Using Induced Ordered Weighted Averaging (IOWA) Operators for Aggregation in Cross-Efficiency Evaluations
This paper proposes an enhancement of the cross-efficiency evaluation through the aggregation of cross-efficiencies by using a particular type of induced ordered weighted averaging IOWA operator. The use of a weighted average of cross-efficiencies for the calculation of the cross-efficiency scores, instead of the usual arithmetic mean, allows us to introduce some flexibility into the analysis. In particular, the main purpose of the approach we present is to provide aggregation weights that reflect the decision maker DM preferences regarding the relative importance that should be attached to the cross-efficiencies provided by the different decision-making units. To do it, an ordering is to b…
Fuzzy temporal random sets with an application to cell biology
Total Internal Reflection Fluorescence Microscopy (TIRFM) greatly facilitates to imaging the first steps of endocytosis, a process whereby cells traffic cargo from the cell surface to endosomes. Using TIRFM, fluorescent-tagged endocytic proteins are observed as overlapped areas forming random clumps of different sizes, shapes and durations. A common procedure to segment these objects consists of thresholding the original gray-level images to produce binary sequences in which a pixel is covered or not by a given fluorescent-tagged protein. This binary logic is not appropriate because it leaves a free tuning parameter to be set by the user which can influence on the conclusions of the statist…
A fuzzy method to repair infeasibility in linearly constrained problems
Abstract In this paper we introduce a fuzzy method to deal with infeasibility in linearly constrained programs. Given an infeasible instance, we determine how much we should perturb the right-hand side coefficients in order to attain feasibility and propose a ‘feasible reformulation’ of the problem. Although we prove that our algorithm always finds such a reformulation the convenience of using it can be decided by the analyst. By this, we mean that the method also provides a simple way to compute lower bounds on the changes on every right-hand side coefficient, and if the decision maker considers that some of the magnitudes are unacceptable, he or she simply stops at this step. We think tha…
Changes in power curve shapes as an indicator of fatigue during dynamic contractions.
The purpose of this study was to analyze exercise-induced leg fatigue during a dynamic fatiguing task by examining the shapes of power vs. time curves through the combined use of several statistical methods: B-spline smoothing, functional principal components and (supervised and unsupervised) classification. In addition, granulometric size distributions were also computed to allow for comparison of curves coming from different subjects. Twelve physically active men participated in one acute heavy-resistance exercise protocol which consisted of five sets of 10 repetition maximum leg press with 120 s of rest between sets. To obtain a smooth and accurate representation of the data, a basis of …
Optimization under Uncertainty and Linear Semi-Infinite Programming: A Survey
This paper deals with the relationship between semi-infinite linear programming and decision making under uncertainty in imprecise environments. Actually, we have reviewed several set-inclusive constrained models and some fuzzy programming problems in order to see if they can be solved by means of a linear semi-infinite program. Finally, we present some numerical examples obtained by using a primal semi-infinite programming method.
On the numerical treatment of linearly constrained semi-infinite optimization problems
Abstract We consider the application of two primal algorithms to solve linear semi-infinite programming problems depending on a real parameter. Combining a simplex-type strategy with a feasible-direction scheme we obtain a descent algorithm which enables us to manage the degeneracy of the extreme points efficiently. The second algorithm runs a feasible-direction method first and then switches to the purification procedure. The linear programming subproblems that yield the search direction involve only a small subset of the constraints. These subsets are updated at each iteration using a multi-local optimization algorithm. Numerical test examples, taken from the literature in order to compar…
Different averages of a fuzzy set with an application to vessel segmentation
Image segmentation is a major problem in image processing, particularly in medical image analysis. A great number of segmentation procedures produce intermediate gray-scale images that can be understood as fuzzy sets. Additionally, some segmentation procedures tend to leave free tuning parameters (very influential in the final binary image) for the user. These different binary images can be easily aggregated (into a fuzzy set) by making use of fuzzy set theory. In any case, a single binary image is required so our interest is to associate a crisp set to a given fuzzy set in an intelligent and unsupervised manner. The main idea of this paper is to define the averages of a given fuzzy set by …
Fuzzy Mathematical Programming for Portfolio Management
