Fuzzy Composite Concepts based on Human Reasoning.

1 downloads 0 Views 643KB Size Report
He is a nice guy, brown hair, tall, and always up for a joke." This quite common description shows how we as humans often describe and decompose.
Fuzzy Composite Concepts based on Human Reasoning. Christian Wagner, Hani Hagras The Computational Intelligence Centre, University of Essex, UK christian AT wagnerweb.net

without referring to any of the characteristics of "nice weather" specifically. This ability to generalize and form composite concepts allows us to make rapid decisions for complex situations and probably even more importantly to reason on and to communicate complex information very easily. The ability of humans to reason based on perceptions and not numbers or symbols as usually done in computing or mathematics has long been recognized and formalized with great success by Zadeh through the introduction of Fuzzy Logic (FL) in general [1] and more recently the Computing With Words (CWW) approach in specific. [2] Since the introduction of Fuzzy Sets 1965 [1], FL has been employed with great success in a vast variety of mostly control related applications due to its ability to model and handle uncertainty both in physical systems and in the logics driving the control of those systems. It is only more recently that the interest in FL as a tool to model human reasoning and decision processes directly has again grown as part of the CWW field (e.g. [3],[4]). In this paper, we will present a technique that allows the formation of composite concepts using fuzzy logic which aims to mimic the composite concepts employed by humans. These Fuzzy Composite Concepts (FCCs) aim to extend the capabilities of standard uses of FL to separately model individual factors (e.g. temperature) using fuzzy sets (and the associated linguistic labels) in order to create high order (composite) concepts which allow straightforward reasoning akin to human reasoning, maintaining a series of advantages such as simplicity, interpretability, robustness and reusability. Section 2 will provide an overview of the philosophical background behind FCCs and review the current mainstream implementation of FL Systems as well as providing an overview of Ambient Intelligence Applications. Section 3 details the actual concept of FCCs followed by examples in Section 4. Section 5 finally presents some conclusions and future work.

Abstract Fuzzy Logic Systems (FLSs) provide a proven toolset in mimicking human reasoning. In this paper, we will present the idea of Fuzzy Composite Concepts (FCCs) which allow for a closer imitation of human reasoning in terms of integrating a large number of parameters into a single concept suitable for higher level reasoning. FCCs are based on standard FLSs and transparently extend them to provide intuitively interpretable rule bases and improve resilience and reusability of the overall FLS. We are providing an overview of the philosophical concepts behind FCCs and discuss their applicability. We describe the implementation of FCCs and demonstrate their benefits using real world examples based on our work in Ambient Intelligent Environments. Keywords: Composite Concepts; Fuzzy Logic Systems, Ambient Intelligence.

1. Introduction Throughout science, a vast number of advances has been achieved is and has been achieved by imitating Nature. As part of this paper we are focusing on imitating a very basic ability which is the ability is to generalize and integrate a wide range of sources of information into a series of simple concepts which can be referred to as composite concepts. This ability is intrinsic to the way that humans (and probably other species) perform reasoning tasks and take decisions on an everyday basis. A simple example of such a composite is the quality of the weather, or simply "the weather" as we refer to it on an everyday basis. Statements such as "Today the weather is nice" are universally understood without having to refer to the intrinsic properties of what "nice weather" actually is. Indeed, as humans we rely on a large body of world knowledge (experience, personal background, education, etc.) which provides us with an understanding of the relevant factors in terms of qualifying for example the weather. As such, by saying: "Today the weather is nice", we refer to and incorporate a wide range of sources of information, from temperature, cloud cover, rain, etc.

IEEE IRI 2010, August 4-6, 2010, Las Vegas, Nevada, USA 978-1-4244-8099-9/10/$26.00 ©2010 IEEE

2. Background In order to provide the basis for our approach , we will in Section 2.1 describe the human reasoning process based on composite concepts as will be referred to

308

more abstract fashion in his work of fuzzy logic based resource management [6].

throughout this paper. In Section 2.2, we will present a brief review of the high level structure of reasoning as is usually performed computationally, for example in fuzzy logic systems. Section 2.3 gives a brief introduction to our work in ambient intelligence as part of which we have developed the concepts presented in this paper and which will serve as the basis for our examples in the latter parts of the paper.

