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Mathematics of
Uncertainty Modeling
in the Analysis of
Engineering and Science
Problems
S. Chakraverty
National Institute of Technology - Rourkela, India
A volume in the Advances in
Computational Intelligence and Robotics
(ACIR) Book Series
Detailed Table of Contents
Preface . ............................................................................................................................................... xiv
Acknowledgment . ............................................................................................................................. xxii
Chapter 1
Interval Mathematics as a Potential Weapon against Uncertainty .......................................................... 1
Hend Dawood, Cairo University, Egypt
This chapter is devoted to introducing the theories of interval algebra to people who are interested in
applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose,
we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of
the interval techniques as they are now practised, provide a historical and epistemological background
of interval mathematics and uncertainty in science and technology, and finally describe some typical
applications that clarify the need for interval computations to cope with uncertainty in a wide variety
of scientific disciplines.
Chapter 2
Uncertainty Modeling Using Expert’s Knowledge as Evidence .......................................................... 40
D. Datta, Bhabha Atomic Research, India
In this paper we discuss the uncertainty modeling using evidence theory. In practice, very often availability
of data is incomplete in the sense that sufficient amount of data which is required may not be possible to
collect. Therefore, uncertainty modeling in that case with this incomplete data set is not possible to carry
out using probability theory or Monte Carlo method. Fuzzy set theory or any other imprecision based
theory is applicable in this case. With a view to this expert’s knowledge is represented as the input data
set. Belief and plausibility are the two bounds (lower and upper) of the uncertainty of this imprecision
based system. The fundamental definitions and the mathematical structures of the belief and plausibility
fuzzy measures are discussed in this chapter. Uncertainty modeling using this technique is illustrated
with a simple example of contaminant transport through groundwater.
Chapter 3
Evidence-Based Uncertainty Modeling ................................................................................................ 55
Tazid Ali, Dibrugarh University, India
Evidence is the essence of any decision making process. However in any situation the evidences that
we come across are usually not complete. Absence of complete evidence results in uncertainty, and
uncertainty leads to belief. The framework of Dempster-Shafer theory which is based on the notion of
belief is overviewed in this chapter. Methods of combining different sources of evidences are surveyed.
Relationship of probability theory and possibility theory to evidence theory is exhibited. Extension of
the classical Dempster-Shafer Structure to fuzzy setting is discussed. Finally uncertainty measurement
in the frame work of Dempster-Shafer structure is dealt with.
Chapter 4
Hybrid Set Structures for Soft Computing ........................................................................................... 75
Sunil Jacob John, National Institute of Technology Calicut, India
Babitha KV, National Institute of Technology Calicut, India
A Major problem in achieving an effective computational systems is the presence of inherent uncertainty
in the computational problem itself. Among various techniques proposed to address this, the technique
of soft computing is of significant interest. Further, Generalized set structures like fuzzy sets, rough
sets, multisets etc. have already proven their role in the context of soft computing. The computational
techniques based on one of these structures alone will not always yield the best results but a fusion of
two or more of them can often give better results. Such structures are regarded as hybrid set structures.
This chapter surveys an analysis of various hybrid set structures which are quite useful tools for soft
computing and shows how this hybridization can help in improving modeling real situations.
Chapter 5
Source and M-Source Distances of Fuzzy Numbers and their Properties ............................................ 95
Majid Amirfakhrian, Islamic Azad University, Iran
In many applications of fuzzy logic and fuzzy mathematics we need (or it is better) to work with the
same fuzzy numbers. In this chapter we present source distance and source distance between two fuzzy
numbers. Also some properties of parametric -degree polynomial approximation operator of fuzzy
numbers. Numerical examples are solved related to the present analysis.
Chapter 6
Construction of Normal Fuzzy Numbers using the Mathematics of Partial Presence . ...................... 109
Hemanta K. Baruah, Gauhati University, India
Every normal law of fuzziness can be expressed in terms of two laws of randomness defined in the measure theoretic sense. Indeed, two probability measures are necessary and sufficient to define a normal
law of fuzziness. Hence, the measure theoretic matters with reference to fuzziness have to be studied
accordingly. In this chapter, we are going to discuss how to construct normal fuzzy numbers using this
concept which is based on our mathematics of partial presence. Three case studies have been presented
with reference to expressing stock prices in terms of fuzzy numbers. We have shown how exactly a
normal fuzzy number appears.
Chapter 7
Numerical Solution of Fuzzy Differential Equations and its Applications ........................................ 127
S. Chakraverty, National Institute of Technology Rourkela, India
Smita Tapaswini, National Institute of Technology Rourkela, India
Theory of fuzzy differential equations is the important new developments to model various science
and engineering problems of uncertain nature because this theory represents a natural way to model
dynamical systems under uncertainty. Since, it is too difficult to obtain the exact solution of fuzzy differential equations so one may need reliable and efficient numerical techniques for the solution of fuzzy
differential equations. In this chapter we have presented various numerical techniques viz. Euler and
improved Euler type methods and Homotopy Perturbation Method (HPM) to solve fuzzy differential
equations. Also application problems such as fuzzy continuum reaction diffusion model to analyse the
dynamical behaviour of the fire with fuzzy initial condition is investigated. To analyse the fire propagation, the complex fuzzy arithmetic and computation are used to solve hyperbolic reaction diffusion
equation. This analysis finds the rate of burning number of trees in bounds where wave variable/ time
are defined in terms of fuzzy. Obtained results are compared with the existing solution to show the efficiency of the applied methods.
Chapter 8
Modeling with Stochastic Fuzzy Differential Equations .................................................................... 150
Marek T. Malinowski, University of Zielona Góra, Poland
In the chapter, the authors consider an approach used in the studies of stochastic fuzzy differential
equations. These equations are new mathematical tools for modeling uncertain dynamical systems.
Some qualitative properties of their solutions such as existence and uniqueness are recalled, and stability properties are shown. Here, the solutions are continuous adapted fuzzy stochastic processes. The
authors consider some examples of applications of stochastic fuzzy differential equations in modeling
real-world phenomena.
Chapter 9
Mathematics of Probabilistic Uncertainty Modeling . ........................................................................ 174
D. Datta, Bhabha Atomic Research, India
This chapter presents the uncertainty modeling using probabilistic methods. Probabilistic method of
uncertainty analysis is due to randomness of the parameters of a model. Randomness of parameters is
characterized by specified probability distribution such as normal, log normal, exponential etc., and the
corresponding samples are generated by various methods. Monte Carlo simulation is applied to explore
the probabilistic uncertainty modeling. Monte Carlo simulation being a statistical process is based on the
random number generation from the specified distribution of the uncertain random parameters. Sample
size is generally very large in Monte Carlo simulation which is required to have small errors in the computation. Latin hypercube sampling and importance sampling are explored in brief. This chapter also
presents Polynomial Chaos theory based probabilistic uncertainty modeling. Polynomial Chaos theory
is an efficient Monte Carlo simulation in the sense that sample size here is very small and dictated by the
number of the uncertain parameters and by choice of the order of the polynomial selected to represent
the uncertain parameter.
Chapter 10
Reaction-Diffusion Problems with Stochastic Parameters Using the Generalized Stochastic Finite
Difference Method .............................................................................................................................. 206
Marcin Kamiński, Technical University of Łódź, Poland
Rafał Leszek Ossowski, Technical University of Łódź, Poland
The main aim of this work is to demonstrate the new stochastic discrete computational methodology
consisting of the generalized stochastic perturbation technique and of the classical Finite Difference
Method for the regular grids to model reaction-diffusion problems with random time series. The generalized stochastic perturbation approach is based on the given order Taylor expansion of all random
variables. A numerical algorithm is implemented here using the Direct Differentiation Method of the
reaction-diffusion equation with respect to the height of a channel in 1D problem; further symbolic
determination of the probabilistic moments and characteristics is completed by the computer algebra
system MAPLE, v. 14. Computational illustration attached proves that it is possible to determine using
this approach up to the fourth order probabilistic moments and coefficients as well as to consider time
series with random coefficients for any dispersion of the input variables. Stochastic fluctuations of the
input uncertainty source are defined here as the power time series with Gaussian random coefficients
having given first two moments.
Chapter 11
MV-Partitions and MV-Powers . ......................................................................................................... 218
C. Drossos, University of Patras, Greece
P. L. Theodoropoulos, Educational Counselor, Greece
In this chapter, the authors generalize the Boolean partition to semisimple MV-algebras. MV-partitions
together with a notion of refinement is tantamount a construction of an MV-power, analogous to Boolean
power construction (Mansfield, 1971). Using this new notion we introduce the corresponding theory
of MV-powers.
Chapter 12
An Algebraic Study of the Notion of Independence of Frames . ........................................................ 240
Fabio Cuzzolin, Oxford Brookes University, UK
In this chapter, the authors discuss the notion of independence of frames in the theory of evidence from
an algebraic point of view, starting from an analogy with standard linear independence. The final goal
is to search for a solution of the problem of conflicting belief function via a generalization of the classical Gram-Schmidt algorithm for vector orthogonalization. Families of frames can be given several
algebraic interpretations in terms of semi-modular lattices, matroids, and geometric lattices. Each of
those structures is endowed with a particular (extended) independence relation, which we prove to be
distinct albeit related to independence of frames.
Chapter 13
Pricing and Lot-Sizing Decisions in Retail Industry: A Fuzzy Chance Constraint Approach............. 269
R. Ghasemy Yaghin, Amirkabir University of Technology, Iran
S.M.T. Fatemi Ghomi, Amirkabir University of Technology, Iran
S.A.Torabi, Amirkabir University of Technology, Iran
Analysis of inventory systems involving market-oriented pricing decisions has recently become an
interesting topic in the field of inventory control. Price and marketing expenditure are considered as
important elements when selling goods and enhancing revenues by manufacturers. The importance of
accounting for uncertainty in such environments spurs an interest to develop appropriate decision making tools to deal with uncertain and ill-defined parameters (such as costs and market function) in joint
pricing and lot-sizing problems. In this research, a fuzzy chance constraint multi-objective programming
model based on p-fractile approach is proposed to determine the optimal price, marketing expenditure
and lot size. Considering pricing, marketing and lot-sizing decisions simultaneously, a possibilistic
programming based on necessity measure is considered to handle imprecise data and constraints. Discount strategy as a fuzzy power function of order quantity is determined. After applying appropriate
strategies to defuzzify the original possibilistic model, the equivalent multi-objective crisp model is then
transformed by a single-objective programming model. A meta-heuristic algorithm is applied to solve
the final crisp counterpart.
