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Year : 2002  Volume
: 4
 Issue : 15  Page : 2744 
Noise annoyance modelling using Fuzzy rule based systems
D Botteldooren^{1}, A Verkeyn^{1}, P Lercher^{2}, ^{1} Ghent University (INTEC), Gent, Belgium ^{2} Institute of Hygiene and Social Medicine, Innsbruck, Austria
Correspondence Address:
D Botteldooren Ghent University (INTEC), StPietersnieuwstraat 41, B9000 Ghent Belgium
Abstract
This paper presents a model that uses a fuzzy rule based engine to predict noise annoyance reported by individuals in a social survey. The rules are proposed by the human expert and are based on linguistic variables. The approach then adapts the sufficiency degree or certainty of a rule to tune the model to a particular survey. Although all possible relations between exposure, attitudinal, emotional, personal, environmental and social variables are not included in the model as yet, the benefits of the new approach are clearly demonstrated. A major limitation that remains is the varying theoretical and empirical basis of the expert for different subset of annoyance determinants. Future applications may include more accurate predictions of noise annoyance for policy support and extraction of knowledge concerning the construct of annoyance from surveys.
How to cite this article:
Botteldooren D, Verkeyn A, Lercher P. Noise annoyance modelling using Fuzzy rule based systems.Noise Health 2002;4:2744

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Botteldooren D, Verkeyn A, Lercher P. Noise annoyance modelling using Fuzzy rule based systems. Noise Health [serial online] 2002 [cited 2022 Oct 5 ];4:2744
Available from: https://www.noiseandhealth.org/text.asp?2002/4/15/27/31790 
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Introduction
As any model, the noise annoyance model presented in this work serves two purposes. Firstly, constructing a model and comparing model outcome to experimental data results in a better understanding of the mechanisms involved. Secondly, a model allows making predictions.
Compared to other fields of scientific research, the use of detailed models has been very rare in noise annoyance research. Most knowledge extraction from large surveys has been based on general multivariate linear or logistic regression techniques or factor analyses. Klaeboe (Klaeboe, 2000) has proposed to use Structural Equation Models, mainly to take into account uncertainties in the variables involved in the model. These models can however also include nonlinearity and more complex interactions between main variables. Artificial Intelligence has been proposed to analyze airport noise annoyance (Boyer and Chaudron, 2000). It is clear however that the use of these types of models has not yet fully matured. The work that is reported in this paper is a next step forward.
For noise policy support purposes a simple dosage annoyance relation where the dose is expressed as Ldn or Lden is becoming more popular (Miedema and Vos, 1998). Other factors can be included as correction on the exposure (Miedema and Vos, 1999) These models have the advantage of calculation simplicity and transparency. However they may oversimplify particular settings. In (Houot, 2000) additional information was introduced in a GIS (geographic information system) noise annoyance mapping exercise using crisp rules. The fuzzy rule based system proposed here may ultimately be used in this context. None of the noise annoyance models described here take into account the dynamics of noise exposure although noise policy necessarily results in a changing environment that can be experienced in a very different way.
The paper is structured in three sections. The first section describes the Fuzzy Rule Based calculation. The second section describes the particular building blocks of a general noise annoyance model that are implemented using the fuzzy rules and the third section summarizes some model results when the model is applied to a particular noise survey.
A fuzzy rule based engine for noise annoyance modelling
[Figure 1] shows the general layout of the fuzzy rule based noise annoyance model. It contains the building blocks commonly found in such systems (Cordon et al., 1999). This model can be regarded as a black box that calculates the (predicted) annoyance term that is associated with the input data. Internally, the knowledge base of the system contains the domain specific intelligence of the model. Using this knowledge, conclusions about the level of annoyance are drawn from the input data by the inference engine. Finally, the result is labeled with an annoyance term in the linguistic approximation part.
Data Base
The knowledge base contains two large groups of information: the database that contains the definitions of the linguistic terms that will be used within the model and the rule base that describes the relations between variables. The system contains rules that are based on many different variables and thus the database contains many definitions. Most of them are constructed as needed and their definition will be given at the appropriate time. However, the terms quantifying the level of annoyance are of such importance in the system that they are deducted more formally.
In the framework of fuzzy rule bases, a linguistic term is usually represented by a fuzzy set on an appropriate universe U, defined by a mapping U[0,1] that is called the membership function. Throughout this paper the same symbol A will be used to denote a linguistic term, the fuzzy set and the membership function. The set of all fuzzy sets over a universe U will be denoted as F(U).
