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Computer aided fuzzy medical diagnosis
Information Sciences 162 (2004) 81–104 www.elsevier.com/locate/ins
Computer aided fuzzy medical diagnosis P.R. Innocent *, R.I. John Centre for Computational Intelligence, Department of Computer Science, De Montfort University, The Gateway, Leicester LE1 9BH, UK
Abstract This paper describes a fuzzy approach to computer aided medical diagnosis in a clinical context. It introduces a formal view of diagnosis in clinical settings and shows the relevance and possible uses of fuzzy cognitive maps and fuzzy logic. A constraint satisfaction method is introduced which uses the temporal uncertainty in symptom durations that may occur with particular diseases. Together with fuzzy symptom descriptions, the method results in an estimate of the stage of a disease if the temporal constraints of the disease in relation to the occurrence of the symptoms are satisfied. The approach is evaluated through simulation experiments showing the effects of symptom ordering, temporal uncertainty and symptom strengths on the diagnosis efficiency. The method is effective and can be developed further using second order (Type 2) fuzzy logic to better represent uncertainty in the clinical context thus improving differential diagnosis accuracy. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy cognitive map; Fuzzy temporal reasoning; Fuzzy sets; Diagnosis
1. Introduction Fuzzy approaches to medical diagnosis have been reviewed by Steimann and Adlassnig [20] and shown to be effective in this domain. This paper considers using fuzzy methods [27] to address the specific problem of disease classification in the presence of uncertain or vague knowledge of a linguistic and
*
Corresponding author. Tel.: +44-1662577484; fax: +44-1662541891. E-mail address: [email protected] (P.R. Innocent).
0020-0255/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.03.003
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temporal nature. The problem context is typified by clinical diagnosis whereby an expert is attempting to classify a patient into a disease category(ies) using limited vague knowledge consisting primarily of elicited linguistic information. This context arises in many problem domains. For example, in the early stages of many classification problems, it is necessary to decide what measurements can and should be made based on a preliminary diagnosis using primarily linguistic information. The medical context is an extreme case whereby most or even all information available is of a linguistic nature when patients are first seen, and the diagnostic problem is severe because of the possibility of confusion between different diseases in their early developmental stages. Our work adopts the ‘Computing with Words’ approach advocated by Zadeh [28] and our particular representation of language in fuzzy sets rather than that adopted by, for example, Wainer and Sandri [25] which uses network models and Zahan et al. [29] who used hierarchial approaches. Our goal is to present a new method for computing a diagnostic support index which uses vague symptom and temporal information in a clinical diagnosis context. In this paper we bring together and extend recent work from Innocent [6–8] in the Soft Computing [10,26] domain of artificial intelligence that has been applied in the area of clinical diagnosis of confusable diseases in their early stages using vague linguistic and temporal knowledge. We first consider the aetiology of diseases and show how fuzzy cognitive maps [14] can encode fuzzy causal structures and so support symptom information elicitation and diagnostic reasoning. Second, we show how fuzzy temporal reasoning and constraints can be used to support diagnosis by providing a quantitative decision support index for diseases which satisfy constraints. We then extend this approach to allow for fuzzy symptoms and present some evaluation results.
2. Fuzzy cognitive maps and medical diagnosis It is clear from the aetiology of some (but not all) diseases [18], that clinicians’ knowledge of disease-symptom relationships can be encoded as statements of causal relationships in the form: Disease A causes symptom S1 in context X Disease A causes symptom S2 in context X Disease B causes condition C in context X condition C causes symptom S2 in context X Disease D negative causes symptom S3 in context X. Fuzzy Cognitive Maps (‘‘FCM’’, [14]) can encode such causal relationships between concepts into a directed graph representation where nodes are con-
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cepts and links are causes [21]. Encoding can be more difficult than simply taking each rule above and transferring into a graph since, as Kosko points out, ‘‘A causes B’’ cannot always be considered equivalent to ‘‘A implies B’’ since the representation of negative concepts and causes becomes problematical. However, causal maps from one context may be combined appropriately with others in other contexts and this makes the representation flexible and useful. In the simplest representation, we are allowed only positive and negative causal links which represent sets of statements like those above. Kosko shows how these links can be extended to fuzzy sets in a particular context using statements like: A is a strong cause of B and A is a weak negative cause of B. He then shows how the fuzzy set operators, e.g. min–max, can be applied to particular sequences of fuzzy values to infer how such a set of concepts can affect other concepts in the FCM. In our example, a clinician might reason about what symptoms would be present and absent given that some diseases are present from a knowledge base such as: Disease A is a positive cause of symptom S1 Disease A is a weak negative cause of symptom S2 Disease B is a weak positive cause of symptom S2 Disease C is a weak positive cause of symptom not-S2 Disease C is a positive cause of condition D condition D is the negative cause of symptom S2. One use for such an approach would be to confirm (or otherwise) a provisional diagnosis made on the basis of how well the symptom set observed matches the fuzzy strength of the symptom set predicted by the FCM. This is a test of ‘‘categorical consistency’’ [25]. Another use is to remind the clinician of the possible range and variation of symptoms that are possible in the disease set information. While the latter use can be relatively easily reliably achieved, the former use requires further refinement of the fuzzy knowledge base and how it is processed. In Kosko’s example, the degree of causality was selected from the ordered set {a little, some, usually, much, very much, a lot}. In our work, we defined a set of appropriate linguistic terms and ordered them on the basis of an analysis of the terms used by a clinician during diagnosis. In eliciting knowledge from a clinician about chest infections, asthma and lung cancer, we found that many terms were used to describe the relation between the condition and the symptoms. Among these were ‘‘always excludes (never)’’, ‘‘sometimes’’, ‘‘occasionally’’, ‘‘often (usually)’’, ‘‘always’’.
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For example: ‘‘chest infection always causes coughing’’. In our knowledge elicitation, we found that some terms were related. ‘‘Always’’, for example, can be construed to be synonymous with ‘‘certainly’’. We modified causal representations by attaching a fuzzy label to the causal link to represent these elicited facts. For example: Disease A is (always) a (very) positive cause of symptom S1. However, the clinician was also providing knowledge gained from practical experience (learned implications) in these sessions in the form of ‘‘symptomrelation-disease’’. For example: ‘‘absence of cough’’ often indicates ‘‘asthma’’ in a particular context (e.g. non-smoker). These indicators are clearly important and posed a problem for our chosen representation since we required knowledge in the form ‘‘A causes B’’. We chose to adopt a weak causal representation of these indicators. For example: Disease A is (sometimes) a (little) weak-negative cause of symptom S2. Occasionally, the clinician indicated comparative information. For example: ‘‘chest pain’’ is less likely for asthma than lung cancer. Again, we have chosen to adopt a weak causal representation of this information in the form: Disease A is (frequently) a stronger negative cause of symptom S1 than Disease B. The comparative information allows some ordering of the linguistic terms and it is clear that some of the other linguistic terms (e.g. ‘‘very’’) can be treated like linguistic hedges which modify the fuzzy label of the causal link. Apart from uses of the semantic network as a simple guide through knowledge structures, the power of the representation is that it is amenable to further processing so that hidden patterns of causal states may be revealed. Typically, we can discover limit cycles or instability (e.g. chaos) as information flows through the network [22]. In our work, we have developed a small prototype to test these ideas using respiratory diseases: influenza, asthma and lung cancer. In test mode, the clinician is presented with a graphical display of the causal network and is allowed to set symptom nodes to be on or off. The FCM interpreter then cycles through the network showing the intermediate nodes states and the disease node states. In development mode, the clinician can highlight the symptoms causally associated with a particular disease. While FCM’s are useful for diagnostic purposes by clinicians using a hypothetico-deductive method, they suffer from being ‘‘shallow’’ in that the
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causal statements are simple representations of deeper knowledge. Thus, confidence in them can be low unless suitable explanations are provided. A second limitation is that FCM’s deal with essentially static descriptions of dynamic processes. This is important since we would like to know the stage of a disease rather than just diagnose the disease itself. Diseases are sometimes misdiagnosed because they are very similar to each other at different stages. We now address each of these limitations. The first limitation can be addressed in many ways; for example, FCM’s can be extended by adding embedded qualitative models which involves ‘‘deeper’’ knowledge at a detailed high granularity. Models can be added which have been learned about human physiology and is both descriptive and procedural in nature. If an explanation is required for a particular causal link such as: Diabetes is (always)a (very) positive cause of high blood sugar. then a deeper appropriate model is used. There are a range of possibilities for the depth of these models ranging from a qualitative to a highly quantitative representation. Given the training of clinicians in the hypothetico-deductive method of diagnosis, we have adopted the principle of parsimony in selecting an appropriate level that implies going from one level to the next deeper level as required for explanation or supporting confidence purposes. Examples of this approach are given in Innocent [8]. There are clear limitations to this approach since diseases usually progress through stages but the FCM and the qualitative models assume stationary values of measured symptoms that do not alter over time. While causal knowledge is useful to reason about many diseases using symptom profiles and qualitative measures alone, there are many diseases which can be easily confused in the early stages because symptom profiles and static measures are not sufficient. There are a number of approaches to deal with this; the principal ones being to make use of local experiential knowledge (e.g. relative frequencies of diseases) and global knowledge about the progression of a disease through different stages. We present our work in overcoming the important temporal limitations by using fuzzy sets and inference as one method of approaching these problems. Other approaches, such as that of Wainer and Sandri [25] are related but adopt different principals and representations (network search and parsimonious covering theory). 2.1. Extending FCM’s directly by using experience and time relationships In our shallow FCM, we have allowed representations of static information relating to the frequency of occurrence in the causal relationship. For example: Disease A is (frequently) a strong negative cause of symptom S1 Disease B is (occasionally) a strong negative cause of symptom S1.
