Functional Cluster Method Research Articles

Distance-based Clustering of Functional Data with Derivative Principal Component Analysis

Functional data analysis (FDA) is an important modern paradigm for handling infinite-dimensional data. An important task in FDA is clustering, which identifies subgroups based on the shapes of measured curves. Considering that derivatives can provide additional useful information about the shapes of functionals, we propose a novel L 2 distance between two random functions by incorporating the functions and their derivative information to determine the dissimilarity of curves under a unified scheme for dense observations. The Karhunen–Loève expansion is used to approximate the curves and their derivatives. Cluster membership prediction for each curve intends to minimize the new distances between the observed and predicted curves through subspace projection among all possible clusters. We provide consistent estimators for the curves, curve derivatives, and the proposed distance. Identifiability issues of the clustering procedure are also discussed. The utility of the proposed method is illustrated via simulation studies and applications to two real datasets. The proposed method can considerably improve cluster performance compared with existing functional clustering methods. Supplementary materials for the article are available online.

Multivariate Functional Clustering with Variable Selection and Application to Sensor Data from Engineering Systems

Multisensor data that track system operating behaviors are widely available nowadays from various engineering systems. Measurements from each sensor over time form a curve and can be viewed as functional data. Clustering of these multivariate functional curves is important for studying the operating patterns of systems. One complication in such applications is the possible presence of sensors whose data do not contain relevant information. Hence, it is desirable for the clustering method to equip with an automatic sensor selection procedure. Motivated by a real engineering application, we propose a functional data clustering method that simultaneously removes noninformative sensors and groups functional curves into clusters using informative sensors. Functional principal component analysis is used to transform multivariate functional data into a coefficient matrix for data reduction. We then model the transformed data by a Gaussian mixture distribution to perform model-based clustering with variable selection. Three types of penalties, the individual, variable, and group penalties, are considered to achieve automatic variable selection. Extensive simulations are conducted to assess the clustering and variable selection performance of the proposed methods. The application of the proposed methods to an engineering system with multiple sensors shows the promise of the methods and reveals interesting patterns in the sensor data. History: Kwok Tsui served as the senior editor for this article. Funding: The research by J. Min and Y. Hong was partially supported by the National Science Foundation [Grant CMMI-1904165] to Virginia Tech. The work by Y. Hong was partially supported by the Virginia Tech College of Science Research Equipment Fund. Data Ethics & Reproducibility Note: The original dataset is proprietary and cannot be shared. The full code to replicate the results in this paper, based on summary statistics of the original data, is available at https://github.com/jiem3/MultiFuncClustering . The code applied to a simplified version is available at https://codeocean.com/capsule/4041000/tree/v1 , which covers the data analysis and part of the simulation scenarios with a single dataset under each scenario using a fixed set of hyperparameters, for reducing computation time, and at https://doi.org/10.1287/ijds.2022.0034 .

Synchronies and asynchronies in the development of COVID-19 pandemic in Italy

the study of the possible determinants of the rise and fall of infections can be of great relevance, as was experienced during the COVID-19 pandemic. One of the methods to understand whether determinants are simultaneous or develop through contiguity between different areas is the study of the diagnostic replication index RDt among regions. to introduce the analysis of RDt variability and the subsequent application of a recently introduced functional clustering method as highly useful procedures for recognizing the presence of clusters with similar trends in epidemic curves. within the considered period, trends in regional RDt are analyzed in detail over four different time intervals. to exemplify this methodology, the study of variability in the period from the end of 2021 to the beginning of 2022 may be of interest. the variability in the regional RDt indices is assessed by means of the correlation coefficient weighted with respect to the populations of the individual regions. The clustering procedure is applied to the time series of absolute RDt values. it emerges that the periods of increasing variability in the RDt correspond to the initial growth or decrease in the number of infections, while functional clustering identifies macro-areas in which the epidemic curves have had similar trends. What caused contagions to increase seems to relate to a factor that is not specific to certain areas, with the contribution in some cases of a contagion dynamic between adjacent areas. the variability in the trend of regional diagnostic replication indices, which are calculated with only a few days delay, is a further indicator for the early detection of major changes in the trend of epidemic curves. The clustering of epidemic index curves may be useful to determine whether determinants act simultaneously or by contiguity between adjacent areas.

