20-07-2012, 03:41 PM
Logic Minimization as an Efficient Means of Fuzzy Structure Discovery
2005Logic Minimization as an Efficient Means.pdf (Size: 957.69 KB / Downloads: 20)
Abstract
Established methods of Boolean minimization have
previously unseen potential as an efficient and unrestricted
means of fuzzy structure discovery, becoming particularly useful
within a design methodology for the automatic development of
fuzzy models. Traditionally used in digital systems design, logic
minimization tools allow us to exploit the fundamental links
between binary (two-valued) and fuzzy (multivalued) logic. In this
paper, we show how logic optimization plays an integral role in a
two-phase fuzzy model design process. Adaptive logic processing
is realized as the discovered Boolean structures are augmented
with fuzzy granules and then refined by adjusting connections of
fuzzy neurons, helping to further capture the numeric details of
the target systems behavior. Accurate and highly interpretable
fuzzy models are the result of the entire development process.
INTRODUCTION
ACROSS many disciplines of science and engineering,
data-driven modelling of complex real-world systems,
concepts, and processes becomes extremely useful. Of course,
the principal goal of the model is to achieve accuracy in its predictive
capabilities, behaving in the same way the target system
would under familiar conditions and generalizing its knowledge
in order to predict behavior in unforeseen circumstances. However,
another important goal, often ignored or undervalued, is
the goal of building a model with truly transparent structure,
allowing us to intuitively interpret and learn the details of its
inner workings. This knowledge is potentially quite valuable,
as obviously an accurate model would reflect the behavioral details
of the target system. Further, this communication through
interpretability is not one-way, as a user could potentially use
prior knowledge to modify the models structure in order to
make corrections or augmentations.
FUZZY MODELLING ARCHITECTURE
The fuzzy model follows the fundamentals of fuzzy (granular)
modelling. As advocated in [21], fuzzy modelling is realized
at the conceptual level formed by a collection of semantically
meaningful information granules defined in each variable.
These are also regarded as linguistic landmarks whose
choice implies a certain point of view of the data (system) under
discussion. When dealing with many variables (that is usually
the case), the fuzzy sets are aggregated and give rise to their
granular manifestations in the form of fuzzy relations (Cartesian
products of contributing fuzzy sets). This gives rise to a
two-level topology of the model that captures the geometry of
data, evidently standing in a one-to-one correspondence to its
logic fabric. The essence of this geometry can be captured in
the form of AND and OR nodes (aggregation operations), as illustrated
in Fig. 1. This figure emphasizes the structural nature
of this construct.
EXPERIMENTAL STUDIES
While the study reported in this paper did not include any
sort of theoretical convergence analysis for the proposed fuzzy
model, it should be stressed that there have been many theoretical
results regarding rule-based approximation of bounded
continuous functions, and those are readily available in the literature
[40], [41]. Note that we do not change the architecture of
the model (which relies on rules) but offer an effective and novel
way for their design. As seen in the remainder of this section, we
carefully investigated the models experimentally, demonstrating
their effectiveness with both synthetic and real-world data. In
Section VII-A we take a finite sample set of a bounded continuous
for meaningful visualization of the results and to allow validation
of the knowledge learnt by the model. In Section VII-B
and VII-C we deal with real-world data, providing a report on
results achieved for several well-known datasets, as well as a
comprehensive case study concerning a very high-dimensional
dataset detailing median housing prices in the United States.
CONCLUSION
In this paper, we have proposed and extensively investigated
the use of established two-valued logic minimization tools in
fuzzy structure optimization. Through understanding the logic
nature of real-world data from the very beginning of the design,
they reveal an intuitive and concise structure that can directly
form a blueprint for a heterogeneous fuzzy neural network
architecture. When combined with these neural augmentations,
a heterogeneous fuzzy model development environment
is formed, able to produce accurate and highly interpretable
models.