2.1. Composite Concepts in Human Reasoning. Figure 1. Human Reasoning based on composite concepts (here shown using three basic concepts).

As part of this paper we are referring to composite concepts as concepts which encapsulate the input from a series of basic concepts. Humans employ such composite concepts naturally and pervasively during reasoning, communication and indeed memorising. Examples range from the aforementioned weather example to beauty, quality (for example of a car), etc. Indeed, it could be argued that most things can be summarized as composites of basic concepts and other composite concepts. For example different people can be modelled as composites to different degrees of specific concepts or qualities, e.g. "Do you know Tom? He is a nice guy, brown hair, tall, and always up for a joke." This quite common description shows how we as humans often describe and decompose other humans in terms of multiple, usually composite concepts. Figure 1 shows a diagram of a composite concept which is based on three basic concepts. Following the example above, the composite concept "nice guy" could for example be composed of the basic concepts of "friendliness", "openness" and "politeness". Or, using the more tangible example of the weather, the basic concepts could be "humidity", "temperature" and "wind". What is intriguing about this composite nature of these concepts is that they allow us to perform very quick and effective decisions on a day to day basis and to communicate about and memorise things or events in our environment very easily. (e.g. "would you like to meet Tom?") Indeed, one could reason that we abstract the world around us and describe it using high level descriptors, e.g. we might remember that the weather was "nice" on our last holiday without qualifying it further. The composite concept of "nice weather" is based on a variety of concepts depending on context (e.g. nice weather on a skiing trip is quite different to nice weather on a beach holiday) but as part of the limited scope in this paper, we will stick to the basic notion of composite concepts as explained above. A more in depth discussion will be provided in a future journal version of this paper. It should also be noted here that the idea of composite concepts employed to describe the world is not new and is being employed for example in the area of description logics [5]. For completeness we would also like to add that Smith employs the term composite concepts in a

2.2. Current Application of Fuzzy Logic Systems. Fuzzy Logic and more specifically Fuzzy Logic Systems (FLSs) are and have been used for decades to perform computational reasoning and control in a variety of fields from industrial engine control [7] to modelling of stock markets and ambient intelligence applications [9]. Traditionally, FLSs focus on the direct transition from inputs to outputs, in other words, a series of inputs is provided to the FLS, the inputs are fuzzified into input fuzzy sets which are processed in a rule base which contains rules following the general format: ܴ‫ ݏ‬ǣ‫ ܦܰܣ ͳܨݏ݅ ͳݔܨܫ‬ǥ ‫ܰܧܪܶ ܲܨݏ݅ ܲݔܦܰܣ‬ ݃ͳ ݅‫ ͳܩݏ‬ǡ ǥ ǡ ݃ܳ ݅‫ ܳܩݏ‬ǡ‫ א ݏ‬ሼͳǡ Ǥ Ǥ Ǥ ǡ ܵሽ

(1)

Where P is the number of FLS inputs, Q the number of FLS outputs, S is the number of rules in the rule base and F and G are input and output sets respectively. What is interesting to note is that this approach of feeding all relevant inputs to a central reasoning component (as depicted conceptually in Figure 2) largely disregards the benefits of composite concepts which are regularly used in the human reasoning process: no composite concepts are used but basic concepts are directly employed for reasoning.

Figure 2. Common, concept based reasoning process in fuzzy logic systems (here shown using three basic concepts). While it can be argued that the concept of composite concepts is not essential for computational reasoning as the basic facts need to be processed "sooner or later" or indeed that the idea of employing composite concepts is already commonplace for example as part of techniques such as hierarchical fuzzy control [9], as part of this paper

309

such, it is important for human users to understand why their environment "acts the way it does", why specific actions are taken etc. (for example: "Why did the light turn on?"). FLSs (type-1 and type-2) provide a unique set of characteristics which allow to model and handle the uncertainty [8], [9] while through their use of simple rules (as shown in Equation (1)) also providing the basis for human interpretability of the system which in turn provides the basis for the user trusting the system.