To get some insights in the precise meaning of several linguistic terms commonly used to denote an annoyance level, an International Annoyance Scaling Study has been conducted (FelscherSuhr et al., 1998). For 21 different annoyance terms, people were asked to put a mark on a continuous line to indicate the level of annoyance they associate with each term, the extreme left meaning no annoyance at all and the extreme right representing the highest possible level of annoyance. After collecting the data for nine different languages, each mark on the line has been converted to a numeric value expressing the distance (in centimeters) from the left side of the 10 centimeters long line. As a result, the study delivered 21 values, ranging over a continuous domain from 0 to 10, for each subject. Although this study was not conducted with fuzzy sets in mind, the results were used to build membership function representations for four linguistic terms that were used in the noise annoyance survey that will be used below to test the model (for the German language, there were 59 records). The membership functions range over the universe U = [0,10]. In the fuzzy literature, one can find several methods for this purpose (e.g. (Medasani et al., 1998)). The specific approach that was adopted here, is described in (Verkeyn et al., 2001b). The result is shown in [Figure 2] Remark that the distributions do not cover the universe to a comparable way. This is one of the grounds why the International Scaling Study recommends to use the German term "betrachtlich" instead of "mittelmaBig" for a fourpoint scale (and "total" instead of "erheblich").
To enhance the practical usability of the resulting "teilweise" and "mittelmaBig" curves they were slightly adapted as shown in [Figure 2]. Those modifications can be informally grounded on the presumption that people asked to rate their level of annoyance using one out of four verbal categories, will tend to distribute the universe of discourse rather evenly, instead of merely looking at the meaning of the given terms.
In the traditional approach, the linguistic annoyance terms are identified using threshold values. Common choices are 72% for "highly annoyed", 50% for "moderately annoyed", 28% for "little annoyed" and everything below this threshold is then denoted as "not at all annoyed". It is obvious that the use of such thresholds is too abrupt to handle the vagueness of the annoyance concept in an appropriate way. However, note the correspondence between the constructed curves and the crisp threshold values: "highly annoyed" becomes dominant at about 7.0, "moderately" has its peak at 5.4 (originally 5.0) and "little" has its peak at 2.5 (originally 2.9).
Rule Base
The rule base contains a collection of linguistic IFTHEN rules of the form IF X = A THEN Y = B, where X and Y are variables over the universes U and V respectively, and A and B are (linguistic terms associated with) possibility distributions over U and V. According to the possibility assignment equation as stated by Zadeh (Zadeh, 1978), the fuzzy sets in the database can be interpreted as possibility distributions over the universe U and used as antecedent or consequent in the rules, e.g.: IF (distance to railway) = small THEN (railway noise annoyance) = high. When the relationship between two variables is described using more than one rule, these are called a set of parallel rules.
The linguistic terms as defined in the data base can also be combined and modified by fuzzy operators in the rules. For each operator in the classical, binary logic, a wide range of fuzzifications that coincide with the classical ones in the crisp case are available. The fuzzy generalization of the conjunction operator (AND) is known as a triangular norm T pointwise defined as a [0,1] x [0,1]→[0,1] mapping that is associative, commutative, monotone and satisfies the boundary condition (∀xε [0, 1]) (T(1, x) = x). The original, still frequently used operator as introduced by Zadeh (Zadeh, 1965) is T(x, y) = min(x, y). Analogous, disjunctive (OR) combinations of linguistics terms are modelled by a triangular conorm S point wise defined as a [0,1] x [0,1]→[0,1] mapping that is associative, commutative, monotone and satisfies the boundary condition (∀xε [0, 1]) (S(0, x) = x). The most typical conorm is defined as S(x, y) = max(x, y). Also the complement (NOT) can be fuzzified with a [0,1]→[0,1] mapping that is monotonically decreasing and satisfies at least the boundary condition N(1) = 0 and N(0) = 1. Usually, the fuzzy complement is defined as N(x) = 1x. If a triangular norm T and a triangular conorm S satisfy the De Morgan law (∀xε [0, 1]) (N(T(x, y)) = S(N(x), N(y)) and N(S(x, y)) = T(N(x), N(y))) there are called dual with respect to the negator N.
Although there are a number of good, wellknown techniques to automatically generate fuzzy rules best suited to model a given data set, such rule deduction was not performed here because the main goal is to construct a model and not to analyze data. Moreover automatic rule extraction tends to fit the rules too tight to the data at hand and therefore makes them less general. Instead, rules used in our model are articulated by experts in the field of acoustics or drawn from the existing noise annoyance literature. This allows introducing existing knowledge or making rule hypothesis sequentially tested on a particular survey.
