Symbolic Interval-valued Data Research Articles

The Unfolding Analysis for Symbolic Objects Based on the Example of the External Car Advertisement Evaluation

Aim: Multidimensional unfolding allows representing both columns (e.g. products, services) and rows (e.g. customers) of the preference matrix on the same low-dimensional map (usually it’s a two or three-dimensional map). The main aim of the paper was to propose how to perform unfolding analysis for symbolic objects. Methodology: The paper describes the possible ways of performing unfolding analysis for symbolic interval-valued data. The external unfolding is described in the details and used in the empirical part of the paper. The data (preferences and dissimilarities) were gathered by using the incomplete method of triads. Results: The empirical part presents an application for unfolding symbolic data to evaluate customers’ preferences, where car advertisements are used as the example. The results presented on a two-dimensional perceptual map allowed to discover seven groups of respondents with different preferences; most of them prefer Skoda, Audi, Volkswagen, and Honda advertisements to Toyota and Volvo. Implications and recommendations: The proposed external approach for symbolic data allows to represent objects as rectangles (on two-dimensional map) or cuboids (in the case of three dimensions). The respondents are represented as points. Further work should focus on creating an algorithm that allows for the presentation of both symbolic objects and preferences expressed by respondents in the form of rectangles or cuboids. Originality/Value: The paper presents an innovative and previously unpresented external unfolding for symbolic data. Besides that it presents how other unfolding approaches could be adapted for symbolic data.

Theoretical derivation of interval principal component analysis

Principal Component Analysis is a well-known method that can be used for dimensionality reduction, a useful technique in the Big Data era. There have been a series of proposed adaptations of the Principal Component Analysis method for interval-valued symbolic data, all of which have the downside of having intermediate steps that deal with conventional data. In this work, we put forward an interval Principal Component Analysis that only deals with symbolic data by developing a theoretical framework that allows for the definition of symbolic principal components. This framework provides the mathematical tools needed to use the symbolic principal components to transform the original data in a way that is mathematically coherent with the remainder of the framework and defines the principal components as solutions to maximisation problems, similarly to what is done in conventional Principal Component Analysis. After the theoretical foundations are laid down, we explore real-world data from the telecommunications sector, in an attempt to detect Internet redirection attacks in real-time. In particular, we use our symbolic method to improve and simplify an anomaly detection method that has been proposed in the literature for conventional data.

Monitoring photochemical pollutants based on symbolic interval-valued data analysis

Monitoring photochemical pollutants based on symbolic interval-valued data analysis

Interval regression model adequacy checking and its application to estimate school dropout in Brazilian municipality educational scenario.

Interval-valued data have been commonly encountered in practice, and Symbolic Data Analysis provides a solution to the statistical treatment of these data. Regression analysis for interval-valued symbolic data is a topic that has been widely investigated in the literature of symbolic data analysis, and several models from different paradigms have been proposed. There are basic regression assumptions, and it is essential to validate them. This paper introduces an approach to check interval regression model adequacy based on residual analysis. Concepts of ordinary and standardized interval residual are presented, and graphical analysis of these residuals is also proposed. To show the usefulness of the proposed approach, an application for estimating school dropout in the scenario of Brazilian municipalities is performed. We observed some outliers from the interval residuals analysis, and interval robust regression models are more suitable for estimating school dropout.

The Measurement of Social Cohesion at Province Level in Poland Using Metric and Interval-Valued Data

The notion of social cohesion is increasingly used in the political, economic and academic debate. Due to its multidimensional, the assessment of social cohesion is not easy, especially if it is conducted at a lower than national level of aggregation. The aim of the study is to assess social cohesion in provinces of Poland in 2018 using the hybrid approach involving multidimensional scaling and linear ordering based on an aggregate measure. This type of study is usually conducted using classic metric data. However, the traditional approach does not account for the variation between lower level units (i.e., districts). The authors propose a methodology which makes this possible. Additionally the results of assessment of the multidimensional phenomenon can be presented in a two-dimensional space. Classic metric data and symbolic interval-valued data (three data types: min-max, 1st decile and 9th decile, 2nd decile and 8th decile) are jointly represented in a single diagram. The consistency of the research method ensures comparability of results of linear ordering. Two criteria were used in the comparative analysis of four rankings of social cohesion. The results of the study clearly showed that the current level of social cohesion at the provinces level is geographically and historically dependent.