The classical portfolio selection problem was formulated by Markowitz in the 1950s as a quadratic programming problem in which the risk variance is minimized. Since then, many other models have been considered and their associated mathematical programming formulations can be viewed as dynamic, stochastic or static decision problems. In our opinion, the model formulation depends essentially on two factors: the data nature and the treatment given to the risk and return goals. In this communication, we consider several approaches to deal with the data uncertainty for different classical formulations of the portfolio problem. We make use of duality theory and fuzzy programming techniques to ana…
Assessing energy forecasting inaccuracy by simultaneously considering temporal and absolute errors
Abstract Recent years have seen a growing trend in wind and solar energy generation globally and it is expected that an important percentage of total energy production comes from these energy sources. However, they present inherent variability that implies fluctuations in energy generation that are difficult to forecast. Thus, forecasting errors have a considerable role in the impacts and costs of renewable energy integration, management, and commercialization. This study presents an important advance in the task of analyzing prediction models, in particular, in the timing component of prediction error, which improves previous pioneering results. A new method to match time series is defined…
Fuzzy Logic for Medical Engineering: An Application to Vessel Segmentation
A fuzzy mathematical programming approach to the assessment of efficiency with DEA models
In many real applications, the data of production processes cannot be precisely measured. This is particularly worrying when assessing efficiency with frontier-type models, such as data envelopment analysis (DEA) models, since they are very sensitive to possible data errors. For this reason, the possibility of having available a methodology that allows the analyst to deal with imprecise data becomes an issue of great interest in these contexts. To that end, we develop some fuzzy versions of the classical DEA models (in particular, the BCC model) by using some ranking methods based on the comparison of α-cuts. The resulting auxiliary crisp problems can be solved by the usual DEA software. We…
Dynamic mean absolute error as new measure for assessing forecasting errors
Abstract Accurate wind power forecast is essential for grid integration, system planning, and electricity trading in certain electricity markets. Therefore, analyzing prediction errors is a critical task that allows a comparison of prediction models and the selection of the most suitable model. In this work, the temporal error and absolute magnitude error are simultaneously considered to assess the forecast error. The trade-off between both types of errors is computed, analyzed, and interpreted. Moreover, a new index, the dynamic mean absolute error, DMAE, is defined to measure the prediction accuracy. This index accounts for both error components: temporal and absolute. Real cases of wind …
Applying logistic regression to relevance feedback in image retrieval systems
This paper deals with the problem of image retrieval from large image databases. A particularly interesting problem is the retrieval of all images which are similar to one in the user's mind, taking into account his/her feedback which is expressed as positive or negative preferences for the images that the system progressively shows during the search. Here we present a novel algorithm for the incorporation of user preferences in an image retrieval system based exclusively on the visual content of the image, which is stored as a vector of low-level features. The algorithm considers the probability of an image belonging to the set of those sought by the user, and models the logit of this prob…
Viability of infeasible portfolio selection problems: A fuzzy approach
Abstract This paper deals with fuzzy optimization schemes for managing a portfolio in the framework of risk–return trade-off. Different models coexist to select the best portfolio according to their respective objective functions and many of them are linearly constrained. We are concerned with the infeasible instances of such models. This infeasibility, usually provoked by the conflict between the desired return and the diversification requirements proposed by the investor, can be satisfactorily avoided by using fuzzy linear programming techniques. We propose an algorithm to repair infeasibility and we illustrate its performance on a numerical example.
New descent rules for solving the linear semi-infinite programming problem
The algorithm described in this paper approaches the optimal solution of a continuous semi-infinite linear programming problem through a sequence of basic feasible solutions. The descent rules that we present for the improvement step are quite different when one deals with non-degenerate or degenerate extreme points. For the non-degenerate case we use a simplex-type approach, and for the other case a search direction scheme is applied. Some numerical examples illustrating the method are given.