we are arguing that closely mimicking the human decision process and its use of composite concepts can provide substantial benefits in computational decision making, in particular in fuzzy systems. These benefits include: x

x

x

Better interpretability of the fuzzy systems. This is essential especially in applications which directly involve human end users such as in ambient intelligence applications. Encapsulation and separation of lower level reasoning (e.g. sensor evaluation) and high level reasoning (e.g. state of the environment). For example the evaluation of multiple light sensors in a room to determine the ambient light level (a composite concept) which we describe in more detail in Section 4). Better performance. As composite concepts are evaluated and established they can be re-used and computed ahead of time.

An in depth description of how composite concepts can be achieved in FLSs will be described in Section 3, followed by a detailed description of the abovementioned benefits of FCCs and practical examples of their application in Section 4. Figure 3. The iSpace, an AIE testbed at the University of Essex, UK

2.3. Ambient Intelligence Applications.

3. Composite Concepts based on Fuzzy Logic Systems.

We are providing a brief overview of our work in Ambient Intelligent Environments (AIEs) which has largely contributed to the development of the idea of composite concepts modelled on human reasoning as described in this paper. Throughout the rest of this paper we will be referring to practical examples based on and around the application of composite concepts in AIEs and pervasive computing applications. In ambient intelligence applications and AIEs such as intelligent homes, one of the applications of computational intelligence techniques such as fuzzy logic is the support of a user in his/her everyday environment such as his/her home through the learning of his/her preferences. For example, an intelligent agent system installed in a home can learn the inhabitant's preferences in terms of indoor lighting, i.e. when, in what situation, at what light level he/she would like which specific lamps or lights in his/her home to be turned ON and to which specific level of brightness. Figure 3 shows the iSpace which is an AIE testbed based at the University of Essex, where the work presented in this paper is being developed and deployed. The iSpace looks like any normal apartment but it is equipped with a large number and variety of sensors, actuators, embedded devices and networks. A special characteristic of intelligent systems in AIEs is the large amount of uncertainty (both in terms of uncertainty about sensor information and about the user behaviour) and the generic requirement for user trust. As

In order to employ composite concepts in computational reasoning, for example using fuzzy logic, we are proposing to directly mimic the human use of composite concepts (Figure 1). Thus, in order to extend the traditional direct computation on basic concepts (Figure 2), we are proposing a multi-tiered FLSs where low-level FLSs are providing low-level reasoning to create Fuzzy Composite Concepts (FCCs) from basic concepts or inputs. The high level reasoning FLSs employ the output of the lower level FLSs as their own input (see Figure 4). As can be seen in Figure 4, FCCs are created and computed using a standard FLS with the standard components of Fuzzifier, Inference Engine, Rule Base and Defuzzifier [10]. The inputs to the FCC component (modelling for example the weather) are crisp inputs representing basic concepts, for example the temperature as perceived by a sensor, the light level by another sensor etc. Additionally, multi-tier FCCs are possible where one FCC is based on the inputs from other FCCs. We are providing concrete examples in Section 4. The outputs of the FCC component(s) are directly fed as inputs into the high level fuzzy reasoning component of the system. It is here that the high level reasoning takes place, for example: "If the weather is nice, I will go for a long walk.".

310

݈ where ‫ܻ א ݕ‬, ‫ څ‬is a t-norm, ‫ܥܥܨܩ‬ is the consequent set ′ of the current rule l and ߤ‫ ݈ ܨ‬ሺ‫ ͳݔ‬ሻ ‫ څ‬ǥ ‫ ݈݌ܨߤ څ‬൫‫݌ݔ‬′ ൯ is the t-

While all the components employed in the system are well known in the field of FLSs, several observations are worth noting: x Figure 4 indicates that the FCC component fuzzy output sets can either be defuzzified or employed directly in the higher level reasoning component through non-singleton fuzzification. This latter option is preferable as it allows the uncertainties (from the FCC component) to flow through the whole fuzzy system and thus be reflected in its final output. x While we describe the overall system by relying on type-1 fuzzy logic for simplicity, the same system is applicable to type-2 fuzzy logic where the potential for better uncertainty handling may provide the potential for more accurate modelling. x For simplification, Figure 4 shows the high level reasoning component with a single FCC as input. Of course, multiple FCCs (and or crisp inputs) can be fed into the high level reasoning, e.g. "If the weather is nice and I have some time....". Also, one FCC may be based on the inputs from other lower level FCCs.