Symbolic interval-valued data analysis for time series based on auto-interval-regressive models

This study considers interval-valued time series data. To characterize such data, we propose an auto-interval-regressive (AIR) model using the order statistics from normal distributions. Furthermore, to better capture the heteroscedasticity in volatility, we design a heteroscedastic volatility AIR (HVAIR) model. We derive the likelihood functions of the AIR and HVAIR models to obtain the maximum likelihood estimator. Monte Carlo simulations are then conducted to evaluate our methods of estimation and confirm their validity. A real data example from the S&P 500 Index is used to demonstrate our method.

Interval Fuzzy c-Regression Models with Competitive Agglomeration for Symbolic Interval-Valued Data

In this study, a novel approach, interval fuzzy c-regression models with competitive agglomeration (IFCRMCA), is proposed to deal with the symbolic interval-valued data. The proposed IFCRMCA approach can identify the partition of the interval-valued data using both the distances to the cluster centers and the errors of interval regression models for each cluster. Due to the concepts of competitive agglomeration is used in the proposed approach, the pre-determination of the cluster number in the proposed IFCRMCA is not necessary. Various real experiments are carried on and the experimentally results shows that the proposed approaches are superior to the existing approaches.

Efficient Estimation of Interval-Valued Symbolic Data Regression Model

In the last two decades, regression analysis with interval-valued type data has received more and more attention. For the interval-valued symbolic data regression, the Minmax method and the center and range (CR) method are two widely used popular estimating approaches. In this paper, to improve the estimating eciency of these two estimating methods, seemingly unrelated regression approach has been applied to take account of the dependence information of two regression models of the Minmax method or the CR method. Finally, real data sets are analysed to examine the performance of our proposed procedure.

Fault detection of uncertain nonlinear process using interval-valued data-driven approach

This paper introduces a new structure kernel principal component analysis (KPCA) that can successfully model symbolic interval-valued data for fault detection. In the proposed structure, interval KPCA (IKPCA) method is proposed to deal with interval-valued data. Two IKPCA models are proposed. The first model is based on the centers and ranges of intervals IKPCACR and the second model is based the upper and lower bounds of intervals IKPCAUL. Residuals are generated and fault detection indices are computed. The aim of using IKPCA is to ensure robustness to false alarm without affecting the fault detection performance. The proposed fault detection approach is carried out using simulation example and Tennessee Eastman Process (TEP). The obtained results demonstrate the effectiveness of the proposed technique.

A fuzzy inference system modeling approach for interval-valued symbolic data forecasting

This paper suggests a fuzzy inference system (iFIS) modeling approach for interval-valued time series forecasting. Interval-valued data arise quite naturally in many situations in which such data represent uncertainty/variability or when comprehensive ways to summarize large data sets are required. The method comprises a fuzzy rule-based framework with affine consequents which provides a (non)linear framework that processes interval-valued symbolic data. The iFIS antecedents identification uses a fuzzy c-means clustering algorithm for interval-valued data with adaptive distances, whereas parameters of the linear consequents are estimated with a center-range methodology to fit a linear regression model to symbolic interval data. iFIS forecasting power, measured by accuracy metrics and statistical tests, was evaluated through Monte Carlo experiments using both synthetic interval-valued time series with linear and chaotic dynamics, and real financial interval-valued time series. The results indicate a superior performance of iFIS compared to traditional alternative single-valued and interval-valued forecasting models by reducing 19% on average the predicting errors, indicating that the suggested approach can be considered as a promising tool for interval time series forecasting.

