulty to separate activities on the basis of a single source of information (e.g. eating a banana vs. eating snack chips, and answering a phone vs. dialing a phone). We use leave-oneperson-out cross-validation evaluation scheme to report the performance of our approach on this dataset.

MSRDailyActivity3D Dataset

The MSRDailyActivity3D dataset [START_REF] Wu | Mining actionlet ensemble for action recognition with depth cameras[END_REF] 3 consists of 16 actions such as: drink, eat, read book, call cellphone, write on a paper, use laptop, use vacuum cleaner, cheer up, sit still, toss paper, play game, lie down on sofa, walk, play guitar, stand up, sit down. Each action is performed by 10 subjects, and each subject performs each action in standing and sitting position, what adds an additional intra-class variation. In total, the dataset contains 320 videos recorded with 640 × 360 pixels spatial resolution. The videos were downsampled to the 320 × 180 pixels spatial resolution. We use leave-one-person-out cross-validation evaluation scheme to report the performance of our approach on this dataset.

HMDB51 Dataset

The HMDB51 dataset [START_REF] Kuehne | HMDB: a large video database for human motion recognition[END_REF] 4 contains 6766 video sequences divided into 51 action categories, each containing a minimum of 101 video clips. All action categories can be divided into 5 groups: (1) general facial actions like smile, laugh, chew, talk, (2) facial actions with object manipulation like eat and drink, (3) general body movements like clap hands and dive, (4) body movements with object interaction like brush hair and ride bike, (5) body movements for human interaction like hug and shake hands. This dataset contains multi person actions and is collected from movies and public datasets such as Prelinger archive and YouTube. It is very challenging due to significant camera/background motion, huge appearance variations of people and actions, not stabilized videos, occlusions, amount of video data and changes in scale, rotation and viewpoint. We use three splits provided by the authors, to report the performance of our approach on this dataset.

Results

The results are available in Table 2 and Table 3. We observe that the VBC descriptor carries complementary information to the HOG descriptor, as fusion of both descriptors consistently outperforms each of them. The HOG descriptor directly models values of given features, whereas the VBC descriptor focuses on their pairwise relations.

The results also show that both descriptors can benefit from PCA. The descriptors with PCA always achieve better results than without it.

Conclusions

We presented a novel, appearance-based descriptor for action recognition, which carries complementary information to the HOG descriptor. In contrast to the HOG (which directly models values of given features), the VBC descriptor focuses on features pairwise relations. The fusion of both descriptors gives a significant advantage in performance. In addition, we showed that further improvement can be made by applying PCA. Our novel descriptor can be applied in complex action recognition methods, which combine appearance-based and motion-based descriptors.

In further work we intend to examine the VBC descriptor using motion features.

Figure 1 .

 1 Figure 1. Low-level appearance features extracted in a video frame. Yellow rectangle indicates a sample patch.

Table 1 .

 1 Results from our experiments presenting the performance of HOG and VBC descriptors individually and together. The VBC descriptor is complementary to the HOG descriptor and improves the action recognition performance.

Table 2 .

 2 Results without PCA.

		HOG	VBC	Fusion
	URADL	69.33%	70%	80.00%
	MSR	42.19%	31.56%	45.31%
	HMDB51	16.71%	11.31%	18.95%

Table 3 .

 3 Results with PCA.

		HOG	VBC	Fusion
	URADL	80.67%	74%	80.67%
	MSR	47.81%	47.81%	54.38%
	HMDB51	26.21%	20.02%	31.07%

The dense trajectories were selected based on their use in the recent literature. However, our approach can be used together with any other algorithm extracting local spatio-temporal patches.

http://research.microsoft.com/en-us/um/people/ zliu/ActionRecoRsrc/

http://serre-lab.clps.brown.edu/resource/ hmdb-a-large-human-motion-database/

Acknowledgment

This work is supported by the Région Provence-Alpes-Côte d'Azur and by the Dem@Care, SafEE, and PANORAMA projects. However, the views and opinions expressed herein do not necessarily reflect those of the financing institution.


Lianmeng Jiao
email: jiaolianmeng@mail.nwpu.edu.cn

Quan Pan

Yan Liang
email: liangyan@nwpu.edu.cn

Xiaoxue Feng

Feng Yang

Combining sources of evidence with reliability and importance for decision making

Keywords: Decision making, Evidence theory, Importance discounting operation

The combination of sources of evidence with reliability has been widely studied within the framework of Dempster-Shafer theory (DST), which has been employed as a major method for integrating multiple sources of evidence with uncertainty. By the fact that sources of evidence may also be different in importance, for example in multi-attribute decision making (MADM), we propose the importance discounting and combination method within the framework of DST to combine sources of evidence with importance, which is composed of an importance discounting operation and an extended Dempster's rule of combination. Three evidence combination axioms are proposed and explored to uncover the differences between reliability and importance in evidence reasoning. Furthermore, a general scheme is proposed for combination of sources of evidence with both reliability and importance. An example of car performance evaluation is studied to show the efficiency of the new general scheme.

Introduction

In many real-life fusion problems, one has to deal with different sources of information arising from human reports, artificial intelligent systems or physical sensors, which are usually imprecise, uncertain, or incomplete. It's essential to combine all the information in a way that one has a better understanding and assessment of the complex problem under consideration for decision-making support. Dempster-Shafer theory (DST) [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF] or evidence theory, as a general extension of Bayesian theory, is a common and efficient method employed to solve the high-level fusion problems in expert reasoning [START_REF] Shafer | Probability judgment in artificial intelligence and expert systems[END_REF][START_REF] Yang | Belief rule-base inference methodology using the evidential reasoning approach -RIMER[END_REF][START_REF] Ghasemi | A novel fuzzy Dempster-Shafer inference system for brain MRI segmentation[END_REF], pattern recognition [START_REF] Denoeux | A k-nearest neighbor classification rule based on Dempster-Shafer theory[END_REF][START_REF] Denoeux | Evidential reasoning in large partially ordered sets[END_REF][START_REF] Tabassian | Combining complementary information sources in the DempstercShafer framework for solving classification problems with imperfect labels[END_REF], data mining [START_REF] Anand | EDM: a general framework for data mining based on evidence theory[END_REF][START_REF] Scotney | Database aggregation of imprecise and uncertain evidence[END_REF][START_REF] Srinivasan | Knowledge discovery in clinical databases with neural network evidence combination[END_REF] and decision making [START_REF] Beynon | An expert system for multi-criteria decision making using Dempster-Shafer theory[END_REF][START_REF] Davis | A software-supported process for assembling evidence and handling uncertainty in decision making[END_REF][START_REF] Xu | An introduction and survey of the evidential reasoning approach for multiple criteria decision analysis[END_REF].

