General Schemes of Combining Rules and the Quality Characteristics - PowerPoint PPT Presentation

General Schemes of Combining Rules and the Quality Characteristics of Combining Alexander Lepskiy National Research University – Higher School of Economics, Moscow, Russia The 3 rd International Conference on Belief Functions, September 26 – 28, 2014, Oxford, UK Alexander Lepskiy (HSE) General Schemes BELIEF 2014 1 / 26

Outline Outline of Presentation 1 General Schemes of Combining Rules Basic Definitions and Notation Combining Rules Combining Rule both the Aggregation of Evidence − Pointwise Aggregation of BFs − Pointwise Aggregation of BPA − Bilinear Aggregation of BFs − Bilinear Normalized Aggregation of BFs 2 Quality Characteristics of Combining Imprecision Index as a Measure of Information Uncertainty Change of Linear Imprecision Index when Evidences are Combined Pessimistic and Optimistic Combining Rules 3 Summary and conclusion Alexander Lepskiy (HSE) General Schemes BELIEF 2014 2 / 26

Combining Rules Basic Definitions and Notation A belief function (BF) g is defined with the help of set function m g ( A ) called the basic probability assignment (BPA): m g ( ∅ ) = 0, � A ⊆ X m g ( A ) = 1. Then � g ( A ) = m g ( B ) . B : B ⊆ A Let Bel ( X ) be a set of all BF on 2 X , M ( X ) be a set of all set functions on 2 X . The BF g can be represented with the help of categorical BF � 1 , B ⊆ A, � η � B � ( A )= 0 , B �⊆ A, A ⊆ X, B � = ∅ . Then g = m g ( B ) η � B � . B ∈ 2 X \{∅} The subset A ∈ 2 X is called a focal element if m g ( A ) > 0. Let A ( g ) be the set of all focal elements related to the BF g . The pair F ( g ) = ( A ( g ) , m g ) is called a body of evidence . Let F ( X ) be the set of all bodies of evidence on X . Alexander Lepskiy (HSE) General Schemes BELIEF 2014 3 / 26

Combining Rules Combining Rules Suppose that we have two bodies of evidence F ( g 1 ) = ( A ( g 1 ) , m g 1 ) and F ( g 2 ) = ( A ( g 2 ) , m g 2 ) which are defined on the set X . In general a combining rule is a some operator ϕ : Bel ( X ) × Bel ( X ) → Bel ( X ). Dempster’s rule 1 � m D ( A )= m g 1 ( A 1 ) m g 2 ( A 2 ) , A � = ∅ , m D ( ∅ ) = 0 , (1) 1 − K A 1 ∩ A 2 = A � K = K ( g 1 , g 2 ) = m g 1 ( A 1 ) m g 2 ( A 2 ) . (2) A 1 ∩ A 2 = ∅ The value K ( g 1 , g 2 ) characterizes the amount of conflict in two information sources which defined with the help of bodies of evidence F ( g 1 ) and F ( g 2 ). If K ( g 1 , g 2 ) = 1 then it means that information sources are absolutely conflicting and Dempster’s rule cannot be applied. Alexander Lepskiy (HSE) General Schemes BELIEF 2014 4 / 26

Combining Rules Combining Rules Discount rule m α ( A ) = (1 − α ) m ( A ) , A � = X, m α ( X ) = α + (1 − α ) m ( X ) . (3) The coefficient α ∈ [0 , 1] characterizes the degree of reliability of information. If α = 0 then it means that information source is absolutely reliable. If α = 1 then it means that information source is absolutely non-reliable. The Dempster’s rule (1) applies after discounting of BPA of two evidences. Yager’s modified Dempster’s rule � m g 1 ( A 1 ) m g 2 ( A 2 ) , A ∈ 2 X , q ( A ) = (4) A 1 ∩ A 2 = A m Y ( A )= q ( A ) , A � = ∅ , X, m Y ( ∅ )= q ( ∅ ) = K, m Y ( X )= m Y ( ∅ )+ q ( X ) . (5) Alexander Lepskiy (HSE) General Schemes BELIEF 2014 5 / 26

