Marco Pota

Transforming probability distributions into membership functions of fuzzy classes: A hypothesis test approach

Abstract In fuzzy Decision Support Systems, methods are strongly required for eliciting knowledge in the form of interpretable fuzzy sets from numerical data. In medical settings, statistical data are often available, or can be obtained from rough data, typically in the form of probability distributions. Moreover, since physicians are used to think and work according to a statistical interpretation of medical knowledge, the definition of fuzzy sets starting from statistical data is thought to be able to significantly reduce the existing lack of familiarity of physicians with fuzzy set theory, with respect to the classical statistical methods. Some methods based on different assumptions transform probability distributions into fuzzy sets. However, no theoretical approach was proposed up to now, for extracting fuzzy knowledge according to a fuzzy class interpretation, which can be used for inference purposes in fuzzy rule based systems. In this paper, a method for transforming probability distributions into fuzzy sets is shown, which generalizes some existing approaches and gives them a justification. It is based on the application of statistical test of hypothesis, and the resulting fuzzy sets are interpretable as fuzzy classes. The method enables the construction of normal fuzzy sets, which can be adapted to have pseudo-triangular or pseudo-trapezoidal shape, both coherently with the corresponding probability distributions, by tuning the method parameters. The properties of this method are illustrated by applying it to simulated probability distributions and its experimental comparison with existing methods is shown. Moreover, an application is performed on a real case study involving the detection of Multiple Sclerosis lesions.

Fuzzy partitioning for clinical DSSs using statistical information transformed into possibility-based knowledge

Fuzzy logic has gained much importance for its applications in Decision Support Systems (DSSs), especially in fields like medicine, where the final user has to handle uncertain data and vague concepts, and needs an intelligible system based on clear rule bases. In medical applications, physicians are often skilled to reason using statistical approaches, since this type of information is often known, or can be extracted from data. However, since decisions have to be applied to single patients, clinical data items have to be classified in order to obtain the plausibility of conclusions, rather than their probability. Therefore, statistical information can be used, in order to define fuzzy sets and rules for constructing DSSs based on fuzzy logic. While the transformation of probability distributions is well known in literature, here, an approach is presented for transforming likelihood functions into fuzzy sets, based on possibility theory, which is actually instanced into four different new methods for knowledge representation. A comparison among different methods is shown, as well as the analysis of transformation properties and resulting fuzzy sets characteristics are considered, by using synthetic and real data. Finally, some considerations about the most suitable method to be used in the context of clinical DSSs are given.

Best Fuzzy Partitions to Build Interpretable DSSs for Classification in Medicine

Decision Support Systems (DSSs) based on fuzzy logic have gained increasing importance to help clinical decisions, since they rely on a transparent and interpretable rule base. On the other hand, probabilistic models are undoubtedly the most effective way to reach high performances. In order to join positive features of both these two approaches, this work proposes a hybrid approach, consisting in transforming the functions describing posterior probabilities, into a combination of orthogonal fuzzy sets approximating them. The resulting fuzzy partition has double hopefulness: since it approximates posterior probabilities, it is able to model information extracted from a dataset in such a form that they can be used to run predictions, and since it is a set of normal, orthogonal and convex fuzzy sets, it can be interpreted as the set of terms of a linguistic variable. As a proof of concept, the method has been applied to a real-life application pertaining the classification of Multiple Sclerosis Lesions. The results show that this method is able to construct, for each one of the variables influencing the classification, interpretable if-then rules, with classification power comparable to that of a classical Bayesian model.

Early prediction of radiotherapy-induced parotid shrinkage and toxicity based on CT radiomics and fuzzy classification

