Controlling the impact of partial supervision on the outcomes of modeling is of uttermost importance in semisupervised fuzzy clustering. Semi-Supervised Fuzzy C-Means (SSFCMeans), a specific model we consider, uses a single hyperparameter called a scaling factor α to weigh the impact of partially labeled data. This concept became widespread and was reused directly in many works building on SSFCMeans, or even applied to other fuzzy clustering algorithms such as Possibilistic C-Means. However, none of the works challenged the original interpretation of α which suggests that the impact of partial supervision is directly proportional to the scaling factor. We fill the above research gap and thoroughly analyze this relationship. We provide novel explanations of the scaling factor α in terms of the key element of fuzzy clustering - the membership values. We prove that the impact of partial supervision is a non-linear function of α. Our approach is rooted in the explainability framework, which distinguishes interpretation from an explanation and treats the latter as superior. Explaining the scaling factor leads to an explainable impact of partial supervision and enables greater control of it. Finally, built on the novel explanations, we propose a unified, analytically justified framework for selecting the value of the hyperparameter α that is based on the crossvalidation approach. We illustrate that the proposed framework enables an extensive analysis of the impact of partial supervision in SSFCMeans with a simulation experiment.

Currently, it is possible to collect a large amount of data from sensors.At the same time, data are often only partially labeled.For example, in the context of smartphonebased monitoring of mental state, there are much more data collected from smartphones than those collected from psychiatrists about the mental state.The approach presented in this paper is designed to examine if unlabeled data can improve the accuracy of classification tasks in the considered case study of classifying a patient's state.First, unlabeled data are represented by clusters membership through Fuzzy C-means algorithm which corresponds to the uncertainty of the patient's condition in this disease.Secondly, the classification is performed using two well-known algorithms, Random Forest and SVM.The obtained results indicate a minimal improvement in the quality of classification thanks to the use of membership in clusters.These results are promising due to both, the accuracy and interpretability.

Semi-supervised learning has gained great interest because of its ability to combine unlabeled data with – potentially few – labeled observations in a training process. However, in some application contexts, one can question whether all available labels are equally valid. For example, in the context of bipolar disorder (BD) remote monitoring, a common practice is to extrapolate the psychiatrist’s assessment onto some fixed time window surrounding the visit, the so-called ground truth period. In consequence, all data from this period are labeled with the same category. Such an approach may potentially result in misguided supervision affecting the model’s performance. In this paper, we consider the problem of label uncertainty, assuming that the labels are crisp, but they may be assigned to particular observations with varying confidence. We propose a novel method called Confidence Path Regularization (CPR) that incorporates this uncertainty into the fuzzy c-means semi-supervised learning. The proposed CPR approach is a novel method for automatic, data-driven handling of label uncertainty. We achieve it by estimating the confidence factor for each labeled observation. In addition, CPR allows for the exploration of potential class-specific patterns in the adjusted confidence. The proposed method is illustrated with experiments on partially labeled data about speech characteristics collected from smartphone application for BD monitoring. In this particular applied scenario, we also use additional contextual data to improve the construction of confidence paths. It is shown that the proposed CPR approach enables to reflect the varying confidence in labels as compared with the nominal approach which assigns the majority of observations to the same class associated with relevant ground truth period

In this work, we provide a statistical procedure to integrate expert preferences towards explanatory variables in stepwise forward regression. The proposed method builds on the traditional stepwise linear regression and goal programming. The procedure is validated experimentally for real-life data from various sources aiming at predicting air pollution. The practical goal is to predict the annual concentrations of two health-related air pollutants, namely PM10 (Particulate Matter that is 10 micrometers or less in diameter) and NO2 (Nitrogen Dioxide). The main finding from this work is that inclusion of expert knowledge leads to more robust and accurate predictive models. Considering the limited size of data from air pollution monitoring stations, additional expert knowledge enabled to select most meaningful explanatory variables, and as the consequence the statistical inference lead to the improved predictions. The main contribution of this work is the proposed simple but solid expert-in-the-loop stepwise forward linear regression method allowing to include expert preferences. Experiments confirm that the proposed procedure is not only more interpretable but also delivers more accurate predictions for the considered air pollutants concentrations.