Chapter 14
A New Approach for Suggesting Takeover Targets Based on Computational Intelligence and
Information Retrieval Methods: A Case Study from the Indian Software Industry............................ 290
Satyakama Paul, University of Johannesburg, South Africa
Andreas Janecek, University of Vienna, Austria
Fernando Buarque de Lima Neto, University of Pernambuco, Brazil
Tshilidzi Marwala, University of Johannesburg, South Africa
In recent years researchers in financial management have shown considerable interest in predicting
future takeover target companies in merger and acquisition (M&A) scenarios. However, most of these
predictions are based upon multiple instances of previous takeovers. Now consider a company that is
at the early stage of its acquisition spree and therefore has only limited data of possibly only a single
previous takeover. Traditional studies on M&A, based upon statistical records of multiple previous
takeovers, may not be suitable for suggesting future takeover targets for this company since the lack of
history data strongly limits the applicability of statistical techniques. The challenge then is to extract
as much knowledge as possible from the single/limited takeover history in order to guide this company
during future takeover selections. Under such an extreme case, the authors present a new algorithmic
approach for suggesting future takeover targets for acquiring companies based on solely one previous
history of acquisition. The approach is based upon methods originating from information retrieval and
computational intelligence. The proposal is exemplified upon a case study using real financial data of
companies from the Indian software industry.
Chapter 15
Fuzzy Finite Element Method in Diffusion Problems ........................................................................ 309
S. Chakraverty, National Institute of Technology, India
S. Nayak, National Institute of Technology, India
Diffusion is an important phenomenon in various fields of science and engineering. It may arise in a variety of problems viz. in heat transfer, fluid flow problem and atomic reactors etc. As such these diffusion
equations are being solved throughout the globe by various methods. It has been seen from literature that
researchers have investigated these problems when the material properties, geometry (domain) etc. are
in crisp (exact) form which is easier to solve. But in real practice the parameters used in the modelled
physical problems are not crisp because of the experimental error, mechanical defect, measurement error
etc. In that case the problem has to be defined with uncertain parameters and this makes the problem
complex. In this chapter related uncertain differential equation of various diffusion problems will be
investigated using finite element method, which may be called fuzzy or interval finite element method.
Chapter 16
Uncertainty Quantification of Aeroelastic Stability . .......................................................................... 329
Georgia Georgiou, University of Liverpool, UK
Hamed Haddad Khodaparast, Swansea University, UK
Jonathan E. Cooper, University of Bristol, UK
The application of uncertainty analysis for the prediction of aeroelastic stability, using probabilistic and
non-probabilistic methodologies, is considered in this chapter. Initially, a background to aeroelasticity and
possible instabilities, in particular “flutter,” that can occur in aircraft is given along with the consideration
of why Uncertainty Quantification (UQ) is becoming an important issue to the aerospace industry. The
Polynomial Chaos Expansion method and the Fuzzy Analysis for UQ are then introduced and a range
of different random and quasi-random sampling techniques as well as methods for surrogate modeling
are discussed. The implementation of these methods is demonstrated for the prediction of the effects that
variations in the structural mass, resembling variations in the fuel load, have on the aeroelastic behavior
of the Semi-Span Super-Sonic Transport wind-tunnel model (S4T). A numerical model of the aircraft is
investigated using an eigenvalue analysis and a series of linear flutter analyses for a range of subsonic
and supersonic speeds. It is shown how the Probability Density Functions (PDF) of the resulting critical
flutter speeds can be determined efficiently using both UQ approaches and how the membership functions of the aeroelastic system outputs can be obtained accurately using a Kriging predictor.
Chapter 17
Uncertain Static and Dynamic Analysis of Imprecisely Defined Structural Systems ........................ 357
S. Chakraverty, National Institute of Technology Rourkela, India
Diptiranjan Behera, National Institute of Technology Rourkela, India
This chapter presents the static and dynamic analysis of structures with uncertain parameters using fuzzy
finite element method. Uncertainties presents in the parameters are modelled through convex normalised
fuzzy sets. Fuzzy finite element method converts the structures into fuzzy system of linear equations
and fuzzy eigenvalue problem for static and dynamic problems respectively. As such method to solve
fuzzy system of linear equations, fully fuzzy system of linear equations and fuzzy eigenvalue problems
are presented. These methods are applied to various structural problems to find out the fuzzy static
and dynamic responses of the structures. Also the chapter analyses the numerical solution of uncertain
fractionally damped spring-mass system. Uncertainties considered in the initial condition of the system.
Compilation of References ............................................................................................................... 383
About the Contributors .................................................................................................................... 412
Index.................................................................................................................................................... 417
240
Chapter 12
An Algebraic Study of the
Notion of Independence
of Frames
Fabio Cuzzolin
Oxford Brookes University, UK
ABSTRACT
The theory of belief functions or “theory of evidence” allows the mathematical representation of uncertain pieces of evidence on which decisions can be based. Frequently, different pieces of evidence belong
to distinct, albeit related, domains or “frames”: for instance, audio and video clues can be combined
to infer the identity of a person from a video. Evidence encoded by different belief functions on separate
frames can be merged on a common frame, a combination which is guaranteed to exist if and only if the
frames are “independent” in the sense of Boolean algebras. In all other cases the evidence conflicts.
Independence of frames and belief function combinability are then strictly related.
INTRODUCTION
The theory of evidence was born as a contribution
towards a mathematically rigorous description of
subjective probability. In subjective probability,
different observers (or “experts”) of the same
phenomenon possess in general different notions
DOI: 10.4018/978-1-4666-4991-0.ch012
of what the decision space is. Mathematically,
this translates into admitting the existence of
several distinct representations of this decision
space at different levels of refinement. Evidence
will in general be available on several of those
domains or frames. In order for those experts to
reach a consensus on the answer to the considered
problem, it is necessary for such frames to be
mathematically related to each other. This idea
is embodied in the theory of evidence by the no-
Copyright © 2014, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited.
An Algebraic Study of the Notion of Independence of Frames
tion of family of frames. The evidence gathered
on distinct elements of the family (corresponding
to different experts or sensors) can be moved to a
common frame in order to be merged.
In this context the notion of independence
of frames IF (Shafer,1976) plays an important
role. If different pieces of evidence are encoded
as different belief functions on distinct frames,
evidence fusion under Dempster’s orthogonal
sum (Dempster, 1967; Dempster, 1968; Dempster,
1969) is guaranteed to take place in all cases if
and only if the involved frames are independent
(Cuzzolin, 2005) in a very precise way, which
comes from the theory of Boolean algebras. As
Dempster’s sum assumes the conditional independence of the underlying probabilities generating
belief functions through multi-valued mappings,
it is not surprising to realize that combinability
(in Dempster’s approach) and independence of
frames (in Shafer’s formulation of the theory of
evidence) are strictly intertwined.
Here we build on the results obtained in (Cuzzolin, 2005) to complete the algebraic analysis
of families of frames and conduct a comparative
study of the notion of independence of frames, so
central in the theory of evidence, in an algebraic
setup. The work is articulated into two parts.
In the first half we prove that families of
frames are endowed with three different algebraic
structures, namely those of: 1. Boolean algebra,
2. upper semi-modular lattice, and 3. lower semimodular lattice. In the second part we study relationships and differences between the different
forms of independence that can be introduced on
such structures, and understand whether IF can
be reduced to one of them.
The contribution of this work is therefore
twofold. On one side, we complete the rich
algebraic description of families of compatible
frames by relating them to semi-modular lattices
and matroids, extending some recent preliminary
results (Cuzzolin, 2005). On the other, we pose
the notion of independence of frames in a wider
context by highlighting its relation with classical
independence in modern algebra. Even though
IF turns out not to be a cryptomorphic form of
matroidal independence, it possesses interesting
relations with several extensions of matroidal
independence to lattices, stressing the need for
a more general, comprehensive definition of this
widespread and important notion.
Eventually, this work may open the way to an
algebraic solution of the conflict problem, where
a pseudo Gram-Schmidt orthogonalization procedure is used to transform any given collection
of belief functions into a new, combinable set by
projection onto a new set of independent frames.
The outline of the Chapter is as follows. We
start by recalling the notions of compatible frames
and independence of frames as Boolean subalgebras. We review a recent result linking frame
independence and combinability with respect to
Dempster’s sum of belief functions: combinability can then be studied in an algebraic setup by
analyzing the algebraic properties of independence
of frames. Next, the example-based pose estimation problem is briefly described, as a typical
application in which (possibly conflicting) belief
functions living on different compatible frames
need to be combined.
After that, the notion of independence on
matroids is recalled. Even though families of
frames endowed with IF do not form a matroid,
matroids are strictly related to other algebraic
structures, such as semi-modular lattices, which
do describe collections of compatible frames.
Subsequently, we prove that families of frames
are indeed both upper and lower semi-modular
lattices, according to which order relation we
pick. On such structures matroidal independence
can be extended, yielding several different relations whose meaning we thoroughly discuss and
whose links with IF we highlight in the final part
of the Chapter.
241
An Algebraic Study of the Notion of Independence of Frames
BACKGROUND
Evidence combination has been widely studied
(Yager, 1987; Zadeh, 1986) in different mathematical frameworks (Dubois & Prade, 1992; Smets,
1990): an exhaustive review would be impossible
here. In particular, work has been done on the
issue of merging conflicting evidence (DeutschMcLeish, 1990; Josang et al, 2003; Lefevre et al,
2002; Wierman, 2001), especially in critical situations in which the latter is derived from dependent
sources (Cattaneo, 2003). Campos and de Souza
(2005) have presented a method for fusing highly
conflicting evidence which overcomes well known
counterintuitive results. Liu (2006) has formally
defined when exactly two basic belief assignments
are in conflict by means of quantitative measures
of both the mass of the combined belief assigned
to the empty set before normalization, and the
distance between betting commitments of beliefs.