Within the framework of DST, the reliability of different sources is usually considered in the evidence aggregation process due to the sources' limitation to provide totally accurate information [START_REF] Hégarat-Mascle | Introduction of neighborhood information in evidence theory and application to data fusion of radar and optical images with partial cloud cover[END_REF][START_REF] Milisavljević | Sensor fusion in anti-personnel mine detection using a two-level belief function model[END_REF]. As for the combination problem of sources of evidence with reliability, usually the classical Shafer's discounting operation [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]) is applied before the classical Dempster's rule of combination [START_REF] Dempster | Upper and lower probabilities induced by multivalued mapping[END_REF]. However, in some decision making problems, especially for MADM, different attributes should be weighted in order to get more reasonable fusion results [START_REF] Hwang | Multiple attribute decision-making: methods and applications[END_REF]. While in some previous works [START_REF] Yang | A general multi-level evaluation process for hybrid multiple attribute decision making with uncertainty[END_REF][START_REF] Beynon | The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modelling[END_REF]Beynon 2005b), the weights of attributes are utilized according to Shafer's discounting operation, which makes no difference with the treatment of reliability. This kind of management fails to follow the common sense rules [START_REF] Yang | On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty[END_REF]) that a rational aggregation process needs to satisfy. Until recently, the importance notion of one source of evidence is proposed to describe the weight of the corresponding attribute [START_REF] Tacnet | Information fusion for natural hazards in mountains[END_REF]. The authors believe that the reliability and importance characterize the properties of a source of evidence in different aspects and thus should be distinguished in the fusion process.

More recently, a method for combining sources of evidence with importance has been established [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF] within the framework of Dezert-Smarandache theory (DSmT), which extends DST by refuting the exclusivity constraint on the frame of discernment. However, it's explored that the proposed method has some problems as follows. Firstly, the combination rule developed in DSmT to integrate the discounted sources of evidence is not associative and hence will become quite computationally demanding as the number of sources of evidence increases [START_REF] Smarandache | Proportional conflict redistribution rules for information fusion[END_REF]. More seriously, the utilized discounting operation transfers all the remaining discounted mass to the empty set. We believe it is theoretically irrational as the empty set is already used to characterize the conflicting information in their operation and the remaining discounted mass essentially represents the indecisiveness rather than the conflicting information.

As discussed above, although distinct notion is presented for importance from reliability, there is still no theoretically sound method to address the evidence reasoning problem with importance in DST framework and the essential differences between reliability and importance is also unclear. Motivated by the above consideration, in this paper, an importance discounting operation is defined that commits the remaining discounted mass not to the empty set but a new set to characterize the indecisiveness that needs to be jointly assigned with other new importance discounted sources. Then, an extended Dempster's rule of combination is developed to combine the importance discounted sources of evidence in DST framework, which is both commutative and associative and thus has a relatively low computing burden. Three evidence combination axioms (independence, consensus and completeness) are proposed and explored, which reveal essential differences between reliability and importance and illustrate the proposed method in more meaningful ways. Furthermore, a reliability-importance discounting and combination method is developed, which provides a general scheme for combination of sources of evidence. Our new general scheme is illustrated via a car performance evaluation example.

This paper is organized as follows. In Sect. 2, we firstly give a brief reminder of the basic notions in DST, and then the reliability discounting and combination method is revisited and analyzed. In Sect. 3, we propose the method for taking into account the importance of sources of evidence in the combination process and then three evidence combination axioms are proposed and explored in Sect. 4. Sect. 5 develops a general scheme for combination of sources of evidence with reliability and importance, and then an example of car performance evaluation is examined in Sect. 6. Finally, Sect. 7 presents some concluding remarks.

Reliability discounting and combination

The combination of sources of evidence with reliability has been widely studied within the framework of DST. To facilitate later discussion, in this section, we first briefly review the basic concepts of DST, and then present the concept of reliability and its acquisition method. Finally, the classical reliability discounting and combination method is revisited and analyzed.

Basic concepts in DST

To begin with, let us introduce some basic concepts in DST [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. In DST, a problem domain is represented by a finite set Θ = {θ 1 , θ 2 , • • • , θn} of mutually exclusive and exhaustive hypotheses called the frame of discernment. A basic belief assignment (BBA) expressing the belief committed to the elements of 2 Θ by a given source of evidence is a mapping function m(•): 2 Θ → [0, 1], such that m(∅) = 0 and

∑ A∈2 Θ m(A) = 1,
where, 2 Θ denotes the power set of Θ, consisting all the 2 n subsets of Θ. Elements A ∈ 2 Θ having m(A) > 0 are called focal elements of the BBA m(•). A BBA m(A) measures the degree of belief exactly assigned to a proposition A and represents how strongly the proposition is supported by evidence. The belief assigned to all the subsets of 2 Θ is summed to unity and there is no belief left to the empty set. The belief assigned to Θ, or m(Θ), is referred to as the degree of global ignorance.