ͳ

norm of the firing strengths of the p inputs to the FCC component. In case of employing singleton fuzzification as part of the high level reasoning component and considering the defuzzified value ‫ ܦܤ‬of the output set B (where ‫ ܦܤ‬has been defuzzified using a defuzzification operation such as the centroid operation [10]), the membership function of the high level component output set O is the union of the individual outputs ߤܱ݅ ሺ‫ݕ‬ሻ (where i represents the ith rule of I rules in the high level fuzzy reasoning component), which can be expressed as: ߤܱ݅ ሺ‫ݕ‬ሻ ൌ ߤ‫݅ ܩ‬

‫ܴܨܮܪ‬

ሺ‫ݕ‬ሻ ‫ څ‬ቂߤ‫ ݅ ܨ‬ሺ‫ ܦͳܤ‬ሻ ‫ څ‬ǥ ‫ ݅ ܨߤ څ‬൫‫ ܦݍܤ‬൯ቃ (3)

where ‫ܻ א ݕ‬, ‫ څ‬is of the current rule i

ͳ

‫ݍ‬

݅ a t-norm, ‫ܴܨܮܪܩ‬ is the consequent set ‫ܦ‬ and ߤ‫ ݅ ܨ‬ሺ‫ ͳܤ‬ሻ ‫ څ‬ǥ ‫ ݅݌ܨߤ څ‬൫‫ ܦݍܤ‬൯ is the tͳ

norm of the firing strengths of the q inputs to the high level fuzzy reasoning component. Note how in (3) q inputs, either from multiple FCC components or FCC components and basic concepts are incorporated. The equations for the non-singleton implementation of the high level fuzzy reasoning component are straightforward to derive considering Equation (2), (3) and [10]. These equations will be included in the journal version of this paper.

4. Practical Examples of the Application of Fuzzy Composite Concepts. As previously mentioned, FCCs aim to closely mimic human reasoning and the human use of composite concepts. In order to illustrate the usefulness and feasibility of the proposed concepts, we are providing two specific examples based closely around our work in AIEs and fuzzy logic in general. Section 4.1 expands on the already mentioned example of the weather while Section 4.2 considers the ambient light level in a home. Each of the examples is shown using diagrams based on the popular Matlab® Fuzzy Logic Toolbox as it allows for a straightforward visualization of the steps with the fuzzy systems. Figure 4. Diagram of a multi-tiered FLS incorporating Fuzzy Composite Concept processing. Note: the dashed arrow implies non-singleton fuzzification.

4.1. The Weather Example. As mentioned previously, the weather or more specifically the "quality" of the weather as perceived by humans is very clearly a composite concept in the sense that nice or bad weather depends on a variety of factors which tend to change according to context. Additionally, the very known sentence "If the weather is nice I will go for a long walk." is frequently used to explain the capability of fuzzy logic to represent concepts in a

Mathematically, the membership function ߤ‫ ܤ‬ሺ‫ݕ‬ሻ of the output set B of the FCC component can be represented as the union of the outputs ߤ‫ ݈ ܤ‬ሺ‫ݕ‬ሻ (where l represents the lth rule of M rules in the FCC component) which in a singleton type-1 fuzzy system can be expressed as [10]: ߤ‫ ݈ ܤ‬ሺ‫ݕ‬ሻ ൌ ߤ‫݈ ܩ‬

