Interval-valued Observations Research Articles

Multiview Classification Through Learning From Interval-Valued Data.

The classification problem concerning crisp-valued data has been well resolved. However, interval-valued data, where all of the observations' features are described by intervals, are also a common data type in real-world scenarios. For example, the data extracted by many measuring devices are not exact numbers but intervals. In this article, we focus on a highly challenging problem called learning from interval-valued data (LIND), where we aim to learn a classifier with high performance on interval-valued observations. First, we obtain the estimation error bound of the LIND problem based on the Rademacher complexity. Then, we give the theoretical analysis to show the strengths of multiview learning on classification problems, which inspires us to construct a new algorithm called multiview interval information extraction (Mv-IIE) approach for improving classification accuracy on interval-valued data. The experiment comparisons with several baselines on both synthetic and real-world datasets illustrate the superiority of the proposed framework in handling interval-valued data. Moreover, we describe an application of Mv-IIE that we can prevent data privacy leakage by transforming crisp-valued (raw) data into interval-valued data.

Domain Adaptation With Interval-Valued Observations: Theory and Algorithms

Domain Adaptation With Interval-Valued Observations: Theory and Algorithms

Nonparametric estimation and forecasting of interval-valued time series regression models with constraints

Nowadays information technology advances allow the collecting and storage of large complex datasets in many areas. Modeling and forecasting interval-valued time series (ITS) has drawn much attention over the last two decades because interval-valued observations contain more information than point-valued observations over the same period and remove undesirable noises in high-frequency data. However, most work mainly focuses on modeling a linear univariate ITS or bivariate point process. This paper proposes nonparametric regression models for interval-valued time series with imposing constraints, e.g., monotonicity. This setting with a monotonic constraint is consistent with the existing literature, which focuses on incorporating valuable empirical information in modeling and forecasts. Two constraint estimators are developed and asymptotic properties are established. Monte Carlo simulation is conducted to show the finite sample performance. An empirical application to equity premium documents that the proposed model yields a better forecast performance than some popular models in the literature.

A Nonparametric Dual Control Algorithm of Multidimensional Objects with Interval-Valued Observations

We focus on the dual interval control problem of multidimensional objects with delay. We propose a new nonparametric algorithm. In such a case, it is not necessary to determine a parametric structure of the investigated object. Another difficulty lies in the complex nature of the decision-making field as it might not be flexible or convenient for decision-makers to exactly quantify their opinions with crisp numbers. Due to this fact, we introduce the interval-valued observations into the algorithm by means of the single-level constraint interval arithmetic. The results of computational experiments illustrate the effectiveness of the algorithm in the case of using intervals instead of crisp values.

Wavelet-based fuzzy clustering of interval time series

We investigate the fuzzy clustering of interval time series using wavelet variances and covariances; in particular, we use a fuzzy c-medoids clustering algorithm. Traditional hierarchical and non-hierarchical clustering methods lead to the identification of mutually exclusive clusters whereas fuzzy clustering methods enable the identification of overlapping clusters, implying that one or more series could belong to more than one cluster simultaneously. An interval time series (ITS) which arises when interval-valued observations are recorded over time is able to capture the variability of values within each interval at each time point. This is in contrast to single-point information available in a classical time series. Our main contribution is that by combining wavelet analysis, interval data analysis and fuzzy clustering, we are able to capture information which would otherwise have not been contemplated by the use of traditional crisp clustering methods on classical time series for which just a single value is recorded at each time point. Through simulation studies, we show that under some circumstances fuzzy c-medoids clustering performs better when applied to ITS than when it is applied to the corresponding traditional time series. Applications to exchange rates ITS and sea-level ITS show that the fuzzy clustering method reveals different and more meaningful results than when applied to associated single-point time series.

Remaining useful life prediction with imprecise observations: An interval particle filtering approach

Particle Filtering (PF) has been widely used for predicting Remaining Useful Life (RUL) of industrial products, especially for those with nonlinear degradation behavior and non-Gaussian noise. Traditional PF is a recursive Bayesian filtering framework that updates the posterior probability density function of RULs when new observation data become available. In engineering practice, due to the limited accuracy of monitoring/inspection techniques, the observation data available for PF are inevitably imprecise and often need to be treated as interval data. In this article, a novel Interval Particle Filtering (IPF) approach is proposed to effectively leverage such interval-valued observations for RUL prediction. The IPF is built on three pillars: (i) an interval contractor that mitigates the error explosion problem when the epistemic uncertainty in the interval-valued observation data is propagated; (ii) an interval intersection method for constructing the likelihood function based on the interval observation data; and (iii) an interval kernel smoothing algorithm for estimating the unknown parameters in the IPF. The developed methods are applied on the interval-valued capacity data of batteries and fatigue crack growth data of railroad tracks. The results demonstrate that the developed methods could improve the performance of RUL predictions based on interval observation data.