Shafer also defines the belief and plausibility functions of A ∈ 2 Θ as follows

Bel(A) = ∑ B⊆A m(B) and Pl(A) = ∑ B∩A̸ =∅ m(B). (1)
Bel(A) represents the exact support to A and its subsets, and Pl(A) represents all the possible support to A and its subsets. The interval [Bel(A), Pl(A)] can be seen as the lower and upper bounds of support to A. The belief functions m(•), Bel(•) and Pl(•) are in one-to-one correspondence.

For decision-making support, the pignistic probability BetP(A) [START_REF] Smets | Decision making in the TBM: the necessity of the pignistic transformation[END_REF]) is commonly used to approximate the unknown probability in [Bel(A), Pl(A)], given by

BetP(A) = ∑ B⊆Θ A∩B̸ =∅ |A ∩ B| |B| m(B), (2)
where, |X| stands for the cardinality of the element X.

Reliability and its acquisition

As in [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF], the definition of reliability for a source of evidence in the context of DST is given as follows.

Definition 1 (reliability of a source of evidence) The reliability, as an objective property of a source of evidence, represents its capability to provide correct measure or assessment of the considered problem.

Remark 1 The reliability of a source of evidence can be characterized by a reliability factor α ∈ [0, 1], where α = 1 and α = 0 represents that the source is fully reliable and totally unreliable, respectively. Besides, according to this definition, the reliability of one source of evidence is determined only by itself and the reliability factors of different sources of evidence involved in the fusion process is independent with each other.

The reliability of a source of evidence does exit in many fusion problems and generally can be acquired in the following two ways:

-The reliability of a source of evidence is prior-known or can be determined from some available properties [START_REF] Haenni | Modeling partially reliable information sources: a general approach based on Dempster-Shafer theory[END_REF]. For example, it may be constructed from the known measurement accuracy of the given sensor. Sometimes, this reliability would be context-dependent [START_REF] Mercier | Refined modeling of sensor reliability in the belief function framework using contextual discounting[END_REF].

For example, if a sensor more possibly does not perform well under bad weather conditions, then the corresponding reliability of information from that sensor will be expected to become smaller. -The reliability of a source of evidence is approximately estimated from the dissimilarity with the other sources of evidence [START_REF] Deng | Combining belief functions based on distance of evidence[END_REF][START_REF] Liu | Combination of sources of evidence with different discounting factors based on a new dissimilarity measure[END_REF]. Generally, for any a source of evidence, its larger dissimilarity with other sources measures the lower reliability.

Shafer's discounting operation and Dempster's rule of combination

The reliability of a source of evidence is generally considered according to Shafer's discounting method [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. Mathematically, Shafer's discounting operation (detonated as ⊗) for taking into account the reliability factor α ∈ [0, 1] of a given source with a BBA m(•) and a frame Θ is defined by

α ⊗ m(A) ∆ = m α (A) = { αm(A), for A ∈ 2 Θ and A ̸ = Θ αm(Θ) + (1 -α), for A = Θ.
(3)

As shown in Eq.( 3), Shafer's discounting method multiplies the masses of focal elements by the reliability factor α, and transfers all the remaining discounted mass 1 -α to the global ignorance Θ. By the fact that αm(Θ) + (1 -α) > αm(Θ) + (1 -α)m(Θ) = m(Θ) in case of m(Θ) < 1 and α < 1, such discounting operation almost always increases the global ignorance.

After the reliability discounting operation, Dempster's rule of combination (the orthogonal sum operation ⊕) is usually used to combine two discounted pieces of evidence on the same frame Θ represented by two BBAs m α1 1 (•) and m α2 2 (•) to generate a new BBA, denoted by m D 12 (•)

m α1 1 ⊕m α2 2 (A) ∆ = m D 12 (A) =        0, for A = ∅ ∑ B,C∈2 Θ ;B∩C=A m α 1 1 (B)m α 2 2 (C) 1- ∑ B,C∈2 Θ ;B∩C=∅ m α 1 1 (B)m α 2 2 (C) , for A ∈ 2 Θ andA ̸ = ∅.
(4) As described in [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]), Dempster's rule of combination is both commutative and associative.

Importance discounting and combination

The importance of a source plays significant role in the process of MADM and it is quite different from the reliability of that source. Specifically, the importance usually reflect the decision maker's subjective preference to this kind of source while the reliability is a objective property representing the source's capability to provide the correct measure or assessment and thus they should be addressed in different ways. This section aims to study the fusion problem of sources of evidence with importance in DST framework.

Importance and its acquisition

Distinct from the concept of reliability, as in [START_REF] Smarandache | Fusion of sources of evidence with different importances and reliabilities[END_REF], the importance of a source of evidence in the context of DST can be defined as follows.

Definition 2 (importance of a source of evidence) The importance, as a subjective property of a source of evidence, represents the weight of that source which reflects the decision maker's subjective preference in the fusion process.

Remark 2 Different from the reliability defined in Sect. 2, the defined importance belongs to a relative notion. In other words, the importance of one source is meaningless without comparing with other sources. It can be characterized by an importance factor, denoted β ∈ [0, 1]. The decision maker will take β = 1 when he/she wants to grant the maximum importance to the source in the fusion process, and will take β = 0 if no importance at all is granted to this source.

The importance of a source of evidence (or the relative weight of the corresponding attribute) can be determined in the following two ways:

-In most cases, weights are selected based on preference information of attributes given by the fusion system designers (FSDs) or the decision makers (DMs). The commonly-used methods include direct rating method [START_REF] Roberts | Weight approximations in multi-attribute decision models[END_REF], ranking ordering method [START_REF] Barron | Decision quality using ranked attribute weights[END_REF], and analytic hierarchy process (AHP) method [START_REF] Ishizaka | How to derive priorities in AHP: a comparative study[END_REF][START_REF] Dijkstra | On the extraction of weights from pairwise comparison matrices[END_REF], which reflect the subjective intuition or the knowledge of FSDs or DMs. -Sometimes the relative weight of the attribute will be determined from some available properties with the respect to the importance of the attribute, for example, the entropy of the decision matrix [START_REF] Xu | A note on the subjective and objective integrated approach to determine attribute weights[END_REF]).