‫ܥܥܨ‬

ሺ‫ݕ‬ሻ ‫ څ‬ቂߤ‫ ݈ ܨ‬൫‫ ʹͳݔ‬൯ ‫ څ‬ǥ ‫ ݈ ܨߤ څ‬൫‫ ʹ݌ݔ‬൯ቃ ͳ

‫݌‬

(2) 311

human-like fashion using fuzzy sets (e.g. encoding nice) rather than relying on crisp numbers such as "If the cloud cover < 30% AND temperature > 20 degrees Celsius etc.". Thus, the quality of the weather makes an ideal example of an FCC and indeed the whole sentence demonstrates the applicability of FCCs as a whole. Consider that three main factors have been determined to be relevant to define the quality of the weather: the amount of wind, the cloud cover and the temperature. We consider each of these factors as basic concepts which feed (as inputs) into the FCC "weather quality". It should be noted that the choice of the basic concepts as well as the selection of the fuzzy sets to model each basic concept and the FCC is determined through standard methods such as by relying on expert knowledge, learning mechanisms, etc. Figure 5 depicts a diagram of the FCC "weather quality" which is a simple, standard FS. Further, Figure 5 shows diagrammatically how the output of the FCC is combined as part of the Higher Level Reasoning (HLR) step with the basic concept SpareTime in order to generate a final output, in this case, WalkingDistance.

Of course the rule bases presented here are not exhaustive but they demonstrate the functionality of FCCs as part of FLSs. Further, by considering the example, several of the previously mentioned benefits of employing FCCs become apparent, specifically: x Better interpretability of the fuzzy system: the high level rules (e.g. IF WeatherQuality is Average AND SpareTime is Small THEN WalkingDistance is Short) are much simpler and more intuitive to interpret than rules encompassing all basic concepts (e.g: IF Wind is Low AND CloudCover is Low AND Temperature is Low AND SpareTime is Small THEN WalkingDistance is Short). x Encapsulation and separation of lower level reasoning: the quality of the weather has been encapsulated and separated from the high level reasoning. This is relevant and highly useful. Consider for example "nice weather" in summer and in winter. By employing the FCC, any changes to what "nice weather" actually means (e.g. high or low temperature,...) is transparent to the HLR and thus the rules in the HLR inference engine can remain unchanged. x Better performance: the quality of the weather, when computed, can be employed as an input to a range of higher level reasoning processes, avoiding having to recompute all the basic cocnepts related to weather (e.g. "If the weather is very nice I will take money to buy ice cream").

4.2. Ambient Light Level.

Figure 5. Diagram of the use of an FCC for weather in the sentence "If the weather is nice I will go for a long walk". While presenting the membership functions for the fuzzy system would exceed the space available in this paper we are providing extracts of the rule bases employed as part of the FCC inference engine and the HLR inference engine:

As a second example, we are providing a more technical example which we employ on an everyday basis as part of our work in AIEs. A general problem in AIEs such as the iSpace shown in Figure 3 is the control of the indoor lighting system. In order to maintain an ambient light level comfortable for the inhabitant, intelligent agents employ light sensors distributed throughout the living space and adjust the lamps according the lighting preferences of the users which have either been learnt or pre-programmed. FLSs provide a great basis for implementing these intelligent agents as they are able to deal with the uncertainty within the system stemming from uncertain sensor information, uncertainties about the user preferences etc., while also providing humanreadable rules. The latter is particularly important in the context of AIEs as it allows for the inhabitant to interpret and understand the systems that control his/her home. An example rule of a standard lighting control FLS as employed in the iSpace traditionally had the following form: x IF KitchenLightSensor is Low AND LivingRLightSensor is Low AND

FCC inference engine: x IF Wind is Low AND CloudCover is Low AND Temperature is Low THEN WeatherQuality is Average. x IF Wind is Low AND CloudCover is Low AND Temperature is High THEN WeatherQuality is Nice. x etc. HLR inference engine: x IF WeatherQuality is Average AND SpareTime is Small THEN WalkingDistance is Short. x IF WeatherQuality is Nice AND SpareTime is Large THEN WalkingDistance is Long. x etc.