Forecasting crude oil price intervals and return volatility via autoregressive conditional interval models

Crude oil prices are of vital importance for market participants and governments to make energy policies and decisions. In this paper, we apply a newly proposed autoregressive conditional interval (ACI) model to forecast crude oil prices. Compared with the existing point-based forecasting models, the interval-based ACI model can capture the dynamics of oil prices in both level and range of variation in a unified framework. Rich information contained in interval-valued observations can be simultaneously utilized, thus enhancing parameter estimation efficiency and model forecasting accuracy. In forecasting the monthly West Texas Intermediate (WTI) crude oil prices, we document that the ACI models outperform the popular point-based time series models. In particular, ACI models deliver better forecasts than univariate ARMA models and the vector error correction model (VECM). The gain of ACI models is found in out-of-sample monthly price interval forecasts as well as forecasts for point-valued highs, lows, and ranges. Compared with GARCH and conditional autoregressive range (CARR) models, ACI models are also superior in volatility (conditional variance) forecasts of oil prices. A trading strategy that makes use of the monthly high and low forecasts is further developed. This trading strategy generally yields more profitable trading returns under the ACI models than the point-based VECM.

Robust aggregation of compositional and interval-valued data: The mode on the unit simplex

We consider calculation of the mode of compositional data, the data related to each other through a linear constraint. Compositional data arises in various extensions of the fuzzy sets theory (type-2 interval-valued, intuitionistic, hesitant fuzzy sets), biomedicine (relative abundance, genome sequencing, activity recognition), and data analytics (various wealth indices, interval-valued observations, traffic congestion, etc.). Mode is a pre-aggregation function in the case of single variable, used as a classical estimator robust to outliers, but its multivariate extensions face the challenges of high computational complexity and potential oversmoothing. In this work we present several novel techniques for mode estimation on the unit k-simplex representing compositional, interval-valued, and general vector-valued data. We highlight the re-weighted k-nearest neighbours algorithm based on the Choquet integral with respect to a 2-additive fuzzy measure, compare its performance against other approaches based on spatial partitioning, and illustrate its applications to aggregation of real-world interval-valued data sets.

RETRACTED: An attribute studentized fuzzy interval-valued chart based on normalized transformation

The attribute data, in general, is as important as variable data for data analysis and quality control which can offer more information and modify the error. And to change the type of multiple observations is applied to enhance the effect of control chart. In this thesis, the approach of normalized transformation would be employed to transfer the attribute data become variable data and extend the data type of interval observation. Using the transfer result construct a new control chart, attribute studentized fuzzy interval-valued (ASFIV) chart, with attribute fuzzy interval-valued data which is used to control the shift of quality characteristics for the manufacturing, electrical industry, traditional industry, education science and management fields where quality characteristics include the education score, sales satisfaction, numbers of defective product and so on. The ASFIV chart with interval observation of quality characteristic can point out the variation and degree of shift for the controlled characteristic in various industries. As the interval-valued quality characteristic, to be more specific, is generated which is faster detection the shift than the approach of traditional control chart. Moreover, the numerical studies are used to explain the significant consequent for application of the new ASFIV chart in this study. As such, the studentized fuzzy chart by attribute interval-valued observations could obtain the effective measurement outcome by the attribute data for the practical phenomena and provide related information for decision making of education policy, performance management and sustainable development of industry etc.

Reliable Fault Detection and Diagnosis of Large-Scale Nonlinear Uncertain Systems Using Interval Reduced Kernel PLS

Kernel partial least squares (KPLS) models are widely used as nonlinear data-driven methods for faults detection (FD) in industrial processes. However, KPLS models lead to irrelevant performance over long operation periods due to process parameters changes, errors and uncertainties associated with measurements. Therefore, in this paper, two different interval reduced KPLS (IRKPLS) models are developed for monitoring large scale nonlinear uncertain systems. The proposed IRKPLS models present an interval versions of the classical KPLS model. The two proposed IRKPLS models are based on the Euclidean distance between interval-valued observations as a dissimilarity metric to keep only the more relevant and informative samples. The first proposed IRKPLS technique uses the centers and ranges of intervals to estimate the interval model, while the second one is based on the upper and lower bounds of intervals for model identification. These obtained models are used to evaluate the monitored interval residuals. The aforementioned interval residuals are fed to the generalized likelihood ratio test (GLRT) chart to detect the faults. In addition to considering the uncertainties in the input-output systems, the new IRKPLS-based GLRT techniques aim to decrease the execution time when ensuring the fault detection performance. The developed IRKPLS-based GLRT approaches are evaluated across various faults of the well-known Tennessee Eastman (TE) process. The performance of the proposed IRKPLS-based GLRT methods is evaluated in terms of missed detection rate, false alarms rate, and execution time. The obtained results demonstrate the efficiency of the proposed approaches, compared with the classical interval KPLS.