An Efficient Signature Verification Method Based on an Interval Symbolic Representation and a Fuzzy Similarity Measure

In this paper, an efficient offline signature verification method based on an interval symbolic representation and a fuzzy similarity measure is proposed. In the feature extraction step, a set of local binary pattern-based features is computed from both the signature image and its under-sampled bitmap. Interval-valued symbolic data is then created for each feature in every signature class. As a result, a signature model composed of a set of interval values (corresponding to the number of features) is obtained for each individual’s handwritten signature class. A novel fuzzy similarity measure is further proposed to compute the similarity between a test sample signature and the corresponding interval-valued symbolic model for the verification of the test sample. To evaluate the proposed verification approach, a benchmark offline English signature data set (GPDS-300) and a large data set (BHSig260) composed of Bangla and Hindi offline signatures were used. A comparison of our results with some recent signature verification methods available in the literature was provided in terms of average error rate and we noted that the proposed method always outperforms when the number of training samples is eight or more.

Constrained center and range joint model for interval-valued symbolic data regression

A constrained center and range joint model to fit linear regression to interval-valued symbolic data is introduced. This new method applies both the center and range of the interval to fit a linear regression model, and avoids the negative value of the range of the predicted dependent interval variable by adding nonnegative constraints. To improve prediction accuracy it adopts overlapping constraints. Using a little algebra, it is constructed as a special case of the least squares with inequality (LSI) problem and is solved with a Matlab routine. The assessment of the proposed prediction method is based on an estimation of the average root mean square error and accuracy rate. In the framework of a Monte Carlo experiment, different data set configurations take into account the rich or lack of error, as well as the slope with respect to the dependent and independent variables. A statistical t-test compares the performance of the new model with that of four previously reported methods. Based on experiment results, it is outlined that the new model has better fitness. An analysis of outliers is performed to determine the effects of outliers on our proposal. The proposed method is illustrated by analyses of data from two real-life case studies to compare its performance with those of the other methods.

NOE TS fuzzy modelling of nonlinear dynamic systems with uncertainties using symbolic interval-valued data

An approach to Nonlinear Output Error (NOE) modelling using Takagi–Sugeno (TS) fuzzy model for a class of nonlinear dynamic systems having variability in their outputs is presented. Furthermore, the approach is compared and graphically illustrated with other alternate approaches on the basis of interval data and interval membership functions. Assuming the identification method can be repeated offline a number of times under similar conditions, multiple input–output time series can be obtained from the underlying system. These time series are pre-processed using the techniques of statistics and probability theory to generate the envelopes of response (curves outlining the upper and lower extremes of response) at each time instant. Two types of envelopes are described in this research: the max–min envelopes and the envelopes based on the confidence intervals provided by extended Chebyshev's inequality. By incorporating interval data in fuzzy modelling and using the theory of symbolic interval-valued data, a TS fuzzy model with interval antecedent and consequent parameters is obtained. This algorithm provides a model for predicting the expected response as well as envelopes. In order to validate the presented model, a simulation case study is devised in this paper. Moreover, it is demonstrated on the real data obtained from an electro-mechanical throttle valve.

Predicting symbolic interval-valued data through symmetrical nonlinear regression

Predicting symbolic interval-valued data through symmetrical nonlinear regression

Predicting symbolic interval-valued data through symmetrical nonlinear regression

We proposed a symmetrical nonlinear regression model to fit interval-valued data. An important feature of this new model is that the estimate and prediction are less sensitive in the presence of outliers than a nonlinear model proposed in the literature. Monte Carlo simulation studies have been developed to investigate the performance of the model on different scenarios in precense of some percentage of outliers. The results based on the mean magnitude of the relative errors are presented and discussed. The model was fitted to one real symbolic dataset with noticeable interval outliers, and the forecast accuracy has been considered.

The Comparison of Fuzzy Clustering Methods for Symbolic Interval-Valued Data

Interval-valued data can find their practical applications in such situations as recording monthly interval temperatures at meteorological stations, daily interval stock prices, etc. The primary objective of the presented paper is to compare three different methods of fuzzy clustering for interval-valued symbolic data, i.e.: fuzzy c-means clustering, adaptive fuzzy c-means clustering and fuzzy k-means clustering with fuzzy spectral clustering. Fuzzy spectral clustering combines both spectral and fuzzy approaches in order to obtain better results (in terms of Rand index for fuzzy clustering). The conducted simulation studies with artificial and real data sets confirm both higher usefulness and more stable results of fuzzy spectral clustering method, as compared to other existing fuzzy clustering methods for symbolic interval-valued data, when dealing with data featuring different cluster structures, noisy variables and/or outliers.