Since all rules are stated as absolute truth relations between antecedent and consequent rather than as modifying operators, a mechanism must be introduced to weigh the impact of each rule. Therefore, each rule is assigned a certainty or sufficiency degree λ ε [0,1]. This degree expresses to what extent it is sufficient that the antecedent is true for having the consequent true. Actually, this certainty degree will be applied to the rule consequent and not to the rule itself. Although both interpretations only coincide when the antecedent is not fuzzy, this is common practice. In (Dubois and Prade, 1991), a certainty qualified proposition "Y = B is λcertain" is linked to a certainty or necessity measure, given "Y = B*":
[INLINE:1]
where I is an implicator. Solving this equation leads to the underlying fuzzy set B*. Choosing for I the reciprocal of a residual implication results in (∀vεV) (B*(v)≤ IS(λ,B(v))) with IS an S implicator defined as (∀(x,y) E [0,1]2)(IS(x,y) = T*(N(x),y)), where T* is the dual conorm of a triangular norm T and N is a negator. For I the reciprocal of the Godel implicator, this results in the KleeneDienes IS implicator: B*(v) = max(1 λ,B(v)). As can be seen from the above formula, the consequent of a rule with λ=1 will remain unchanged (B*=B). On the other hand, a completely uncertain rule (λ=0) will not have any effect because the consequent will have possibility 1 over the whole universe (concluding from the antecedent that everything is possible and thus providing no additional information).
Inference
As primary inference engine, the generalized modus ponens is used, where B', the inferred possibility distribution of Y, is calculated with
the compositional rule of inference:
[INLINE:2]
where R is the representation of the rule and A' is the fuzzy input given to the rule.
Semantic analysis of fuzzy IFTHEN rules, revealed three different rule interpretations, depending on the operator used to construct the rule representation R (Dubois and Prade, 1991; Dubois and Prade, 1996). It was shown that the relation R is lower bounded when a triangular norm is used (conjunction model), and upper bounded when a fuzzy extension of the classical implication operator is applied (implication model). The former case leads to possibility qualifying rules, to be interpreted as "the more X is A, the more possible B is a range for Y". As a consequence, when a rule is not triggered at all, the resulting possibility distribution is 0 over the whole universe. The latter can be seen as truth qualifying (or gradual) rules, "the more X is A, the more Y is B", in case of a residual implicator. For an Simplicator, one gets certainty qualifying rules with associated interpretation "the more X is A, the more certain Y is B". In both interpretations of the implication model, everything is possible (all possibility degrees are 1), when a rule is not triggered at all.
The rules in the noise annoyance model are formulated with a "certainty" interpretation in mind: the antecedent is required to create certainty about the consequent (e.g. high annoyance), not to create possibility. Therefore, R should be modelled with the KleeneDienes implicator, a prominent member of Simplicators.
Expected annoyance as derived from rules based on different input variables can be interpreted as expert opinion that should be aggregated to form a consensus. To exploit the maximum amount of available information a product norm is used as an aggregator. The product has the desired property that a possibility distribution with 1 everywhere (thus providing no information at all) has no effect on the aggregated result, which is intuitively a good thing.
Inference based on rules constructed using the KleeneDienes implicator turns out to be a numerically costly operation. In (Verkeyn et al., 2001), we prove that performance of the noise annoyance model presented here does not degrade when a Mamdani norm is used in conjunction with the product aggregator as long as a normalization step is added before the aggregating results of inference. At the same time calculation speed increases by almost a factor 5.
Linguistic Approximation
The final step in the fuzzy system is the linguistic approximation phase, in which the aggregated possibility distribution B' that resulted from the inference engine is matched to one of the four annoyance terms. The optimal term is determined by calculating an approximate descriptor D, defined as a mapping from F(V) to F(T) that measures the degree of similarity between a fuzzy set on a universe V and a collection of linguistic terms on the same universe (Dubois et al., 1999). The "upper approximation" descriptor that measures the consistency of B' with each term Ti, is used. It is defined as
[INLINE:3]
The term with the highest descriptor value is selected as the most plausible, and thus the outcome of the system. When the difference between matching values is too small to make an adequate selection, the result of the model is declared unknown.
The noise annoyance model
General annoyance model
Although the theoretical basis for the concept of annoyance is rather vague and annoyance has different meanings for different researchers annoyance measures have been the primary response variable in social noise surveys and a large body of data about this response variable has been accumulated. From these archival data we know that transportation noises differ substantially in their annoyance response (Fidell et al., 1991; Miedema and Vos, 1998). Typically, aircraft noise triggering the strongest and rail noise the modest reaction with highway noise falling in between. Therefore, the analyses are presented separately for rail and highway noise. The doseresponse can differ largely from community to community (Fields, 1988; Berglund and Lindvall, 1995; Gjestland, 1998; Berglund et al., 1999) and variations in personal, situational, environmental, and cultural factors have been put forward as explanation (Fields, 1993; Lercher, 1996; Miedema and Vos, 1999; Guski, 1999; Stallen, 1999). However, the sources attributable to the large differences are still poorly understood. This provides the basic rationale for this work.