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The following knowledge may formally be deduced from these statements: Disease A is (always) a stronger negative cause of symptom S1 than Disease B. Such a shallow deduction can be explained through the causal links and used to support differential diagnosis. However, a statistical approach may compute the relative likelihood of a disease using a method for handling uncertainty such as Baye’s rule [3] given a profile of symptoms and a set of conditional probabilities. Ross [17] describes how a fuzzy Baye’s rule can be derived and used. These methods are knowledge intensive but have been shown to be successful in a limited domain. They use information about both the presence and absence of symptoms. However, information about the duration of symptoms (here called ‘‘temporal’’ knowledge) is not used and we propose that this is important in making a differential diagnosis in the early stages of diseases. In the examples above, temporal knowledge acquired from experience could be incorporated into the representation as follows: For example: Disease A (always) causes symptom S1 in time interval t1 to t2. Disease B (occasionally) causes symptom S2 in time interval t3 to t4. The first statement means that the clinician is expecting to see a particular symptom (S1) in the interval t1 to t2 (measured form presumed onset of a disease) and the ‘‘always’’ indicates that if S1 is not observed in this time, the disease cannot be Disease A. The crisp temporal intervals t1 to t2 allow logical temporal reasoning within, for example, formal logical procedures [12]. These can be implemented in expert systems using appropriate reasoning methods [9] and uncertainty measures of rule inference strengths. This supports clinicians in using inductive logic: ‘‘given a particular disease(s) onset at time t, what are the expected symptom profiles to time t1 and do any of these match a current patient symptom profile?’’. FCM’s have been extended in a number of ways to allow for some temporal processing. Park and Kim [16] take the simplest case where the time domain is made discrete and extra nodes are introduced into the FCM which correspond to different time points. They give rules on how such nodes are to be introduced and processed. Tsadiras and Margaritis [24] extended FCM’s (EFCM) in a more general case where nodes are allowed to take continuous truth values between [0. . .1] and these values are allowed to decay with time unless reinforced by positive causal links (the converse is also true). Thus, we would be able to use the EFCM to provide symptom profiles as lists of real numbers for given sets of diseases which are developing over time. Both of these techniques
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may be combined to provide information which would enable a clinician to see the dynamics of multiple related diseases in terms of symptom changes over time. In non-linear cases, chaos might ensue which would be hard to see by any other means. In terms of diagnostic efficiency, however, clinicians do not only use a combination of deductive and inductive logic in their hypothetico-deductive approach, they also use constraint satisfaction knowledge. For example, if a symptom does not occur at a certain stage when it is expected for a specific disease, then the disease is eliminated. We therefore consider a constraint satisfaction approach using fuzzy temporal logic as a complementary approach to support diagnosis. Such an approach should also enable questions such as: ‘‘given S1 at time t1, and S2 at time t2 . . ., what is the most likely disease from a given range of possible diseases?’’. There is usually vagueness associated with temporal knowledge of the symptoms with respect to a given disease. A symptom occurrence over time for a particular disease is therefore better represented as a fuzzy set and we reason by using fuzzy inferences on these sets [23]. A successful non-temporal example of this approach is described by Kovalerchuk [13] in the general issue edited by Steimann [19] on fuzzy set theory in medicine. There is also a case for using this approach more generally to model a clinicians reasoning given by Esogbue [2] which gives some support for continuing the approach further. This is presented in the following section of this paper.
2.2. Extending FCM’s by embedding a lightweight fuzzy process which uses causation and time relationships Temporal knowledge acquired from experience or written sources could be incorporated into the FCM causal representation statements [4] as follows: Disease D (CR) causes symptom S in time interval t1 to t2. The statement means that the clinician is expecting to see symptom (S) with causal relevance (CR) in the interval t1 to t2 (measured from disease onset) given that the hypothesis that the patient has disease D is true. Temporal constraint reasoning is then possible. The following example makes this clear: For example: Influenza (always) causes symptom fever in day 1 to day 3. The ‘‘always’’ indicates that if fever is not observed in the interval of days (1,3) measured form onset (day 0), the disease cannot be influenza. Thus, decision making is within a more constrained temporal context. In our approach, the temporal knowledge base relating the class of disease, D, to the general symptom set, S, uses fuzzy representations of time
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Fig. 1. Fuzzy sets for two symptoms of influenza.
information [23] since, in medical literature, we commonly find linguistic descriptions such as: ‘‘ fever usually develops between the 3rd and 4th day of influenza and lasts for approximately 3 days’’. By using a linguistic translation, we use the causal relevance term ‘‘always’’ to be synonymous with ‘‘usually’’ and use fuzzy sets for the time information. This may be represented using fuzzy set notation as: For example: Influenza (always) causes fever at time fever_set where fever_set is a discrete fuzzy set such as those shown in Fig. 1 and designated by the symbol, QðtÞ. The interpretation of these sets is that they define the possibility distribution of the intervals of time within which a symptom may be observed. They should not be interpreted to mean that a symptom occurs to a particular degree at a particular time. Thus at this point, a symptom observation is assumed to be crisp in its duration and starting time. A graphic representation of the fuzzy sets for an example disease with example common symptoms is shown in Fig. 1. These have been constructed after interaction with medical expertise as an exercise to demonstrate our approach.
3. Fuzzy temporal inference In order to test the feasibility of using fuzzy sets for temporal reasoning, we have developed a simple diagnostic aid based on fuzzy inferences and con-
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straint satisfaction using fuzzy set representations of temporal knowledge for diseases which have a subset of temporally overlapping common symptom sets. We chose chest infection, measles and scarlet fever since the disease histories are well known and readily available in, for example, [1]. This results in about 60 different symptoms and about 90 different symptom descriptions in a data base where each symptom is related to a temporal fuzzy set description for each disease. Two examples of the discrete fuzzy set, QT1 ðtÞ, are shown in Fig. 1 where T ¼ 14. The granularity reflects the clinician’s use of ‘‘days’’ when relating to symptoms for these diseases. The first and last indices of the non-zero membership grade of the discrete fuzzy set, QT1 ðtÞ, have special significance for the computation of our support index and is denoted by low Q (e.g. day 1 in Fig. 1 for vertigo) and high Q (e.g. day 11 in Fig. 1 for vertigo) respectively. Diseases (which are not fuzzy) are said to start on day 0 (onset) although a symptom may not be observable until after this day. The basic goal of the inference engine is to establish the stage of a disease at the current day given observations of symptoms in the crisp form: ‘‘Symptom occurred from M days ago up to and including N days ago’’ where M > N >¼ 0. A value of 0 is interpreted to mean the current day. For example: ‘‘fever started 3 days ago and is still present’’ is represented as: ‘‘fever occurred during time interval (3,0)’’. We place the observed symptom into the history and estimation of the current disease by checking to see if the observed symptom duration is possible in QT1 ðtÞ and, if so, determine the lower and upper boundaries of the position of the current day in QT1 ðtÞ for each symptom expected for each disease. The lower boundary is set to be low Q. The upper boundary is the result of a simple search for the best fit of the observed symptom interval into the full range of possible intervals. Observation of another causally relevant symptom for the disease under test produces another set of boundaries and these are combined with the older estimate using fuzzy min/max operators. If this combination results in a null interval, then we assume that the temporal dependencies for that disease are not fulfilled. Otherwise, we have a revised estimate of the upper and lower boundaries of the current day for that disease. This is repeated for all observed symptoms for each disease. The outcome of this process is an estimate of the stage of each disease presented as Dj Dj Dj Dj the interval Tstage ¼ ½nulllow ; nullhigh , where nulllow is the lower bound and Dj nullhigh is the upper bound for the day of the jth disease, Dj , since onset. This information is then used together with the known disease/symptom profiles to provide clinical decision support information.
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3.1. Clinical decision support information In order to provide diagnostic information, we now use the estimated stages of each disease, the temporal sets and the causal relevance labels (always, normally, often, sometimes, never) to compute an estimate of ‘‘goodness of fit’’ called an index of support (Support) following the use of this term in the CADIAG system (referred to in Steimann and Adlassnig [20]) for each disease. In our system, support may be seen as a measure of compatibility which expresses the degree to which a particular diagnosis logically follows from the medical evidence using the FCM. We now introduce the basis for the computation of this support index. Suppose we have a set of N symptoms, Sym ¼ fS1; S2; . . . ; SN g which has been generated from causal information relating a disease, D, to the symptoms, S. The Sym set can be partitioned into two exclusive subsets; one containing the symptoms which have been observed (Sobs) and the other which contains those which have not been observed (Sunobs). Thus Sobs [ Sunobs ¼ Sym and Sobs \ Sunobs ¼ /. If there are m observed symptoms out of the expected N symptoms, the supporting evidence for D can be simply expressed as pðDÞ ¼ cardinalityðSobsÞ=cardinalityðSymÞ ¼ m=N and similarly, evidence against D as pðDÞ ¼ n=N where N ¼ m þ n. This basic representation is minimal but, as mentioned in Section 2.1, provided we have additional statistical information available on, for example, prior and conditional probabilities, we could compute the joint probability for pðDÞ which uses pðDÞ and pðDÞ using various methods such as Bayes rule. We may increase the usefulness of these methods if temporal information is taken into account. In this paper, we wish to demonstrate a method which uses fuzzy causal and symptom information and fuzzy temporal sets. We will demonstrate this method using the weakest statistical assumptions. The more powerful but knowledge intensive methods, such as Bayes, may be seen as complementary to ours in the FCM application although we would hope that future work would involve incorporating our new method into them. In order to use the causal and temporal information, we apply the idea of ‘fuzzy sets’. In fuzzy sets, objects belong to a set to some degree, k, rather than all or nothing. We first fuzzify our sets of observed and unobserved symptoms by allowing a symptom to belong to appropriate sets to some degree. Let DSup be the fuzzy set relating the observed symptoms in the set Sobs to the degree of support for disease, D. The membership grades of DSup are denoted k þ , i.e., DSup ¼ fðS; k þ ðSÞÞjS 2 Sobsg. The cardinality of DSup is m. Let DunSup be the fuzzy set relating the unobserved but expected symptoms in the set Sunobs to the degree of evidence against disease, D. The membership grades are denoted by k , i.e., DunSup ¼ fðS; k ðSÞÞjS 2 Sunobsg. The cardinality of DunSup is n.