Functional Data Clustering Method Based on Shape Information and Functional Mahalanobis Distance

Function-type data can provide insights into the internal structure of the data and facilitate the extraction of data features from the perspective of interactive derivative functions. The paper proposes a nonparametric clustering method for function-based data that incorporates first-order and second-order derivative function information into the Marginal distance for clustering function-based data. The method is based on the traditional K-means algorithm and is designed to cluster function-based data that contains rich shape information. The paper evaluated the performance of the algorithm of the text by comparing its purity and adjusted Rand coefficients against six other clustering algorithms on three different datasets. The results show that the algorithm of the text outperforms the other algorithms, demonstrating its outstanding performance, wide applicability, and practical significance in solving real-world problems.

Functional Data Clustering Via Functional Mahalanobis Distance

As an exploratory data analysis method, functional data clustering aims to identify the underlying features of the observed data. In this context, this paper proposes a functional data clustering method based on functional Mahalanobis distance. As a distance-based non-parametric clustering model, the proposed method can effectively avoid the disadvantages of generative models and has excellent properties of decoupling and dimension standardization. Compared with other functional data clustering models, this method has lower computational complexity. In addition, the proposed method can be applied to any distance-based multivariate clustering method, thus generalizing it to the case of functional data. In practical data analysis, this paper compares the performance of this method with some other functional clustering methods, using k-means clustering as an example, and finds that it has better performance in terms of purity, adjusted Rand index, and computational speed. Finally, the idea of using Mahalanobis distance for functional data distance measurement can also be extended to construct kernel functions for measuring similarity between functional data samples, thus developing non-linear functional data analysis methods based on reproducing kernel theory.

Functional data analysis to characterize disease patterns in frequent longitudinal data: application to bacterial vaginal microbiota patterns using weekly Nugent scores and identification of pattern-specific risk factors

BackgroundTechnology advancement has allowed more frequent monitoring of biomarkers. The resulting data structure entails more frequent follow-ups compared to traditional longitudinal studies where the number of follow-up is often small. Such data allow explorations of the role of intra-person variability in understanding disease etiology and characterizing disease processes. A specific example was to characterize pathogenesis of bacterial vaginosis (BV) using weekly vaginal microbiota Nugent assay scores collected over 2 years in post-menarcheeal women from Rakai, Uganda, and to identify risk factors for each vaginal microbiota pattern to inform epidemiological and etiological understanding of the pathogenesis of BV.MethodsWe use a fully data-driven approach to characterize the longitudinal patters of vaginal microbiota by considering the densely sampled Nugent scores to be random functions over time and performing dimension reduction by functional principal components. Extending a current functional data clustering method, we use a hierarchical functional clustering framework considering multiple data features to help identify clinically meaningful patterns of vaginal microbiota fluctuations. Additionally, multinomial logistic regression was used to identify risk factors for each vaginal microbiota pattern to inform epidemiological and etiological understanding of the pathogenesis of BV.ResultsUsing weekly Nugent scores over 2 years of 211 sexually active and post-menarcheal women in Rakai, four patterns of vaginal microbiota variation were identified: persistent with a BV state (high Nugent scores), persistent with normal ranged Nugent scores, large fluctuation of Nugent scores which however are predominantly in the BV state; large fluctuation of Nugent scores but predominantly the scores are in the normal state. Higher Nugent score at the start of an interval, younger age group of less than 20 years, unprotected source for bathing water, a woman’s partner’s being not circumcised, use of injectable/Norplant hormonal contraceptives for family planning were associated with higher odds of persistent BV in women.ConclusionThe hierarchical functional data clustering method can be used for fully data driven unsupervised clustering of densely sampled longitudinal data to identify clinically informative clusters and risk-factors associated with each cluster.

Interval-valued functional clustering based on the improved Euclidean distance with application to air quality index

Interval function clustering is a statistical method used to classify functional data based on interval number similarity measurements. However, existing similarity measurements focus on measuring the similarity of the curves in terms of numerical distance. This ignores the changing characteristics of the curve shape, which may lead to unreasonable clustering results when clustering interval-valued functional data. To address this issue, an improved Euclidean-distance-based interval-valued functional clustering method is proposed in this study. By deducing a specific calculation formula for the Euclidean distance of the interval function under the basis function and derivative information, the absolute difference in the numerical value was reflected based on the distance of the basis function, and the curve shape difference was reflected based on the distance of the derivative function information. Furthermore, a similarity measurement method combining numerical distance and curve shape was constructed to reflect the change characteristics of the functional data more completely and improve the K-means clustering process. Finally, the air quality index of different cities is taken as an example to cluster, and the differences and variation characteristics of different types of air quality are analyzed, which verifies the effectiveness of the proposed method.