Murphy (2000), on her side, has studied a related
problem: the failure to balance multiple sources
of evidence. The notion of conflicting evidence
is well known in the context of sensor fusion
(Carlson & Murphy, 2005): the matter has been
recently surveyed by Sentz and Ferson (2002).
In opposition, though, not much work has
been done on the properties of the families of
compatible frames and their link with evidence
combination. In (Shafer et al, 1987) an analysis
of the collections of partitions of a given frame in
the context of the hierarchical representation of
belief can nevertheless be found, while in (Kohlas
& Monney, 1995) both the lattice-theoretical interpretation of families of frames and the meaning
of the concept of independence were discussed.
In (Cuzzolin, 2005) these themes were reconsidered: the structure of Birkhoff lattice of a family
of frames was proven, and the relation between
Dempster’s combination and independence of
frames highlighted.
TOWARDS AN ALGEBRAIC
APPROACH TO CONFLICT
Independence of Frames and
Dempster’s Combination
In the theory of evidence (Dempster, 1967; Shafer,
1976), the mathematical representation of subjective probability is not a standard probability
measure, but a belief function. Belief functions
are defined by distributing non-zero masses to
elements of the power set of the domain or frame
(rather than elements of the domain itself), masses
that need to normalize to 1. In this interpretation
they can be considered as finite random sets
(Nguyen, 1978). Obviously, they include probability measures in the special case in which
masses are given only to singletons. They can also
be interpreted as lower bounds to convex sets of
probabilities (Cuzzolin, 2008).
Different sources of evidence on the same
problem can generate two or more distinct belief functions. These functions have then to be
merged to take into account the full available
evidence. Several different operators have been
proposed. Historically the first such proposal is
due to Dempster (1968, Section 2.1). In addition,
such belief functions might not be defined on the
same domain, but on different domains which all
relate to the same decision or estimation problem.
This idea is encoded by the notion of family of
compatible frames. It turns out that belief functions defined on different compatible frames are
guaranteed to be combinable if and only if such
frames are independent is a way derived from
Boolean algebras. Albeit not equivalent to independence of sources in the original formulation of
Dempster’s combination, independence of frames
is then strictly intertwined with combinability.
A basic probability assignment (b.p.a.) over
a finite set or frame (Shafer, 1976) Θ is a function
m : 2˜ → [0, 1] on its power set 2˜ = {A ⊆ ˜ }
such that m (∅) = 0 ,
∑m (A) = 1 .
A⊆˜
242
An Algebraic Study of the Notion of Independence of Frames
The belief function (b.f.) b : 2˜ → [0, 1] associated with a basic probability assignment m
on Θ is defined as b (A) = ∑m(B ) . The focal
B ⊆A
elements of a belief function are the subsets A for
which the b.p.a. is non-zero m(A) ≠ 0 . Different
belief functions representing different pieces of
evidence on the same decision space ˜ can be
combined by Dempster’s orthogonal sum (Dempster, 1968).
Definition 1: The orthogonal sum or Dempster’s
sum of two belief functions b1, b2 : 2˜ → [0, 1]
is a new belief function b1 ⊕ b2 : 2˜ → [0, 1]
with b.p.a.
mb ⊕b (A) =
1
2
∑
∑
∩
mb (B ) mb (C )
B C =A
∩
1
2
mb (B ) mb (C )
B C ≠∅
1
,
where ρ(A) Θ' is called a refinement of Θ , Θ a coarsening
of ˜ ' . Shafer calls a structured collection F of
frames a family of compatible frames of discernment ((Shafer, 1976), pages 121-125: see the
Appendix for a formal definition). In such a family, in particular, every pair of frames has a common refinement, i.e., a frame which is a refinement
of both. Each finite collection of compatible frames
has many such common refinements. One of these
is particularly simple.
Proposition 1: If {Θ1, …, Θn } are elements of a
family of compatible frames F then there
exists a unique common refinement ˜ ∈ F
of them such that for all i = 1, …, n such
that
(1)
2
{θ} = ρ ({θ }) ∩… ∩ρ ({θ }), 1
where mb denotes the b.p.a. of bi .
i
When the denominator of Equation (1) is zero
the two functions are said to be non-combinable,
and their orthogonal sum simply does not exist. What happens when the belief functions to
combine are defined on different frames? If such
frames are compatible, in a way that was given a
mathematical characterization by Shafer (1976),
then such a combination is still possible.
Given two frames (finite sets) Θ and Θ' , a
'
map ρ : 2Θ → 2Θ is a refining if it maps the elements of Θ to a disjoint partition of Θ' :
( ) ∩ρ ({θ }) = ∅ ∀θ, θ
ρ {θ}
'
'
( )
∈ Θ; ∪ρ {θ} = Θ' ,
θ∈Θ
∪ρ({θ}) for all A ⊆ Θ . The frame
θ∈A
1
n
n
(2)
where ρi denotes the refining between Θi and
Θ . This unique frame is called the minimal refinement Θ1 ⊗ … ⊗ Θn of {Θ1, …, Θn } .
In the example of Figure 1 we want to find out
the position of a target point in an image. We can
pose the problem on a frame Θ1 = {c1, …, c5 }
obtained by partitioning the column range of the
image into 5 intervals, or partition it into 10 intervals, yielding Θ2 = {c11, c12 , …, c51, c52 } .
The row range can also be divided in, say, 6
intervals Θ3 = {r1, …, r6 } . All those frames belong to a family of compatible frames, with the
collection of cells Θ = {e1, …, e60 } depicted in
Figure 1-left as a common refinement, and refinings shown in Figure 1-right.
It is easy to verify that Θ meets condition (2)
for the frames Θ2 , Θ3 as, for example,
243
An Algebraic Study of the Notion of Independence of Frames
Figure 1. An example of a family of compatible frames. Different discrete quantizations of row and
column ranges of an image have as common refinement the set of cells on the left. The refinings ρ1, ρ2 , ρ3
between all those frames are shown on the right.
{e } = ρ (c ) ∩ρ (r ) . Hence Θ is the minimal
41
2
11
3
4
refinement of Θ2 , Θ3 .
The notion of compatible frames is crucial
when facing real-world problems, as the available pieces of evidence are typically defined on
different (albeit related) domains. For instance, in
computer vision inference on the objects present
in a scene is typically based upon a number of
salient measurements or features extracted from
the image(s). These features can be of various
natures and belong to different spaces. Nevertheless, they all say something on, for example, the
class of the object we are observing. Therefore,
when representing evidence via belief functions,
is important to be able to combine them even
when they belong to different frames of a family.
It has been proven that this is guaranteed to
be possible (via Dempster’s rule) if and only if
such frames are independent in the following way.
Definition 2: Let {Θ1, …, Θn } be elements of a
family of compatible frames, and the corresponding refinings to their minimal refinement. We say that {Θ1, …, Θn } are independent (Shafer, 1976) if
244
(
) ∩…∩ρ ({A }) ≠ ∅ ρ1 {A1 }
n
n
(3)
whenever ∅ ≠ Ai ⊆ Θi for i = 1, …, n (see
Figure 2).
In particular, it is easy to see that if
∃j ∈ 1, …, n  s.t. Θ j is a coarsening of some
other frame Θi , Θi > 1, then {Θ1, …, Θn } are
not independent. Mathematically, families of
compatible frames are collections of Boolean
sub-algebras of their common refinement ((Sikorski, 1964), see Appendix), as Equation (3) is
nothing but the independence condition (23) for
the associated sub-algebras.
We denote in the following by IF the independence relation introduced in Definition 2. It can
be proven that {Θ1, …, Θn } are IF iff (Cuzzolin,
2005)
Θ1 ⊗ … ⊗ Θn = Θ1 ×…× Θn , (4)
i.e., their minimal refinement is their Cartesian
product. More importantly, belief functions defined on a number of frames {Θ1, …, Θn } are
An Algebraic Study of the Notion of Independence of Frames
Figure 2. Independence of frames
surely combinable through Dempster’s rule iff
{Θ1, …, Θn } are IF .
Proposition 2 (Cuzzolin, 2005): Let {Θ1, …, Θn }
be a set of compatible frames. Then they are
independent iff all the possible collections
of b.f.s b1, …, bn defined on Θ1, …, Θn ,
respectively, are combinable through Dempster’s sum (1) over their minimal refinement
Θ1 ⊗ … ⊗ Θn . The notion of independence of frames is then
intertwined with that of combinability.
A Typical Application: ExampleBased Pose Estimation
An example application in which the combinability
of belief functions defined on different elements
of a family of compatible frames is central is
provided by the field of computer vision.
Pose estimation (Agarwal & Triggs, 2006;
Mori & Malik, 2006) is a well-studied problem
there. Given an image sequence capturing the
motion and evolution of an object of interest, the
problem consists in estimating the position and
orientation of the object at each time instant along
the sequence, along with its internal configuration
or pose. Such estimation is typically based on two
pillars: salient measurements extracted from the
images (features), and, when present, a model of
the structure and kinematics of the moving body.
Example based methods, on the other hand,
explicitly store a set of training examples whose
3D poses are known, and estimate the object’s pose
by searching for training image(s) similar to the
given input image and interpolating the associated poses (Athitsos et al, 2004; Shakhnarovich et
al, 2003). No prior structure of the pose space is
incorporated in the estimation. Let us assume that:
•
The available evidence comes in the form
of a training set of images containing sample poses of an unspecified object (see
Figure 3(a)); The nature of the object is not
necessarily known; we only know that its
configuration can be described by a vector
q∈ Q ⊂ RD in a pose space Q which is a
subset of RD ; we assume, however, that an
245
An Algebraic Study of the Notion of Independence of Frames
•
“oracle” exists which provides for each
training image I k the configuration qk of
the object portrayed in the image;
Also, we assume that the location of the
object within each training image is roughly known, in the form of a bounding box
containing the object of interest: features
are then extracted only within the bounding box.
During training, the object explores its range
of possible configurations, and a set of poses is
collected to form a finite approximation Q of the
pose space:
Q = {qk , k = 1, …,T } (5)
At the same time a number N of distinct features
are extracted from the available image(s):
Y = {yi (k ), k = 1, …,T }, i = 1, …, N . (6)
To acquire Q we need a source of “ground
truth” (an “oracle”) to tell us what pose the object
is in at each instant k of the training session. One
option is to use a motion capture system as in
(Cuzzolin and Gong, 2013; Rosales and Sclaroff,
2000) for the human body tracking problem (Figure 3(a)). In particular, if we apply a number of
reflective markers in fixed positions of the body,
the system is able to provide (through some triangulation algorithm) the 3D locations of the
markers during the motion in the training stage.