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If we assume that the degree of support for a disease is the equivalent of the complement of the degree of evidence against a disease, we can infer membership values for the implied support for D by the unobserved symptoms in Sunobs. We generate a new fuzzy set of membership values in set notDunSup where notDunSup ¼ fðS; ~k ÞÞjS 2 Sunobsg and ~k ¼ 1 k . notDunSup is the complementary fuzzy set of DunSup, i.e., notDunSup ¼ DunSup. We then calculate the average membership values for both the sets DSup and notDunSup to represent the degree of support for a disease. (We are aware of other possible methods for this which use, for example, fuzzy operators.) The average of the n membership values of k for computing the degree of support for D from notDunSup is: averageð~k Þ ¼ 1 averageðk Þ. We now use a centre of gravity (‘‘cog’’) computation applied to the averages of the fuzzy set membership values for DSup and notDunSup. Thus, the degree of support (Sup) for D may be computed using (1): SupðDÞ ¼
m averageðk þ Þ þ n ð1 averageðk ÞÞ mþn
ð1Þ
In this expression, we assume that the presence and absence of a symptom has equal effect. However, it is more reasonable to weight the effects of symptoms according to their causal relevance. If we assume weights, wi , represent the causal relevance of symptom Si to disease D, then we modify the computation of the support for D as follows in (2) where the ‘+’ suffix indicates weights related to Sobs and the ‘)’ suffix indicates weights which are related to Sunobs: Pm þ Pn averageðk þ Þ þ ð1 averageðk ÞÞ i wi i wi Pm þ P SupðDÞ ¼ n i wi þ i wi ð2Þ The weights in our approach are determined by the causal relevance (CR) of the symptom, Si , to the disease Dj and is a linguistic term from the set {never, sometimes, often, always, normally}. We translate these terms into a numeric weight using two functions (see Fig. 2). The ‘wþ ’ function is for observed symptoms in Sobs and the w function is for the unobserved symptoms in Sunobs. The non-observation of an expected symptom, Sk , implies that the disease is not supported and thus the size of the weights, w , are determined by the ‘S’ curve shown in Fig. 2. In contrast, if an expected symptom is observed, it carries full weight of ‘1’. The tuning of these functions is a topic for future research. Presently, they are assumed to be independent of the diseases under consideration. We now address how the membership values, k, may be estimated. There are many possibilities including using expert assessments. In this paper, we are going to estimate k using temporal information on the symptom. We compute
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Fig. 2. Causal relevance to weight relation.
the area under the temporal fuzzy set, Q, using the sigma count over the temporal range of the symptom. Thus, the larger the area, the larger the membership value and the greater the support for the diagnosis. In order to compute this area, suppose the ith symptom, Si , was observed in time interval DTs ¼ ½tilow ; tihigh , and is causally related to the disease Dj . We now use DTs in the appropriate temporal fuzzy set in discrete form, QT1 ðtÞ, (which is diagrammatically represented in Fig. 1 for example) for calculating the sigma count which reflects the support for the particular disease, Dj . Thus, for a particular symptom relating to a disease, k þ is estimated using (3). X kþ ¼ QðtÞ ð3Þ DTs
For a particular disease, Dj , we now take each causally relevant symptom associated with it in turn and compute the support from the possibility distributions using sigma count and causal relevance weights. Using this definition of k þ , Fjþ is the weighted average of DSup, and is computed using all the expected symptoms which have been observed (denoted by ‘m’ - the number of symptom ‘manifestations’) and hence support Dj . Fjþ is computed using (4) Dj Dj where Dti is the corrected DTs interval ½ðnulllow þ tilow Þ; ðnullhigh þ tihigh Þ. The correction allows for the estimate of the current day of the disease using temporal constraints on symptoms previously described at the start of this section. Pm þj P j w Q ðtÞ i i i¼1 Dti Fjþ ¼ ð4Þ Pm þj i¼1 wi Similarly, Fj , the weighted average of all ð1 k Þ values in notDunSup for evidence against a particular disease Dj is given in (5) where there are ‘n’ unobserved but expected symptoms. As before, we convert the CR linguistic term into a numeric weight, wj k , for symptoms which have not been observed
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(see Fig. 2). Fj may be computed using (5) where DTk is the corrected interval Dj low j Dj high j ½minðnulllow ; Qk Þ; minðnullhigh ; Qk Þ. P Pn j DTk Qjk ðtÞ k¼1 wk ð5Þ Fj ¼ Pn j k¼1 wk As previously derived in (2), Fjþ and Fj are combined into an index, ‘‘SupportðDj Þ’’, using (6) which indicates the overall degree of support for a particular disease Dj given ‘‘m’’ observed symptoms and ‘‘n’’ unobserved symptoms expected for that disease. Pm þj Pn j Fjþ þ ð1 Fj Þ i¼1 wi k¼1 wk Pm þj Pn ð6Þ SupportðDj Þ ¼ j þ i¼1 wi k¼1 wk We now present some results to show the method in operation. 3.2. Crisp symptom experiments A series of simple tests were made with the prototype which implemented the ideas presented above for the diseases of influenza and scarlet fever. We wished to show how the support index changes as symptom information is collected in the early stages (i.e., the first few days since onset) of a disease. In all the tests, we entered symptom observations in the same order and with the same temporal details (all symptoms assumed to start 3 days ago and are still present). The symptoms and ordering was based on what would be ideally expected for the disease of influenza (see Table 1). Figs. 3 and 4 show the results of one of the tests. There is an ordered left to right interpretation in that as each symptom is considered the support index is computed, i.e., symptom Table 1 Symptom descriptions and ordering Symptom
Causal relevance to:
Index
Description
Influenza
Scarlet Fever
1 2 3 4 5 6 7 8 9 10 11
Fever Headache Vertigo Hills Back pains Muscle pains Collapse Cough Running eyes Running nose Sore throat
Always Always Always Always Always Normally Sometimes Often Often Often Often
Always Sometimes Never Sometimes Sometimes Never Never Never Never Never Normally
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Fig. 3. Support for influenza as a function of symptom knowledge accumulation.
Fig. 4. Support for scarlet fever as a function of symptom knowledge accumulation.
index 1 (refer to Table 1 to see that this means ‘fever’) is the first symptom elicited and results in about 20% support for influenza. This is followed by eliciting symptom index 2 (headache) which improves the support for influenza to about 32%. It is noticeable from Fig. 3 that closure (i.e., all symptoms for this disease whose causal relevance is ‘always’ have been observed) is reached for influenza after the first four symptoms that are always present for influenza and, in this special case, could have been used to correctly make a diagnosis. However, there is no guarantee in general that the order of the symptoms arises in this optimal manner. The negative evidence goes to zero after the input of the last symptom necessary to make a contribution to the disease support information. This indicates completeness for that disease. In contrast, the support graph for scarlet fever below shows neither closure nor completeness. Finally, it is clear from Fig. 3 that the stage of influenza narrows as particular symptoms are known. We would expect this given that we are optimal for this disease. In contrast, Fig. 4 for scarlet fever shows no narrowing of the initial guess.
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The tests enabled us to compare the degree of support for each disease for the same set of symptoms as each symptom is observed. It is assumed that the duration and start of the symptoms are all the same and started one day ago and are all present. Fig. 4 shows the results. In this case, scarlet fever would be supported more than influenza for the symptoms observed in the first stages. After closure of the influenza symptoms, influenza is gradually supported more over scarlet fever. The slope of the influenza graph is greater than for the scarlet fever graph indicating faster positive evidence accumulation in favour of influenza as expected for this optimal symptom selection and presentation order. However, it is clear that these diseases can be confused using symptoms presence and temporal durations alone (Fig. 5). Three further sets of tests were carried out with the same conditions as those just described with the purpose of finding if the method behaved as expected when the duration of the symptoms varied from 1 to 3 days. We expected that as the symptom duration increases, the better the discrimination between the diseases. The first set assumed symptoms all started 1 day ago and are still
Fig. 5. Example of optimum support aggregation for influenza.
Fig. 6. Support as a function of symptom duration.
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present ð1; 0Þ. The second set assumed symptoms all started 2 days ago and are still present ð2; 0Þ. The third set assumed symptoms all started 3 days ago and are still present ð3; 0Þ. Fig. 6 shows the degree of support for each disease as the symptom duration increases. It clearly shows that the difference between the support index values for each disease increases with symptom duration. This implies that, as time proceeds, our support index enables better discrimination. 3.3. Extending the index to include fuzzy symptom observations There are a number of possible ways to introduce the idea of fuzzy observed symptoms into our work. The simplest method is to allow the symptoms to be defined at a higher linguistic granularity, such as: ‘‘(high) fever occurred during (approximate) time interval (3,0)’’. This must then be treated as a different symptom from: ‘‘(mild) fever occurred during (exact) time interval (3,0)’’. If we use this approach, then for every disease, and for all variations of fever and duration, there must be a set of temporal possibilities ðQðtÞÞ and causal relevance’s generated as in Fig. 1. This is a major knowledge acquisition and user interface problem and we propose an alternative approach. In this paper we propose introducing the ‘‘strength’’ ðPi Þ of the ith observed symptom. Strength of a particular observed manifestation is similar to the manifestation ‘intensity’ defined by Wainer and Sandri [25] and could be used in a similar way, i.e., the FCM can include intensity information to match against strength values. We allow our strength values to be in the range ½0; 1 and use particular strength values of particular observed symptoms to modify the computation of Fjþ as given in (7). P Pm þj j w P Q ðtÞ i i i i¼1 Dti ð7Þ Fjþ ¼ Pm þj w i¼1 i The computation of the index of support then continues as before using (6). 3.4. Estimating symptom strength The strength of an observed symptom depends on a number of factors. In this example, we define strength in terms of an assumed relationship with the observability (O) of a symptom as in Fig. 7. Thus weakly observable symptoms have a low strength. The relationship between O and P is assumed linear above a particular threshold in this work. Future work can tune this relationship appropriately using feedback from actual cases.