Review of Clustering Methods for Functional Data

Functional data clustering is to identify heterogeneous morphological patterns in the continuous functions underlying the discrete measurements/observations. Application of functional data clustering has appeared in many publications across various fields of sciences, including but not limited to biology, (bio)chemistry, engineering, environmental science, medical science, psychology, social science, and so on. The phenomenal growth of the application of functional data clustering indicates the urgent need for a systematic approach to develop efficient clustering methods and scalable algorithmic implementations. On the other hand, there is abundant literature on the cluster analysis of time series, trajectory data, spatio-temporal data, and so on, which are all related to functional data. Therefore, an overarching structure of existing functional data clustering methods will enable the cross-pollination of ideas across various research fields. We here conduct a comprehensive review of original clustering methods for functional data. We propose a systematic taxonomy that explores the connections and differences among the existing functional data clustering methods and relates them to the conventional multivariate clustering methods. The structure of the taxonomy is built on three main attributes of a functional data clustering method and therefore is more reliable than existing categorizations. The review aims to bridge the gap between the functional data analysis community and the clustering community and to generate new principles for functional data clustering.

Analysis of the Influencing Factors of Crystalline Blockages in Mountain Tunnel Drainage Systems Based on Decision Analysis Methods

Crystalline blockages in mountain tunnel drainage systems are becoming a common environmental problem. Considering the lack of research on the influence degree of the factors affecting crystalline blockages in mountain tunnel drainage systems, this paper classified and evaluated the importance of relevant factors through decision analysis methods. Our purpose is to provide a comprehensive understanding of the primary factors causing crystalline blockages in tunnels. The influence factors are selected and categorized through a literature review, and then the influence factors are screened twice by the expert scoring method and the gray-whitening weighted function clustering method to eliminate the less important influence factors. Finally, the influence factors are evaluated systematically according to the hierarchical analysis method. The results indicate that the factors affecting the crystalline blockage of the drainage system can be divided into five categories: hydrology, geology, shotcrete materials, drainage facilities, and the cave environment. Among these factors, shotcrete materials are the key factors affecting the problem of crystalline blockages. Specifically, the density of shotcrete and the content of calcium in cement have a significant impact on the crystalline blockages, which have the following comprehensive weights: 0.221 and 0.152, respectively. Since the shotcrete materials are human controllable factors, they can be taken as the key research objects to solve the problem of crystalline blockages.

Quantile-based Clustering for Functional Data via Modelling Functional Principal Components Scores

Clustering tasks of functional data arise naturally in many applications, and efficient classification approaches are needed to find groups. The current paper combines the quantile-based model with the principal component analysis of functional data (FPCA). In our proposed procedures, the projection of functional data is first approximated based on (rotated) FPCA. The quantile-based model is then implemented on the space of rotated scores to identify the potential features of underlying clusters. The proposed method overcomes the limitation of using direct basis function expansion such as Fourier, B-spline, or linear fitting, besides representing a nonparametric clustering alternative based on a quantile approach. The proposed method’s performance has been evaluated in a comprehensive simulation study and afterward compared with existing functional and non-functional clustering methods. The simulation study results showed that the proposed method performs well in terms of correct classification rate and computing time average. Finally, a real-world application concerning temporal wind speed data has been analyzed to demonstrate the proposed method’s advantages and usefulness.

Functional clustering on a sphere via Riemannian functional principal components

We propose the functional clustering algorithm applicable to the sphere‐valued random curves, called ‐centres Riemannian functional clustering (kCRFC). It is based on Riemannian functional principal component scores and ‐centres functional clustering algorithm; thus, we can obtain accurate clustering results by reflecting the geometry of the sphere. Our method shows better clustering performances than existing multivariate functional clustering methods in various simulation settings. We apply the proposed method to the migration trajectories of Egyptian Vultures in the Middle East and East Africa and fruit fly behaviours, containing the curves lied on two‐dimensional and three‐dimensional sphere, respectively.