Since we do not know the parameter space of the
Figure 3. (a) Top: training image of a person standing still and moving his right arm. Bottom: training
image of the person walking inside a rectangle on the floor. Motion capture equipment is used to provide
a ground truth pose vector qk for each training image I k . (b) Feature extraction: the object of interest
is color segmented and the bounding box containing the foreground is detected to simulate localization
annotation. The row and column indices of the vertices of this bounding box are collected in a feature
vector y k = yi (k ) . (c) How a Mixture of Gaussian learned via EM clustering from the features col

lected in the training stage defines an implicit partition on the set of training poses Q . The Gaussian
densities Γ j , j = 1, …, n on the feature range Y define a partition of Y into regions Y j . Each region
{
}
Y j is in correspondence with the set Q j of sample poses qk whose feature value y(qk ) falls inside Y j .
246
An Algebraic Study of the Notion of Independence of Frames
object, it is then reasonable to adopt the markers’
locations as body configuration. The choice of
the “sample” motions here is typically based on
intuition, or gives a preference to “categorized”
motions (like dancing or walking in human body
tracking), perhaps as a legacy of the recognition
problem.
Based on this training evidence, in the testing
stage:
•
A localization algorithm (trained in the
training stage, e.g. (Felzenszwalb et al,
2010)) is employed to locate the object
within the test image: features are only
extracted (roughly) within the resulting
bounding box;
Such features are exploited to produce an
•
estimate q of its configuration, together
with a measure of how reliable this estimate is.
Learning Feature-To-Pose Maps
Learning a model from a set of samples necessarily implies some sort of approximation of the
involved feature spaces as well: this has non-trivial
consequences on the performance of the resulting
estimation.
Given the training data (5), (6), EM clustering
(McLachlan & Peel, 2000) provides a fast and reliable method to automatically build feature-pose
maps that can eventually be used for examplebased pose estimation.
Consider the N sequences of feature values
{yi (k ), k = 1, …,T }, i = 1, …, N , acquired in the
training session. EM clustering can be applied
separately to each one of them, yielding a Mixture
of Gaussian (MoG) approximation
{Γ , j = 1, …, n } , Γ
j
i
i
j
i
∼ N (µij , Σij ) (7)
with ni components (Gaussian pdfs) of each
d
feature space (the range Yi ⊂ R i of the unknown
feature function yi : I → Yi on the set of all images I ). The MoG (7) induces an implicit partition
{
Θi Yi 1, …, Yi
ni
}
(8)
o f t h e i - t h fe a t u r e r a n ge , w h e r e
Yi j = {y ∈ Yi s.t. “ ij (y ) > “ li (y ) ∀l ≠ j } is the
region of Yi in which the j-th Gaussian component
dominates all the others (Figure 3(c)). The number n of clusters is a critical parameter of the EM
algorithm, and is not known a priori (as, in fact,
there might be no definite or unique answer). An
estimate of n can be obtained from the data by
cross-validation (Burman, 1989). As a result, the
number of clusters ni will in general be different
for each feature space.
We call the finite partition ˜ i (8) of the range
Yi of the i-th feature function the i-th approximate
feature space. In virtue of the fact that features
are computed in the training stage in synchronous
with the true poses provided by the oracle, each
element Yi j of the discrete feature space is associated with the set of training poses whose i-th
feature value falls in Yi j (Figure 3(c) again):
j


ρi : Yi j Q i = qk ∈ Q : yi (k ) ∈ Yi j  . 



(9)
The feature-pose maps (9) are inherently multivalued, i.e., they map elements of an approximate
feature space ˜ i to subsets of the collection of
training poses Q . Given new feature evidence,
we need to exploit the maps (9) to make inferences
at the level of the pose.
247
An Algebraic Study of the Notion of Independence of Frames
Evidential Modeling
Summarizing, in the training stage the body moves
freely in front of the camera(s), exploring its
parameter space, while a sequence of training
poses Q = {qk , k = 1, …,T } is provided by an
oracle. The sample images are annotated by a
bounding box indicating the location of the object
within each image. At the same time:
1. For each time instant k, a number of features
are computed from the region of interest of
all the available images:
Y = {yi (k ), k = 1, …,T }, i = 1, …, N ;
2. EM clustering is applied to each feature
sequence {yi (k ), k = 1, …,T } (after choosing the appropriate number of clusters ni
by cross validation), yielding:
3. N a p p r o x i m a t e f e a t u r e s p a c e s
n
Θi {Yi 1, …, Yi i } , i.e., the implicit partitions of the feature ranges Yi associated with
the EM clusters;
4. N refinings (9) mapping clusters in the feature
space to sets of sample training poses in the
approximate pose space Q ,
So that the latter is a common refinement of
the collection of approximate feature spaces
{Θ1, …, Θn } . T h e c o l l e c t i o n o f FO D s
Q, {Θ1, …, Θn } along with their refinings
{ρ , …, ρ } is characteristic of both the object to
1
n
track and the feature functions y1, …, yN designed
and selected in the training stage: we call it the
evidential model (Cuzzolin & Gong, in press) of
the body (Figure 4).
Figure 4. Evidential model architecture. EM clustering on each set of feature values collected in the
n
training stage yields an approximate feature space Θi {Yi 1, …, Yi i } . Refining maps ρi between each
approximate feature space and the approximate parameter space Q = {qk , k = 1, …,T } are learned,
allowing the fusion on Q of the evidence measured on {Θ1, …, Θn } .
248
An Algebraic Study of the Notion of Independence of Frames
Tracking Algorithm
5. These “feature” belief functions are projected
How can this structure be used to estimate the pose
of the object, given new feature measurements?
Assuming that each of those features yi gen-
onto Q by vacuous extension;
6. Their Dempster’s combination is computed
via (1);
7. The pignistic transform (Smets, 1990) is
erates a belief function bi on the related feature
space Θi , the feature pose maps (9) can be used
to transfer the individual pieces of feature evidence
onto the pose space and combine them there.
If Ω is a refinement of Θ with refining ρ ,
any belief function defined on Θ can be “moved”
to its refinement Ω by vacuous extension: a belief
function b ' on Ω is called the vacuous extension
of a b defined on Θ iff its focal elements are the
images (via the refining ρ ) of focal elements of
b.
Since Q is a common refinement of all the
feature spaces, the belief functions associated with
new image measurements y1 (t ), …, yN (t ) can be
projected on it and there combined via Dempster’s
rule (1).
Summarizing, given an evidential model of
the moving body with N feature spaces, and
given at time t one or more test images, possibly
coming from different cameras:
1. The object detector learned in the training
stage is applied to the test image(s), returning for each of them a bounding box roughly
containing the object of interest;
2. The N feature values are extracted from the
resulting bounding boxes, as in the training
stage;
3. The likelihoods “ ij (yi (t )) of each feature
value yi (t ) with respect to the appropriate
learned Mixture of Gaussian distribution on
Yi are computed, according to (7);
4. For each feature i = 1, …, N a belief function bi (t ) is built from these likelihood (cfr.
Cuzzolin and Gong, in press);
applied to the resulting belief function b(t ) ,
yielding a distribution on Q from which a
pose estimate q(t ) can be computed
(Cuzzolin and Gong, in press).
Handling Conflict
The mass assigned to the empty set by (1)
represents the amount of evidence which is in
contradiction, as it is assigned by the two belief
functions to contradictory (disjoint) events. In
our pose estimation scenario, conflict can arise
in the test stage for basically two reasons: 1- the
object is localized but in an imprecise way (due
to limitations of the detector learned during
training): as a consequence background features
which conflict with foreground information are
also extracted; 2- occlusions are present, causing
conflict for similar reasons.
By Proposition 2, belief functions coming
from different image features are guaranteed to
be combinable only in the ideal case in which all
approximate feature spaces are IF independent (4).
In all practical cases, the largest group of coherent, non-conflicting features need to be sought:
under the assumption that most features will in
any case come from the foreground, this will lead
to the integration of foreground information only,
bringing robustness to localization and occlusions.
An analysis of the notion of conflict, and efficient
algorithms for the detection of non-conflicting
groups of belief functions are a crucial part of the
framework (Cattaneo, 2011; Liu, 2006).
249
An Algebraic Study of the Notion of Independence of Frames
An Algebraic Study of
Independence of Frames
In (Cuzzolin, 2005), starting from an analogy between independence of frames and linear independence, we conjectured a possible algebraic solution
to the conflict problem, based on a mechanism
similar to the classical Gram-Schmidt algorithm
for the orthogonalization of vectors. Indeed, one
can observe that the independence condition (3)
resembles the condition under which a collection of vector spaces has maximal span (see also
Equation (4)):
v1 + … + vn ≠ ∅ ∀vi ∈ Vi
(
) ∩…∩ρ ({A }) ≠ ∅
ρ1 {A1 }
(
n
n
∀Ai ⊆ Θi
span {V1, …,Vn }
= V1 ×…×Vn
Θ1 ⊗ … ⊗ Θn
≡
= Θ1 ×…× Θn .
≡
10)
Let us call a number of subspaces {V1, …,Vn }
of a vector space V “independent” iff each collection of vectors {vi ∈ Vi , i = 1, …, n } is linearly independent. It follows that while a number
of compatible frames {Θ1, …, Θn } are IF iff each
˜
selection of their representatives Ai ∈ 2 i has
non-empty intersection, a collection of vectors
subspaces {V1, …,Vn } is independent iff for each
choice of vectors vi ∈ Vi their sum is non-zero.
The collection of all the subspaces of a vector
space or projective geometry forms a modular
lattice. As we prove here, families of frames can
be given the algebraic structure of semi-modular
lattice, providing a partial explanation of this analogy. The goal of this Chapter is to go further and
analyze the notion of independence of frames from
an algebraic point of view, in order to understand
whether it possesses any meaningful relations with
other forms of independence.
The purpose is two-fold: on one side, as independence of frames is formally independence
of Boolean sub-algebras, this is a contribution
250
towards a better understanding of the notion in
different fields of modern algebra. On the other
hand, it provides the basis for an eventual algebraic
proposal to the solution of the conflict problem
in subjective probability.