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Fig. 7. The assumed relationship between strength and observability of a symptom.
Fig. 8. Observability as a function of the intensity and certainty of duration of a symptom.
Observability is assumed to be dependent on the intensity ðIÞ and the certainty of the duration ðDÞ of the observed symptom. We assume that both of these attributes are fuzzy in nature since they are usually reported from the recollections of patients. If crisp information from test results is known, it would have a maximum strength value. To determine the observability (and hence strength) we propose simple fuzzy relations of the form shown in Fig. 8 where I ¼ flow; medium; highg and D ¼ flow; medium; highg and O ¼ fweak; easy; strongg. Thus, for a given symptom, we require the user to provide an estimate of the intensity and duration certainty as well as the duration, e.g. (high) fever occurred during (low certainty) interval (3,0). This statement is treated as if it were:
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(easily observable) fever occurred during interval (3,0)and hence translated as: fever occurred during interval (3,0) with strength 0.75. Clearly this method requires development of the appropriate relations and it may be possible to develop type 2 rules [11] from these relations that would allow greater precision in estimating the strength of a symptom. In a recent paper, Innocent et al. [5], have used a Genetic Algorithm to successfully search for suitable type 2 relations which satisfy rules in a supervised learning situation which gives reasonable cause that such an approach is feasible here. 3.5. Fuzzy symptom experiments A series of simple tests was made with the prototype which implemented the ideas presented above for the diseases of influenza and scarlet fever since the disease histories are well known and readily available in, for example, Condon [1]. In all the tests, we entered symptom observations in the same order and with the same temporal details (all symptoms started X days ago and are still present where X varied from 1 to 3). The symptoms and ordering was based on what would be ideally expected for the disease of influenza as shown in Table 1 for that disease. Fig. 9 shows the sensitivity of the support index values for influenza and scarlet fever to full strength (high intensity, zero uncertainty) symptoms when the duration is varied from 2 to 4 days. The graphs indicate that influenza support is less sensitive than scarlet fever and that as the duration of the symptom increases, so the support for scarlet fever is reduced. Fig. 9 also
Fig. 9. Confusability as a function of symptom duration.
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shows we get single crossover points indicating when the support for a particular disease exceeds the other. For example, if the symptoms have been present at full strength for 4 days, we can predict that influenza is the most likely disease (has more support) after the presentation of just three symptoms. If the symptoms are present for 3 days, then we require four symptoms and for just 2 days then we require five symptoms. However, the values of the support index do not approach maximum for influenza until all causally relevant symptoms are present. Before these minimal numbers of symptoms are observed, we may infer incorrectly that the disease is likely to be scarlet fever, i.e., the diseases are confused with each other. However, because the symptoms are all at full strength and the same duration, the pattern is linear with one cross over point for each duration. In a further set of experiments, we fix the duration so all symptoms started 3 days ago and are still present and fix the strength of all the symptoms to be the same. Figs. 10 and 11 shows the results of computing the support index for influenza and scarlet fever with different fixed strengths of 0.5 (weakly observable), 0.75 (easily observable) and 1.0 (strongly observable). It is clear that the strength of an observed symptom has a large effect on influenza support since every observed symptom is deliberately causally relevant in this test whereas this is not true for scarlet fever. The effect is to move the crossover points from four symptoms at full strength (1.0) so that they are at five symptoms (0.75 strength) and eight symptoms (0.5 strength). This is a larger effect than for time differences and has to be validated in a real context. Fig. 13 shows perhaps a more realistic possibility for a differential diagnosis based on the support index. It is constructed from an artificially generated
Fig. 10. Support for influenza as a function of symptom strength.
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Fig. 11. Support for scarlet fever as a function of symptom strength.
Fig. 12. A symptom strength profile.
pattern of symptoms of different strength (see Fig. 12). Fig. 13 shows that up to symptom 5, we prefer scarlet fever over influenza, but from symptom 6 to 8 the reverse is true. At symptom 9, there is no difference, then for one symptom we marginally favour scarlet fever before symptom 11 tells us that we should prefer influenza. Since the observer has control over the order and how many symptoms are observed and their strength, the pattern is likely to be complicated and reinforces the notion that the method proposed here should be embedded in an expert system which provides suitable context and advice for its use.
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Fig. 13. Support for diseases from the strength profile of Fig. 12.
We are aware of the difficulties of explanation of the index and guidance in assessing symptom observability for a physician. We would expect this lightweight fuzzy process to be embedded within a larger FCM system to produce an expert system within which some explanation and guidance capability exists and which would make use of the closure information.
4. Discussion and future work There are a number of sources of uncertainty in the method we propose. A major source occurs in the knowledge base provided by clinicians in the fuzzy sets which describe the temporal possibility sets of each symptom for every disease (as in Fig. 1). We would like to explore the use of type 2 fuzzy sets [11] to deal with these uncertainties by placing a ‘footprint of uncertainty’ [15] on these sets (see Fig. 14) and to process them to take vagueness into account. The result of this work will be an estimate of support for each disease that includes upper and lower bounds. A second source of uncertainty exists in the point estimate of the strength of a symptom using type 2 fuzzy relations. We recognise that there will be a corresponding linguistic ‘footprint of uncertainty’ associated with the interpretation of the words used to estimate the duration and strength of a symptom. We will develop methods for computing bounded estimates for the strength of a symptom using type 2 fuzzy approaches. A third source of uncertainty arises from the assumed relationships between the negative weights and the causal relevance (see Fig. 2). An ‘S’ shaped function has been assumed for all diseases/symptom combinations and causal
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Fig. 14. The ‘footprint of uncertainty’ in the knowledge base.
relevance factors. However, it is likely that these parameters require specialisation to disease/symptom combinations within determined bounds for optimisation of the computational results. Given the solutions proposed for handling various sources of uncertainty, it is clear that we need to develop a theoretically sound method for combining these and presenting a comprehensible index of support for the user through the FCM interface.
5. Conclusion This paper has outlined the principles, existing work and possible future research and development of fuzzy methods using FCM’s in a clinical diagnostic application. We have successfully demonstrated our simple approach with theoretical data although we have yet to formally evaluate it in context. Clearly, our methods have a large number of parameters which need to be ‘tuned’ to enable successful usage in the clinical context. This requires a major effort in the establishment of a data base of empirical evidence and a formal evaluation process. The explanatory power afforded by the use of FCM’s and embedded lightweight fuzzy processes is clearly an advantage over nonquantitative approaches.
Acknowledgements Ms. J. Kwiat for eliciting information and developing FCM structures and software; Mr. T. Goeckler for developing prototype temporal fuzzy reasoning programs. Dr. A. Sharpe for valuable medical information in general practice.
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References [1] E.J. Condon (Ed.), Virtues Family Physician, Published by Trinity Press, 1977 Edition, London. [2] A.O. Esogbue, R.C. Elder, Fuzzy sets and the modelling of physician decision processes, Part (II): Fuzzy diagnosis decision models, Fuzzy Sets and Systems 3 (1) (1980) 1–9. [3] R.A. Frost, Introduction to knowledge based systems, Collins, Chapter 7 ‘Theories for dealing with uncertainty’, Baye’s Rule, 1986, p. 399, ISBN-0-00-383114-0. [4] J. Gamper, W. Nejdl, Abstract temporal diagnosis in medical domains, Artificial in Medicine 10 (1997) 209–234. [5] P.R. Innocent, I. Belton, I. Finlay, D.B.L., R.I. John, Type 2 Fuzzy representations of lung scans to predict pulmonary emboli, in: Conference Proceedings of NAFIPS, Vancouver, July, 2001. [6] P.R. Innocent, Fuzzy symptoms and a decision support index for the early diagnosis of confusable diseases, in: RASC Conference, Dept. of Computing Science, De Montfort University, July 2000. [7] P.R. Innocent, A lightweight fuzzy process to support early diagnosis of confusable diseases using causation and time relationships, in: Conference Proceedings of FUZZIEEE2000, vol. 2, May 2000, San Antonio, Texas, 2000, pp. 516–521. [8] P.R. Innocent, Clinical diagnosis using FCM and fuzzy temporal reasoning, Paper in Conference Proceedings of SCB99 (ICSC), Rochester, NY, 1999, pp. 490–496. [9] P. Jackson, Introduction to Expert Systems, Addison-Wesley, 1990, ISBN 0-201-1758-9. [10] J.-S.R. Jang, C_T. Sun, E. Mizutani, Neuro-Fuzzy and Soft Computing, Prentice-Hall, 1997, ISBN 0-13-287467-9. [11] R.I. John, Type-2 fuzzy sets: an appraisal of theory and applications, International Journal of Uncertainty, Fuzziness and Knowledge Based Systems 6 (6) (1998) 563–576. [12] E.T. Keravnou, Temporal reasoning in medicine, Artificial Intelligence in Medicine 8 (1996) 187–191. [13] B. Kovalerchuk, E. Triantaphyllou, J.F. Ruiz, J. Clayton, Fuzzy logic in computer aided breast cancer diagnosis: analysis of lobulation, Artificial Intelligence in Medicine 11 (1997) 75– 85. [14] B. Kosko, Fuzzy cognitive maps, International Journal of Man-Machine Studies 24 (1986) 65– 75. [15] J.M. Mendel, Uncertain/Rule Based Fuzzy Logic Systems, Prentice-Hall, 2001, ISBN 0-13040969-3. [16] K.S. Park, S.H. Kim, Fuzzy cognitive maps considering time relationships, IJ HCS 42 (1995) 157–168. [17] T.J. Ross, Fuzzy Logic with Engineering Applications, McGraw-Hill, 1995, ISBN 0-07053917-0. [18] K. Sadegh-Zadeh, Fundamentals of clinical methodology: 2. Aetiology, Artificial Intelligence in Medicine 12 (1998) 227–270. [19] F. Steimann, Fuzzy set theory in Medicine, Artificial Intelligence in Medicine 11 (1997) 1–7. [20] F. Steimann, K.-P. Adlassnig, Fuzzy Medical Diagnosis, 2000. url: citeseer.nj.nec.com/ 160037.html. [21] W.R. Taber, M.A. Siegel, Estimation of expert credibility weights using FCM, in: Proceedings of the IEEE 1st International Conference on Neural Nets, San Diego, CA, 1987. [22] R. Taber, Knowledge Processing with FCM, Expert Systems With Applications 2 (1991) 83–87. [23] H. Thiele, S. Kalenka, On fuzzy temporal logic, in: Proceedings of the 2nd International Conference on Fuzzy Systems, vol. II, San Francisco, 1993, pp. 1027–1032. [24] A.K. Tsadiras, K.G. Margaritis, Cognitive mapping and certainty neuron FCM, Information Sciences 101 (1997) 109–130.