Using mathematical morphology for unsupervised classification of functional data
This paper is concerned with the unsupervised classification of functional data by using mathematical morphology. Different morphological operators are used to extract relevant structures of the functions (considered as sets through their subgraph representations). These operators can be considered as preprocessing tools whose outputs are also functional data. We explore some dissimilarity measures and clustering methods for the classification of the transformed data. Our approach is illustrated through a detailed analysis of two data sets. These techniques, which have mainly been used in image processing, provide a flexible and robust toolbox for improving the results in unsupervised funct…
Looking for representative fit models for apparel sizing
This paper is concerned with the generation of optimal fit models for use in apparel design. Representative fit models or prototypes are important for defining a meaningful sizing system. However, there is no agreement among apparel manufacturers and each one has their own prototypes and size charts i.e. there is a lack of standard sizes in garments from different apparel manufacturers. We propose two algorithms based on a new hierarchical partitioning around medoids clustering method originally developed for gene expression data. We are concerned with a different application; therefore, the dissimilarity between the objects has to be different and must be designed to deal with anthropometr…
Compatibility of the Different Tuning Systems in an Orchestra
Focusing on the daily practice of musicians, we give flexibility to the mathematical treatment of musical notes, tuning systems and the relations between them. This allows us to connect the theory and the practice of music. Using the techniques of fuzzy logic, we describe the concepts with fuzzy sets and introduce the α-compatibility as a degree of interchangeability between tuning systems. To show how our proposal works, we use a fragment of Haydn and analyze the compatibility of the notes taken from 48 recordings for the tuning systems of Pythagoras, Zarlino and Equal Temperament of 12 notes.
Solving a class of fuzzy linear programs by using semi-infinite programming techniques
This paper deals with a class of Fuzzy Linear Programming problems characterized by the fact that the coefficients in the constraints are modeled as LR-fuzzy numbers with different shapes. Solving such problems is usually more complicated than finding a solution when all the fuzzy coefficients have the same shape. We propose a primal semi-infinite algorithm as a valuable tool for solving this class of Fuzzy Linear programs and, we illustrate it by means of several examples.
Mathematics and Soft Computing in Music
Mathematics is the fundamental tool for dealing with the physical processes that explain music but it is also in the very essence of this art. Musical notes, the first elements which music works with, are defined for each tuning system as very specific frequencies; however, instrumentalists know that small changes in these values do not have serious consequences. In fact, sometimes consensus is only reached if the entire orchestra alters the theoretical pitches. The explanation for this contradiction is that musicians implicitly handle very complex mathematical processes involving some uncertainty in the concepts and this is better explained in terms of fuzzy logic. Modelling the notes as f…
Obtaining the Compatibility between Musicians Using Soft Computing
Modeling the musical notes as fuzzy sets provides a flexible framework which better explains musicians’ daily practices. Taking into account one of the characteristics of the sound: the pitch (the frequency of a sound as perceived by human ear), a similarity relation between two notes can be defined. We call this relation compatibility. In the present work, we propose a method to asses the compatibility between musicians based on the compatibility of their interpretations of a given composition. In order to aggregate the compatibilities between the notes offered and then obtain the compatibility between musicians, we make use of an OWA operator. We illustrate our approach with a numerical e…
Analyzing Protein-Protein Spatial-Temporal Dependencies from Image Sequences Using Fuzzy Temporal Random Sets
Total Internal Reflection Fluorescence Microscopy (TIRFM) allows us to image fluorescenttagged proteins near the plasma membrane of living cells with high spatial-temporal resolution. Using TIRFM imaging of GFP-tagged clathrin endocytic proteins, areas of fluorescence are observed as overlapping spots of different sizes and durations. Standard procedures to measure protein-protein colocalization of dual labeled samples threshold the original graylevel images to segment areas covered by different proteins. This binary logic is not appropriate as it leaves a free tuning parameter which can influence the conclusions. Moreover, these procedures rely on simple statistical analysis based on corre…
A purification algorithm for semi-infinite programming
Abstract In this paper we present a purification algorithm for semi-infinite linear programming. Starting with a feasible point, the algorithm either finds an improved extreme point or concludes with the unboundedness of the problem. The method is based on the solution of a sequence of linear programming problems. The study of some recession conditions has allowed us to establish a weak assumption for the finite convergence of this algorithm. Numerical results illustrating the method are given.