The paradigm of abstract independence in
modern algebra is represented by the notion of
matroid, introduced by Whitney (1935). Whitney,
van der Waerden (1937), Mac Lane (1938) and
Teichmuller (1936) all recognized that several
apparently different notions of dependence (Beutelspacher & Rosenbaum, 1998; Harary & Tutte,
1969) in algebra (circuits in graphs, flats in affine
geometries) have many properties in common with
linear dependence of vectors.
Definition 3: A matroid M = (E, I ) is a pair
formed by a ground set E, and a collection
of independent sets I ∈ 2E , such that:
1. ∅ ∈ I;
2. If I ∈ I and I' ⊆ I then I' ∈ I ;
3. If I1 and I2 are in I, and | I1| < | I2|, then
there is an element e of I2 − I1 such that
I 1 ∪e ∈ I .
Condition (3) is called augmentation axiom,
and is the foundation of the notion of matroidal
independence. The name was coined by Whitney
(1935) because of a fundamental class of matroids
which arise from the collections of linearly independent (in the ordinary sense) sets of columns of
a matrix, called “vector matroid”. Unfortunately,
Theorem 1: A family of compatible frames F
endowed with Shafer’s independence IF is
not a matroid.
Proof: In fact, IF does not meet the augmentation
axiom (3) of Definition 3. Consider two
independent frames I = {Θ1, Θ2 } . If we
pick another arbitrary frame Θ3 of the fam-
ily, the collection I ' = {Θ3 } is trivially IF.
Suppose that Θ3 ≠ Θ1, Θ2 . Then, since
An Algebraic Study of the Notion of Independence of Frames
|I|>|I’|, by augmentation we can form a new
pair of independent frames by adding any
of Θ1, Θ2 to Θ3 . But it is easy to find a
counterexample, for instance by picking as
Θ3 the common coarsening of Θ1 and Θ2
(remember the remark after Definition 2).
Matroidal independence, however, extends to
similar relations in other algebraic structures, in
particular semi-modular and geometric lattices
(Stern, 1999). Even though families of frames are
not matroids, they form semi-modular lattices:
therefore, IF inherits interesting relations with a
number of extensions of matroidal independence
to semi-modular lattices (Birkhoff, 1935).
Eventually, IF is in fact opposed to matroidal
independence.
The Lattice of Frames
Collections of compatible frames (see Appendix)
are collections of Boolean sub-algebras of (the
power set of) their minimal refinement. In addition, as it has been proven in (Cuzzolin, 2005),
they possess the structure of lattice. Two different
order relations between frames can be defined.
According to the chosen ordering, the resulting
lattice will be either upper or lower semi-modular.
This allows us to introduce a number of different
extensions of matroidal independence to compatible frames, as we will see later on.
A partially ordered set or poset is a set P endowed with a binary relation such that, for all x,
y, z in P the following conditions hold:
x ≤ x; if x ≤ y and y ≤ x then x = y; if x ≤ y
and y ≤ z then x ≤ z.
In a poset we say that x covers y (x y) if x
≥ y and there is no intermediate element in the
chain (collection of consecutive elements) linking
them. An example is the power set 2Θ of a set ˜
together with the inclusion relation ⊂ . A poset
has finite length if the length of all its chains is
bounded. Given two elements x,y ∈ P of a poset
P their least upper bound supP (x,y) = x ∨ y is the
smallest element of P that is bigger than both x
and y. Their greatest lower bound infP (x,y) = x
∧ y is the biggest element of P that is smaller than
both x and y. In the case of L = (2Θ , ⊆) “sup” is
the usual set-theoretic union, A ∨ B = A ∪ B,
while “inf” is the usual intersection A ∧ B = A
∩ B. By induction, sup and inf can be defined
for arbitrary finite collections too.
However, not any pair of elements of a poset,
in general, is guaranteed to admit inf and/or sup.
Definition 4: A lattice L is a poset in which each
pair of elements admits both inf and sup.
When each arbitrary (even not finite) collection of elements of L admits both inf and sup, L
is said complete. In this case there exist 0 ≡ ∧
L, 1 ≡ ∨ L called respectively initial and final
element of L. 2Θ is complete, with 0 = ∅ and 1
= {˜ } . The height h(x) of an element x in L is
the length of the maximal chain from 0 to x. For
the power set 2Θ , the height of a subset A ∈ 2Θ
is simply its cardinality |A|.
Families of Frames as Lattices
In a family of compatible frames one can define
two distinct order relations on pairs of frames,
both associated with the notion of refining:
Θ1
Θ1 ≤* Θ2 ⇔ ∃ρ : 2
Θ
→ 2 2 refining (11)
( Θ1 is a coarsening of Θ2 ), or
Θ2
Θ1 ≤ Θ2 ⇔ ∃ρ : 2
Θ
→ 2 1 refining (12)
i.e., Θ1 is a refinement of Θ2 . Relation (12) is
clearly the inverse of (11). It makes sense to distinguish them explicitly as they generate two
251
An Algebraic Study of the Notion of Independence of Frames
distinct algebraic structures, in turn associated
with different extensions of the notion of matroidal independence.
As it has been proven in (Cuzzolin, 2005), a
family of frames F is a poset with respect to both
(11) and (12). More precisely, after introducing
the notion of maximal coarsening as the largest
cardinality common coarsening Θ1 ⊕ …⊕Θn of
x x ∧ y x ∨ y y ⇒ x ∨* y * x
y * x ∧* y. (13)
For lattices of finite length, upper and lower
semi-modularity together imply modularity. In
this sense semi-modularity is indeed “one half”
of modularity.
a given collection of frames, {Θ1 … Θn } , we can
prove that:
Upper and Lower SemiModular Lattices of Frames
Proposition 3: Both (F , ≤) and (F , ≤* ) , where
F is a family of compatible frames of discernment, are lattices, where
Families of frames possess indeed the structure
of semi-modular lattice.
∧i Θi = ⊗i Θi , ∨i Θi =
⊕i Θi , ∧i* Θi = ⊕i Θi , ∨i* Θi = ⊗i Θi .
Upper and Lower Semi-Modularity
A special class of lattices (modular lattices) arises
from projective geometries, i.e. collections L(V)
of all subspaces of a vector space V. Modular
lattices, as many authors have shown, are also
related to abstract independence. This quality is
retained by a wider class of lattices called semimodular lattices.
Definition 5: A lattice L is upper semi-modular
if for each pair x, y of elements of L, x x ∧ y
implies x ∨y y . A lattice L is lower semimodular if for each pair x, y of elements of
L, x ∨ y y implies x x ∧ y .
If L is upper semi-modular with respect to an
order relation ≤ , than the corresponding dual
lattice with order relation ≤* is lower semimodular, as
252
Theorem 2: (F , ≤) is an upper semi-modular
lattice; (F , ≤* ) is a lower semi-modular
lattice.
Proof: We just need to prove the upper semimodularity with respect to ≤ .
Consider two compatible frames Θ, Θ ' , and
suppose that Θ covers their minimal refinement
Θ ⊗ Θ ' , (their inf with respect to ≤ ). The proof
articulates into steps depicted in Figure 5.
•
•
As Θ covers Θ ⊗ Θ ' we have that
| Θ |=| Θ ⊗ Θ ' | +1 ;
This means that there exists a single element p ∈ Θ which is refined into a pair of
elements {p1, p2 } of Θ ⊗ Θ ' , while all
other elements of Θ are left unchanged:
{p1, p2 } = ρ (p );
•
•
•
This in turn implies that p1, p2 each belong
to the image of a different element of Θ '
(otherwise Θ would itself be a refinement
of Θ ' , and we would have Θ ⊗ Θ' = Θ ):
p1 ∈ ρ ' (p '1 ) , p2 ∈ ρ ' (p '2 ) ;
Now, if we merge p '1, p '2 we obviously
have a coarsening Θ '' of Θ ' :
{p '1, p '2 } = ρ '' (p '')
An Algebraic Study of the Notion of Independence of Frames
Figure 5. Proof of the upper semi-modularity of (F , ≤)
•
But Θ '' is a coarsening of Θ , too, as we
can build the refining σ : Θ '' → 2Θ :
σ (q ) = ρ ' ρ '' (q ) where ρ ' ρ '' (q ) is a
(
)
(
)
subset of Θ for all q ∈ ˜ '' :
◦◦
If
q = p '' ,
σ (q )
{p} ∪(ρ ' (p ' ) {p })∪(ρ ' (p ' ) {p }) ;
If q ≠ p '' , ρ ' (ρ '' (q )) is also a set of
1
◦◦
is
1
2
2
elements of Θ , as all elements of Θ
but p are left unchanged by ρ .
•
•
As Θ'' = Θ' − 1 we have that Θ'' is the
maximal
coarsening
of
Θ, Θ ' :
''
Θ = Θ ⊕Θ ' ;
Hence Θ ' covers Θ ⊕ Θ ’ , which is the sup
of Θ, Θ ' in (F, ≤) .
The lower semi-modularity with respect to ≤*
comes then from (13).
Theorem 2 strengthens the main result of
(Cuzzolin, 2005), where we proved that finite
families of frames are Birkhoff. A lattice is Birkhoff if x ∧ y ≺ x , y implies x , y ≺ x ∨ y . (Upper)
semi-modularity implies the Birkhoff property,
but not vice-versa.
We will here focus on finite families of frames.
Given a set of compatible frames {Θ1, …, Θn }
we can consider the set P(Θ) of all the partitions
of their minimal refinement Θ = Θ1 ⊗ … ⊗ Θn .
As the independence condition (Definition 2)
involves only partitions of Θ1 ⊗ … ⊗ Θn , we can
conduct our analysis there. We denote by
L (Θ) = (P (Θ), ≤) , L* (Θ) = (P (Θ), ≤* ) the
two lattices associated with the set P(Θ) of partitions of Θ , with order relations (11), (12) respectively.
Example: The partition lattice P4. Consider for
example the partition lattice associated with
a frame of size 4: Θ = {1, 2, 3, 4} , depicted
in Figure 6, with order relation ≤* . Each
edge indicates here that the bottom partition
covers the top one.