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[25] J. Wainer, S. Sandri, Fuzzy temporal/categorical information in diagnosis, Journal of Intelligent Information Systems 13 (1–2) (1999) 9–26. [26] L.A. Zadeh, Fuzzy Logic, neural networks and soft computing, One page course announcement of CS-294-4, Spring 1993, UC Berkeley, Nov. 1992. [27] L.A. Zadeh, Fuzzy logic and its application to approximate reasoning, Information Processing 74 (1974) 591–594. [28] L.A. Zadeh, Fuzzy logic ¼ computing with words, IEEE Transactions on Fuzzy Systems 4 (2) (1996) 103–111. [29] S. Zahan, C. Michael, S. Nikolakeas, A fuzzy hierarchical approach to medical diagnosis, in: Proceedings of FUZZ-IEEE, 1997, pp. 319–324.
Computer aided fuzzy medical diagnosis P.R. Innocent *, R.I. John Centre for Computational Intelligence, Department of Computer Science, De Montfort University, The Gateway, Leicester LE1 9BH, UK
Abstract This paper describes a fuzzy approach to computer aided medical diagnosis in a clinical context. It introduces a formal view of diagnosis in clinical settings and shows the relevance and possible uses of fuzzy cognitive maps and fuzzy logic. A constraint satisfaction method is introduced which uses the temporal uncertainty in symptom durations that may occur with particular diseases. Together with fuzzy symptom descriptions, the method results in an estimate of the stage of a disease if the temporal constraints of the disease in relation to the occurrence of the symptoms are satisfied. The approach is evaluated through simulation experiments showing the effects of symptom ordering, temporal uncertainty and symptom strengths on the diagnosis efficiency. The method is effective and can be developed further using second order (Type 2) fuzzy logic to better represent uncertainty in the clinical context thus improving differential diagnosis accuracy. Ó 2004 Elsevier Inc. All rights reserved. Keywords: Fuzzy cognitive map; Fuzzy temporal reasoning; Fuzzy sets; Diagnosis
1. Introduction Fuzzy approaches to medical diagnosis have been reviewed by Steimann and Adlassnig [20] and shown to be effective in this domain. This paper considers using fuzzy methods [27] to address the specific problem of disease classification in the presence of uncertain or vague knowledge of a linguistic and
*
Corresponding author. Tel.: +44-1662577484; fax: +44-1662541891. E-mail address: [email protected] (P.R. Innocent).
0020-0255/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2004.03.003
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temporal nature. The problem context is typified by clinical diagnosis whereby an expert is attempting to classify a patient into a disease category(ies) using limited vague knowledge consisting primarily of elicited linguistic information. This context arises in many problem domains. For example, in the early stages of many classification problems, it is necessary to decide what measurements can and should be made based on a preliminary diagnosis using primarily linguistic information. The medical context is an extreme case whereby most or even all information available is of a linguistic nature when patients are first seen, and the diagnostic problem is severe because of the possibility of confusion between different diseases in their early developmental stages. Our work adopts the ‘Computing with Words’ approach advocated by Zadeh [28] and our particular representation of language in fuzzy sets rather than that adopted by, for example, Wainer and Sandri [25] which uses network models and Zahan et al. [29] who used hierarchial approaches. Our goal is to present a new method for computing a diagnostic support index which uses vague symptom and temporal information in a clinical diagnosis context. In this paper we bring together and extend recent work from Innocent [6–8] in the Soft Computing [10,26] domain of artificial intelligence that has been applied in the area of clinical diagnosis of confusable diseases in their early stages using vague linguistic and temporal knowledge. We first consider the aetiology of diseases and show how fuzzy cognitive maps [14] can encode fuzzy causal structures and so support symptom information elicitation and diagnostic reasoning. Second, we show how fuzzy temporal reasoning and constraints can be used to support diagnosis by providing a quantitative decision support index for diseases which satisfy constraints. We then extend this approach to allow for fuzzy symptoms and present some evaluation results.
2. Fuzzy cognitive maps and medical diagnosis It is clear from the aetiology of some (but not all) diseases [18], that clinicians’ knowledge of disease-symptom relationships can be encoded as statements of causal relationships in the form: Disease A causes symptom S1 in context X Disease A causes symptom S2 in context X Disease B causes condition C in context X condition C causes symptom S2 in context X Disease D negative causes symptom S3 in context X. Fuzzy Cognitive Maps (‘‘FCM’’, [14]) can encode such causal relationships between concepts into a directed graph representation where nodes are con-
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cepts and links are causes [21]. Encoding can be more difficult than simply taking each rule above and transferring into a graph since, as Kosko points out, ‘‘A causes B’’ cannot always be considered equivalent to ‘‘A implies B’’ since the representation of negative concepts and causes becomes problematical. However, causal maps from one context may be combined appropriately with others in other contexts and this makes the representation flexible and useful. In the simplest representation, we are allowed only positive and negative causal links which represent sets of statements like those above. Kosko shows how these links can be extended to fuzzy sets in a particular context using statements like: A is a strong cause of B and A is a weak negative cause of B. He then shows how the fuzzy set operators, e.g. min–max, can be applied to particular sequences of fuzzy values to infer how such a set of concepts can affect other concepts in the FCM. In our example, a clinician might reason about what symptoms would be present and absent given that some diseases are present from a knowledge base such as: Disease A is a positive cause of symptom S1 Disease A is a weak negative cause of symptom S2 Disease B is a weak positive cause of symptom S2 Disease C is a weak positive cause of symptom not-S2 Disease C is a positive cause of condition D condition D is the negative cause of symptom S2. One use for such an approach would be to confirm (or otherwise) a provisional diagnosis made on the basis of how well the symptom set observed matches the fuzzy strength of the symptom set predicted by the FCM. This is a test of ‘‘categorical consistency’’ [25]. Another use is to remind the clinician of the possible range and variation of symptoms that are possible in the disease set information. While the latter use can be relatively easily reliably achieved, the former use requires further refinement of the fuzzy knowledge base and how it is processed. In Kosko’s example, the degree of causality was selected from the ordered set {a little, some, usually, much, very much, a lot}. In our work, we defined a set of appropriate linguistic terms and ordered them on the basis of an analysis of the terms used by a clinician during diagnosis. In eliciting knowledge from a clinician about chest infections, asthma and lung cancer, we found that many terms were used to describe the relation between the condition and the symptoms. Among these were ‘‘always excludes (never)’’, ‘‘sometimes’’, ‘‘occasionally’’, ‘‘often (usually)’’, ‘‘always’’.
