33 research outputs found
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
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
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
Machine Learning Friendly Set Version of Johnson-Lindenstrauss Lemma
In this paper we make a novel use of the Johnson-Lindenstrauss Lemma. The
Lemma has an existential form saying that there exists a JL transformation
of the data points into lower dimensional space such that all of them fall into
predefined error range .
We formulate in this paper a theorem stating that we can choose the target
dimensionality in a random projection type JL linear transformation in such a
way that with probability all of them fall into predefined error
range for any user-predefined failure probability .
This result is important for applications such a data clustering where we
want to have a priori dimensionality reducing transformation instead of trying
out a (large) number of them, as with traditional Johnson-Lindenstrauss Lemma.
In particular, we take a closer look at the -means algorithm and prove that
a good solution in the projected space is also a good solution in the original
space. Furthermore, under proper assumptions local optima in the original space
are also ones in the projected space. We define also conditions for which
clusterability property of the original space is transmitted to the projected
space, so that special case algorithms for the original space are also
applicable in the projected space.Comment: 38 pages, 6 Figure
Restricted Causal Inference Algorithm
This paper proposes a new algorithm for recovery of belief network structure
from data handling hidden variables. It consists essentially in an extension of
the CI algorithm of Spirtes et al. by restricting the number of conditional
dependencies checked up to k variables and in an extension of the original CI
by additional steps transforming so called partial including path graph into a
belief network. Its correctness is demonstrated.Comment: M.A. K{\l}opotek: Restricted Causal Inference Algorithm. [in:] B.
Pehrson, I. Simon Eds.: Proc. World Computer Congress of IFIP . Hamburg 28
August - 2 September 1994, Vol.1, Elsevier Scientific Publishers
(North-Holland), Amsterdam, pp. 342-34
Beliefs and Probability in Bacchus' l.p. Logic: A~3-Valued Logic Solution to Apparent Counter-intuition
Fundamental discrepancy between first order logic and statistical inference
(global versus local properties of universe) is shown to be the obstacle for
integration of logic and probability in L.p. logic of Bacchus. To overcome the
counterintuitiveness of L.p. behaviour, a 3-valued logic is proposed.Comment: Draft for the conference M.A. K{\l}opotek: Beliefs and Probability in
Bacchus' l.p. Logic: A 3-Valued Logic Solution to Apparent Counter-intuition.
[in:] R. Trappl Ed,: Cybernetics and Systems Research. Proc. 11 European
Meeting on Cybernetics and System Research EMCSR'92, Wien, Osterreich, 20.
April 1992. World Scientific Singapore, New Jersey, London, HongKong Vol. 1,
pp. 519-52