Space, Time, and Ambient Intelligence: Papers from the AAAI 2013 Workshop
A Fuzzy Set Approach to Representing
Spatio-Temporal and Environmental Context
– Preliminary Considerations –
Hans W. Guesgen
School of Engineering and Advanced Technology, Massey University, Palmerston North, New Zealand
h.w.guesgen@massey.ac.nz
Illingworth et al., 2007; Tapia et al., 2004]. Although the
reported successes are promising, determining the correct
behaviour from sensor data alone is often impossible, since
sensors can only provide very limited information and
human behaviours are inherently complex.
Several researchers have realised that context
information, in particular spatial and temporal information,
can be useful to improve the behaviour recognition process
[Aztiria et al., 2008; Guesgen and Marsland, 2010; Jakkula
and & Cook, 2008; Tavenard et al., 2007]. For example,
breakfast usually occurs in the kitchen in the morning,
which means that if something is happening at 07:00 in the
kitchen, it is more likely to be breakfast than taking a
shower.
In this paper, we discuss how context information can be
represented and how it can be used in the behaviour
recognition process. We argue that using a probabilistic
approach has shortfalls, due to incomplete context
information, and that using an approach based on the
concept of beliefs fails due to its complexity. We then
make a case for fuzzy logic, which not only provides a
simple and robust mechanism for reasoning about context
information but also provides a means to represent
imprecise information.
Abstract
This paper aims at providing a preliminary discussion on
how to deal with spatio-temporal information in the context
of behaviour recognition. It draws comparison with how
humans reason in other areas, such as law, and discusses
some of the pros and cons of formalisms for handling
uncertainty, starting with probability theory, continuing with
the Dempster-Shafer theory, and concluding with fuzzy
logic.
Introduction
Most developed countries around the world are facing the
problem of an aging population. Life expectation is higher
than ever before, and so is the expectation to live a highquality, independent lifestyle throughout the entire life,
independent of age or illnesses such as Parkinson’s or
Alzheimer’s disease. Unfortunately this expectation is not
always met.
Rather than moving to a nursing home or employing the
continuous support of a caretaker, we will soon have the
technology to set up smart homes, which use sensors to
monitor the person’s activities in a non-obtrusive way.
However, without an intelligent reasoning engine, the
information obtained from the sensors is of little use. The
reasoning engine has to determine which activity currently
takes place and whether this activity is a normal behaviour
or poses a threat to the person living in the smart home.
The approaches to behaviour recognition that have been
developed over the last ten years range from logic-based
approaches to probabilistic machine learning approaches
[Augusto and Nugent, 2004; Chua et al., 2011; Duong et
al., 2005; Gopalratnam and Cook, 2004; Rivera-
A Case for Fuzzy Logic
Behaviours often take place in particular contexts, but
there is usually no one-to-one relationship between a
behaviour and the context it occurs in. Rather, given a
certain behaviour, context information is determined
according to some probability distribution. If B is a
behaviour (e.g., making breakfast) and C some context
information (e.g., in the kitchen), then P(C|B) is the
probability that C is true if B occurs (e.g., the behaviour
Copyright © 2013, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.
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formulate uncertainty, where classical probability lies
between belief and plausibility:
takes place in the kitchen if we know that the behaviour is
making breakfast).
Given the conditional probabilities P(C|B), we can
calculate conditional probabilities P(B|C) using Bayes’
rule.
This assumes that we know the conditional
probability for each context and behaviour. Moreover, we
not only need P(B|C) but also P(B|C1,…,Cn), since context
information is usually correlated. Assuming that we have
knowledge of all the necessary conditional probabilities is
unrealistic. In addition, the result of using conditional
probabilities might be counterintuitive, as it does not align
with how humans combine evidence.
In [2012], Clermont puts forward an argument against
probabilistic reasoning in law. Combining evidence on a
probabilistic basis is not in line with the common practice
of law, leading to the infamous conjunction paradox
[Clermont, 2012]:
Bel(B|C) ≤ P(B|C) ≤ Pl(B|C)
Belief and plausibility is defined in the Dempster-Shafer
theory on the basis of a mass function, which assigns basic
probabilities to the power set of the frame of discernment.
Although this solves the problem of evidence not adding
up to 100%, it is not a solution to reasoning about context
information in general, since the Dempster-Shafer theory is
an even more complex framework than probability theory.
In the next section, we discuss a simpler framework, which
is not as rigorous from the mathematical point of view but
which provides a simple and robust way to deal with
context information.
We purport to decide civil cases according to a moreprobable-than-not standard of proof. We would
expect this standard to take into account the rule of
conjunction, which states that the probability of two
independent events occurring together is the product
of the probability of each event occurring separately.
The rule of conjunction dictates that in a case
comprised of two independent elements the plaintiff
must prove each element to a much greater degree
than 50%: only then will the plaintiff have shown that
the probability that the two elements occurred
together exceeds 50%. Suppose, for example, that a
plaintiff must prove both causation and fault and that
these two elements are independent. If the plaintiff
shows that causation is 60% probable and fault is 60%
probable, then he apparently would have failed to
satisfy the civil standard of proof because the
probability that the defendant both acted negligently
and caused injury is only 36%.