Using Functional Clustering to Diagnose Person Misfit

The purpose of this study is to demonstrate clustering methods within a functional data analysis (FDA) framework for identifying subgroups of individuals that may be exhibiting categories of misfit. Person response functions (PRFs) estimated within a FDA framework (FDA-PRFs) provide graphical displays that can aid in the identification of persons that have responded unexpectedly to items comprising an achievement test. Typical person fit statistics are also useful for detecting unexpected response patterns, but they do not provide insight into the underlying behaviors responsible for those responses. However, different responding behaviors tend to produce FDA-PRFs of different shapes, and may provide additional information regarding the reasons for misfit. Functional clustering methods are useful for categorizing respondents into subgroups based on the shapes of their FDA-PRFs. In this study, a small simulation illustrates the potential of clustering FDA-PRFs for identifying persons displaying common types of responding behaviors. The methodology is also applied to data from a high school biology assessment and a mathematics achievement test. Clustering FDA-PRFs offers a promising methodology for operationalizing person fit evaluations in large-scale assessments, and may be a valuable step in person fit assessment when used in conjunction with traditional indices of psychometric quality.

Conditional functional clustering for longitudinal data with heterogeneous nonlinear patterns

In studies of cognitive aging, it is crucial to distinguish subtypes of longitudinal cognition change while accounting for the effects of given covariates. The longitudinal cognition trajectories and the covariate effects can both be nonlinear with heterogeneous shapes that do not follow a simple parametric form, where flexible functional methods are preferred. However, most functional clustering methods for longitudinal data do not allow controlling for the possible functional effects of covariates. Although traditional mixture-of-experts methods can include covariates and be extended to the functional setting, using nonlinear basis functions, satisfactory parsimonious functional methods required for robust functional coefficient estimation and clustering are still lacking. In this paper we propose a novel latent class functional mixed-effects model in which we assume the covariates have fixed functional effects, and the random curves follow a mixture of Gaussian processes that facilitates a model-based conditional clustering. A transformed penalized B-spline approach is employed for parsimonious modeling and robust model estimation. We propose a new iterative-REML method to choose the penalty parameters in heterogeneous data. The new method is applied to the latest data from the Religious Orders Study and Rush Memory and Aging Project, and four novel subtypes of cognitive changes are identified.

Interval-valued functional clustering based on the Wasserstein distance with application to stock data

Interval-valued functional clustering is a novel approach for functional data analysis where each observation is represented by an interval. Existing interval-valued functional clustering methods typically rely on the interval-valued functional Hausdorff distance, which considers the upper and lower bounds of interval-valued functions but ignores the distribution of the original interval-valued data. This study proposes an interval-valued functional clustering method based on the Wasserstein distance to address this shortcoming. Taking into account the data distribution (i.e. the distribution’s centre and spread), this distance fully utilises the underlying data patterns. Furthermore, this study proposes a practical application of the interval-valued functional entropy method when multiple variables are used to comprehensively describe a phenomenon. Finally, an empirical example from the stock market demonstrates the proposed method’s effectiveness and superiority.

Clustering-based simultaneous forecasting of life expectancy time series through Long-Short Term Memory Neural Networks

In this paper, we apply a functional clustering method to the multivariate time series of life expectancy at birth of the female populations collected in the Human Mortality Database. We reconstruct the functional form of life expectancy from the available discrete observations and derive the curves through non-parametric smoothing. Once the clustering is realized, we perform the life expectancy simultaneous forecasting of the countries inside each cluster implementing a multivariate Long-Short Term Memory neural network. Although functional clustering has already been used in the actuarial literature, in this work it is applied for the first time to the study of life expectancy. The originality of the work also lies in the combination of a functional clustering approach with simultaneous forecasting obtained through the Long-Short Term Memory. We point out that such a combination provides a more informative outlook of the evolution of life expectancy, allowing us to depict country-specific longevity consistently with acknowledged mortality profiles. The results show that the evolution of developed countries follows a homogeneous pattern and supports the persisting homogeneity within the high longevity cluster over time. Moreover, we find a remarkable cross-country heterogeneity in the medium-low longevity cluster. By exploiting the cluster information, we improve the simultaneous forecasting of life expectancy time series using Long Short Term Memory neural networks and compare the error forecast of our approach with those of the classical VAR model, showing a better performance of the former when considering the cluster average errors.