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For example: ‘‘chest infection always causes coughing’’. In our knowledge elicitation, we found that some terms were related. ‘‘Always’’, for example, can be construed to be synonymous with ‘‘certainly’’. We modified causal representations by attaching a fuzzy label to the causal link to represent these elicited facts. For example: Disease A is (always) a (very) positive cause of symptom S1. However, the clinician was also providing knowledge gained from practical experience (learned implications) in these sessions in the form of ‘‘symptomrelation-disease’’. For example: ‘‘absence of cough’’ often indicates ‘‘asthma’’ in a particular context (e.g. non-smoker). These indicators are clearly important and posed a problem for our chosen representation since we required knowledge in the form ‘‘A causes B’’. We chose to adopt a weak causal representation of these indicators. For example: Disease A is (sometimes) a (little) weak-negative cause of symptom S2. Occasionally, the clinician indicated comparative information. For example: ‘‘chest pain’’ is less likely for asthma than lung cancer. Again, we have chosen to adopt a weak causal representation of this information in the form: Disease A is (frequently) a stronger negative cause of symptom S1 than Disease B. The comparative information allows some ordering of the linguistic terms and it is clear that some of the other linguistic terms (e.g. ‘‘very’’) can be treated like linguistic hedges which modify the fuzzy label of the causal link. Apart from uses of the semantic network as a simple guide through knowledge structures, the power of the representation is that it is amenable to further processing so that hidden patterns of causal states may be revealed. Typically, we can discover limit cycles or instability (e.g. chaos) as information flows through the network [22]. In our work, we have developed a small prototype to test these ideas using respiratory diseases: influenza, asthma and lung cancer. In test mode, the clinician is presented with a graphical display of the causal network and is allowed to set symptom nodes to be on or off. The FCM interpreter then cycles through the network showing the intermediate nodes states and the disease node states. In development mode, the clinician can highlight the symptoms causally associated with a particular disease. While FCM’s are useful for diagnostic purposes by clinicians using a hypothetico-deductive method, they suffer from being ‘‘shallow’’ in that the
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causal statements are simple representations of deeper knowledge. Thus, confidence in them can be low unless suitable explanations are provided. A second limitation is that FCM’s deal with essentially static descriptions of dynamic processes. This is important since we would like to know the stage of a disease rather than just diagnose the disease itself. Diseases are sometimes misdiagnosed because they are very similar to each other at different stages. We now address each of these limitations. The first limitation can be addressed in many ways; for example, FCM’s can be extended by adding embedded qualitative models which involves ‘‘deeper’’ knowledge at a detailed high granularity. Models can be added which have been learned about human physiology and is both descriptive and procedural in nature. If an explanation is required for a particular causal link such as: Diabetes is (always)a (very) positive cause of high blood sugar. then a deeper appropriate model is used. There are a range of possibilities for the depth of these models ranging from a qualitative to a highly quantitative representation. Given the training of clinicians in the hypothetico-deductive method of diagnosis, we have adopted the principle of parsimony in selecting an appropriate level that implies going from one level to the next deeper level as required for explanation or supporting confidence purposes. Examples of this approach are given in Innocent [8]. There are clear limitations to this approach since diseases usually progress through stages but the FCM and the qualitative models assume stationary values of measured symptoms that do not alter over time. While causal knowledge is useful to reason about many diseases using symptom profiles and qualitative measures alone, there are many diseases which can be easily confused in the early stages because symptom profiles and static measures are not sufficient. There are a number of approaches to deal with this; the principal ones being to make use of local experiential knowledge (e.g. relative frequencies of diseases) and global knowledge about the progression of a disease through different stages. We present our work in overcoming the important temporal limitations by using fuzzy sets and inference as one method of approaching these problems. Other approaches, such as that of Wainer and Sandri [25] are related but adopt different principals and representations (network search and parsimonious covering theory). 2.1. Extending FCM’s directly by using experience and time relationships In our shallow FCM, we have allowed representations of static information relating to the frequency of occurrence in the causal relationship. For example: Disease A is (frequently) a strong negative cause of symptom S1 Disease B is (occasionally) a strong negative cause of symptom S1.
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The following knowledge may formally be deduced from these statements: Disease A is (always) a stronger negative cause of symptom S1 than Disease B. Such a shallow deduction can be explained through the causal links and used to support differential diagnosis. However, a statistical approach may compute the relative likelihood of a disease using a method for handling uncertainty such as Baye’s rule [3] given a profile of symptoms and a set of conditional probabilities. Ross [17] describes how a fuzzy Baye’s rule can be derived and used. These methods are knowledge intensive but have been shown to be successful in a limited domain. They use information about both the presence and absence of symptoms. However, information about the duration of symptoms (here called ‘‘temporal’’ knowledge) is not used and we propose that this is important in making a differential diagnosis in the early stages of diseases. In the examples above, temporal knowledge acquired from experience could be incorporated into the representation as follows: For example: Disease A (always) causes symptom S1 in time interval t1 to t2. Disease B (occasionally) causes symptom S2 in time interval t3 to t4. The first statement means that the clinician is expecting to see a particular symptom (S1) in the interval t1 to t2 (measured form presumed onset of a disease) and the ‘‘always’’ indicates that if S1 is not observed in this time, the disease cannot be Disease A. The crisp temporal intervals t1 to t2 allow logical temporal reasoning within, for example, formal logical procedures [12]. These can be implemented in expert systems using appropriate reasoning methods [9] and uncertainty measures of rule inference strengths. This supports clinicians in using inductive logic: ‘‘given a particular disease(s) onset at time t, what are the expected symptom profiles to time t1 and do any of these match a current patient symptom profile?’’. FCM’s have been extended in a number of ways to allow for some temporal processing. Park and Kim [16] take the simplest case where the time domain is made discrete and extra nodes are introduced into the FCM which correspond to different time points. They give rules on how such nodes are to be introduced and processed. Tsadiras and Margaritis [24] extended FCM’s (EFCM) in a more general case where nodes are allowed to take continuous truth values between [0. . .1] and these values are allowed to decay with time unless reinforced by positive causal links (the converse is also true). Thus, we would be able to use the EFCM to provide symptom profiles as lists of real numbers for given sets of diseases which are developing over time. Both of these techniques
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may be combined to provide information which would enable a clinician to see the dynamics of multiple related diseases in terms of symptom changes over time. In non-linear cases, chaos might ensue which would be hard to see by any other means. In terms of diagnostic efficiency, however, clinicians do not only use a combination of deductive and inductive logic in their hypothetico-deductive approach, they also use constraint satisfaction knowledge. For example, if a symptom does not occur at a certain stage when it is expected for a specific disease, then the disease is eliminated. We therefore consider a constraint satisfaction approach using fuzzy temporal logic as a complementary approach to support diagnosis. Such an approach should also enable questions such as: ‘‘given S1 at time t1, and S2 at time t2 . . ., what is the most likely disease from a given range of possible diseases?’’. There is usually vagueness associated with temporal knowledge of the symptoms with respect to a given disease. A symptom occurrence over time for a particular disease is therefore better represented as a fuzzy set and we reason by using fuzzy inferences on these sets [23]. A successful non-temporal example of this approach is described by Kovalerchuk [13] in the general issue edited by Steimann [19] on fuzzy set theory in medicine. There is also a case for using this approach more generally to model a clinicians reasoning given by Esogbue [2] which gives some support for continuing the approach further. This is presented in the following section of this paper.
2.2. Extending FCM’s by embedding a lightweight fuzzy process which uses causation and time relationships Temporal knowledge acquired from experience or written sources could be incorporated into the FCM causal representation statements [4] as follows: Disease D (CR) causes symptom S in time interval t1 to t2. The statement means that the clinician is expecting to see symptom (S) with causal relevance (CR) in the interval t1 to t2 (measured from disease onset) given that the hypothesis that the patient has disease D is true. Temporal constraint reasoning is then possible. The following example makes this clear: For example: Influenza (always) causes symptom fever in day 1 to day 3. The ‘‘always’’ indicates that if fever is not observed in the interval of days (1,3) measured form onset (day 0), the disease cannot be influenza. Thus, decision making is within a more constrained temporal context. In our approach, the temporal knowledge base relating the class of disease, D, to the general symptom set, S, uses fuzzy representations of time
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Fig. 1. Fuzzy sets for two symptoms of influenza.
information [23] since, in medical literature, we commonly find linguistic descriptions such as: ‘‘ fever usually develops between the 3rd and 4th day of influenza and lasts for approximately 3 days’’. By using a linguistic translation, we use the causal relevance term ‘‘always’’ to be synonymous with ‘‘usually’’ and use fuzzy sets for the time information. This may be represented using fuzzy set notation as: For example: Influenza (always) causes fever at time fever_set where fever_set is a discrete fuzzy set such as those shown in Fig. 1 and designated by the symbol, QðtÞ. The interpretation of these sets is that they define the possibility distribution of the intervals of time within which a symptom may be observed. They should not be interpreted to mean that a symptom occurs to a particular degree at a particular time. Thus at this point, a symptom observation is assumed to be crisp in its duration and starting time. A graphic representation of the fuzzy sets for an example disease with example common symptoms is shown in Fig. 1. These have been constructed after interaction with medical expertise as an exercise to demonstrate our approach.
3. Fuzzy temporal inference In order to test the feasibility of using fuzzy sets for temporal reasoning, we have developed a simple diagnostic aid based on fuzzy inferences and con-
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straint satisfaction using fuzzy set representations of temporal knowledge for diseases which have a subset of temporally overlapping common symptom sets. We chose chest infection, measles and scarlet fever since the disease histories are well known and readily available in, for example, [1]. This results in about 60 different symptoms and about 90 different symptom descriptions in a data base where each symptom is related to a temporal fuzzy set description for each disease. Two examples of the discrete fuzzy set, QT1 ðtÞ, are shown in Fig. 1 where T ¼ 14. The granularity reflects the clinician’s use of ‘‘days’’ when relating to symptoms for these diseases. The first and last indices of the non-zero membership grade of the discrete fuzzy set, QT1 ðtÞ, have special significance for the computation of our support index and is denoted by low Q (e.g. day 1 in Fig. 1 for vertigo) and high Q (e.g. day 11 in Fig. 1 for vertigo) respectively. Diseases (which are not fuzzy) are said to start on day 0 (onset) although a symptom may not be observable until after this day. The basic goal of the inference engine is to establish the stage of a disease at the current day given observations of symptoms in the crisp form: ‘‘Symptom occurred from M days ago up to and including N days ago’’ where M > N >¼ 0. A value of 0 is interpreted to mean the current day. For example: ‘‘fever started 3 days ago and is still present’’ is represented as: ‘‘fever occurred during time interval (3,0)’’. We place the observed symptom into the history and estimation of the current disease by checking to see if the observed symptom duration is possible in QT1 ðtÞ and, if so, determine the lower and upper boundaries of the position of the current day in QT1 ðtÞ for each symptom expected for each disease. The lower boundary is set to be low Q. The upper boundary is the result of a simple search for the best fit of the observed symptom interval into the full range of possible intervals. Observation of another causally relevant symptom for the disease under test produces another set of boundaries and these are combined with the older estimate using fuzzy min/max operators. If this combination results in a null interval, then we assume that the temporal dependencies for that disease are not fulfilled. Otherwise, we have a revised estimate of the upper and lower boundaries of the current day for that disease. This is repeated for all observed symptoms for each disease. The outcome of this process is an estimate of the stage of each disease presented as Dj Dj Dj Dj the interval Tstage ¼ ½nulllow ; nullhigh , where nulllow is the lower bound and Dj nullhigh is the upper bound for the day of the jth disease, Dj , since onset. This information is then used together with the known disease/symptom profiles to provide clinical decision support information.