Importance discounting operation

In this subsection, we propose a solution to consider the importance of sources of evidence in DST framework. Our definition of importance discounting operation draws inspiration from the weight addressing technique in ER algorithm for MAD-M [START_REF] Yang | On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty[END_REF][START_REF] Xu | The ER approach for multi-attribute decision analysis under interval uncertainties[END_REF][START_REF] Yang | Introduction to the ER rule for evidence combination[END_REF]. We define this operation in the viewpoint of "importance"as the counterpart of Shafer's discounting operation for reliability of sources of evidence. It commits the remaining discounted mass to a new set Ω rather than the global ignorance set Θ as done by Shafer in reliability discounting presented in Sect. 2.

Definition 3 (importance discounting operation)

The importance discounting operation (detonated as ⊙) of a source of evidence having the importance factor β ∈ [0, 1] and associated BBA m(•) can be defined as

β ⊙ m(A) ∆ = m β (A) = { βm(A), for A ∈ 2 Θ 1 -β, for A = Ω (5)
where, Ω is the power set of Θ, i.e. Ω = 2 Θ .

Remark 3 In this new discounting operation, m β (Ω) is not interpreted as the mass committed to the conflicting information (as m α (Θ) in reliability discounting), nor as the mass committed to some unknown elements (as m(∅) in Smets' open-word assumption [START_REF] Smets | The combination of evidence in the transferable belief model[END_REF])), but only the the indecisiveness among the subset of 2 Θ that needs to be jointly assigned with other new importance discounted sources of evidence. Further, the indecisiveness introduced in this paper is different from the non-specificity also discussed in DST literature (Beynon 2005b). It is clear that indecisiveness is defined with respect to different focal elements A ⊆ Θ (or A ∈ Ω = 2 Θ ), while non-specificity is defined with respect to different hypotheses

θ ∈ Θ.
It is worth noting that, after the importance discounting operation, the original BBA is extended with new belief assignment m β (Ω). To facilitate later discussion, we define this kind of extended BBA as follows.

Definition 4 (importance basic belief assignment) An importance basic belief assignment (IBBA, for short) in DST framework is a function m

β (•) : 2 Θ ∪ {Ω} ∆ = 2 Θ + → [0, 1], satisfying { m β (∅) = 0 ∑ A∈2 Θ + m β (A) = ∑ A∈2 Θ m β (A) + m β (Ω) = 1. ( 6
)
An IBBA m β (A) represents the importance discounted degree of belief assigned to a proposition A and measures how strongly the proposition is supported by evidence in the sense of the relative importance. The belief assigned to Ω, or m β (Ω), is referred to as the degree that is not assigned to any subset of 2 Θ and needs to be jointly assigned with other new IBBAs.

Extended Dempster's rule of combination

After all the sources of evidence are discounted according to the importance discounting operation defined in the above subsection, a combination rule is needed to integrate all the discounted sources to get the fusion result. As new belief assignment m β (Ω) is introduced in IBBA, so the classical Dempster's rule of combination needs to be extended to combine different IBBAs.

m β1 1 ⊕m β2 2 (A) ∆ = m ED 12 (A) =          0, for A = ∅ ∑ B,C∈2 Θ + ;B∩C=A m β 1 1 (B)m β 2 2 (C) 1- ∑ B,C∈2 Θ + ;B∩C=∅ m β 1 1 (B)m β 2 2 (C) , for A ∈ 2 Θ + andA ̸ = ∅. ( 7
)
Similar with the Dempster's rule of combination, the proposed extended Dempster's rule of combination is also based on the orthogonal sum operation ⊕, so the extended Dempster's rule of combination is also both commutative and associative. As a result, for multiple sources of evidence, the combination process can proceed in a recursive way to avoid high computation burden.

Lemma 1 The combination result with the extended Dempster's rule of combination is an IBBA. In other words, for the combination result, m ED (∅) = 0 and

∑ A∈2 Θ + m ED (A) = 1. Proof See Appendix A.
Since we usually need to work with normal BBA for decision-making support, after combining all the importance discounted IBBAs using the extended Dempster's rule of combination, the fusion result m ED (•) will be normalized by redistributing the mass of belief committed to the set Ω to the other focal elements proportionally to their masses as follows

m norm (A) = m ED (A) 1 -m ED (Ω) , ∀A ∈ 2 Θ . ( 8
)
4 Analysis of reliability and importance

In the previous section, the importance of a source of evidence is defined differently from the reliability and then a method is proposed to combine sources of evidence with importance. However, the necessity of distinguishing reliability and importance in the evidence combination process is still unclear. In this section, three evidence combination axioms are proposed and explored to uncover the essential differences between reliability and importance and thus lay theoretical foundation for the proposed method. Inclined to developing theoretically sound methods for dealing with MADM problems under uncertainty, [START_REF] Yang | On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty[END_REF] proposed four self-evident rules, with which a rational attribute aggregation process needs to satisfy. Inspired by these ideas, here we refine and extend these rules within the framework of DST and propose three evidence combination axioms as follows (Suppose

m i (•) (i = 1, • • • , L)
are L basic sources of evidence's BBAs on the same frame of discernment Θ and m(•) is the fusion result).

Axiom 1 (independence) If no evidence assigns mass to one non-global ignorance set or its supersets, then the combination result will not assign mass to the same set

either. Mathematically, ∀S ∈ 2 Θ \ Θ, if ∀S + ⊇ S, m i (S + ) = 0 for all i = 1, • • • , L, then m(S) = 0.
Axiom 2 (consensus) If all evidence assigns full mass to one non-global ignorance set, then the combination result will assign full mass to the same set too. Mathemati-

cally, ∀S ∈ 2 Θ \ Θ, if m i (S) = 1 for all i = 1, • • • , L, then m(S) = 1.
Axiom 3 (completeness) If all evidence assigns full mass to the subsets of one nonglobal ignorance set, the combination result will also assign full mass to the subsets of

the same set. Mathematically, ∀S ∈ 2 Θ \Θ, if ∑ S -⊆S m i (S -) = 1 for all i = 1, • • • , L, then ∑ S -⊆S m(S -) = 1.
Remark 4 In the three axioms defined above, the global ignorance set Ω is not taken into consideration mainly because as a special set it may take other explanation (e.g. conflicting information in reliability discounting) besides global ignorance. So, the explanation ambiguity of set Ω will destroy the rationality of the three axioms.