MOTIVATIONnPatients under radiotherapy for head-and-neck cancer often suffer of long-term xerostomia, and/or consistent shrinkage of parotid glands. In order to avoid these drawbacks, adaptive therapy can be planned for patients at risk, if the prediction is obtained timely, before or during the early phase of treatment. Artificial intelligence can address the problem, by learning from examples and building classification models. In particular, fuzzy logic has shown its suitability for medical applications, in order to manage uncertain data, and to build transparent rule-based classifiers. In previous works, clinical, dosimetric and image-based features were considered separately, to find different possible predictors of parotid shrinkage. On the other hand, a few works reported possible image-based predictors of xerostomia, while the combination of different types of features has been little addressed.nnnOBJECTIVEnThis paper proposes the application of a novel machine learning approach, based on both statistics and fuzzy logic, aimed at the classification of patients at risk of i) parotid gland shrinkage and ii) 12-months xerostomia. Both problems are addressed with the aim of individuating predictors and models to classify respective outcomes.nnnMETHODSnKnowledge is extracted from a real dataset of radiotherapy patients, by means of a recently developed method named Likelihood-Fuzzy Analysis, based on the representation of statistical information by fuzzy rule-based models. This method enables to manage heterogeneous variables and missing data, and to obtain interpretable fuzzy models presenting good generalization power (thus high performance), and to measure classification confidence. Numerous features are extracted to characterize patients, coming from different sources, i.e. clinical features, dosimetric parameters, and radiomics-based measures obtained by texture analysis of Computed Tomography images. A learning approach based on the composition of simple models in a more complicated one allows to consider the features separately, in order to identify predictors and models to use when only some data source is available, and obtaining more accurate results when more information can be combined.nnnRESULTSnRegarding parotid shrinkage, a number of good predictors is detected, some already known and confirmed here, and some others found here, in particular among radiomics-based features. A number of models are also designed, some using single features and others involving models composition to improve classification accuracy. In particular, the best model to be used at the initial treatment stage, and another one applicable at the half treatment stage are identified. Regarding 12-months toxicity, some possible predictors are detected, in particular among radiomics-based features. Moreover, the relation between final parotid shrinkage rate and 12-months xerostomia is evaluated. The method is compared to the naïve Bayes classifier, which reveals similar results in terms of classification accuracy and best predictors. The interpretable fuzzy rule-based models are explicitly presented, and the dependence between predictors and outcome is explained, thus furnishing in some cases helpful insights about the considered problems.nnnCONCLUSIONnThanks to the performance and interpretability of the fuzzy classification method employed, predictors of both parotid shrinkage and xerostomia are detected, and their influence on each outcome is revealed. Moreover, models for predicting parotid shrinkage at initial and half radiotherapy stages are found.

From likelihood uncertainty to fuzziness: a possibility-based approach for building clinical DSSs

For data classification, in fields like medicine, where vague concepts have to be considered, and where, at the same time, intelligible rules are required, research agrees on utility of fuzzy logic. In this ambit, if statistical information about the problem is known, or can be extracted from data, it can be used to define fuzzy sets and rules. Statistical knowledge can be acquired in terms of probability distributions or likelihood functions. Here, an approach is proposed for the transformation of likelihood functions into fuzzy sets, which considers possibility measure, and different methods arising from this approach are presented. By using real data, a comparison among different methods is performed, based on the analysis of transformation properties and resulting fuzzy sets characteristics. Finally, the best method to be used in the context of clinical decision support systems (DSSs) is chosen.

Transformation of probability distribution into fuzzy set interpretable with likelihood view

Recent research agrees on the utility of fuzzy reasoning for the development of Decision Support Systems, which help to classify clinical data. In this context, methods or techniques for representing fuzzy terms in the form of interpretable fuzzy sets obtained from numerical data are strongly required. Typically, in medical settings, statistical data are available or can be obtained from rough data, in the form of probability distributions or likelihood functions. Until now, no theoretical approach was proposed for transforming a probability distribution into a likelihood view fuzzy set. In this paper, a method is developed which generalizes some existing approaches by giving them a theoretical justification. The method enables the construction of normal fuzzy sets, which can be chosen to have a triangular or trapezoidal shape where lateral edges are adapted depending on the input probability distribution. The method was assessed through its application to a simulated normal probability distribution and to real case study pertaining the classification of Multiple Sclerosis lesions.

