46 research outputs found
Reasoning From Data in the Mathematical Theory of Evidence
Mathematical Theory of Evidence (MTE) is known as a foundation for reasoning
when knowledge is expressed at various levels of detail. Though much research
effort has been committed to this theory since its foundation, many questions
remain open. One of the most important open questions seems to be the
relationship between frequencies and the Mathematical Theory of Evidence. The
theory is blamed to leave frequencies outside (or aside of) its framework. The
seriousness of this accusation is obvious: no experiment may be run to compare
the performance of MTE-based models of real world processes against real world
data.
In this paper we develop a frequentist model of the MTE bringing to fall the
above argument against MTE. We describe, how to interpret data in terms of MTE
belief functions, how to reason from data about conditional belief functions,
how to generate a random sample out of a MTE model, how to derive MTE model
from data and how to compare results of reasoning in MTE model and reasoning
from data.
It is claimed in this paper that MTE is suitable to model some types of
destructive processesComment: presented as poster M.A. K{\l}opotek: Reasoning from Data in the
Mathematical Theory of Evidence. [in:] Proc. Eighth International Symposium
On Methodologies For Intelligent Systems (ISMIS'94), Charlotte, North
Carolina, USA, October 16-19, 1994. arXiv admin note: text overlap with
arXiv:1707.0388
Transferrable Plausibility Model - A Probabilistic Interpretation of Mathematical Theory of Evidence
This paper suggests a new interpretation of the Dempster-Shafer theory in
terms of probabilistic interpretation of plausibility. A new rule of
combination of independent evidence is shown and its preservation of
interpretation is demonstrated.Comment: Pre-publication version of: M.A. K{\l}opotek: Transferable
Plausibility Model - A Probabilistic Interpretation of Mathematical Theory of
Evidence O.Hryniewicz, J. Kacprzyk, J.Koronacki, S.Wierzcho\'{n}: Issues in
Intelligent Systems Paradigms Akademicka Oficyna Wydawnicza EXIT, Warszawa
2005 ISBN 83-87674-90-7, pp.107--11
Learning Belief Network Structure From Data under Causal Insufficiency
Though a belief network (a representation of the joint probability
distribution, see [3]) and a causal network (a representation of causal
relationships [14]) are intended to mean different things, they are closely
related. Both assume an underlying dag (directed acyclic graph) structure of
relations among variables and if Markov condition and faithfulness condition
[15] are met, then a causal network is in fact a belief network. The difference
comes to appearance when we recover belief network and causal network structure
from data.
A causal network structure may be impossible to recover completely from data
as not all directions of causal links may be uniquely determined [15].
Fortunately, if we deal with causally sufficient sets of variables (that is
whenever significant influence variables are not omitted from observation),
then there exists the possibility to identify the family of belief networks a
causal network belongs to [16]. Regrettably, to our knowledge, a similar result
is not directly known for causally insufficient sets of variables. Spirtes,
Glymour and Scheines developed a CI algorithm to handle this situation, but it
leaves some important questions open.
The big open question is whether or not the bidirectional edges (that is
indications of a common cause) are the only ones necessary to develop a belief
network out of the product of CI, or must there be some other hidden variables
added (e.g. by guessing). This paper is devoted to settling this question.Comment: A short version of this paper appeared in [Klopotek:94m] M.A.
K{\l}opotek: Learning Belief Network Structure From Data under Causal
Insufficiency. [in:] F. Bergadano, L.DeRaed Eds.: Machine Learning ECML-94 ,
Proc. 13th European Conference on Machine Learning, Catania, Italy, 6-8 April
1994, Lecture Notes in Artificial Intelligence 784, Springer-Verlag, 1994,
pp. 379-38
Beliefs in Markov Trees - From Local Computations to Local Valuation
This paper is devoted to expressiveness of hypergraphs for which uncertainty
propagation by local computations via Shenoy/Shafer method applies. It is
demonstrated that for this propagation method for a given joint belief
distribution no valuation of hyperedges of a hypergraph may provide with
simpler hypergraph structure than valuation of hyperedges by conditional
distributions. This has vital implication that methods recovering belief
networks from data have no better alternative for finding the simplest
hypergraph structure for belief propagation. A method for recovery
tree-structured belief networks has been developed and specialized for
Dempster-Shafer belief functionsComment: Preliminary versioin of conference paper: M.A. K{\l}opotek: Beliefs
in Markov Trees - From Local Computations to Local Valuation. [in:] R.
Trappl, Ed.: Cybernetics and Systems Research , Proc. 12th European Meeting
on Cybernetics and System Research, Vienna 5-8 April 1994, World Scientific
Publishers, Vol.1. pp. 351-35
An Aposteriorical Clusterability Criterion for -Means++ and Simplicity of Clustering
We define the notion of a well-clusterable data set combining the point of
view of the objective of -means clustering algorithm (minimising the centric
spread of data elements) and common sense (clusters shall be separated by
gaps). We identify conditions under which the optimum of -means objective
coincides with a clustering under which the data is separated by predefined
gaps.
We investigate two cases: when the whole clusters are separated by some gap
and when only the cores of the clusters meet some separation condition.