In a final stage a noise annoyance model could include the complex relations between many variables shown in [Figure 3]. In this figure clear forward paths as well as adaptation (internal and external) are shown in full lines while paths that are considered more hypothetical and could ultimately turn out to follow a completely different route are marked in dashed lines. Although the fuzzy rule based system presented here is theoretically capable of handling this degree of complexity, it is at this stage restricted by a lack of fundamental knowledge (good linguistic rules) and accurate data to test it on. Paths (rules) included in the model that are important for the analyses of model performance, which is the main goal of the present paper, are described in more detail below.
Exposure
Exposure to noise at home includes all physical characteristics of the intruding noise and the background noise level. The basic exposure variable is a calculated Aweighted daynight sound level. Two approaches were tested to implement the relation between L dn and reported annoyance level. The first approach uses dosage response relations based on meta analyses that are reported in literature (Miedema and Vos, 1998) as a first step. The calculated annoyance level is then translated to a possibility distribution using a set of parallel rules. However a more straightforward system of four parallel rules on L dn performs equally well. [Figure 4] shows the possibility distributions of linguistic terms that are used as antecedent.
Sound level calculations do not include the dwelling of the respondent. Additional exposure information can be extracted from the orientation of the house. The view from the living room main window and the main bedroom window provide some clue. The antecedent universe UF in this case equals {quiet street, highway, through road, railway, and quiet backyard}.
Several fuzzy sets defined on UF are shown in [Figure 5]. Two sets of rules are introduced IF [living room, bedroom] window faces = the source THEN annoyance = high IF [living room, bedroom] window faces = a quiet area THEN annoyance = not very high.
Definition of "high" and "very high" annoyance used in these rules are shown in [Figure 6].
Average day night sound level does not take into account specific variations of the sound level. The distance to the source can provide additional information. Therefore two rules are added:
IF distance to source = close THEN
annoyance = not not
IF distance to source = far THEN
annoyance = not high.
Possibility distributions for far and close can be found in [Figure 7]. Note that the above rules can include other than purely exposurerelated effects.
Exposure to noise from the source under consideration can be masked by background noise. The fuzzy noise annoyance model can incorporate the combination of exposure and masking in different ways. Two possibilities were investigated: applying the "and" operation to the antecedent or applying it to the consequent. The latter seems more easily implemented, and is used in the remainder of this work. The proposed rules for predicting masking are all based on the generic rule:
IF masking = high THEN annoyance = not very high.
The important feature here is that these rules only include a statement about high annoyance, not about moderate or slight annoyance (Botteldooren et al., 2000). The possibility distribution for 'not very high' is shown in [Figure 6]. Since masking requires a loud continuous background, only sound from highways and main roads is considered. Several indicators for masking can be found and the final choice will depend on the available data:
Difference between Ldn,road and Ldn,rail small or very small (MLD)[Figure 8]Short distance to masker (MD) ("close" in [Figure 7]High masker noise level (L dn ) (ME) ("at least high" in [Figure 4]Windows (sleeping room and living room) facing the masker (MF) [Figure 5]
Sensitivity
Noise sensitivity is found to be significantly linked with noise annoyance in most noise surveys (Ohrstrom et al., 1988; Job, 1988; Stansfeld, 1992; Fields, 1993; Nivison and Endresen, 1993; Halpern, 1995; Lercher, 1996; Miedema and Vos, 1999; Stansfeld et al., 2000).
Particularly high noise sensitivity seems to be a relative stable personality trait over time (Stansfeld, 1988; Stansfeld, 1992). Therefore it is introduced as an explicit variable in the model. The relation between noise sensitivity and annoyance is translated into a set of two rules:
IF sensitivity to noise = high THEN annoyance = not not (or at least somewhat)
IF sensitivity to noise = not THEN annoyance = not high
With the linguistic terms related to sensitivity defined by the possibility distributions in [Figure 2] but with annoyance universe replaced by sensitivity universe.