Besides, the three axioms are extended from Yang's self-evident rules [START_REF] Yang | On the evidential reasoning algorithm for multiple attribute decision analysis under uncertainty[END_REF], in which they also didn't take into account the global ignorance set in the attribute aggregation process.

In order to explain the three evidence combination axioms in a more comprehensible way, a simple MADM example is given below.

Example 1 (car performance evaluation). As in Fig. 1, we consider the problem of evaluating the performance of three types of car (decision alternatives, DAs), namely, A, B, and C. In the DST terminology used above, {A, B, C} is the frame of discernment Θ. The overall performance of each type of car is evaluated based on three major attributes: quality of engine, operation, and comfortableness. Now, the three axioms can be explained from the aspect of MADM as follows.

-Suppose no attribute assigns belief to car A as the best car, then the attribute aggregation result will not assign any belief to car A. -Suppose all attributes assign full belief to car A as the best car, then the attribute aggregation result will also assign full belief to car A. (consensus) -Suppose all attributes assign full belief to car B or C as the best car, then the attribute aggregation result will also assign full belief to car B or C.

(completeness)

The above axioms characterize three commonsense properties of the evidence combination process. While, if the reliability of sources of evidence is considered, some of them may not be satisfied as shown in the following theorem.

Theorem 1 The reliability discounting and combination method, displayed as Eqs.( 3)-( 4), satisfies the independence axiom, but fails to satisfy the consensus and completeness axioms.

Proof See Appendix B. This theorem is straightforward if one has a deep understanding of the reliability. Now, we consider the consensus axiom in Example 1. Suppose all attributes assign full belief to car A as the best car, considering the decision maker is unfamiliar with the engine, wrong assessment may be provided (maybe car B has the best engine performance in reality), and so the attribute aggregation result should not assign full belief to car A as the best car.

Then, with different concept from reliability, it's interesting to know whether the importance discounting and combination method proposed in Sect. 3 satisfies the three axioms. The following theorem presents the conclusion.

Theorem 2 The proposed importance discounting and combination method, displayed as Eqs.( 5),( 7)-( 8), satisfies the independence, consensus and completeness axioms.

Proof See Appendix C.

As illustrated in the above theorem, distinct from the reliability, the consensus and completeness axioms are still preserved if the importance is considered in the evidence combination process. Here, we also take the consensus axiom for example to explain it in a reasonable way. Suppose all attributes assign full belief to car A as the best car, as the importance is a subjective property of a source of evidence and it does no matter with the source's capability, so the importance discounting only assign weights to each attributes rather than change their assessments, and thus it's intuitively reasonable that the attribute aggregation result will also assign full belief to car A.

In short, due to the fact that reliability and importance are distinct notions, they should be managed in different ways and consequently the two discounting and combination methods differ in properties. Furthermore, considering that reliability and importance may coexist in some cases, a general scheme for combination of sources of evidence will be developed in next section.

A general scheme for combination of sources of evidence

In this section, we would like to establish a general scheme for combination of sources of evidence. Now, we examine the possibility to take into account both the reliability factor α and the importance factor β of a given source of evidence characterized by its BBA m(•). Apparently, it's a natural way to discount the original BBA m(•) by α with reliability discounting operation and then discount the result m α (•) by β with importance discounting operation. This kind of realization is given as follows.

Definition 6 (reliability-importance discounting operation) We define the reliabilityimportance discounting operation of a source of evidence having the reliability factor α ∈ [0, 1], the importance factor β ∈ [0, 1] and the associated BBA m(•) in the frame of discernment Θ by

β ⊙ (α ⊗ m(A)) ∆ = m α,β (A) =    αβm(A), for A ∈ 2 Θ , A ̸ = Θ αβm(Θ) + (1 -α)β, for A = Θ 1 -β, for A = Ω. ( 9
)
where, Ω has the same meaning as in Definition 3.

As the result of reliability-importance discounting operation is an IBBA, so the proposed extended Dempster's rule of combination in Sect. 3 can be used to combine multiple reliability-importance discounted IBBAs.

This kind of reliability-importance discounting and combination method provides a general scheme for combination of sources of evidence to take into account both the reliability and importance. It can be seen that, when β = 1 (the source has full importance), the reliability-importance discounting and combination method simplifies to the reliability discounting and combination method presented in Sect. 2; when α = 1 (the source has full reliability), the importance discounting and combination method proposed in Sect. 3 is obtained; and when both α = 1 and β = 1 (the source has both full importance and reliability), the original BBA m(•) keeps unchanged and the classical Dempster's rule of combination is used to integrate the sources of evidence.

Remark 5 Because the reliability is considered in the evidence combination process, it's easy to know that the proposed general scheme also only satisfies the independence axiom, and it doesn't satisfy the consensus and completeness axioms except that all the sources of evidence take full reliability (α i = 1, i = 1, • • • , L), in which case the general scheme reduces to the importance discounting and combination method.

6 Case study

Problem presentation

Here we conduct a comprehensive study of the car performance evaluation problem described in Example 1. For this MADM problem, as described in Sect. 3, the relative importance of different attributes plays significant role in the process of decision making. Besides, the knowledge level of decision maker (DM) may vary in the three corresponding fields (i.e. Engine, Operation, and Comfortableness), and so the reliability of different knowledge matrices provided by the DM should also be considered in the decision process.