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3.1. Clinical decision support information In order to provide diagnostic information, we now use the estimated stages of each disease, the temporal sets and the causal relevance labels (always, normally, often, sometimes, never) to compute an estimate of ‘‘goodness of fit’’ called an index of support (Support) following the use of this term in the CADIAG system (referred to in Steimann and Adlassnig [20]) for each disease. In our system, support may be seen as a measure of compatibility which expresses the degree to which a particular diagnosis logically follows from the medical evidence using the FCM. We now introduce the basis for the computation of this support index. Suppose we have a set of N symptoms, Sym ¼ fS1; S2; . . . ; SN g which has been generated from causal information relating a disease, D, to the symptoms, S. The Sym set can be partitioned into two exclusive subsets; one containing the symptoms which have been observed (Sobs) and the other which contains those which have not been observed (Sunobs). Thus Sobs [ Sunobs ¼ Sym and Sobs \ Sunobs ¼ /. If there are m observed symptoms out of the expected N symptoms, the supporting evidence for D can be simply expressed as pðDÞ ¼ cardinalityðSobsÞ=cardinalityðSymÞ ¼ m=N and similarly, evidence against D as pðDÞ ¼ n=N where N ¼ m þ n. This basic representation is minimal but, as mentioned in Section 2.1, provided we have additional statistical information available on, for example, prior and conditional probabilities, we could compute the joint probability for pðDÞ which uses pðDÞ and pðDÞ using various methods such as Bayes rule. We may increase the usefulness of these methods if temporal information is taken into account. In this paper, we wish to demonstrate a method which uses fuzzy causal and symptom information and fuzzy temporal sets. We will demonstrate this method using the weakest statistical assumptions. The more powerful but knowledge intensive methods, such as Bayes, may be seen as complementary to ours in the FCM application although we would hope that future work would involve incorporating our new method into them. In order to use the causal and temporal information, we apply the idea of ‘fuzzy sets’. In fuzzy sets, objects belong to a set to some degree, k, rather than all or nothing. We first fuzzify our sets of observed and unobserved symptoms by allowing a symptom to belong to appropriate sets to some degree. Let DSup be the fuzzy set relating the observed symptoms in the set Sobs to the degree of support for disease, D. The membership grades of DSup are denoted k þ , i.e., DSup ¼ fðS; k þ ðSÞÞjS 2 Sobsg. The cardinality of DSup is m. Let DunSup be the fuzzy set relating the unobserved but expected symptoms in the set Sunobs to the degree of evidence against disease, D. The membership grades are denoted by k , i.e., DunSup ¼ fðS; k ðSÞÞjS 2 Sunobsg. The cardinality of DunSup is n.
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If we assume that the degree of support for a disease is the equivalent of the complement of the degree of evidence against a disease, we can infer membership values for the implied support for D by the unobserved symptoms in Sunobs. We generate a new fuzzy set of membership values in set notDunSup where notDunSup ¼ fðS; ~k ÞÞjS 2 Sunobsg and ~k ¼ 1 k . notDunSup is the complementary fuzzy set of DunSup, i.e., notDunSup ¼ DunSup. We then calculate the average membership values for both the sets DSup and notDunSup to represent the degree of support for a disease. (We are aware of other possible methods for this which use, for example, fuzzy operators.) The average of the n membership values of k for computing the degree of support for D from notDunSup is: averageð~k Þ ¼ 1 averageðk Þ. We now use a centre of gravity (‘‘cog’’) computation applied to the averages of the fuzzy set membership values for DSup and notDunSup. Thus, the degree of support (Sup) for D may be computed using (1): SupðDÞ ¼
m averageðk þ Þ þ n ð1 averageðk ÞÞ mþn
ð1Þ
In this expression, we assume that the presence and absence of a symptom has equal effect. However, it is more reasonable to weight the effects of symptoms according to their causal relevance. If we assume weights, wi , represent the causal relevance of symptom Si to disease D, then we modify the computation of the support for D as follows in (2) where the ‘+’ suffix indicates weights related to Sobs and the ‘)’ suffix indicates weights which are related to Sunobs: Pm þ Pn averageðk þ Þ þ ð1 averageðk ÞÞ i wi i wi Pm þ P SupðDÞ ¼ n i wi þ i wi ð2Þ The weights in our approach are determined by the causal relevance (CR) of the symptom, Si , to the disease Dj and is a linguistic term from the set {never, sometimes, often, always, normally}. We translate these terms into a numeric weight using two functions (see Fig. 2). The ‘wþ ’ function is for observed symptoms in Sobs and the w function is for the unobserved symptoms in Sunobs. The non-observation of an expected symptom, Sk , implies that the disease is not supported and thus the size of the weights, w , are determined by the ‘S’ curve shown in Fig. 2. In contrast, if an expected symptom is observed, it carries full weight of ‘1’. The tuning of these functions is a topic for future research. Presently, they are assumed to be independent of the diseases under consideration. We now address how the membership values, k, may be estimated. There are many possibilities including using expert assessments. In this paper, we are going to estimate k using temporal information on the symptom. We compute
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Fig. 2. Causal relevance to weight relation.
the area under the temporal fuzzy set, Q, using the sigma count over the temporal range of the symptom. Thus, the larger the area, the larger the membership value and the greater the support for the diagnosis. In order to compute this area, suppose the ith symptom, Si , was observed in time interval DTs ¼ ½tilow ; tihigh , and is causally related to the disease Dj . We now use DTs in the appropriate temporal fuzzy set in discrete form, QT1 ðtÞ, (which is diagrammatically represented in Fig. 1 for example) for calculating the sigma count which reflects the support for the particular disease, Dj . Thus, for a particular symptom relating to a disease, k þ is estimated using (3). X kþ ¼ QðtÞ ð3Þ DTs
For a particular disease, Dj , we now take each causally relevant symptom associated with it in turn and compute the support from the possibility distributions using sigma count and causal relevance weights. Using this definition of k þ , Fjþ is the weighted average of DSup, and is computed using all the expected symptoms which have been observed (denoted by ‘m’ - the number of symptom ‘manifestations’) and hence support Dj . Fjþ is computed using (4) Dj Dj where Dti is the corrected DTs interval ½ðnulllow þ tilow Þ; ðnullhigh þ tihigh Þ. The correction allows for the estimate of the current day of the disease using temporal constraints on symptoms previously described at the start of this section. Pm þj P j w Q ðtÞ i i i¼1 Dti Fjþ ¼ ð4Þ Pm þj i¼1 wi Similarly, Fj , the weighted average of all ð1 k Þ values in notDunSup for evidence against a particular disease Dj is given in (5) where there are ‘n’ unobserved but expected symptoms. As before, we convert the CR linguistic term into a numeric weight, wj k , for symptoms which have not been observed
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(see Fig. 2). Fj may be computed using (5) where DTk is the corrected interval Dj low j Dj high j ½minðnulllow ; Qk Þ; minðnullhigh ; Qk Þ. P Pn j DTk Qjk ðtÞ k¼1 wk ð5Þ Fj ¼ Pn j k¼1 wk As previously derived in (2), Fjþ and Fj are combined into an index, ‘‘SupportðDj Þ’’, using (6) which indicates the overall degree of support for a particular disease Dj given ‘‘m’’ observed symptoms and ‘‘n’’ unobserved symptoms expected for that disease. Pm þj Pn j Fjþ þ ð1 Fj Þ i¼1 wi k¼1 wk Pm þj Pn ð6Þ SupportðDj Þ ¼ j þ i¼1 wi k¼1 wk We now present some results to show the method in operation. 3.2. Crisp symptom experiments A series of simple tests were made with the prototype which implemented the ideas presented above for the diseases of influenza and scarlet fever. We wished to show how the support index changes as symptom information is collected in the early stages (i.e., the first few days since onset) of a disease. In all the tests, we entered symptom observations in the same order and with the same temporal details (all symptoms assumed to start 3 days ago and are still present). The symptoms and ordering was based on what would be ideally expected for the disease of influenza (see Table 1). Figs. 3 and 4 show the results of one of the tests. There is an ordered left to right interpretation in that as each symptom is considered the support index is computed, i.e., symptom Table 1 Symptom descriptions and ordering Symptom
Causal relevance to:
Index
Description
Influenza
Scarlet Fever
1 2 3 4 5 6 7 8 9 10 11
Fever Headache Vertigo Hills Back pains Muscle pains Collapse Cough Running eyes Running nose Sore throat
Always Always Always Always Always Normally Sometimes Often Often Often Often
Always Sometimes Never Sometimes Sometimes Never Never Never Never Never Normally
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Fig. 3. Support for influenza as a function of symptom knowledge accumulation.
Fig. 4. Support for scarlet fever as a function of symptom knowledge accumulation.
index 1 (refer to Table 1 to see that this means ‘fever’) is the first symptom elicited and results in about 20% support for influenza. This is followed by eliciting symptom index 2 (headache) which improves the support for influenza to about 32%. It is noticeable from Fig. 3 that closure (i.e., all symptoms for this disease whose causal relevance is ‘always’ have been observed) is reached for influenza after the first four symptoms that are always present for influenza and, in this special case, could have been used to correctly make a diagnosis. However, there is no guarantee in general that the order of the symptoms arises in this optimal manner. The negative evidence goes to zero after the input of the last symptom necessary to make a contribution to the disease support information. This indicates completeness for that disease. In contrast, the support graph for scarlet fever below shows neither closure nor completeness. Finally, it is clear from Fig. 3 that the stage of influenza narrows as particular symptoms are known. We would expect this given that we are optimal for this disease. In contrast, Fig. 4 for scarlet fever shows no narrowing of the initial guess.