Statistical inference for the Arrhenius-Weibull accelerated life testing model with imprecision based on the likelihood ratio test

In this article, we present a new imprecise statistical inference method for accelerated life testing data, where nonparametric predictive inferences at normal stress levels are integrated with a parametric Arrhenius-Weibull model. The method includes imprecision based on the likelihood ratio test which provides robustness with regard to the model assumptions. We use the likelihood ratio test to obtain an interval for the parameter of the Arrhenius link function providing imprecision into the method. The imprecision leads to observations at increased stress levels being transformed into interval-valued observations at the normal stress level, where the width of an interval is larger for observations from higher stress levels. If the model fits well, our method has relatively little imprecision. However, if the model fits poorly, it leads to more imprecision. Simulation studies are presented to investigate the performance of the proposed method.

An exponential-type kernel robust regression model for interval-valued variables

The presence of outliers is very common in regression problems and the use of robust regression methods is strongly recommended such that the bad fitted observations not affect the parameter estimates of the model. Interval-valued variables are becoming common in data analysis problems since this type of data represents either the uncertainty existing in an error measurement or the natural variability present in the data. Regarding the presence of outliers in interval-valued data sets, few robust regression methods have been proposed in literature. This paper introduces a new robust regression method for interval-valued variables that penalizes the presence of outliers in the midpoints and/or in the ranges of interval-valued observations through the use of exponential-type kernel functions. Thus, the weight given to the midpoint and range of each interval-valued observation is updated at each iteration in order to optimize a suitable objective function. The convergence of the parameter estimation algorithm is guaranteed with a low computational cost. A comparative study between the proposed method against some previous robust regression approaches for interval-valued variables is also considered. The performance of these methods are evaluated based on the bias and mean squared error (MSE) of the parameter estimates for the midpoints and ranges of the intervals, considering synthetic data sets with X-space outliers, Y-space outliers and leverage points, different sample sizes and percentage of outliers in a Monte Carlo framework. The results suggest that the proposed approach presents a competitive performance (or best), in comparison with the previous approaches, on interval-valued outliers scenarios that are comparable to those found in practices. Applications to real interval-valued data sets corroborates the usefulness of the proposed method.

Distance-based linear discriminant analysis for interval-valued data

Interval-valued observations arise in many real-life situations as either the precise representation of the objective entity or the representation of incomplete knowledge. Thus given p features observed over a sample of objects belonging to one of two possible classes, each observation can be perceived as a non-empty closed and bounded hyperrectangle on Rp. The aim of the paper is to suggest a p-dimensional classification method for random intervals when two or more classes are considered, by the generalization of Fisher’s procedure for linear discriminant analysis. The idea consists of finding a directional vector which maximizes the ratio of the dispersion between the classes and within the classes of the observed hyperrectangles. A classification rule for new observations is also provided and some simulations are carried out to compare the behavior of the proposed classification procedure with respect to other methods known from the literature. Finally, the suggested methodology are applied on a real-life situation example.

A mathematical programming approach to sample coefficient of variation with interval-valued observations

The coefficient of variation is a useful statistical measure, which has long been widely used in many areas. In real-world applications, there are situations where the observations are inexact and imprecise in nature and they have to be estimated. This paper investigates the sample coefficient of variation (CV) with uncertain observations, which are represented by interval values. Since the observations are interval-valued, the derived CV should be interval-valued as well. A pair of mathematical programs is formulated to calculate the lower bound and upper bound of the CV. Originally, the pair of mathematical programs is nonlinear fractional programming problems, which do not guarantee to have global optimum solutions. By model reduction and variable substitutions, the mathematical programs are transformed into a pair of quadratic programs. Solving the pair of quadratic programs produces the global optimum solutions and constructs the interval of the CV. The given example shows that the proposed model is indeed able to help the manufacturer select the most suitable manufacturing process with interval-valued observations.