The DS/AHP framework [START_REF] Beynon | The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modelling[END_REF][START_REF] Beynon | DS/AHP method: A mathematical analysis, including an understanding of uncertainty[END_REF]) is used here to conduct the analysis of the above MADM problem. Compared with standard analytic hierarchy process (AHP) method, the inclusion of DST allows for uncertainty in the judgements. More details about this framework can be found in [START_REF] Beynon | The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modelling[END_REF]. Within the DS/AHP framework, it is assumed that with respect to each attribute the "uncertain"knowledge matrices for decision alternatives (DAs) are given by the DM based on his/her knowledge as Table 1.

A B C Θ A 1 0 a 0 3 B 0 1 0 1/2 C 0 0 1 1 Θ 1/3 2 1 1 Operation A B C Θ A 1 0 0 2 B 0 1 0 5 C 0 0 1 1 Θ 1/2 1/5 1 1 Comfortableness A B C Θ A 1 0 0 1/2 B 0 1 0 1 C 0 0 1 2 Θ 2 1 1/2 1
a The zero's appearing in the knowledge matrix indicate there is no necessity to compare different individual DAs (e.g., A to B) directly; this assertion can be made indirectly through knowledge of the favorability of A to Θ and B to Θ relatively.

Derivation of BBAs, reliability and importance factors

The derivation of BBAs is the basis for evidence combination in DST. Here, the method in [START_REF] Beynon | The Dempster-Shafer theory of evidence: an alternative approach to multicriteria decision modelling[END_REF] is used to construct the BBAs from the above knowledge matrices. The normalized principal eigenvectors of the knowledge matrices displayed in Table 1 are derived as the corresponding BBAs for the three attributes engine, operation and comfortableness as follows.

m 1 (•) : m 1 (A) = 0.4814, m 1 (B) = 0.0802, m 1 (C) = 0.1605, m 1 (Θ) = 0.2779 m 2 (•) : m 2 (A) = 0.2055, m 2 (B) = 0.5138, m 2 (C) = 0.1028, m 2 (Θ) = 0.1780 m 3 (•) : m 3 (A) = 0.0956, m 3 (B) = 0.1911, m 3 (C) = 0.3823, m 3 (Θ) = 0.3310
The reliability factors will be estimated if the DM's knowledge level in the three corresponding fields is known. For example, if it is known prior that the DM is familiar with engine, expert in operation but has a low sensitivity for the comfortableness of a car, then the reliability factors for the above three BBAs can be empirically set as α 1 = 0.9, α 2 = 1.0 and α 3 = 0.5, respectively.

As for the importance factors, the AHP method mentioned in Sect. 3 is utilized here. Suppose the DM believes that comfortableness is two times more important than the quality of engine, three times more important than the quality of operation and the quality of engine is two times more important than the quality of operation, then the preference matrix among the three attributes is displayed as

  Engine Operation Comfortableness Engine 1 2 1/2 Operation 1/2 1 1/3 Comfortableness 2 3 1   .
Its normalized principal eigenvector w = [0.30, 0.16, 0.54] can be used as the corresponding importance factors.

Numerical results

In this numerical study, to make a comparison, the reliability discounting and combination method (RDC, only consider the reliability), the importance discounting and combination method (IDC, only consider the importance), and the reliability-importance discounting and combination method (RIDC, consider both the reliability and importance) are tested to integrate the BBAs with different reliability and importance factors derived in the previous subsection. For each method, the BBAs for the three attributes are firstly discounted and then combined to get the combination result. The final decision is then can be made based on the combination result according to some decision criteria. Table 2 shows the combination results of the three pieces of evidence using different methods (Eqs.(3)-(4) for RDC, Eqs.( 5),( 7)-(8) for IDC, Eqs.( 9),( 7)-(8) for RIDC, respectively). Usually, the DM is concerned with a single choice among the three cars. The decision can be made using the maximum of pignistic probability (BetP) criteria defined as Eq.( 2) over the raw combination results obtained above. Table 3 shows the BetP criteria of the DAs (Rows 2-4) as well as the decision results based on these criteria (Row 5). For comparison, apart from the three methods (RDC, IDC, and RIDC), the decision results based on each individual attribute (Engine, Operation, and Comfortableness) are also given. It can be seen that the decision results are different for the three methods mainly because different kinds of additional information are considered in the decision process. For RDC method, the decision result is the same with that based on the Operation attribute, since its corresponding knowledge matrix has the largest reliability. For IDC method, the decision result about the best car is the same with that based on the Comfortableness attribute, since the DM assigns the greatest importance to this attribute. While, for RIDC method, the decision result is the same with that based on the Engine attribute which behaves not bad both in reliability and importance. It's believed that the RIDC method gives the most reasonable decision result as both the reliability of the knowledge matrices and the importance of attributes are considered.

Conclusions

Real-world fusion problems are complex in that they often involve multiple sources of evidence with uncertainty. Further, different sources of evidence to be combined may not have the same reliability or importance, while the two properties make no difference in some previous works. In this paper, the concepts of reliability and importance are clarified and the importance discounting and combination method composed of an importance discounting operation and an extended Dempster's rule of combination is proposed within the framework of DST to integrate sources of evidence with importance. We show some interesting axioms satisfied by the new method, which further reveal the essential differences between reliability and importance. A general scheme is developed to combine sources of evidence to take into account both the importance and reliability. An example of car performance evaluation is studied and the numerical results show the efficiency of the new general scheme considering both the reliability of the knowledge matrices and the importance of different attributes.

Noting that the importance of a source of evidence is still a new concept in DST framework, the importance discounting and combination method proposed in this paper is just one kind of realization. That is, other importance discounting and combination methods in future works may also be reasonable if some axioms (such as independence, consensus and completeness) reflecting the nature of importance are satisfied.