312

WindowLightSensor is Low AND WallLightSensor is Low THEN LivingRoomCeilingLamp is Bright. While this and similar rules are and were functional, they presented a series of problems including difficult interpretation as well as a need for frequent adjustments to the entire rulebase (e.g. if furniture was moved and the WallLightSensor was covered all the rule bases for all the lamps had to be adjusted). Incidentally, as humans we simply perceive light level as a common, integral property and not as the result of a series of sensors. While relying on the concept of FCCs detailed in this paper we have established a new FCC referred to as "AmbientLightLevel". Figure 6 shows a the simple structure of the FCC based on only two light sensors (to simplify the visualization).

mechanism to mimic human reasoning more closely using FLSs. We have provided details on the philosophical background on FCCs and reviewed how the current standard implementations of FLSs fail to incorporate the existence of composite concepts in human reasoning. We have presented in detail how FCCs can be implemented in current FLSs, noting their applicability to both type-1 and type-2 FLSs. Finally, we have provided practical examples from our background in Ambient Intelligence Applications which demonstrate the benefits of employing FCCs. In the future we are aiming to expand our investigation into the use of FCCs in terms of modelling accuracy, and computational cost as well as establishing their improved interpretability compared to standard FLSs through real world experiments involving lay users. Additionally, we feel that the use of the output sets of the FCCs as inputs using non-singleton fuzzification will provide interesting results and is worth investigating.

7. References [1] L.A. Zadeh, "Fuzzy sets", Information and Control, 8 (3), 1965, pp. 338—353. [2] L.A. Zadeh, “The Concept of a Linguistic Variable and Its Application to Approximate Reasoning-I”, Information Sciences, 1975, vol. 8, no. 3, pp. 199-249. [3] J. M. Mendel, “An architecture for making judgments using computing with words,” Int. J. Appl. Math. Comput. Sci., 2002, vol. 12, No. 3, pp. 325-335. [4] J. M. Mendel, “Computing with words and its relationships with fuzzistics,” Information Sciences, 2007, vol. 177, pp. 998-1006. [5] P. Lambrix, J. Maleki, "Learning Composite Concepts in Description Logics: A First Step", Foundations of Intelligent Systems, LNAI, Springer, 1996, vol. 1079, pp. 68-77. [6] J. F. Smith III, "Multi-agent Fuzzy Logic Resource Manager", Intelligent Data Engineering and Automated Learning, LNCS, Springer, 2002, vol. 2412, pp. 231-236. [7] C. Lynch, H. Hagras and V. Callaghan “Using Uncertainty Bounds in the Design of an Embedded RealTime Type-2 Neuro-Fuzzy Speed Controller for Marine Diesel Engines”, Proceeding of the International Conference on Fuzzy Systems, 2006, pp. 1446-1453, Vancouver. [8] F. Doctor, H. Hagras, and V. Callaghan, “A type-2 fuzzy embedded agent to realise ambient intelligence in ubiquitous computing environments,” Journal of Information Sciences, 2005, vol. 171, no. 4, pp. 309-334. [9] H. Hagras, "A Hierarchical type-2 fuzzy logic control architecture for autonomous mobile robots,” IEEE Transactions On Fuzzy Systems, 2004, vol. 12 No. 4, pp. 524-539. [10] J. Mendel, Uncertain Rule-Based Fuzzy Logic Systems: Introduction and New Directions. Upper Saddle River, NJ, Prentice Hall, 2001.

Figure 6. FCC for AmbientLightLevel with two (heterogeneous) sensors and two sample inputs. The output set is shown in the bottom right corner. The output of the FCC (as shown in Figure 6) is subsequently employed for HLR of the lamps in the room using rules in the following format: x IF AmbientLightLevel is Low THEN LivingRoomCeilingLamp is Bright. Again, it is straightforward to see how the use of FCCs has improved interpretability by mimicking the human reasoning process about indoor light levels. Figure 6 also shows an example of an output set which can be employed using non-singleton fuzzification in the HLR component, thus allowing to incorporate the uncertainty information from the sensor evaluation throughout the fuzzy system. Finally, considering the examples in 4.1 and 4.2, it can be seen how multiple FCCs can be combined for HLR (e.g. establish FCC LightLevel from sensors, establish FCC weather incorporating LightLevel, proceed to HLR.)

6. Conclusions In this paper, we have introduced the concept of Fuzzy Composite Concepts which provides an (we find elegant)

313