The link between exposure and sensitivity was already shown in [Figure 3] and is probably due to migration from noisy areas or avoidance of moving into noisy areas (external adaptation). Because noise sensitive people are known to react stronger and adapt more slowly to noise and to asses noises as more threatening and out of their control it is reasonable to assume this mechanism (Stansfeld, 1992). Much less consistent information can be found in literature on other variables that determine noise sensitivity. There is however some evidence that noise sensitivity increases with age particularly in women (Langdon, 1976; Thomas and Jones, 1982; Nivison and Endresen, 1993). (Nivison and Endresen, 1993) also reported an increased "sensitization" over time in women who lived a long time (and greater time spend) at the same place of residence. Furthermore, a number of personality constructs (neuroticism, critical tendency, trait anxiety, negative affectivity) have been inconsistently associated with noise sensitivity (Stansfeld, 1992; Stansfeld et al., 2000). Persons with such a personality outfit may feel more vulnerable and/or perceive environmental degradation more threatening. It is assumed here that the path between these primary variables and annoyance passes, at least to some extent through noise sensitivity and may be moderated by other factors that increase the vulnerability of these persons (families with young children, unfavourable housing etc). Unfortunately, research into these situational and environmental factors that may aggravate annoyance/health effects has not been carried out.
In cases where reported noise sensitivity is not available or when particular paths from primary variables to annoyance are studied it may be worthwhile to include some of this available knowledge. The proposed rules used to model noise sensitivity are:
IF age = young/old THEN sensitivity to noise = not high/not high
IF gender = female THEN sensitivity to noise = not (or at least somewhat)
IF number of children = none/few/many THEN sensitivity to noise = less than average/high/not high
The possibility distributions relating to age are shown in [Figure 9]. Possibility distributions for the number of children are given in [Figure 10].
Expectation
The degree of expectation about the quality of the environment has often be quoted as a neglected determinant of annoyance (Langdon, 1987; Job, 1988; Job, 1993), however rarely been studied. However, it is the rationale for allowing different noise levels in different categories of land use. To our knowledge, only one study has studied and confirmed this claim (Kastka et al., 1978). Other studies did show the effects of the residential appearance on annoyance (Langdon, 1976; Kastka and Hangartner, 1986). Studies on the effect of the type of house have revealed mixed results
(Bradley and Jonah, 1979; Fields, 1993; Sato et al., 1999).
In this model it is assumed that people having high expectations regarding their living environment have a higher probability of being annoyed by noise. This is expressed in the rule: IF expectation = very high THEN annoyance = not
Possibility distributions for the expectation variable are the same as for the annoyance variable with the annoyance universe replaced by the expectation universe. Due to the lack of literature, it is neither a simple task to ask directly for expectation in a survey or to deduce it indirectly from indicators of environmental quality. Seven indicators are combined in a fuzzy way to estimate expectation:
The type of houseThe ownership of the house (being an owner or a renter)The presence of a gardenThe size of the city (number of inhabitants)The general attractiveness of the areaThe living quality of the neighborhoodThe availability of leisure facilities
Noise exposure influences expectation through an external path that is reflected also in the indicators. Moreover, some of the indicators (e.g. type of house or city size) are correlated strongly with noise exposure, thus further complicating the variable. The most important shortcoming in this variable is related to a conflicting variable that was labeled "context elasticity".
Coping
Coping, actively (e.g. closing windows), emotionally (e.g. (not) feeling helpless), communicative (e.g. talking to neighbors about it), or politically (e.g. signing a petition), correlates strongly with annoyance (Fields and Hall, 1986; van Kamp, 1990; Lercher and Kofler, 1996; Lercher, 1998). The direction of the effect is coming from both sides. Having to close your window to be able to sleep may be annoying. However, you may develop feelings of helplessness or anger when doing nothing. Thus, annoyance can trigger you to close windows or move sleeping room. Therefore, it is not surprising that active coping with noise can reduce the adverse effect of noise on health and increase reported annoyance at the same time (Lercher and Kofler, 1993; Lercher, 1996; Lercher, 1998).
The very strong feedback is modelled explicitly for this variable. The level of coping is deduced from the calculated annoyance level and a number of modifiers using the rules:
IF calculated annoyance = vaguely above average THEN coping level = vaguely above average (λ=0.8)
IF age = young/middle/old THEN coping level = not high/not less than average/not high (λ=0.7/0.9/0.9)
IF house owner/renter THEN coping level = high/less than average (λ=0.5)
IF education level = low/high THEN coping level = high/low (λ=0.5)
The possibility distributions for the linguistic terms used on the coping level universe are shown in [Figure 6]. The certainty or sufficiency degree λ is given between brackets and is not part of the optimization described below. The causal influence of coping on annoyance is expressed by the rule:
IF coping level = very high THEN annoyance = high
The inclusion of feedback in the calculation introduces a loop that is followed just once for the moment.