In addition, this paper mainly focuses on the combination of sources of evidence with reliability and importance encountered in general MADM problems. While, to the more complicated group multi-attribute decision making (GMADM) problems (Beynon 2005a), besides the attributes' reliability and importance considered in general MADM, a rational evidence aggregation process also needs to take into account the importance of individuals in the group. In future work, we will extend the reliability-importance discounting and combination method developed in this paper to address the evidence aggregation problem in GMADM.

A Proof of Lemma 1

According to Definition 5, m ED 12 (∅) = 0 is satisfied directly. Besides,

∑ A∈2 Θ + m ED 12 (A) = ∑ A∈2 Θ + ,A̸ =∅ m ED 12 (A) + m ED 12 (∅) = ∑ A∈2 Θ + ,A̸ =∅ m ED 12 (A) = ∑ A∈2 Θ + ,A̸ =∅    ∑ B,C∈2 Θ + ;B∩C=A m β 1 1 (B)m β 2 2 (C)    1- ∑ B,C∈2 Θ + ;B∩C=∅ m β 1 1 (B)m β 2 2 (C) . ( 10
)
Because,

∑ A∈2 Θ + ,A̸ =∅   ∑ B,C∈2 Θ + ;B∩C=A m β 1 1 (B)m β 2 2 (C)   + ∑ B,C∈2 Θ + ;B∩C=∅ m β 1 1 (B)m β 2 2 (C) = ∑ A∈2 Θ +   ∑ B,C∈2 Θ + ;B∩C=A m β 1 1 (B)m β 2 2 (C)   = ∑ B∈2 Θ +   m β 1 1 (B) ∑ C∈2 Θ + m β 2 2 (C)   = ∑ B∈2 Θ + m β 1 1 (B) = 1,
so, Eq.( 10) equals 1. Therefore, the combination result m ED 12 (•) with the extended Dempster's rule of combination is an IBBA.

B Proof of Theorem 1

Suppose m i (•) (i = 1, • • • , L) are L basic sources of evidence's BBAs on the same frame of discernment Θ with reliability factors α i ∈ [0, 1] (i = 1, • • • , L).
Denote m(•) the integrated BBA with the reliability discounting and combination method.

As for the independence axiom, ∀S ∈ 2 Θ \ Θ, suppose ∀S + ⊇ S, m i (S + ) = 0 for all i = 1, • • • , L. Discount all the L BBAs with their corresponding reliability factors α i using Shafer's discounting operation displayed as Eq.( 3), we can get the reliability discounted BBAs assigned to S +

m i α i (A) = { 0, for A ∈ 2 Θ , A ⊇ S, A ̸ = Θ 1 -α i , for A = Θ for i = 1, • • • , L. (11)
Then, the Dempster's rule of combination displayed as Eq.( 4) will be used to get the integrated BBAs assigned to S m(S) =

∑ X i ∈2 Θ ; L ∩ i=1 X i =S L ∏ i=1 m α i i (X i ) /       1 - ∑ X i ∈2 Θ ; L ∩ i=1 X i =∅ L ∏ i=1 m α i i (X i )       . (12) Since L ∩ i=1 X i = S, so X i ⊇ S. According to Eq.(11), if X i ̸ = Θ, m α i i (X i ) = 0 for all i = 1, • • • , L. Because it's impossible for all X i to take Θ satisfying L ∩ i=1 X i = S, so, L ∏ i=1 m α i
i (X i ) will equal 0 in any cases. Hence, m(S) in Eq.( 12) equals 0. That is, the reliability discounting and combination method satisfies the independence axiom.

As for the consensus axiom, ∀S ∈ 2 Θ \ Θ, suppose m i (S) = 1 for all i = 1, • • • , L. Discount all the L BBAs with their corresponding reliability factors α i using Shafer's discounting operation displayed as Eq.( 3), we can get the reliability discounted BBAs

m i α i (A) =    α i , for A = S 0, for A ∈ 2 Θ , A ̸ = S, A ̸ = Θ for i = 1, • • • , L. 1 -α i , for A = Θ
Then, the Dempster's rule of combination displayed as Eq.( 4) will be used to integrate the reliability discounted BBAs

m(A) =    0, for A ∈ 2 Θ , A ̸ = S, A ̸ = Θ L ∏ i=1 (1 -α i ), for A = Θ. So, m(S) = 1 - ∑ A∈2 Θ ,A̸ =S,A̸ =Θ m(A) -m(Θ) = 1 - L ∏ i=1 (1 -α i ).
Thus, the reliability discounting and combination method only satisfies the consensus axiom when at least one source of evidence takes full reliability (∃k ∈ {1, • • • , L}, α k = 1).

As for the completeness axiom, ∀S ∈ 2 Θ \ Θ, suppose ∑ S -⊆S m i (S -) = 1 for all i = 1, • • • , L. We can know that ∀S ++ ⊃ S, m i (S ++ ) = 0. Discount all the L BBAs with their corresponding reliability factors α i using Shafer's discounting operation displayed as Eq.(3), we can get the reliability discounted BBAs assigned to S ++

m i α i (A) = { 0, for A ∈ 2 Θ , A ⊃ S, A ̸ = Θ 1 -α i , for A = Θ for i = 1, • • • , L. (13)
Then, the Dempster's rule of combination displayed as Eq.( 4) will be used to get the integrated BBAs assigned to S ++ m(S ++ ) =

∑ X i ∈2 Θ ; L ∩ i=1 X i =S ++ L ∏ i=1 m α i i (X i ) /       1 - ∑ X i ∈2 Θ ; L ∩ i=1 X i =∅ L ∏ i=1 m α i i (X i )       . (14) Since L ∩ i=1 X i = S ++ , so X i ⊇ S ++ ⊃ S. Now, we consider it for two cases. If S ++ ̸ = Θ,
according to Eq.( 13), m

α i i (X i ) = 0 for all i = 1, • • • , L. So, m(S ++ ) in Eq.(14) equals 0. If S ++ = Θ, it's easy to get m(S ++ ) = m(Θ) = L ∏ i=1 (1 -α i ) / (1 -k). So, ∑ S -⊆S m(S -) = 1 - ∑ S ++ ⊃S m(S ++ ) = 1 - ∑ S ++ ⊃S,S ++ ̸ =Θ m(S ++ ) -m(Θ) = 1 -m(Θ) = 1 - L ∏ i=1 (1 -α i ) / (1 -k).
Therefore, the reliability discounting and combination method only satisfies the completeness axiom when at least one source of evidence takes full reliability (∃k ∈ {1, • • • , L}, α k = 1).