Intelligent systems for the medical domain often require processing data streams that evolve over time and are only partially labeled. At the same time, the need for explanations is of utmost importance not only due to various regulations, but also to increase trust among systems’ users. In this work, an online data-driven learning method with focus on the explainability of evolving models equipped with incremental semi-supervised learning algorithms is considered. The proposed method combines: (i) the Dynamic Incremental Semi-Supervised Fuzzy C-Means (DISSFCM) algorithm to incrementally classify subsets of data; with (ii) Linguistic Summarization, which provides explanations of the classification results in terms of short sentences in a natural language. The approach has been illustrated for streaming data collected from voice calls of patients affected by Bipolar Disorder. The results show the effectiveness of the proposed method in classifying instances belonging to healthy and affective states, and explaining the approximate reasoning behind the classification of new acoustic data related to patients.

Smartphones enable to collect large data streams about phone calls that, once combined with Computational Intelligence techniques, bring great potential for improving the monitoring of patients with mental illnesses. However, the acoustic data streams recorded in uncontrolled environments are dynamically changing due to various sources of uncertainty. In addition, such acoustic data are usually difficult to interpret by psychiatrists. Within this study, we propose an approach based on Linguistic Summaries with Fuzzy Clustering (LS-FC) aiming at the development of human-consistent and easily interpretable summaries about relations between acoustic data and mental state of a patient affected by Bipolar Disorder, e.g., Most calls in the state of hypomania have low loudness compared to the state of euthymia [T = 1]. To capture the dynamics of acoustic data streams, we apply a dynamic incremental semi-supervised fuzzy clustering that synthesizes data into clusters. These clusters are represented by prototypes which are used for the construction of the membership functions describing linguistic terms e.g., low loudness, and then, linguistic summaries. The main contribution of this paper is the incorporation of information about clusters’ prototypes in the generation of linguistic summaries. The primary goal of this research is explainability. The semi-supervised learning algorithm is used mainly for deriving clusters and building improved linguistic summaries. Numerical results indicate that linguistic summaries provide intuitive and clear information about voice features in a patient’s affective state and they are consistent with clinical observation. In particular, during most calls in hypomania/mania both the quality of the patient’s voice and the dynamics of change in the spectrum signal reflected in spectral flux are low compared to euthymia. The proposed approach enables to summarize large data streams into meaningful descriptions that, although relatively simple, offer information granules that are very intuitive for clinicians and are promising to support the smartphone-based monitoring of bipolar disorder patients to inform about the potential change of mental state.

We introduce an approach called PLENARY (exPlaining bLack-box modEls in Natural lAnguage thRough fuzzY linguistic summaries), which is an explainable classifier based on a data-driven predictive model. Neural learning is exploited to derive a predictive model based on two levels of labels associated with the data. Then, model explanations are derived through the popular SHapley Additive exPlanations (SHAP) tool and conveyed in a linguistic form via fuzzy linguistic summaries. The linguistic summarization allows translating the explanations of the model outputs provided by SHAP into statements expressed in natural language. PLENARY accounts for the imprecision related to model outputs by summarizing them into simple linguistic statements and for the imprecision related to the data labeling process by including additional domain knowledge in the form of middle-layer labels. PLENARY is validated on preprocessed speech signals collected from smartphones from patients with bipolar disorder and on publicly available mental health survey data. The experiments confirm that fuzzy linguistic summarization is an effective technique to support meta-analyses of the outputs of AI models. Also, PLENARY improves explainability by aggregating low-level attributes into high-level information granules, and by incorporating vague domain knowledge into a multi-task sequential and compositional multilayer perceptron. SHAP explanations translated into fuzzy linguistic summaries significantly improve understanding of the predictive modelling process and its outputs.