Context elasticity
There has been some evidence that people living in a generally pleasing environment may tolerate more noise before feeling annoyed or reporting this annoyance (Kastka and Hangartner, 1986; Sabadin et al., 1991). However, also opposite effects on annoyance have been reported (Lercher et al., 1999; Flindell and Stallen, 1999; Lercher and Brauchle, 2000). This makes it especially difficult to find appropriate rules that combine both possible effects. The rule included to take into account the higher elasticity in a more pleasant context is
IF context elasticity = high/low THEN annoyance = below average/not low
Three indicators are identified to extract knowledge about this variable
The general attractiveness of the areaThe living quality of the neighborhoodThe availability of leisure facilities
Health
General health status has typically shown a variable but weak relationship with annoyance (McLean and Tarnopolsky, 1977; Fields and Hall, 1986) The inconsistent findings result mainly from inherent differences in population structure and morbidity among surveys and from the mostly indirect pathways (via sensitivity or fear of health effects) to annoyance. Also the problem of reverse causality or circularity is a potential problem here (noisehealthannoyance). Therefore, in a more complete model that includes sociodemographic factors and noise sensitivity information on health status should be included.
IF health = very good/poor THEN annoyance = not high/not not
Life style
A number of variables are catalogued as lifestyle variables where live style is defined in the broadest possible sense. No combined variable is introduced. All rules related to these variables are used in parallel in the system.
IF age = young/middle/old THEN annoyance = low/high/low [Figure 9]
IF number of children = none/few/many THEN annoyance = less than average/high/not high [Figure 10]
IF crowding = few/many THEN annoyance = less than average / high [Figure 11]
Testing the model on a noise annoyance survey
Fitting rule sufficiency
In the fuzzy rule based model a certainty degree λi is introduced for each rule i. This certainty degree to some extend weighs the relative importance of each rule in predicting noise annoyance and is used to tune the model. The λi weights are optimized to minimize the prediction error on the database using a genetic algorithm (Goldberg, 1989), because of the highly multimodal search space. The annoyance model results in the approximate descriptor D, a fuzzy set on T. Assume that the respondent of a survey marked the linguistic term Tk. Then the model prediction is correct if D(Tk) > D(Ti),∀i#k. The performance of the model on the database is expressed as the weighted percentage of correct predictions. The weights equalize the occurrence of different elements of T in the database. In the optimization a more subtle definition of the prediction error is used. The strength of the model prediction is taken into account in this definition. If the prediction is correct then a strong belief (expressed as a sharply peaked D) is appreciated while this strong belief should be punished in case of an incorrect prediction. One should however be careful not to create an indecisive model (expressed as rather flat D). These considerations resulted in a prediction error e defined as
[INLINE:4]
where w Tk are the weights mentioned above and a is a constant that is experimentally determined for optimal performance. The summations in the first term run over all subjects in the database, the summation in the second term runs over incorrectly predicted annoyance levels in the database only.
In the genetic algorithm the set of rule λi's under optimization is represented as an array of real values where the allele set of each gene is bounded to the unit interval [0,1], and discretized in steps of 0.01. The steady state genetic algorithm was configured with a population size of 50 individuals. The crossover operator is a standard, twoparent uniform array crossover. The size of mutations follows a Gaussian probability distribution with µ = 0 and 6 = 0.02. The mutation probability is 0.1. The fitness is scaled linearly to keep selection pressure optimal during the whole convergence process.
Noise annoyance survey
As part of an ongoing environmental health impact assessments of a new rail track in the Austrian part of the Alps near Innsbruck, a representative phone survey was conducted within the about 40 km area of the new rail track. This mainly rural area consists of small towns and villages with a mix of industrial, small business and agricultural activities. The primary noise sources are road and rail traffic. In total, 2007 inhabitants were interviewed. Initially, 1500 inhabitants (aged 18 to 75) were sampled at random from the whole Innvalley area (sample 1). This sample was enriched by another random sampling of 500 residents living within 150 m of the existing rail track and the highway or within 50 m of local roads (sample 2) to guarantee a sufficient number of people with higher exposure to noise and vibrations. The overall response was 83 %. The standardized interview (typical length 20 minutes) covered sociodemographic data, housing, satisfaction with public services and the environment, general annoyance, interference, coping with noise and health.
Noise exposure was assessed first by modelling (Soundplan) according to Austrian guidelines (OAL Nr 28+30, ONORM S 5011). Afterwards calibration was conducted and corrections were applied to the modelled data based on the recordings of 31 measuring stations. Based on both data sources approximate daynight levels (L dn ) were calculated for each respondent to ease comparison with typical doseresponse data.
Recently, a standardized annoyance question has been proposed for various languages based on several semantic studies (Fields et al., 2001). The survey data utilized for this analyses is already based on this standard question, however, in a four category response format due to difficulties with the German semantic of the fifth response category at that time (Guski et al., 1998).