C Proof of Theorem 2

Suppose m i (•) (i = 1, • • • , L) are L basic sources of evidence's BBAs on the same frame of discernment Θ with importance factors β i ∈ [0, 1] (i = 1, • • • , L). Denote m(•) the integrated BBA with the proposed importance discounting and combination method.

As for the independence axiom, ∀S ∈ 2 Θ \ Θ, suppose ∀S + ⊇ S, m i (S + ) = 0 for all i = 1, • • • , L. Discount all the L BBAs with their corresponding importance factors β i using the importance discounting operation displayed as Eq.( 5), we can get the importance discounted IBBAs assigned to S + m i β i (A) = 0, ∀A ⊇ S,

for i = 1, • • • , L. ( 15
)
Then, the extended Dempster's rule of combination displayed as Eq.( 7) will be used to get the integrated IBBAs assigned to S m ED (S) = So, the importance discounting and combination method satisfies the consensus axiom.

∑ X i ∈2 Θ + ; L ∩ i=1 X i =S L ∏ i=1 m β i i (X i ) /       1 - ∑ X i ∈2 Θ + ; L ∩ i=1 X i =∅ L ∏ i=1 m β i i (X i )       . ( 16
As for the completeness axiom, ∀S ∈ 2 Θ \ Θ, suppose ∑ S -⊆S m i (S -) = 1 for all i = 1, • • • , L. We can know that ∀S ++ ⊃ S, m i (S ++ ) = 0. Discount all the L BBAs with their corresponding importance factors β i using the importance discounting operation displayed as Eq.( 5), we can get the importance discounted IBBAs assigned to S ++ Then, the extended Dempster's rule of combination displayed as Eq.( 7) will be used to get the integrated IBBAs assigned to S ++ m ED (S ++ ) =

∑ X i ∈2 Θ + ; L ∩ i=1 X i =S ++ L ∏ i=1 m β i i (X i ) /       1 - ∑ X i ∈2 Θ + ; L ∩ i=1 X i =∅ L ∏ i=1 m β i i (X i )       . ( 18

Fig. 1 A

 1 Fig. 1 A car performance evaluation example.

X

  i = S, so X i ⊇ S. According to Eq.(15), mβ i i (X i ) = 0 for all i = 1, • • • , L. Hence, m ED (S) in Eq.(16) equals 0. It's straightforward that m(S) = m ED (S) /( 1 -m ED (Ω) ) = 0.That is, the importance discounting and combination method satisfies the independence axiom.As for the consensus axiom, ∀S ∈ 2 Θ \ Θ, suppose m i (S) = 1 for all i = 1, • • • , L. Discount all the L BBAs with their corresponding importance factors β i using the importance discounting operation displayed as Eq.(5), we can get the importance discountedA ∈ 2 Θ , A ̸ = S for i = 1, • • • , L. 1 -β i , for A = ΩThen, the extended Dempster's rule of combination displayed as Eq.(7) will be used to integrate the importance discounted IBBAsm ED (A) = 0, ∀A ∈ 2 Θ , A ̸ = S. According to Lemma 1, it holds that ∑ A∈2 Θ + m ED (A) = ∑ A∈2 Θ ,A̸ =S m ED (A)+m ED (S)+m ED (Ω) = m ED (S)+m ED (Ω) = 1.Furthermore, via the normalization in Eq.(8

  m i β i (A) = 0, ∀ A ⊃ S, for i = 1, • • • , L. (17)

X

  i = S ++ , so X i ⊇ S ++ ⊃ S. According to Eq.(17), mβ i i (X i ) = 0 for all i = 1, • • • , L. Hence, m ED (S ++ ) in Eq.(18) equals 0. It's straightforward that m(S ++ ) = m ED (S ++ ) /( 1 -m ED (Ω) ) = 0. So, ∑ S -⊆S m(S -) = 1 -∑ S ++ ⊃S m(S ++ ) = 1.That is, the importance discounting and combination method satisfies the completeness axiom.

Definition 5 (extended Dempster's rule of combination)

  With two pieces of evidence with importance factors β 1 and β 2 represented by two IBBAs m β1

	and m β2 2 (•) defined on 2 Θ +	1 (•) , the classical Dempster's rule of combination can be
	extended as

Table 1

 1 Knowledge matrices for different attributes Engine

Table 2

 2 Combination results with different methods(RDC, IDC, and RIDC)

		m 1	m 2	m 3	RDC	IDC	RIDC
	{A}	0.4814	0.2055	0.0956	0.3354	0.2260	0.1905
	{B}	0.0802	0.5138	0.1911	0.4046	0.2161	0.1674
	{C}	0.1605	0.1028	0.3823	0.1618	0.2974	0.1740
	Θ	0.2779	0.1780	0.3310	0.0982	0.2605	0.4681

Table 3

 3 BetP criteria and decision results based on BetP

	Cars	Engine	Operation Comfortableness RDC	IDC	RIDC
	A	0.5740	0.2648	0.2059	0.3681	0.3128	0.3465
	B	0.1729	0.5731	0.3015	0.4374	0.3030	0.3234
	C	0.2531	0.1621	0.4926	0.1945	0.3842	0.3301
	Decision	A≻C≻B	B≻A≻C	C≻B≻A	B≻A≻C	C≻A≻B	A≻C≻B

Extended Dempster's rule of combination

This work is partially supported by China Natural Science Foundation (No. 61135001, 61075029) and the Doctorate Foundation of Northwestern Polytechnical University (No. CX201319).