Combining Rules Combining Rules Zhang’s center combination rule � r ( A 1 , A 2 ) m g 1 ( A 1 ) m g 2 ( A 2 ) , A ∈ 2 X , m Z ( A ) = A 1 ∩ A 2 = A where r ( A 1 , A 2 ) is a measure of intersection of sets A 1 and A 2 . Dubois and Prade’s disjunctive consensus rule � m g 1 ( A 1 ) m g 2 ( A 2 ) , A ∈ 2 X . m DP ( A ) = (6) A 1 ∪ A 2 = A Alexander Lepskiy (HSE) General Schemes BELIEF 2014 6 / 26

Combining Rule both the Aggregation of Evidence Combining Rule both the Aggregation of Evidence We will consider an operator ϕ : Bel 2 ( X ) → Bel ( X ) that is called the aggregation of two BFs g 1 , g 2 ∈ Bel ( X ) in one BF g = ϕ ( g 1 , g 2 ) ∈ Bel ( X ). We have g ↔ m g = ( m g ( B )) B ⊆ X . Therefore there is an aggregation of BPA m g = Φ( m g 1 , m g 2 ) for any aggregation of BFs g = ϕ ( g 1 , g 2 ) and vice versa. We consider some special cases of aggregation of BFs and we will give the descriptions of aggregation operators in these special cases. Alexander Lepskiy (HSE) General Schemes BELIEF 2014 7 / 26

Combining Rule both the Aggregation of Evidence 1. Pointwise Aggregation of Belief Functions The new value of BF g ( A ) = ϕ ( g 1 ( A ) , g 2 ( A )) is associated with every pair ( g 1 ( A ) , g 2 ( A )) of BFs on the same set A ∈ 2 X . We consider the finite differences for description of aggregation operator: s � ( − 1) s − k � ∆ s ϕ ( x ; ∆ x 1 , ..., ∆ x s )= ϕ ( x + ∆ x i 1 + ... + ∆ x i k ) , k =0 1 ≤ i 1 <...<i k ≤ s where ∆ x 1 , ..., ∆ x s ∈ [0 , 1] 2 ( x + ∆ x 1 + ... + ∆ x k ∈ [0 , 1] 2 ∀ k = 1 , ..., s . Theorem The function ϕ : [0 , 1] 2 → [0 , 1] defines the aggregation operator of BFs by the rule g ( A ) = ϕ ( g 1 ( A ) , g 2 ( A )) , A ∈ 2 X , g 1 , g 2 ∈ Bel ( X ) iff it satisfies the conditions: 1 ϕ ( 0 ) = 0 , ϕ ( 1 ) = 1 ; 2 ∆ k ϕ ( x ; ∆ x 1 , ..., ∆ x k ) ≥ 0 , k = 1 , 2 , ... for all x , ∆ x 1 , ..., ∆ x k ∈ [0; 1] 2 , x + ∆ x 1 + ... + ∆ x k ∈ [0 , 1] 2 . Alexander Lepskiy (HSE) General Schemes BELIEF 2014 8 / 26

Combining Rule both the Aggregation of Evidence 2. Pointwise Aggregation of BPA The new BPA m g ( A ) = Φ( m g 1 ( A ) , m g 2 ( A )) is associated with every pair ( m g 1 ( A ) , m g 2 ( A )) of BPA ∀ A ∈ 2 X . Theorem The continuous function Φ : [0 , 1] 2 → [0 , 1] defines the aggregation operator of BPA by the rule m g ( A ) = Φ( m g 1 ( A ) , m g 2 ( A )) , A ∈ 2 X , g 1 , g 2 ∈ Bel ( X ) iff it satisfies the condition Φ( s, t ) = λs + (1 − λ ) t , λ ∈ [0 , 1] . This result is a generalization of the corresponding result for probability measures [K.J. McConway 1981] . Alexander Lepskiy (HSE) General Schemes BELIEF 2014 9 / 26