Fuzzy logic helps to integrate music theory and practice
Among the many practices of composers, instrumentalists and singers which clearly correspond with fuzzy logic our focus here is on tuning. Different criteria have been used to select the sounds that music uses. A set containing these sounds (musical notes) is called a tuning system. Several tuning systems coexist in a classical orchestra. The pitches of the notes are different and very precisely defined for each system; however the consequences of small deviations from these theoretical frequencies are not serious. Actually, the orchestra members are aware of the necessity of reaching a consensus and adjust their instruments to tune well. Because of this, many musicians feel that the mathem…
Cross-Efficiency in Fuzzy Data Envelopment Analysis (FDEA): Some Proposals
Different techniques have been proposed in the literature to rank decision making units (DMUs) in the context of Fuzzy Data Envelopment Analysis. In our opinion, those that result from using a ranking method to order the fuzzy efficiencies obtained are susceptible to a serious criticism: they are not based on objective criteria. Cross-efficiency evaluation was introduced as an extension of DEA aimed at ranking the DMUs. This methodology has found a significant number of applications and has been extensively investigated. In this chapter, we discuss some difficulties that arise with the definition of fuzzy cross-efficiencies and we propose a fuzzy cross-efficiency evaluation based on the FDE…
IOWA Operators and Its Application to Image Retrieval
This paper presents a relevance feedback procedure based on logistic regression analysis. Since, the dimension of the feature vector associated to each image is typically larger than the number of evaluated images by the user, different logistic regression models have to be fitted separately. Each fitted model provides us with a relevance probability and a confidence interval for that probability. In order to aggregate these set of probabilities and confidence intervals we use an IOWA operator. The results will show the success of our algorithm and that OWA operators are an efficient and natural way of dealing with this kind of fusion problems.
A method for detecting malfunctions in PV solar panels based on electricity production monitoring
In this paper a new method is developed for automatically detecting outliers or faults in the solar energy production of identical sets (sister arrays) of photovoltaic (PV) solar panels. The method involves a two-stage unsupervised approach. In the first stage, "in control" energy production data are created by using outlier detection methods and functional principal component analysis in order to remove global and local outliers from the data set. In the second stage, control charts for the "in control" data are constructed using both a parametric method and three non-parametric methods. The control charts can be used to detect outliers or faults in the production data in real-time or at t…
A fuzzy framework to explain musical tuning in practice
A theoretical tuning system is a set of pitches that can be used to play music. It is a fact that the human ear perceives notes with very close frequencies as if they were the same note. Therefore, in our approach a musical note and its pitch sensation are modeled as L-R fuzzy numbers with a modal interval and a bounded support. We pay particular attention to the 12-tone equal temperament (12-TET) for being the most widely used tuning system and we define the fuzzy 12-TET composed of 12 fuzzy notes. A similarity relation between a fuzzy note and a theoretical note can be defined, and subsequently a similarity class associated to each one of the fuzzy notes in the fuzzy 12-TET arises. Finall…
Introducing the Temporal Distortion Index to perform a bidimensional analysis of renewable energy forecast
Abstract Wind has been the largest contributor to the growth of renewal energy during the early 21st century. However, the natural uncertainty that arises in assessing the wind resource implies the occurrence of wind power forecasting errors which perform a considerable role in the impacts and costs in the wind energy integration and its commercialization. The main goal of this paper is to provide a deeper insight in the analysis of timing errors which leads to the proposal of a new methodology for its control and measure. A new methodology, based on Dynamic Time Warping, is proposed to be considered in the estimation of accuracy as attribute of forecast quality. A new dissimilarity measure…
A multi-local optimization algorithm
The development of efficient algorithms that provide all the local minima of a function is crucial to solve certain subproblems in many optimization methods. A “multi-local” optimization procedure using inexact line searches is presented, and numerical experiments are also reported. An application of the method to a semi-infinite programming procedure is included.
A Morphological Clustering Method for daily solar radiation curves
Abstract We present a new method based on Mathematical Morphology techniques for the classification of solar radiation curves that we call MfCM. The main advantage of using MfCM as opposed to daily clearness index distributions is that it allows us to keep the dynamics of the solar radiance curves in the analysis: both cloud transitions and variability in direct radiation are simultaneously taken into account. To illustrate our proposal, we use a set of real radiation data collected in a location sited in southern Spain.
Measuring Dissimilarity Between Curves by Means of Their Granulometric Size Distributions
The choice of a dissimilarity measure between curves is a key point for clustering functional data. Functions are usually pointwise compared and, in many situations, this approach is not appropriate. Mathematical Morphology provides us with a toolbox to overcome this problem. We propose some dissimilarity measures based on morphological granulometries and their performance is evaluated on some functional datasets.