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The tests enabled us to compare the degree of support for each disease for the same set of symptoms as each symptom is observed. It is assumed that the duration and start of the symptoms are all the same and started one day ago and are all present. Fig. 4 shows the results. In this case, scarlet fever would be supported more than influenza for the symptoms observed in the first stages. After closure of the influenza symptoms, influenza is gradually supported more over scarlet fever. The slope of the influenza graph is greater than for the scarlet fever graph indicating faster positive evidence accumulation in favour of influenza as expected for this optimal symptom selection and presentation order. However, it is clear that these diseases can be confused using symptoms presence and temporal durations alone (Fig. 5). Three further sets of tests were carried out with the same conditions as those just described with the purpose of finding if the method behaved as expected when the duration of the symptoms varied from 1 to 3 days. We expected that as the symptom duration increases, the better the discrimination between the diseases. The first set assumed symptoms all started 1 day ago and are still
Fig. 5. Example of optimum support aggregation for influenza.
Fig. 6. Support as a function of symptom duration.
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present ð1; 0Þ. The second set assumed symptoms all started 2 days ago and are still present ð2; 0Þ. The third set assumed symptoms all started 3 days ago and are still present ð3; 0Þ. Fig. 6 shows the degree of support for each disease as the symptom duration increases. It clearly shows that the difference between the support index values for each disease increases with symptom duration. This implies that, as time proceeds, our support index enables better discrimination. 3.3. Extending the index to include fuzzy symptom observations There are a number of possible ways to introduce the idea of fuzzy observed symptoms into our work. The simplest method is to allow the symptoms to be defined at a higher linguistic granularity, such as: ‘‘(high) fever occurred during (approximate) time interval (3,0)’’. This must then be treated as a different symptom from: ‘‘(mild) fever occurred during (exact) time interval (3,0)’’. If we use this approach, then for every disease, and for all variations of fever and duration, there must be a set of temporal possibilities ðQðtÞÞ and causal relevance’s generated as in Fig. 1. This is a major knowledge acquisition and user interface problem and we propose an alternative approach. In this paper we propose introducing the ‘‘strength’’ ðPi Þ of the ith observed symptom. Strength of a particular observed manifestation is similar to the manifestation ‘intensity’ defined by Wainer and Sandri [25] and could be used in a similar way, i.e., the FCM can include intensity information to match against strength values. We allow our strength values to be in the range ½0; 1 and use particular strength values of particular observed symptoms to modify the computation of Fjþ as given in (7). P Pm þj j w P Q ðtÞ i i i i¼1 Dti ð7Þ Fjþ ¼ Pm þj w i¼1 i The computation of the index of support then continues as before using (6). 3.4. Estimating symptom strength The strength of an observed symptom depends on a number of factors. In this example, we define strength in terms of an assumed relationship with the observability (O) of a symptom as in Fig. 7. Thus weakly observable symptoms have a low strength. The relationship between O and P is assumed linear above a particular threshold in this work. Future work can tune this relationship appropriately using feedback from actual cases.
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Fig. 7. The assumed relationship between strength and observability of a symptom.
Fig. 8. Observability as a function of the intensity and certainty of duration of a symptom.
Observability is assumed to be dependent on the intensity ðIÞ and the certainty of the duration ðDÞ of the observed symptom. We assume that both of these attributes are fuzzy in nature since they are usually reported from the recollections of patients. If crisp information from test results is known, it would have a maximum strength value. To determine the observability (and hence strength) we propose simple fuzzy relations of the form shown in Fig. 8 where I ¼ flow; medium; highg and D ¼ flow; medium; highg and O ¼ fweak; easy; strongg. Thus, for a given symptom, we require the user to provide an estimate of the intensity and duration certainty as well as the duration, e.g. (high) fever occurred during (low certainty) interval (3,0). This statement is treated as if it were:
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(easily observable) fever occurred during interval (3,0)and hence translated as: fever occurred during interval (3,0) with strength 0.75. Clearly this method requires development of the appropriate relations and it may be possible to develop type 2 rules [11] from these relations that would allow greater precision in estimating the strength of a symptom. In a recent paper, Innocent et al. [5], have used a Genetic Algorithm to successfully search for suitable type 2 relations which satisfy rules in a supervised learning situation which gives reasonable cause that such an approach is feasible here. 3.5. Fuzzy symptom experiments A series of simple tests was made with the prototype which implemented the ideas presented above for the diseases of influenza and scarlet fever since the disease histories are well known and readily available in, for example, Condon [1]. In all the tests, we entered symptom observations in the same order and with the same temporal details (all symptoms started X days ago and are still present where X varied from 1 to 3). The symptoms and ordering was based on what would be ideally expected for the disease of influenza as shown in Table 1 for that disease. Fig. 9 shows the sensitivity of the support index values for influenza and scarlet fever to full strength (high intensity, zero uncertainty) symptoms when the duration is varied from 2 to 4 days. The graphs indicate that influenza support is less sensitive than scarlet fever and that as the duration of the symptom increases, so the support for scarlet fever is reduced. Fig. 9 also
Fig. 9. Confusability as a function of symptom duration.
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shows we get single crossover points indicating when the support for a particular disease exceeds the other. For example, if the symptoms have been present at full strength for 4 days, we can predict that influenza is the most likely disease (has more support) after the presentation of just three symptoms. If the symptoms are present for 3 days, then we require four symptoms and for just 2 days then we require five symptoms. However, the values of the support index do not approach maximum for influenza until all causally relevant symptoms are present. Before these minimal numbers of symptoms are observed, we may infer incorrectly that the disease is likely to be scarlet fever, i.e., the diseases are confused with each other. However, because the symptoms are all at full strength and the same duration, the pattern is linear with one cross over point for each duration. In a further set of experiments, we fix the duration so all symptoms started 3 days ago and are still present and fix the strength of all the symptoms to be the same. Figs. 10 and 11 shows the results of computing the support index for influenza and scarlet fever with different fixed strengths of 0.5 (weakly observable), 0.75 (easily observable) and 1.0 (strongly observable). It is clear that the strength of an observed symptom has a large effect on influenza support since every observed symptom is deliberately causally relevant in this test whereas this is not true for scarlet fever. The effect is to move the crossover points from four symptoms at full strength (1.0) so that they are at five symptoms (0.75 strength) and eight symptoms (0.5 strength). This is a larger effect than for time differences and has to be validated in a real context. Fig. 13 shows perhaps a more realistic possibility for a differential diagnosis based on the support index. It is constructed from an artificially generated
Fig. 10. Support for influenza as a function of symptom strength.
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Fig. 11. Support for scarlet fever as a function of symptom strength.
Fig. 12. A symptom strength profile.
pattern of symptoms of different strength (see Fig. 12). Fig. 13 shows that up to symptom 5, we prefer scarlet fever over influenza, but from symptom 6 to 8 the reverse is true. At symptom 9, there is no difference, then for one symptom we marginally favour scarlet fever before symptom 11 tells us that we should prefer influenza. Since the observer has control over the order and how many symptoms are observed and their strength, the pattern is likely to be complicated and reinforces the notion that the method proposed here should be embedded in an expert system which provides suitable context and advice for its use.
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Fig. 13. Support for diseases from the strength profile of Fig. 12.
We are aware of the difficulties of explanation of the index and guidance in assessing symptom observability for a physician. We would expect this lightweight fuzzy process to be embedded within a larger FCM system to produce an expert system within which some explanation and guidance capability exists and which would make use of the closure information.
4. Discussion and future work There are a number of sources of uncertainty in the method we propose. A major source occurs in the knowledge base provided by clinicians in the fuzzy sets which describe the temporal possibility sets of each symptom for every disease (as in Fig. 1). We would like to explore the use of type 2 fuzzy sets [11] to deal with these uncertainties by placing a ‘footprint of uncertainty’ [15] on these sets (see Fig. 14) and to process them to take vagueness into account. The result of this work will be an estimate of support for each disease that includes upper and lower bounds. A second source of uncertainty exists in the point estimate of the strength of a symptom using type 2 fuzzy relations. We recognise that there will be a corresponding linguistic ‘footprint of uncertainty’ associated with the interpretation of the words used to estimate the duration and strength of a symptom. We will develop methods for computing bounded estimates for the strength of a symptom using type 2 fuzzy approaches. A third source of uncertainty arises from the assumed relationships between the negative weights and the causal relevance (see Fig. 2). An ‘S’ shaped function has been assumed for all diseases/symptom combinations and causal
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Fig. 14. The ‘footprint of uncertainty’ in the knowledge base.
relevance factors. However, it is likely that these parameters require specialisation to disease/symptom combinations within determined bounds for optimisation of the computational results. Given the solutions proposed for handling various sources of uncertainty, it is clear that we need to develop a theoretically sound method for combining these and presenting a comprehensible index of support for the user through the FCM interface.
5. Conclusion This paper has outlined the principles, existing work and possible future research and development of fuzzy methods using FCM’s in a clinical diagnostic application. We have successfully demonstrated our simple approach with theoretical data although we have yet to formally evaluate it in context. Clearly, our methods have a large number of parameters which need to be ‘tuned’ to enable successful usage in the clinical context. This requires a major effort in the establishment of a data base of empirical evidence and a formal evaluation process. The explanatory power afforded by the use of FCM’s and embedded lightweight fuzzy processes is clearly an advantage over nonquantitative approaches.
Acknowledgements Ms. J. Kwiat for eliciting information and developing FCM structures and software; Mr. T. Goeckler for developing prototype temporal fuzzy reasoning programs. Dr. A. Sharpe for valuable medical information in general practice.
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