Combining Rule both the Aggregation of Evidence 3. Bilinear Aggregation of Belief Functions In this case the aggregation function ϕ should be linear for each argument so ϕ ( αg 1 + (1 − α ) g 2 , g 3 ) = αϕ ( g 1 , g 3 ) + (1 − α ) ϕ ( g 2 , g 3 ) , α ∈ [0 , 1] . (7) Since we have g i = � B ∈ 2 X \{∅} m g i ( B ) η � B � ∈ Bel ( X ), i = 1 , 2, then every bilinear function on Bel 2 ( X ) has the form � ϕ ( g 1 , g 2 ) = m g 1 ( A ) m g 2 ( B ) γ A,B , (8) A,B ∈ 2 X \{∅} is some set function on 2 X . � � where γ A,B = ϕ η � A � , η � B � Alexander Lepskiy (HSE) General Schemes BELIEF 2014 10 / 26

Combining Rule both the Aggregation of Evidence We consider the non-empty set B ( X ) ⊆ Bel 2 ( X ) which satisfies the � � condition: if ( g 1 , g 2 ) ∈ B ( X ) then ∈ B ( X ) for all A ∈ A ( g 1 ), η � A � , η � B � B ∈ A ( g 2 ). Theorem The bilinear set function ϕ : B ( X ) → M ( X ) of the form (8) determines � � � � ∈ Bel ( X ) for all ∈ B ( X ) . the BF iff γ A,B = ϕ η � A � , η � B � η � A � , η � B � The Dubois and Prade’s disjunctive consensus rule and Dempster’s rule (Yager’s rule) for non conflicting evidences are the examples of bilinear aggregation functions of the form (8). Alexander Lepskiy (HSE) General Schemes BELIEF 2014 11 / 26

Combining Rule both the Aggregation of Evidence 4. Bilinear Normalized Aggregation of Belief Functions ϕ ( g 1 , g 2 ) ϕ 0 ( g 1 , g 2 ) = ϕ ( g 1 , g 2 )( X ) , (9) where ϕ ( g 1 , g 2 ) is a bilinear aggregation function, γ A,B ( C ) ≥ 0 ∀ A, B, C ∈ 2 X \{∅} . The function ϕ 0 is determined on the set ( g 1 , g 2 ) ∈ Bel 2 ( X ) | ∃ A i ∈A ( g i ) , ϕ � � � � B ϕ ( X )= η � A 1 � , η � A 2 � ( X ) � =0 . Theorem Let ϕ be a bilinear aggregation function and ϕ 0 : B ϕ ( X ) → M ( X ) has the form (9). Then ϕ 0 determines the BF iff γ A,B / γ A,B ( X ) ∈ Bel ( X ) , � � � � γ A,B = ϕ for all ∈B ϕ ( X ) . η � A � , η � B � η � A � , η � B � The Dempster’s rule and Zhang’s center combination rule are the examples of bilinear normalized aggregation functions of the form (9). Alexander Lepskiy (HSE) General Schemes BELIEF 2014 12 / 26

Research of Quality Characteristics of Combining Research of Quality Characteristics of Combining 1. A priori characteristics that estimate the quality of information sources: the reliability of sources in discount rule; the conflict measure of evidence in Dempster’s rule, Yage’s rule etc.; the degree of independence of evidence; etc. 2. A posteriori characteristics which estimate the result of combining. The amount of change of ignorance after the use of combining rule is the most important a posteriori characteristic. Alexander Lepskiy (HSE) General Schemes BELIEF 2014 13 / 26

Research of Quality Characteristics of Combining Change of Ignorance when Evidences are Combin Change of Ignorance when Evidences are Combined Let we have two sources of information and this information is described by BFs g 1 , g 2 ∈ Bel ( X ) respectively. Let some rule ϕ be used for combining of these BFs. We will get the new BF g = ϕ ( g 1 , g 2 ) ∈ Bel ( X ). Main questions 1. How the measure of information uncertainty can be estimated? 2. How much the value of measure of information uncertainty will change after combining of evidence? Alexander Lepskiy (HSE) General Schemes BELIEF 2014 14 / 26