We overcome a major obstacle in using clusterability criteria due to the fact
that known approaches to clusterability checking had the disadvantage that they
are related to the optimal clustering which is NP hard to identify.
Compared to other approaches to clusterability, the novelty consists in the
possibility of an a posteriori (after running -means) check if the data set
is well-clusterable or not. As the -means algorithm applied for this purpose
has polynomial complexity so does therefore the appropriate check.
Additionally, if -means++ fails to identify a clustering that meets
clusterability criteria, with high probability the data is not
well-clusterable.Comment: 58 page
Independence, Conditionality and Structure of Dempster-Shafer Belief Functions
Several approaches of structuring (factorization, decomposition) of
Dempster-Shafer joint belief functions from literature are reviewed with
special emphasis on their capability to capture independence from the point of
view of the claim that belief functions generalize bayes notion of probability.
It is demonstrated that Zhu and Lee's {Zhu:93} logical networks and Smets'
{Smets:93} directed acyclic graphs are unable to capture statistical
dependence/independence of bayesian networks {Pearl:88}. On the other hand,
though Shenoy and Shafer's hypergraphs can explicitly represent bayesian
network factorization of bayesian belief functions, they disclaim any need for
representation of independence of variables in belief functions.
Cano et al. {Cano:93} reject the hypergraph representation of Shenoy and
Shafer just on grounds of missing representation of variable independence, but
in their frameworks some belief functions factorizable in Shenoy/Shafer
framework cannot be factored.
The approach in {Klopotek:93f} on the other hand combines the merits of both
Cano et al. and of Shenoy/Shafer approach in that for Shenoy/Shafer approach no
simpler factorization than that in {Klopotek:93f} approach exists and on the
other hand all independences among variables captured in Cano et al. framework
and many more are captured in {Klopotek:93f} approach.%Comment: 1994 internal repor
Identification and Interpretation of Belief Structure in Dempster-Shafer Theory
Mathematical Theory of Evidence called also Dempster-Shafer Theory (DST) is
known as a foundation for reasoning when knowledge is expressed at various
levels of detail. Though much research effort has been committed to this theory
since its foundation, many questions remain open. One of the most important
open questions seems to be the relationship between frequencies and the
Mathematical Theory of Evidence. The theory is blamed to leave frequencies
outside (or aside of) its framework. The seriousness of this accusation is
obvious: (1) no experiment may be run to compare the performance of DST-based
models of real world processes against real world data, (2) data may not serve
as foundation for construction of an appropriate belief model.
In this paper we develop a frequentist interpretation of the DST bringing to
fall the above argument against DST. An immediate consequence of it is the
possibility to develop algorithms acquiring automatically DST belief models
from data. We propose three such algorithms for various classes of belief model
structures: for tree structured belief networks, for poly-tree belief networks
and for general type belief networks.Comment: An internal report 199
Spectral Analysis of Laplacians of an Unweighted and Weighted Multidimensional Grid Graph -- Combinatorial versus Normalized and Random Walk Laplacians
In this paper we generalise the results on eigenvalues and eigenvectors of
unnormalized (combinatorial) Laplacian of two-dimensional grid presented by
Edwards:2013 first to a grid graph of any dimension, and second also to other
types of Laplacians, that is unoriented Laplacians, normalized Laplacians, and
random walk Laplacians. While the closed-form or nearly closed form solutions
to the eigenproblem of multidimensional grid graphs constitute a good test suit
for spectral clustering algorithms for the case of no structure in the data,
the multidimensional weighted grid graphs, presented also in this paper can
serve as testbeds for these algorithms as graphs with some predefined cluster
structure. The weights permit to simulate node clusters not perfectly separated
from each other. This fact opens new possibilities for exploitation of
closed-form or nearly closed form solutions eigenvectors and eigenvalues of
graphs while testing and/or developing such algorithms and exploring their
theoretical properties. Besides, the differences between the weighted and
unweighted case allow for new insights into the nature of normalized and
unnormalized Laplacians.Comment: 73 pages, 18 figure
Rigid Body Structure and Motion From Two-Frame Point-Correspondences Under Perspective Projection
This paper is concerned with possibility of recovery of motion and structure
parameters from multiframes under perspective projection when only points on a
rigid body are traced. Free (unrestricted and uncontrolled) pattern of motion
between frames is assumed. The major question is how many points and/or how
many frames are necessary for the task. It has been shown in an earlier paper
{Klopotek:95b} that for orthogonal projection two frames are insufficient for
the task. The paper demonstrates that, under perspective projection, that total
uncertainty about relative position of focal point versus projection plane
makes the recovery of structure and motion from two frames impossible.Comment: arXiv admin note: text overlap with arXiv:1705.0398
Reconstruction of~3-D Rigid Smooth Curves Moving Free when Two Traceable Points Only are Available
This paper extends previous research in that sense that for orthogonal
projections of rigid smooth (true-3D) curves moving totally free it reduces the
number of required traceable points to two only (the best results known so far
to the author are 3 points from free motion and 2 for motion restricted to
rotation around a fixed direction and and 2 for motion restricted to influence
of a homogeneous force field). The method used is exploitation of information
on tangential projections. It discusses also possibility of simplification of
reconstruction of flat curves moving free for prospective projections