To understand how inf and sup work in the
frame lattice, pick the partitions
x = {1 / 2, 3, 4} , x ' = {1 / 2, 3 / 4} .
253
An Algebraic Study of the Notion of Independence of Frames
According to the diagram the partition x ∨* x '
which refines both and has smallest size is
Θ = {1 / 2 / 3 / 4} itself. Their inf x ∧* x ' is x ,
as x ' is a refinement of x . If we pick instead the
pair of par titions y = {1, 2 / 3 / 4} and
y ' = {1, 3 / 2 / 4} , we can notice that both y and
y ' cover their inf y ∧* y ' = {1, 2, 3, 4} but in turn
their sup y ∨* y ' = ˜ = {1 / 2 / 3 / 4} does not
cover them. Therefore, (P (Θ), ≤* ) is not upper
semi-modular but lower semi-modular.
On the atoms of a lattice, i.e., the elements of
the lattice covering 0 (think of one-dimensional
subspaces of a vector space V) it is possible to
define a matroidal independence relation. In
particular, for each upper semimodular lattice L
there exists a collection I of sets of atoms such
that (A, I ) is a matroid. As families of frames
form semi-modular lattices, we can explore the
relations between IF and several extensions of
matroidal independence to both L (Θ) and L* (Θ) .
INDEPENDENCE ON LATTICES VS.
INDEPENDENCE OF FRAMES
Atom Matroid of a SemiModular Lattice
are linearly independent iff ∑αi vi = 0 αi = 0
i
for all i.
This classical definition can be given several
equivalent formulations:
I2 :
v j ≠ span(vi , i ≠ j ) ∀j = 1, …, n ;
v j ∩span (v1, …, v j −1 ) = 0, ∀j = 2, …, n ;
(
)
I3 : dim span (v1, …, vn ) = n. 254
Definition 6: The following relations on the elements of a semi-modular lattice with initial
element 0 can be defined:
{v1, …, vn } are I1 if
(l
(14)
j
) (
∨ li ≡ l j ∧ ∨ li ≠ l j
i≠j
i≠j
)
∀j = 1, …, n;
{v1, …, vn } are I2 if
l j ∧ ∨ li = 0
∀j = 2, …, n;
i<j
{v1, …, vn } are I3 if
h
Consider again the classical example of linear
independence of vectors. By definition {v1, …, vn }
I1 :
Remember that the one-dimensional subspaces of a vector space V are the atoms of the
lattice L(V) all the linear subspaces of V, for which
∨= span, ∧ = ∩ , dim = h and 0 = 0. Following
this intuition we can extend the relations (14) to
collections of arbitrary (not necessarily atomic)
non-zero elements of an arbitrary semi-modular
lattice with initial element 0.
( l ) = ∑h (l ).
i i
i
i
These relations have been studied by several
authors in the past: our purpose here is to understand their relation with independence of frames
in the semimodular lattice of frames.
For arbitrary elements of a lattice, I1, I2 and
I3 are distinct, and none of them generates a
matroid. However, when defined on the atoms of
an upper semimodular lattice with initial element
they do coincide, and form a matroid (Szasz,
1963).
An Algebraic Study of the Notion of Independence of Frames
Proposition 4: The restrictions of the above relations to the set of the atoms A of an upper
semi-modular lattice L with initial element
are equivalent, namely I1 = I2 = I3 = I
on A, and (A, I ) is a matroid.
As the partition lattice has both an upper L (Θ)
and lower L* (Θ) semi-modular form, we can
introduce two forms I1, I2 , I3 and I1* , I2* , I*3 of
the above relations associated with L (Θ) and
L* (Θ) , respectively. These constitute different
valid extensions of matroidal independence to
elements of semi-modular lattices. As families of
compatible frames form semi-modular lattices in
their own right, it is natural to wonder what their
relationship is, if any, with Shafer’s independence
of frames IF.
Boolean and Lattice Independence
in the Upper Semi-Modular Lattice
L (Θ)
Form of the relations. In L (Θ) the relations introduced in Definition 6 assume the forms:
only if no frame Θ j is a refinement of the maximal coarsening of all the others. They are I2 if
and only if, for all j = 2, …, n , Θ j does not have
a non-trivial common refinement with the maximal coarsening of all its predecessors.
The interpretation of I3 is perhaps more interesting. The latter is equivalent to say that the
coarsening that generates ⊕ni =1 Θi can be broken
up into n steps of the same length of the coarsenings that generate each of the frames Θi starting
from Θ. .
Namely: first Θ1 is obtained from Θ by merging Θ − | Θ1 | elements, then Θ − | Θ2 | elements of this new frame are merged, and so on
until we get to a frame of cardinality ⊕ni =1 Θi .
We will return on this when discussing the dual
relation on the lower semi-modular lattice L* (Θ) .
To study the logical implications between
these lattice-theoretic relations and independence
of frames, and between themselves, we first need
a useful Lemma.
Lemma 1: If {Θ1 … Θn } are IF, n > 1, then
{v1, …, vn } ∈ I1 iff
Θ j ⊗ ( ⊕i ≠ j Θi ) ≠ Θ j ∀j = 1, …, n, They read as follows: {Θ1 … Θn } are I1 if and
(15)
⊕ni =1 Θi = 0F .
Proof: We prove Lemma 1 by induction. For n =
2, let us suppose that {Θ1, Θ2 } are IF. Then
(
Θ j ⊗ ( ⊕i < j Θi ) = Θ
∀j = 2, …, n, (16)
2
for
all
denotes as usual the refining from Θi to
(
)
Θ − ⊕ni =1 Θi = ∑ Θ − | Θi | , i =1
2
∅ ≠ A1 ⊆ Θ1, ∅ ≠ A2 ⊆ Θ2 (where ρi
{v1, …, vn } ∈ I3 iff
n
) ∩ρ ({A }) ≠ ∅
ρ1 {A1 }
{v1, …, vn } ∈ I2 iff
Θ1 ⊗ Θ2 ). Suppose by absurd that their
common coarsening contains more than a
single element, Θ1 ⊕ Θ2 = {a, b} . But then
(17)
forinthelattice L (Θ) wehave Θi ∧ Θ j = Θi ⊗ Θ j ,
Θi ∨ Θ j = Θi ⊕ Θ j , h (Θi ) = Θ − | Θi |, and
(
) ∩ρ (ρ (b )) = ∅
ρ1 ρ1 (a )
2
2
0 = Θ.
255
An Algebraic Study of the Notion of Independence of Frames
where ρi denotes the refining between Θ1 ⊕ Θ2
5. and Θi ), going against the hypothesis.
{Θ , Θ } ∈ IF and {Θ , Θ } ∈ ℑ
1
2
1
2
3
iff
Θi = 0F and Θ j = Θ. .
Induction step: Suppose that the thesis is true
for n − 1. We know that if {Θ1 … Θn } are
Proof:
1. If {Θ1, Θ2 } are IF then Θ1 is not a
refinement of Θ2 , and vice-versa, un-
IF we have that any sub-collection {Θi , i ≠ j }
is also IF. By inductive hypothesis, the latter
implies:
less one of them is 0F . But then they
are I1 ( Θ1 ⊗ Θ2 ≠ Θ1, Θ2 ).
2. We can give a counterexample as in
Figure 7 in which {Θ1, Θ2 } are I1 (as
⊕i ≠ j Θi = 0F ∀j = 1, …, n.
none of them is refinement of the other
one) but their minimal refinement
Θ1 ⊗ Θ2
has
cardinality
Then, since 0F is a coarsening of Θ j for all
4 ≠| Θ1 | ⋅ | Θ2 |= 6 (hence they are
not IF).
3. Trivial.
4. I3 I1
is equivalent to
j, Θ j ⊕ (⊕i ≠ j Θi ) = Θ j ⊕ 0F = 0F .
Pairs of frames: Let us consider first the special
case of collections of just two frames. For
n = 2 the three relations I1, I2 , I3 read respectively as
not I1 not I3 . But {Θ1, Θ2 } not
I1 reads as Θ1 ⊗ Θ2 = Θi , which is
(18)
in turn equivalent to Θ1 ⊕ Θ2 = Θ j .
I.E.,
5. As {Θ1, Θ2 } are IF, Θ1 ⊕ Θ2 = 1 so
It is interesting to remark that if {Θ1, Θ2 } are
lent to | Θ | + | 1 |=| Θ1 | + | Θ2 | .
Now, by definition
Θ1 ⊗ Θ2 ≠ Θ1, Θ2 , Θ1 ⊗ Θ2 = Θ, Θ + Θ1 ⊕ Θ2 =| Θ1 | + Θ2 .
I1 then Θ1, Θ2 ≠ Θ. .
We can prove the following logical implications.
Theorem 3:
1. If {Θ1, Θ2 } are IF then {Θ1, Θ2 } are
2. 3. I1 if Θ1, Θ2 ≠ 0F .
{Θ , Θ } ∈ ℑ d o e s n ot i m p ly
{Θ , Θ } ∈ IF.
{Θ , Θ } ∈ ℑ implies {Θ , Θ } ∈ ℑ
1
2
1
2
1
2
1
2
1
iff Θ1, Θ2 ≠ Θ .
4. {Θ , Θ } ∈ ℑ
1
2
3
1
implies {Θ1, Θ2 } ∈ ℑ1
iff Θ1, Θ2 ≠ Θ .
256
2
that {Θ1, Θ2 } ∈ ℑ3 , which is equiva-
| Θ |≥| Θ1 ⊗ Θ2 |=| Θ1 || Θ2 |
(the last passage holding as those frames are IF).
{Θ , Θ } ∈ ℑ
≡ | Θ | + | Θ j |=| Θi | + | Θ j | ≡ | Θ |=| Θi | .
1
2
3
Butthen {Θ1, Θ2 } ∈ ℑ3 implies {Θ1, Θ2 } ∈ ℑ1
iff Θ1, Θ2 ≠ Θ. .
An Algebraic Study of the Notion of Independence of Frames
Figure 7. Counterexample for the conjecture “ I1
implies IF” of Theorem 3.
stronger than the weakest form I1 . Some of those
features are retained by the general case too.
General Case n > 2
The situation is somehow different in the general
case of a collection of n frames. IF and I1 , in
particular, turn out to be incompatible.
Theorem 4: If {Θ1 … Θn } are IF, n > 2 then
{Θ1 … Θn } are not I1 .