Functional clustering methods for resistance spot welding process data in the automotive industry

AbstractIn the automotive industry, quality assessment of resistance spot welding (RSW) joints of metal sheets is typically based on costly and lengthy offline tests, which are unfeasible in the full‐scale production on a large scale. However, the massive industrial digitalization triggered by the Industry 4.0 framework makes online measurements of RSW process parameters available for every joint produced. Among these, the so‐called dynamic resistance curve (DRC) is recognized as the full technological signature of the spot welds. Motivated by this context, this article intends to show the potentiality and practical applicability of clustering methods to data represented by curves, and, in general, to functional data. In this way, the task of separating DRCs into homogeneous groups pertaining to spot welds with common mechanical and metallurgical properties can be performed without the need for arbitrary and problem‐specific feature extraction. We provide a hands‐on overview of the most promising functional clustering methods, and apply them to DRCs collected during RSW lab tests at Centro Ricerche Fiat. The identified groups of DRCs emerge to be strictly linked with the wear status of the electrodes, which, in turn, is conjectured to impact the RSW joint final quality. The analysis code, developed in the software environment , accompanied by an essential tutorial and the ICOSAF project data set containing DRC measurements, are openly available online at https://github.com/unina‐sfere/funclustRSW/.

Ambulatory blood pressure profile and stroke recurrence

ObjectivesTo establish a new ambulatory blood pressure (ABP) parameter (24-hour ABP profile) and evaluated its performance on stroke outcome in ischaemic stroke (IS) or transient ischaemic attack (TIA) patients.MethodsThe prospective...

Functional clustering methods for longitudinal data with application to electronic health records.

We develop a method to estimate subject-level trajectory functions from longitudinal data. The approach can be used for patient phenotyping, feature extraction, or, as in our motivating example, outcome identification, which refers to the process of identifying disease status through patient laboratory tests rather than through diagnosis codes or prescription information. We model the joint distribution of a continuous longitudinal outcome and baseline covariates using an enriched Dirichlet process prior. This joint model decomposes into (local) semiparametric linear mixed models for the outcome given the covariates and simple (local) marginals for the covariates. The nonparametric enriched Dirichlet process prior is placed on the regression and spline coefficients, the error variance, and the parameters governing the predictor space. This leads to clustering of patients based on their outcomes and covariates. We predict the outcome at unobserved time points for subjects with data at other time points as well as for new subjects with only baseline covariates. We find improved prediction over mixed models with Dirichlet process priors when there are a large number of covariates. Our method is demonstrated with electronic health records consisting of initiators of second-generation antipsychotic medications, which are known to increase the risk of diabetes. We use our model to predict laboratory values indicative of diabetes for each individual and assess incidence of suspected diabetes from the predicted dataset.

Green efficiency performance analysis of the logistics industry in China: based on a kind of machine learning methods

This paper aims to analyze the green efficiency performance of the logistics industry in China’s 30 provinces from 2008 to 2017. We first evaluate the green efficiency of the logistics industry through the non-directional distance function method. Then, we use the functional clustering method funHDDC, which is one of the popular machine learning methods, to divide 30 provinces into 4 clusters and analyze the similarities and differences in green efficiency performance patterns among different groups. Further, we explore the driving factors of dynamic changes in green efficiency through the decomposition method. The main conclusions of this paper are as follows: (1) in general, the level of green efficiency is closely related to the geographical location. From the clustering results, we can find that most of the eastern regions belong to the cluster with higher green efficiency, while most of the western regions belong to the cluster with lower green efficiency. However, the green efficiency performance in several regions with high economic levels, such as Beijing and Shanghai, is not satisfactory. (2) Based on the analysis of decomposition results, the innovation effect of China’s logistics industry is the most obvious, but the efficiency change still needs to be improved, and technical leadership should be strengthened. Based on these conclusions, we further propose some policy recommendations for the green development of the logistics industry in China.

A generalization of functional clustering for discrete multivariate longitudinal data.

This paper presents a new model-based generalized functional clustering method for discrete longitudinal data, such as multivariate binomial and Poisson distributed data. For this purpose, we propose a multivariate functional principal component analysis (MFPCA)-based clustering procedure for a latent multivariate Gaussian process instead of the original functional data directly. The main contribution of this study is two-fold: modeling of discrete longitudinal data with the latent multivariate Gaussian process and developing of a clustering algorithm based on MFPCA coupled with the latent multivariate Gaussian process. Numerical experiments, including real data analysis and a simulation study, demonstrate the promising empirical properties of the proposed approach.