Designing rule-based fuzzy systems for classification in medicine

Decision Support Systems (DSSs) based on Fuzzy Logic are gaining increasing research interest in order to solve classification problems in a wide range of application fields, especially in medicine, where the chance of presenting classification results together with a clear explanation and with a measure of the associated uncertainty is highly appealing. However, designing a fuzzy system is a thorny process, requiring many steps to be accomplished, from the knowledge extraction and representation, to the inference process, until the presentation of results. Therefore, this paper proposes a general procedure for constructing rule-based fuzzy classifiers, according to the system characteristics of performance and interpretability required by the specific application, which can be used with any type of data, and is particularly useful for the medical field requirements. The proposed procedure is based on the nave Bayes approximation, therefore, the optimization of necessary parameters is performed only once and separately for each variable, thus resulting computationally fast, while later steps of the procedure enable to calculate more complicate models and choose the best one, without any further optimization. Moreover, the choices of all degrees of freedom of the design, associated with the variables constituting the model, their fuzzy partitions, the rule base construction, and the inference process, are suggested in this paper. Some of them are motivated by general considerations regarding systems applied in the medical ambit. Some other design choices depend on the dataset and on the application. In order to provide an objective way for choosing these degrees of freedom, some parameters for defining the required trade-off between performance and interpretability are proposed here. The application of the proposed procedure is guided by showing a running example, using data of the Wisconsin Breast Cancer Dataset. For different values of the trade-off parameters, optimal interpretability, or first-rate performance, or acceptable interpretability and performance are obtained, with respect to the best fuzzy systems applied on the same dataset. Finally, the procedure is applied on a number of benchmark datasets, and outstanding results are achieved in terms of performance, with respect to the best classification methods of the state-of-the-art.

Interpretability indexes for Fuzzy classification in cognitive systems

Classification systems based on Fuzzy Logic are of particular importance in the ambit of cognitive systems, due to their ability of managing uncertainty and presenting interpretable knowledge bases by emulating human cognition processes. However, the notion of interpretability is not yet exhaustively defined. In this work, the properties assessing the interpretability of a fuzzy classifier are discussed, and on this basis two indexes are proposed, which numerically evaluate readability and semantic interpretability, respectively. These indexes are calculated for a benchmark dataset, showing how they can be used to quantitatively compare systems with different characteristics.

Likelihood-fuzzy analysis of parotid gland shrinkage in radiotherapy patients.

In head-and-neck radiotherapy, an early detection of patients who will undergo parotid glands shrinkage during the treatment is of primary importance, since this condition has been found to be associated with acute toxicity. In this work, a recently proposed approach, here named Likelihood-Fuzzy Analysis, based on both statistical learning and Fuzzy Logic, is proposed to support the identification of early predictors of parotid shrinkage from Computed Tomography images acquired during radiotherapy. For this purpose, a set of textural image parameters was extracted and considered as candidate of parotid shrinkage prediction; for all these parameters and combinations of maximum three of them, a fuzzy rule base was extracted, gaining very good results in terms of accuracy, sensitivity and specificity. The performance of classification was also compared to a classical Fishers Linear Discriminant Analysis and found to provide better results. Moreover, the use of Fuzzy Logic allowed obtaining an interpretable description of the relations between textural features and the shrinkage process.

Approximation of Statistical Information with Fuzzy Models for Classification in Medicine

Fuzzy logic has gained increasing importance in Decision Support Systems (DSSs), in particular in medical field, since it allows to build a transparent and interpretable knowledge base. However, in order to obtain a general description of a system, probabilistic approaches undoubtedly offer the most significant information. Moreover, a good classifier to be used for medical scopes should be able to: (i) classify data items which are lacking of some input features; (ii) extract knowledge from incomplete datasets; (iii) consider categorical features; (iv) give responses in terms of a set of possible classes with respective degrees of plausibility. The approach here proposed pursues and achieve these objectives by approximating probabilistic information from incomplete datasets with an interpretable fuzzy system for classifying medical data. Resulting fuzzy sets can be interpreted as the terms of the involved linguistic variables, corresponding to numerical and/or categorical features, while weighted rules model probabilistic information. Rules are presented in two forms: the first is a set of one-dimensional models, which can be used if only one input feature is known; the second is a multidimensional combination of them, which can be used if more input features are known. As a proof of concept, the method has been applied for the detection of Multiple Sclerosis Lesions. The results show that this method is able to construct, for each one of the variables influencing the classification, an interpretable fuzzy partition, and very simple if-then rules. Moreover, multidimensional rule bases can be constructed, by means of which improved results are obtained, also with respect to naive Bayes classifier.