General result
As mentioned, the weighted percentage of correctly predicted noise annoyance response is used as a measure of performance of the model. This differs from the more commonly used explained variance in two important aspects. Firstly, surveys based on a random sample of the population result in higher numbers of subjects reporting lower annoyance. Therefore a model correctly predicting situations where annoyance is low will be favoured more by optimizing explained variance than by optimizing the weighted performance measure used here. Secondly, once a prediction is incorrect performance decreases independently of the prediction itself. An advantage over explained variance is that randomly answered surveys influence results less. Since this performance measure is less common, results of a linear regression model expressed in this performance measure were calculated and are presented for reference.
If no knowledge is available on the subjects taking part in the survey, performance is 25%. Traditionally a linear relation between reported level of annoyance and LAdn is often assumed. If such a linear relation is fitted to the noise annoyance data at hand, 30.2% of responses are predicted correctly for railway noise and 29.5% for road noise. This means that L dn explains only a very small part of individual annoyance variations, when used in a traditional way. Remark that the definition of correct prediction includes 4 levels of annoyance: "iiberhaupt nicht", "gering oder teilweise", "mittelmal3ig", and "stark/erheblich". If only high annoyance is to be predicted, success rates rise well over 80%. When variables reported by the subjects of a noise annoyance survey are included in the model one must be very careful when analyzing performance. Indeed, if an answer includes a certain degree of subjectiveness, one may as well be sampling an underlying variable also contributing to the answer on the annoyance question. Therefore a distinction is made between variables that can be measured without the cooperation of the subject and variables that require information that can only be provided by the subject within the current model. The model could eventually be provided with more knowledge in order model these variables "from scratch" also. [Table 1] summarizes results. The label "no input by subjects" refers to the situation where all input to the model can be measured objectively. When the label "input by subjects" is used, additional rules rely on data obtained by asking the subjects (e.g. sensitivity to noise or the attractiveness of their environment). Also in this case, variables that correlate in a trivial way to the reported noise annoyance (e.g. speech disturbance) are carefully omitted. The table also includes the results of a multivariate linear regression based on the same input variables and the relative increase of performance purely attributed to the use of the fuzzy rule based model.
It is interesting to observe that the fuzzy rule based model succeeds better in predicting extreme levels of annoyance correctly than it does for moderate levels of annoyance. This is illustrated in [Figure 12] and [Figure 13] for road and railway noise. The diameter of the circles in this chart represent the relative number of subjects categorized in each particular combination of reported and calculated annoyance term. The terms are labeled 0 to 3. For both sources the categories "no annoyance" and "high annoyance" seem to be predicted fairly well. Moreover subjects reporting "moderate" or "high" annoyance are practically never predicted to have lower annoyance and subjects reporting "no" or "little" annoyance are practically never predicted to have "high" annoyance. This difficulty in predicting was already recognized in early annoyance studies (Schultz, 1978).
An indication of the importance of a particular rule in the optimized fuzzy rule base system can be gained from studying the sufficiency and the adaptability of each rule. Sufficiency was defined above and can be interpreted as the degree to which the particular rule dominates the outcome of the system if fired. Adaptability as defined in (Klir and Yuan, 1995) indicates in a fuzzy way how often and to what extend a rule is fired or triggered when evaluating the database or in other words for which percentage of the population it results in some kind of clue on predicted annoyance. However, sufficiency and adaptability give only a first indication. There are several reasons for this. Firstly, rules with low sufficiency can still be important if they provide information on a region of the annoyance universe where information is scarce (e.g. they allow distinguishing between "teilweise"and "mittelmaBig"). Secondly, rules are in general not orthogonal. This means that more than one rule in the system describes the same underlying (possibly more dimensional) mechanism. The choice between such rules that is made by the optimization is often not very stable and can not be used as decisive about which rule is best. A more thorough analysis of rule importance will be presented in the following subsections for particular cases.
[Figure 14] shows sufficiency and adaptability for the exposure rules in a model for railway noise including input by the subjects. When the ten first rules are considered it can be seen that rules based on low exposure or very high exposure, short distance and direction of the living and sleeping rooms get high sufficiency after optimization. Since these rules are largely overlapping (nonorthogonal) the low sufficiency found for a rule based on living far away from the railroad or being highly exposed can be explained by the fact that their counterpart (high exposure and living close to the railway) performs better on this database.
For railway noise the rules labeled "masking" assume that the noise from a highway or main road is the masker. Except for masking described by exposure ME (L dn level), all masking rules get a reasonable sufficiency. The high sufficiency of the masking rules based on the direction of the living room and sleeping rooms are remarkable. They state that railway noise annoyance is not high when your living or sleeping room window is facing a highway or a through road.