Therefore {Θ1, Θ2 } ∈ IF and {Θ1, Θ2 } ∈ ℑ3
together imply
Θ1 + Θ2 ≥ Θ1 Θ2 + 1
(
)
≡ Θ1 − 1 ≥ Θ1 Θ2 − Θ2 = Θ2 Θ1 − 1
≡ Θ2 ≤ 1,
which holds iff the equality holds, i.e., Θ2 = 0F .
The latter implies Θ = Θ1 ⊗ Θ2 = Θ1 .
In the singular case Θ1 = 0F , Θ2 = Θ by
definition (18) the pair {0F , Θ} is both I2 and
Proof: If {Θ1 … Θn } are IF then any collection
formed by some of those frames is IF (otherwise we could find empty intersections in
Θ1 ⊗ … ⊗ Θn ). But then, by Lemma 1,
⊕i ∈L⊂{1,…,n } Θi = 0F
for all subsets L of {1, …, n } with at least 2 elements: |L| > 1.
But then, as L = {i ≠ j, i ∈ {1, …, n }} has
cardinality n − 1 > 1 (as n > 2 ) we have
⊕i ≠ j Θi = 0F for all j ∈ {1, …, n } . Therefore
I3 , but not I1 . Besides, two frames can be both
Θ j ⊗ (⊗i ≠ j Θi ) = Θ j ⊗ 0F = Θ j
I2 and IF without being singular in the above
sense. The pair of frames {y,y’} of Figure 6 provides such an example, as y ⊗ y ' = Θ ( I2 ) and
they are IF.
Finally, it well known that (Szasz, 1963) on
an upper-semi-modular lattice (such as L (Θ) ):
and {Θ1 … Θn } are not I1 .
Proposition 5: I2 I3 .
Proof: If {Θ1 … Θn } are IF then {Θ1 … Θk −1 }
are IF for all k = 3,…,n.
Independence of frames and the most demanding form I3 of extended matroidal independence
relation to frames as elements of an upper semimodular lattice are mutual exclusive, and are both
∀j ∈ {1, …, n }
Theorem 5: If {Θ1 … Θn } are IF, n > 2, then
{Θ1 … Θn } are not I2 .
But by Lemma 1 this implies ⊕i <k Θi = 0F ,
so that:
Θk ⊗ ⊕i <k Θi ≠ Θk ∀k > 2.
257
An Algebraic Study of the Notion of Independence of Frames
Now, {Θ1 … Θn } ∈ IF with n > 2 implies
Θk ≠ Θ for all k. That holds because, as n > 2 ,
there is at least one frame Θi in the collection
{Θ1 … Θn } which is distinct from 0F , and
clearly {Θi , Θ} are not IF (as Θi is a non-trivial
coarsening of Θ ). Hence
Θk ⊗ ⊕i <k Θi ≠ Θ ∀k > 2,
which is in fact a much stronger condition than
¬I2 .
A special case is that in which one of the frames
is Θ itself. By Definitions (15) and (16) of I1
and I2 , if there exists j such that Θ j = Θ then
{Θ1 … Θn } ∈ ℑ2 implies {Θ1 … Θn } not I1 .
By Proposition 5 it follows that
Corollary 1: If {Θ1 … Θn } are IF, n > 2, then
{Θ … Θ } are not
1
n
I3 .
Putting together the results of Theorems 3 and
5 and Corollary 1 we get that IF and I3 are incompatible in all significant cases.
Corollary 2: If {Θ1 … Θn } are IF then they are
not
I3 , unless n = 2 , Θ1 = 0F and
Θ2 = Θ .
Boolean and Lattice Independence
in the Lower Semi-Modular Lattice
{v1, …, vn } ∈ I2* iff
(
)
Θ j ⊕ ⊗ij=−11 Θi = 0F ∀j = 2, …, n, (20)
n
i =1
Θi ∧* Θ j = Θi ⊕ Θ j , , ˜ i ∨* ˜
h (˜
*
i
)=
˜
i
{v1, …, vn } ∈ I iff
Θ j ⊕ (⊗i ≠ j Θi )≠ Θ j ∀j = 1, …, n, (19)
258
(21)
j
= ˜ i ⊗ ˜ j, ,
− 1, , and 0 = 0F . .
These relations also have quite interesting
semantics. The frames {Θ1 … Θn } are I1* iff none
of them is a coarsening of the minimal refinement
of all the others. In other words, there is no
proper subset of {Θ1 … Θn } which has still
Θ1 ⊗ … ⊗ Θn as a common refinement. They are
I2* iff for all j > 1 the frame Θ j does not have a
non-trivial common coarsening with the minimal
refinement of its predecessors.
Finally, the third form I*3 of extended matroidal independence relation possesses a very interesting semantics in terms of probability spaces.
As the dimension of the polytope of probability
measures definable on a domain of size k is k − 1,
, {Θ1 … Θn } are I*3 iff the dimension of the probability polytope associated with the minimal refinement is the sum of the dimensions of the
polytopes associated with the individual frames:
{Θ1 … Θn } ∈I*3 ≡ dim P˜
1
⊗…⊗˜
L (Θ)
*
1
)
for in the lattice L (Θ) we have
*
Semantics of extended matroidal independence:
Analogously, the extended matroidal independence relations associated with the lower semimodular lattice L* (Θ) read as
(
{v1, …, vn } ∈ I*3 iff ⊗ni =1 Θi − 1 = ∑ Θi − 1 ,
n
= ∑dim P˜ .
i
(22)
It is interesting to point out the following analogy between independence of frames and I*3 .
While condition (4) for IF
Θ1 ⊗ … ⊗ Θn = Θ1 ×…× Θn
An Algebraic Study of the Notion of Independence of Frames
Figure 6. The partition (lower) semi-modular lattice L* (˜ ) for a frame ˜ of size 4. Partitions of ˜ are
denoted by {A1 / … / Ak } . Partitions with the same number of elements are arranged on the same
level. An edge between two nodes indicates that the bottom partition covers the top one.
states that the minimal refinement is the Cartesian
product of the individual frames, Equation (22)
for I*3 states that the probability simplex of the
minimal refinement is a Cartesian product of the
individual ones.
General Case
{˜ i , i ≠ j} ∈ IF, for all θ ∈ ⊗i ≠ j Θi there exist
θi ∈ Θi , i ≠ j s.t.
{θ} = ∩ρ ' (θ ),
i≠j
i
i
where ρ 'i is the refining to ⊗i ≠ j Θi . Now, θ
Theorem 6: If {Θ1 … Θn } ∈ IF and Θ j ≠ 0F
for all j, then {Θ1 … Θn } ∈ ℑ1* .
Proof: Let us suppose that {Θ1 … Θn } ∈ IF but
not I1* , i.e., there exist j such that Θ j is a
coarsening of ⊗i ≠ j Θi . We need to prove
that ∃A1 ⊂ Θ1, …, An ⊂ Θn such that
belongs to a certain element A of the partition Π j
. By hypothesis ( Θ j ≠ 0F for all j) Π j contains
at least two elements. But then we can choose
θj = ρ−1 (B ) with B another element of Π j . In
this case we obviously get
ρj (θj ) ∩∩ρi (θi ) = ∅,
i≠j
(
) ∩…∩ρ ({A }) = ∅,
ρ1 {A1 }
n
n
where ρi denotes the refining from Θi to
Θ1 ⊗ … ⊗ Θn .
Since Θ j is a coarsening of ⊗i ≠ j Θi , there
exists a partition Π j of ⊗i ≠ j Θi associated with
Θ j , and a refining ρ from Θ j to ⊗i ≠ j Θi . As
which implies that {Θ1 … Θn } are not IF, against
the hypothesis.
Does IF imply I1* even when ∃˜ i = 0F ? The
answer is negative. {Θ1 … Θn } ∈ ¬I1* means that
there exists j s.t. Θ j is a coarsening of ⊗i ≠ j Θi .
But if Θi = 0F then Θi is a coarsening of ⊗i ≠ j Θi .
The reverse implication does not hold: IF and
*
I1 are distinct.
259
An Algebraic Study of the Notion of Independence of Frames
Theorem 7: {Θ1 … Θn } ∈ I1* does not imply
{Θ1 … Θn } ∈ IF.
Proof: We need a simple counterexample. Consider two frames Θ1 and Θ2 in which Θ1
Θ j ⊗ ⊗i ≠ j Θi = Θ j × (×i ≠ j Θi ),
Then Θ1, Θ2 ≠ Θ1 ⊗ Θ2 but it easy to find
which is equivalent to {Θ j , ⊗i ≠ j Θi } ∈ IF. But
then by Lemma 1 we get as desired.
It follows from Theorems 6 and 9 that, unless
some frame is unitary,
an example (see Figure 8) in which Θ1, Θ2
are not IF.
Corollary
is not a coarsening of Θ2 ( Θ1, Θ2 are I1* ).
Besides, like in the upper semi-modular case,
I2* does not imply I1* .
Theorem 8: {Θ1 … Θn } ∈ I2* does not imply
{Θ1 … Θn } ∈ I1* .
Proof: Figure 9 shows a counterexample to the
conjecture I2* I1* . Given Θ1 ⊗ … ⊗ Θ j −1
and Θ j , one possible choice of Θ j +1 such
that {Θ1 … Θ j +1 } are I2* but not I1* is shown.
3:
{˜ 1 … ˜ n } ∈
IF
{˜ 1 … ˜ n } ∈ I ∧ I (i.e., independence
of frames is a more demanding requirement
than both the first two forms of latticetheoretic independence)
*
1
*
2
Note that the converse is false. Think of a pair
of frames (n = 2), for which
(
)
Θ1 ⊕ Θ2 ≠ Θ1, Θ2 Θ1, Θ2 ∈ ℑ1* , (
*
2
)
Θ1 ⊕ Θ2 = 0F Θ1, Θ2 ∈ ℑ .
Independence of frames IF is a stronger condition than I2* as well.
Such conditions are met for instance in Figure
8, where the two frames are not IF.
Theorem 9: {Θ1 … Θn } ∈ IF {Θ1 … Θn } ∈ ℑ2* .