Voice features from everyday phone conversations are regarded as a sensitive digital marker of mood phases in bipolar disorder. At the same time, although acoustic data collected from smartphones are relatively large, their psychiatric labelling is usually very limited, and there is still a need for intelligent and interpretable approaches to process such multiple data streams with a low percentage of labelling. Furthermore, both acoustic data and psychiatric labels are subject to several sources of uncertainty (e.g., irregular phone usage, background noises, subjectivity in psychiatric evaluation). To cope with these characteristics of an acoustic data stream, this paper introduces an intelligent qualitative and quantitative analysis based on the Dynamic Incremental Semi-Supervised Fuzzy C-Means algorithm (DISSFCM) for supporting bipolar disorder monitoring. The proposed approach is illustrated with real-life data collected from smartphones and psychiatric assessments of a bipolar disorder patient. Analysis of the dynamics of data streams basing on the cluster prototypes from fuzzy semi-supervised learning is a highly novel approach. It is also showed that the DISSFCM algorithm obtains relatively high classification performance (accuracy ranging from 0.66 to 0.76) already with 25% labelling percentage, thanks to the splitting mechanism that is adapting the number of clusters to the structure of data.

Streams of data collected from sensors are usually large and inhomogeneous in time. In this paper, we consider the case when data consist of subsegments of different lengths governed by possibly different probability distributions. The data describing consecutive subsegments are presented in the form of histograms. Next, these subsegments are grouped in larger segments whose characteristics, such as measures of location or variability, are used in further analysis. We present a possibilistic method for the aggregation of subsegment data represented by histograms into segment data represented by possibilistic distributions. The performance of the proposed method is illustrated in monitoring of bipolar disorder patients using their voice data collected from smartphones.

In this article, we propose two new resampling algorithms for the simulation of bootstrap-like samples of interval-valued fuzzy numbers (IVFNs). These methods (namely, the d-method and the s-method) reuse a primary sample (an initial set) of IVFNs to generate a secondary sample, which also consists of this type of fuzzy numbers, and simultaneously utilize existing dependencies in pairs of some characteristic points of IVFNs. During a corresponding resampling step, a nonparametric approach is used. Additionally, we apply a widely used assumption about the Gaussian kernel densities. The proposed methods in some way resemble Efron's bootstrap, but, contrary to this classical approach, they generate “not exactly the same as previous” IVFNs, so it leads to a greater diversity of the obtained secondary sample. We also numerically check the quality of the introduced methods using a few more statistically oriented approaches together with four similarity measures and three types of IVFNs.

In this paper, we propose two new nonparametric resampling methods for the simulation of bootstrap-like samples of fuzzy numbers. The generated secondary samples are based on an input set (i.e., a primary sample) consisting of left–right fuzzy numbers. The proposed approaches utilize random simulations in a way which, to some extent, resembles a bootstrap. However, contrary to the classical bootstrap approach, the proposed methods are based on alpha-cuts of fuzzy numbers, which are generated in a new nonparametric way. Therefore, these procedures give us an opportunity to create ”not exactly the same as previous” fuzzy numbers and also lead to greater diversity of the obtained output. Moreover, we check whether the introduced methods can be successfully applied in two statistical tests about the mean value of a population of fuzzy numbers.

Nonstandard probability theories have been developed for modeling random systems in complex spaces, such as, quantum systems. One of these theories, the MV-algebraic probability theory, involves the notions of state and observable, which were introduced by abstracting the properties of the Kolmogorovian probability measure and the classical random variable, as well as the notion of independence. Although within these nonstandard probability theories, many important theorems, including the strong law of large numbers (SLLN) for sequences of independent and identically distributed observables, have been considered, some practical applications require their further development. This paper is devoted to the development of the IVM-probability theory for the data described by interval-valued fuzzy random sets (IVM-events). The generalizations of Marcinkiewicz-Zygmund SLLN and Brunk-Prokhorov SLLN for independent IVM-events have been proved within this new theory. Our results open new possibilities in the theoretical analysis of imprecise random events in more complex spaces.

Interval-valued fuzzy sets were introduced in 1970s as an extension of Zadeh’s fuzzy sets. For interval-valued fuzzy events, (IV-events for short) IV-probability theory has been developed. In this paper, we prove central limit theorems for triangular arrays of IV-observables within this theory. We prove the Lindeberg CLT and the Lyapunov CLT, assuming that IV-observables are not necessary identically distributed. We also prove the Feller theorem for null arrays of IV-observables. Furthermore, we present examples of applications of the aforementioned theorems. In particular, we study the convergence in distribution of scaled sums of identically distributed IV-observables.