Principal component histograms from interval-valued observations

The focus of this paper is to propose an approach to construct histogram values for the principal components of interval-valued observations. Le-Rademacher and Billard (J Comput Graph Stat 21:413---432, 2012) show that for a principal component analysis on interval-valued observations, the resulting observations in principal component space are polytopes formed by the convex hulls of linearly transformed vertices of the observed hyper-rectangles. In this paper, we propose an algorithm to translate these polytopes into histogram-valued data to provide numerical values for the principal components to be used as input in further analysis. Other existing methods of principal component analysis for interval-valued data construct the principal components, themselves, as intervals which implicitly assume that all values within an observation are uniformly distributed along the principal components axes. However, this assumption is only true in special cases where the variables in the dataset are mutually uncorrelated. Representation of the principal components as histogram values proposed herein more accurately reflects the variation in the internal structure of the observations in a principal component space. As a consequence, subsequent analyses using histogram-valued principal components as input result in improved accuracy.

Linear regression of interval-valued data based on complete information in hypercubes

Recent years have witnessed an increasing interest in interval-valued data analysis. As one of the core topics, linear regression attracts particular attention. It attempts to model the relationship between one or more explanatory variables and a response variable by fitting a linear equation to the interval-valued observations. Despite of the well-known methods such as CM, CRM and CCRM proposed in the literature, further study is still needed to build a regression model that can capture the complete information in interval-valued observations. To this end, in this paper, we propose the novel Complete Information Method (CIM) for linear regression modeling. By dividing hypercubes into informative grid data, CIM defines the inner product of interval-valued variables, and transforms the regression modeling into the computation of some inner products. Experiments on both the synthetic and real-world data sets demonstrate the merits of CIM in modeling interval-valued data, and avoiding the mathematical incoherence introduced by CM and CRM.

Symbolic Covariance Principal Component Analysis and Visualization for Interval-Valued Data

This article proposes a new approach to principal component analysis (PCA) for interval-valued data. Unlike classical observations, which are represented by single points in p-dimensional space ℜp, interval-valued observations are represented by hyper-rectangles in ℜp, and as such, have an internal structure that does not exist in classical observations. As a consequence, statistical methods for classical data must be modified to account for the structure of the hyper-rectangles before they can be applied to interval-valued data. This article extends the classical PCA method to interval-valued data by using the so-called symbolic covariance to determine the principal component (PC) space to reflect the total variation of interval-valued data. The article also provides a new approach to constructing the observations in a PC space for better visualization. This new representation of the observations reflects their true structure in the PC space. Supplementary materials for this article are available online.

CIPCA: Complete-Information-based Principal Component Analysis for interval-valued data

Principal Component Analysis (PCA) has long been used as a tool in exploratory data analysis and for making predictive models. Recent years have witnessed the continuous emergence of huge-volume data from various computerized industries, which triggers the call for more efficient and effective PCA methods. In light of this, in this paper, we work on interval-valued data and propose a new PCA method called CIPCA. CIPCA discriminates itself from various well-established methods, e.g., VPCA and CPCA, in that it can capture the complete information in interval-valued observations. Taking a hypercube view with infinitely dense points uniformly distributed within the hypercubes, CIPCA defines the inner product of interval-valued variables, and transforms the PCA modeling into the computation of some inner products in the covariance matrix. Both comparative experiments with VPCA and CPCA on the synthetic data sets and applications on real-world data demonstrate the merits of CIPCA in modeling interval-valued data. In particular, CIPCA provides an efficient and effective way for conducting PCA for large-scaled numerical data, and can find the meaningful structure information hidden in massive data.

CAUTIOUS RELIABILITY ANALYSIS OF MULTI-STATE AND CONTINUUM-STATE SYSTEMS BASED ON THE IMPRECISE DIRICHLET MODEL

Cautious reliability estimates of multi-state and continuum-state systems are studied in the paper under condition that initial data about reliability of components are given in the form of interval-valued observations, measurements or expert judgments. The interval-valued information is processed by means of a set of the imprecise Dirichlet model which can be regarded as a set of Dirichlet distributions. The developed model of reliability provides cautious reliability measures when the number of observations or measurements is rather small. It can be viewed as an extension of models based on random set theory and robust statistical models. A numerical example illustrates the proposed model and an algorithm for computing the system reliability.

Descriptive statistics for interval-valued observations in the presence of rules

While symbolic data exist in their own right, contemporary datasets can be too large to analyse using traditional statistical methodologies. Aggregation of these large datasets into sets of more managable size perforce produce datasets whose entries are symbolic data. This paper studies the derivation of basic description statistics, in particular, histograms and mean and variances plus joint histograms for interval-valued datasets when logical dependency rules are present. Algorithms for calculating these histograms are also provided.