Theorem 10: If a collection {Θ1 … Θn } of com-
Proof: We first need to show that {Θ1 … Θn } are
IF iff for all j = 1,…,n the pair {Θ j , ⊗i ≠ j Θi }
is IF. As a matter of fact (4) can be written
as
patible frames is IF then it is not I*3 , unless
n = 2 and one of the frames is the trivial
partition.
Proof: According to Equation (4), {Θ1 … Θn }
are IF iff | ⊗Θi |= ∏ | Θi | , while accordi
Figure 9. A counterexample to I2* I1*
260
An Algebraic Study of the Notion of Independence of Frames
Figure 8. A counterexample in which I1* does not imply IF
ing
to
(21)
they
are
I*3
iff
⊗Θi − 1 = ∑( Θi − 1) . Those conditions
i
are both met iff
∑Θ
i
i
− ∏ Θi = n − 1, i
which happens only if n = 2 and either
Θ1 = 0F or Θ2 = 0F .
Instead of being algebraically related notions,
independence of frames and matroidicity work
against each other. As independence of frames
derives from independence of Boolean subalgebras of a Boolean algebra (Sikorski, 1964), this
is likely to have interesting wider implications on
the relationship between independence in those
two fields of mathematics.
FUTURE RESEARCH DIRECTIONS
On Independence
Figure 10 illustrates what we have learned so
far about the relations between independence of
frames and the various extensions of matroidal
independence to semi-modular lattices, in both
the upper (left) and lower (right) semi-modular
lattice of frames.
Only the general case of a collection of more
than two non-atomic frames is shown for sake of
simplicity: special cases ( Θi = 0F for L* (Θ) ,
Θi = Θ for L (Θ) ) are also neglected.
In the upper semi-modular case, minding the
special case in which one of the frames is ˜ itself,
independence of frames IF is mutually exclusive
with all lattice-theoretic relations I1, I2 and I3
(Theorems 4, 5 and Corollary 1) unless we consider two non-atomic frames, for which IF implies
I1 (Theorem 3). In fact they are the negation of
each other in the case of atoms of L (Θ) (frames
of size n − 1), when I = I1 = I2 = I3 is trivially true for all frames, while IF is never met. The
exact relation between I1, I2 and I3 is not yet
understood, but we know that the latter imply the
former when dealing with pairs.
In the lower semi-modular case IF is a stronger condition than both I1* and I2* (Theorems 6,
9). On the other side, notwithstanding the analogy coming from Equation (22), IF is mutually
261
An Algebraic Study of the Notion of Independence of Frames
Figure 10. Left: relations between independence of frames IF and all the different extended forms of
matroidal independence on the upper semi-modular lattice L (Θ) . Right: relations on the lower semimodular lattice L* (Θ) .
exclusive with the third independence relation
even in its lower semi-modular incarnation.
Some common features emerge: the first two
forms of lattice independence are always trivially
met by atoms of the related lattice. More, independence of frames and the third form of lattice
independence are mutually exclusive in either case.
The lower semi-modular case is clearly the
most interesting. Indeed, on L (Θ) independence
of frames and lattice-theoretic independence are
basically unrelated (see Figure 10-left). Their
lower-semimodular counterparts, instead, although distinct from IF, have meaningful links
with it (right).
The knowledge of which collections of frames
are I1* , I2* and I*3 tells us much about IF frames,
as collections of Boolean independent frames are
in
¯
I1* ∩ I2* ∩ I*3 .
We know that IF is strictly included in I1* ∩
I , but the possibility that independence of frames
*
2
may indeed coincide with I
to be explored.
*
1
262
∩
*
2
I
¯
*
3
∩I
is still
An Algebraic Solution to
the Conflict Problem
We can also exploit the analogy between vector spaces and families of compatible frames to
conjecture an elegant algebraic solution to the
conflict problem.
Given a collection of arbitrary elements of
a vector space, the well-known Gram-Schmidt
algorithm generates a different collection of independent vectors, spanning the same subspace.
The ingredients of the algorithm are: 1- the independence condition, and 2- the existence of a
projection operator mapping vectors onto linear
subspaces.
The possibility exists of designing a “pseudo
Gram-Schmidt” algorithm which, starting from a
given set of belief functions defined over a finite
collection of frames (e.g., the discrete feature
spaces in the pose estimation problem, Figure 4)
would generate a different collection of independent frames belonging to the same family of frames
{Θ1, …, Θn } → {Θ'1 … Θ'm }
An Algebraic Study of the Notion of Independence of Frames
with, in general, m ≠ n , and the same minimal
refinement
Θ1 ⊗ … ⊗ Θn = Θ '1 ⊗ … ⊗ Θ'm .
Once projected the n original b.f.s b1, …, bn
onto the new set of frames we would achieve a
set of surely combinable belief functions
b '1, …, b 'm equivalent, in some sense, to the
previous one. We would then be able to combine
them on their common refinement: in the pose
estimation problem this would allow us to extract
an object pose estimate from a set of feature values without resorting to a conflict graph. A formal
definition of this equivalence relation needs to be
introduced, possibly involving their Dempster’s
combination.
CONCLUSION
In this paper we gave a rather exhaustive description of families of compatible frames in terms of
the algebraic structures they form: Boolean subalgebras, upper, and lower semi-modular lattices.
Each of those comes with a characteristic form of
independence. We compared them with Shafer’s
notion of independence of frames, in a pursuit
for an algebraic interpretation of independence
in the theory of evidence. Even though IF cannot be explained in terms of classical matroidal
independence, it possesses interesting relations
with its extended forms on semimodular lattices.
It turns out that independence of frames is actually opposed to matroidal independence, a rather
surprising result. Even though this can be seen as
a negative in the original perspective of finding
an algebraic solution to the problem of merging
conflicting belief functions on non-independent
frames, we now understand much better where
independence of frames stands from an algebraic
point of view.
New lines of research remain open, for instance
in what concerns an explanation of independence
of frames as independence of flats in a geometric
lattice (Cuzzolin, 2008).
We believe the prosecution of this study could
in the future shed some more light on both the
nature of independence of sources in the theory
of subjective probability, and the relationship
between matroidal and Boolean independence in
discrete mathematics, pointing out the necessity
of a more general, comprehensive definition of
this very important notion.
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KEY TERMS AND DEFINITIONS
Belief Function: A non-additive measure, it
generalizes standard probability measures; mathematically is described by a random set.
Conflict: Two belief functions are said to
conflict when they do not admit Dempster’s
combination.
Dempster’s Rule: A generalization of Bayes’
rule, allows us to combine belief functions generated by independent sources of evidence.
Frame: The finite domain of a belief function.
Independent Frames: Collections of frames
which are independent as Boolean sub-algebras
of their common refinement.
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An Algebraic Study of the Notion of Independence of Frames
Lattice: A partially ordered element in which
all elements admit both inf and sup.
Matroid: A set endowed with a collection of
independent subsets.
Pose Estimation: The problem of inferring the
configuration or “pose” of an articulated object
from one or more images.
Semi-Modular Lattice: A lattice in which if
an element covers the inf, the sup covers the other
element or vice-versa. Families of compatible
frames are semi-modular lattices.
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An Algebraic Study of the Notion of Independence of Frames
APPENDIX
Definition 7. A non-empty collection of finite non-empty sets F is a family of compatible frames of
discernment (Shafer, 1976; pages 121-125) with refinings R , where R is a non-empty collection
of refinings between pairs of frames in F , if F and R meet the following requirements:
Θ
Θ
Θ
Θ
1. Composition of Refinings: If ρ1 : 2 1 → 2 2 and ρ2 : 2 2 → 2 3 are in R , then ρ1 ρ2 ∈ R
.
'
'
Θ
Θ
2. Identity of Coarsenings: If ρ1 : 2 1 → 2Θ and ρ2 : 2 2 → 2Θ are in R and ∀θ1 ∈ Θ1
∃θ2 ∈ Θ2 such that ρ1 (θ1 ) = ρ2 (θ2 ) then Θ1 = Θ2 and ρ1 = ρ2 .
'
'
3. Identity of Refinings: If ρ1 : 2Θ → 2Θ and ρ2 : 2Θ → 2Θ are in R , then ρ1 = ρ2 .
4. Existence of Coarsenings: If Θ ∈ F and A1, …, An is a disjoint partition of Θ , then there
is a coarsening ˜ ' of Θ in F which corresponds to this partition, i.e. ∀Ai there exists an
element of Θ ' whose image under the appropriate refining is Ai .
'
5. Existence of Refinings: if θ ∈ Θ ∈ F and n ∈ N then there exists a refining ρ : 2Θ → 2Θ
in R and Θ ' ∈ F such that ρ(θ) has n elements.
6. Existence of Common Refinements: every pair of elements in F has a common refinement
in F .
Definition 8: A Boolean algebra is a non-empty set B provided with three internal operations
∩
:
∪
:
¬
B×B → B
A, B A∩B
B×B → B
A, B A∪B
: B → B
A ¬A
called respectively meet, join and complement, characterized by the following properties:
A∪B = B ∪A, A∩B = B ∩A;
(A1)
A∪ (B ∪C ) = (A∪B ) ∪C , A∩ (B ∩C ) = (A∩B ) ∩C ;
(A2)
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An Algebraic Study of the Notion of Independence of Frames
(A∩B ) ∪B = B, (A∪B ) ∩B = B;
(A3)
A∩ (B ∪C ) = (A∩B ) ∪(A∩C ), A∪ (B ∩C ) = (A∪B ) ∩(A∪C );
(A4)
(A∩¬A) ∪B = B, (A∪¬A) ∩B = B.
(A5)
'
For instance, the power set 2Θ of a set Θ is a Boolean algebra. X = 2Θ with Θ' a disjoint partition of
Θ is then a Boolean sub-algebra (Sikorski, 1964) of 2Θ . A collection of compatible frames {Θ1, …, Θn }
Θ
Θ
Θ1 ⊗…⊗Θn
corresponds then to a collection of Boolean sub-algebras {2 1 , …, 2 n } of the power set 2
their minimal refinement Θ1 ⊗ … ⊗ Θn .
Now, a collection of Boolean sub-algebras X1, …, X n is independent (IB) if
∩A
i
i
≠∧
∀Ai ∈ X i ,
where ∧ ∩B is the initial element of the Boolean algebra B .
For a collection of compatible frames (23) is expressed as (3).
268
of