Using Symbolic Data in Gravity Model of Population Migration to Reduce Modifiable Areal Unit Problem (MAUP)

Abstract Spatial analyses suffer from modifiable areal unit problem (MAUP). This occurs while operating on aggregated data determined for high-level territorial units, e.g. official statistics for countries. Generalization process deprives the data of variation. Carrying out research excluding territorial distribution of a phenomenon affects the analysis results and reduces their reliability. The paper proposes to use symbolic data analysis (SDA) to reduce MAUP. SDA proposes an alternative form of individual data aggregation and deals with multivariate analysis of inter-valvalued, multi-valued and histogram data. The paper discusses the scale effect of MAUP which occurs in a gravity model of population migrations and shows how SDA can deal with this problem. Symbolic interval-valued data was used to determine the economic distance between regions which served as a separation function in the model. The proposed approach revealed that economic disparities in Poland are lower than official statistics show but they are still one of the most important factors of domestic migration flows.

Using symbolic data in gravity model of population migration to reduce modifiable areal unit problem (MAUP)

Spatial analyses suffer from modifiable areal unit problem (MAUP). This occurs while operating on aggregated data determined for high-level territorial units, e.g. official statistics for countries. Generalization process deprives the data of variation. Carrying out research excluding territorial distribution of a phenomenon affects the analysis results and reduces their reliability. The paper proposes to use symbolic data analysis (SDA) to reduce MAUP. SDA proposes an alternative form of individual data aggregation and deals with multivariate analysis of interval valued, multi-valued and histogram data. The paper discusses the scale effect of MAUP which occurs in a gravity model of population migrations and shows how SDA can deal with this problem. Symbolic interval-valued data was used to determine the economic distance between regions which served as a separation function in the model. The proposed approach revealed that economic disparities in Poland are lower than official statistics show but they are still one of the most important factors of domestic migration flows.

A Novel Approach to T-S Fuzzy Modeling of Nonlinear Dynamic Systems with Uncertainties using Symbolic Interval-Valued Outputs

A novel approach to Takagi-Sugeno (T-S) fuzzy modeling of a class of nonlineardynamic systems having variability in their outputs for the Nonlinear Output Error (NOE)case is addressed in this article. Multiple input-output datasets were obtained by repeating the identification experiment. The variability in the output time series is captured by defining the envelops of response at each time instant. These envelops actually provide the confidence interval based upper and lower bounds of the output time series using the extended Chebyshev’s Inequality. Different from the previous approach, in which two independent T-S fuzzy models were used for identifying each bound, a single T-S fuzzy model is identified in this work, which resulted in interval parameters for the antecedent and consequent variables. This is accomplished by first transforming the bounds into the symbolic interval-valued data and then using this data for identification. In order to get the expected value of the response, the estimated lower and upper bound time series of the identified T-S fuzzy model were averaged out at each time instant, as permitted by the extended Chebyshev’s Inequality. The proposed approach is demonstrated on an industrial diesel-engine electro-mechanical throttle valve.

Exploratory data analysis of interval-valued symbolic data with matrix visualization

Symbolic data analysis (SDA) has gained popularity over the past few years because of its potential for handling data having a dependent and hierarchical nature. Amongst many methods for analyzing symbolic data, exploratory data analysis (EDA: Tukey, 1977) with graphical presentation is an important one. Recent developments of graphical and visualization tools for SDA include zoom star, closed shapes, and parallel-coordinate-plots. Other studies project high dimensional symbolic data into lower dimensional spaces using symbolic data versions of principal component analysis, multidimensional scaling, and self-organizing maps. Most graphical and visualization approaches for exploring symbolic data structure inherit the advantages of their counterparts for conventional (non-symbolic) data, but also their disadvantages. Here we introduce matrix visualization (MV) for visualizing and clustering symbolic data using interval-valued symbolic data as an example; it is by far the most popular symbolic data type in the literature and the most commonly encountered one in practice. Many MV techniques for visualizing and clustering conventional data are converted to symbolic data, and several techniques are newly developed for symbolic data. Various examples of data with simple to complex structures are brought in to illustrate the proposed methods.