Context Fuzzy Sets
Unlike traditional sets, fuzzy sets allow their elements to
belong to the set with a certain degree. Rather than
deciding whether an element d does or does not belong to a
set A of a domain D, we determine for each element of D
the degree with which it belongs to the fuzzy set Ã. In
other words, a fuzzy subset à of a domain D is a set of
ordered pairs, (d, ȝÃ(d)), where d ∈ D and ȝÃ: D → [0, 1]
is the membership function of Ã. The membership
function replaces the characteristic function of a classical
subset A ⊂ D.
Rather than asking the question of what is the
probability of a certain behaviour occurring in a particular
context, we now pose the question as follows. Given some
context information C, to which degree is a particular
behaviour a C-behaviour. For example, if C is the day of
the week, then we can ask for the degree of the behaviour
to be a Monday behaviour, Tuesday behaviour, and so on.
In terms of fuzzy sets, we define D as the set of the seven
days of the week and ȝÃ as the membership function that
determines to which degree the behaviour occurs on a
particular day. For example, we might want to define a
fuzzy set that reflects the degree to which a restaurant visit
falls on a particular day. The membership function of such
a fuzzy set is shown graphically in Figure 1. Unlike
probabilities, the membership grades do not need to add up
to one.
A similar argument can be made for reasoning about
behaviours. If we know that it is 60% probable for a
behaviour to occur at the given time and 60% probable for
it to occur at the observed location, then the combined
probability would only be 36%.
Another shortcoming of probability theory is that
probabilities have to add up to 100%. In cases where we
have perfect information, this is not a problem. However,
when we do not have complete knowledge, this may lead
to counterintuitive results. For example, if we do not have
any information that a behaviour normally occurs on the
weekend, we cannot assign a probability of 0% to P(B|C),
since this would imply that P(¬B|C) equals 100%, which
would means that the behaviour has to occur on a
weekday.
The Dempster-Shafer theory [Dempster, 1967; Shafer,
1976] offers a way out of this dilemma. It uses the concept
of belief (Bel) and plausibility (Pl) instead of probability to
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membership grade of at least α is called the α-level set of
Ã:
( • α}
Aα = {d ∈ D | ȝÃ(d)
y greater than α, then the
If the membership grade is strictly
set is referred to as a strong α-leveel set.
Reasoning with Conteext Fuzzy Sets
To avoid the problems with accu
umulating values, which
we encounter in probability theeory and the DempsterShafer theory, we choose onee of the schemes for
combining fuzzy sets that was originally proposed by
y sets Ã1 and Ã2 with
Zadeh [1965]. Given two fuzzy
membership functions ȝÃ1(d) and ȝÃ2(d), respectively, then
the membership function of the intersection Ã3 = Ã1 ∩ Ã2
is pointwise defined by:
membership
Fig. 1: Graphical representation of a m
functions that determines the degree to whiich a restaurant
visit falls on a particular day of thee week.
In the example above, the context infoormation is still
crisp information, despite the fact that it is used in a fuzzy
set: for any restaurant visit, we can determiine precisely on
which day of the week it occurs.
Other context
information might not be precise, but ratheer conveys some
vague information. For example, if a behaaviour occurs on
a cold day, it is not clear what it means foor a day to be a
cold day.
In this case, we represennt the context
information itself as a fuzzy set, as illustrateed in Figure 2.
d), ȝÃ2(d)}
ȝÃ3(d) = min{ȝÃ1(d
nction of the union Ã3 =
Analogously, the membership fun
Ã1 ∪ Ã2 is pointwise defined by:
d), ȝÃ2(d)}
ȝÃ3(d) = max{ȝÃ1(d
omplement of a fuzzy set
The membership grade for the co
Ã, denoted as ¬Ã, is defined in
n the same way as the
complement in probability theory:
ȝ¬Ã (d) = 1 – ȝÃ(d)
Fig. 2: A fuzzy set that maps temperatuures to the
qualitative values cold, medium, warm
m, and hot.
he min/max combination
In [1965], Zadeh stresses that th
scheme is not the only scheme for defining intersection and
union of fuzzy sets, and that it depends on the context
which scheme is the most approprriate. While some of the
schemes are based on empirical investigations,
i
others are
the result of theoretical considerattions [Dubois and Prade,
1980; Klir and Folger, 1988]. However,
H
[Nguyen et al.,
1993] proved that the min/max operations are the most
robust operations for combining fuzzy sets, where
robustness is defined in terms of how much impact
uncertainty in the input has on the error in the output.
Similarly, we can define a fuzzy set that expresses
distances by rounding them to the closesst half metre –
something we as humans often do wheen we perceive
distances, although not necessarily alwayys on the same
scale (see Figure 3).
Fig. 2: A fuzzy set that approximates disttances with a
granularity of half a metre.
n
Conclusion
This paper tries to offer some preeliminary insights in how
context information can be repressented and how it can be
used in the behaviour recognition
n process. Since context
information is often incomp
plete and imprecise,
probabilistic approaches are impraactical and therefore often
not useful in real-world application
ns. Fuzzy logic can offer
As the examples have shown, fuzzy sets can be used for
associating behaviours with context inforrmation and for
representing imprecise context informatioon. Fuzzy set
theory also provides us with a means to convert fuzzy sets
back to crisp sets, which is achieved with tthe notion of an
α-level set. Let à be a fuzzy subset in D, then the (crisp)
set of elements that belong to the fuzzyy set à with a
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a way out of this problem, as it provides robust
mechanisms for dealing with uncertainty.
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