Railway noise annoyance near highways and through roads
As an example of a more detailed analyses using the fuzzy rule based model, the annoyance caused by railway noise near highways and through roads was studied (Botteldooren et al., 2000). It was observed that railway noise annoyance is lower than expected based on the exposure close to highways. Two possible explanations for this were proposed: physiological masking and external adaptation of the population to the noisy situation (sensitive people do not come and live there). This typical analysis is based on comparison of the performance of 4 partial models. [Table 2] shows the error e (between brackets) and the difference to the best value in the table. The rules based on reported sensitivity (decrease 13) are more general than the rules based on physiological masking (decrease 8) as expected. However a clear overlap between both sets of rules is observed. The sensitivitybased rules predict most of the variation also predicted by the masking rules. This leads to the conclusion that near highways and through roads annoyance caused by railway noise is lower because sensitive people move out or do not choose to live there. If this conclusion holds, it should also be observed that reported sensitivity is lower than expected under conditions that correspond to masking. This will be shown in the next subsection.
The methodology followed above to compare two sets of effects is not independent of the other rules present in the model. Some indirect effect may be present that is not recognized at first. In the case of masking versus sensitivity the rules based on age were present and may influence the results in a sense that the effect of sensitivity rules is underestimated (see also the next section).
Reported sensitivity and sensitivity model
Noise sensitivity was assessed by a short direct question with the same four categories semantic as the annoyance question. 19% assessed themselves as very sensitive. Rules based on reported noise sensitivity obtain sufficiency 0.64 and 0.80 for "not sensitive" and "very sensitive" respectively when optimized in the database at hand (railway noise annoyance). However quantifying reported noise sensitivity requires input from the subject. Therefore a linguistic rule based model for noise sensitivity is more attractive in many situations. In constructing such a model the variables age, gender, "number of children" and "crowding" (number of persons per room) were used. In the survey noise sensitivity was categorized using the labels "tiberhaupt nicht", "gering", "mittelmassig", and "stark". For the purpose of this model the possibility distributions for these terms are identified with the 4 terms used for describing noise annoyance. After optimization a correct prediction is obtained for 30% of the subjects (again referring to 25% as the no knowledge situation). Sufficiency obtained for the rules involved are shown in [Figure 15]. At least one of the rules based on any of the four variables seem to be important. A clear degradation in performance was also observed when omitting one of the variables, thus confirming the importance of all variables.
External adaptation to exposure was mentioned as a possible feedback path while studying railway noise annoyance near highways. When a rule "IF distance to highway is short THEN noise sensitivity is not very high" is added to the fuzzy rule based model, predictability of noise sensitivity increases to 31%. This confirms the conclusion of the previous section.
Expectation and context elasticity
Two examples of combined variables used in the model are the expectation and the context elasticity variable. These variables are constructed using a fuzzy aggregation based on several primary variables. The value of the aggregated fuzzy variable is obtained as a possibility distribution for every subject in the database. A linguistic approximation can be used to translate this value back to for example 4 terms that can then be used in classical analyses. The combined variables considered in this subsection interact strongly since they include a number of common basic variables.
[Table 3] shows relative and absolute prediction errors for the four relevant combinations of including rules related to these variables.
Rules related to context elasticity increase model performance considerably (30 units in absence of expectation). However including the expectation rules decreases performance, even after optimization. Remark that a sufficiency X=0 would ultimately remove the rule and therefore not decrease performance. However the optimization does not find this trivial solution. In fact, the local optimum shown here is so strong that it is found as a result of several independent optimizations. Adding context elasticity removes most of the adverse effect of the expectation variable, but does not result in a general performance increase as was hoped for.
Conclusions
A fuzzy rule based model was constructed to predict reported noise annoyance in a social survey. The model takes into account the vagueness in the concept of annoyance and allows taking into account uncertainty in both input data and relations (rules).
This paper shows that such a fuzzy rule based model works and demonstrates how it can be used to extract knowledge from a noise annoyance survey. It is very hard to prove however that this type of model is considerably better than any other approach. In fact it may be far easier to prove that "there is no free lunch" in a sense that there are disadvantages to any numerical approach and that there is no single best. We did clearly show however that the fuzzy rule based model is far better in predicting the subjects answer in a noise annoyance survey than a linear regression model even if the same set of modifying variables are included.
Compared to other more elaborate and nonlinear techniques such as neural networks, the fuzzy rule based system has the advantage that the knowledge included in its database can easily be read or written by a human expert since it is expressed using common language. A major limitation remains: the theoretical and empirical basis for the expert varies for the different subset of annoyance determinants.[55]
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