Measure and integral with purely ordinal scales

We develop a purely ordinal model for aggregation functionals for lattice valued functions, comprising as special cases quantiles, the Ky Fan metric and the Sugeno integral. For modeling findings of psychological experiments like the reflection effect in decision behaviour under risk or uncertainty, we introduce reflection lattices. These are complete linear lattices endowed with an order reversing bijection like the reflection at $0$ on the real interval $[-1,1]$. Mathematically we investigate the lattice of non-void intervals in a complete linear lattice, then the class of monotone interval-valued functions and

Lasso Estimation of an Interval-Valued Multiple Regression Model

A multiple interval-valued linear regression model considering all the cross-relationships between the mids and spreads of the intervals has been introduced recently. A least-squares estimation of the regression parameters has been carried out by transforming a quadratic optimization problem with inequality constraints into a linear complementary problem and using Lemke's algorithm to solve it. Due to the irrelevance of certain cross-relationships, an alternative estimation process, the LASSO (Least Absolut Shrinkage and Selection Operator), is developed. A comparative study showing the differences between the proposed estimators is provided

An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives

This article proposes a framework to handle multiattribute group decision making problems with incomplete pairwise comparison preference over decision alternatives where qualitative and quantitative attribute values are furnished as linguistic variables and crisp numbers, respectively. Attribute assessments are then converted to interval-valued intuitionistic fuzzy numbers (IVIFNs) to characterize fuzziness and uncertainty in the evaluation process. Group consistency and inconsistency indices are introduced for incomplete pairwise comparison preference relations on alternatives provided by the decision-makers (DMs). By minimizing the group inconsistency index under certain constraints, an auxiliary linear programming model is developed to obtain unified attribute weights and an interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS). Attribute weights are subsequently employed to calculate distances between alternatives and the IVIFPIS for ranking alternatives. An illustrative example is provided to demonstrate the applicability and effectiveness of this method

Asymptotic properties for a class of partially identified models

We propose inference procedures for partially identified population features for which the population identification region can be written as a transformation of the Aumann expectation of a properly defined set valued random variable (SVRV). An SVRV is a mapping that associates a set (rather than a real number) with each element of the sample space. Examples of population features in this class include sample means and best linear predictors with interval outcome data, and parameters of semiparametric binary models with interval regressor data. We extend the analogy principle to SVRVs, and show that the sample analog estimator of the population identification region is given by a transformation of a Minkowski average of SVRVs. Using the results of the mathematics literature on SVRVs, we show that this estimator converges in probability to the identification region of the model with respect to the Hausdorff distance. We then show that the Hausdorff distance between the estimator and the population identification region, when properly normalized by vn, converges in distribution to the supremum of a Gaussian process whose covariance kernel depends on parameters of the population identification region. We provide consistent bootstrap procedures to approximate this limiting distribution. Using similar arguments as those applied for vector valued random variables, we develop a methodology to test assumptions about the true identification region and to calculate the power of the test. We show that these results can be used to construct a confidence collection, that is a collection of sets that, when specified as null hypothesis for the true value of the population identification region, cannot be rejected by our test.Partial Identification, Confidence Collections, Set-Valued Random Variables.

Fault Detection of Single and Interval Valued Data Using Statistical Process Monitoring Techniques

Principal component analysis (PCA) is a linear data analysis technique widely used for fault detection and isolation, data modeling, and noise filtration. PCA may be combined with statistical hypothesis testing methods, such as the generalized likelihood ratio (GLR) technique in order to detect faults. GLR functions by using the concept of maximum likelihood estimation (MLE) in order to maximize the detection rate for a fixed false alarm rate. The benchmark Tennessee Eastman Process (TEP) is used to examine the performance of the different techniques, and the results show that for processes that experience both shifts in the mean and/or variance, the best performance is achieved by independently monitoring the mean and variance using two separate GLR charts, rather than simultaneously monitoring them using a single chart. Moreover, single-valued data can be aggregated into interval form in order to provide a more robust model with improved fault detection performance using PCA and GLR. The TEP example is used once more in order to demonstrate the effectiveness of using of interval-valued data over single-valued data

Shrinkage linear regression for symbolic interval-valued variables

This paper proposes a new approach to fit a linear regression for symbolic internal-valued variables, which improves both the Center Method suggested by Billard and Diday in \cite{BillardDiday2000} and the Center and Range Method suggested by Lima-Neto, E.A. and De Carvalho, F.A.T. in \cite{Lima2008, Lima2010}. Just in the Centers Method and the Center and Range Method, the new methods proposed fit the linear regression model on the midpoints and in the half of the length of the intervals as an additional variable (ranges) assumed by the predictor variables in the training data set, but to make these fitments in the regression models, the methods Ridge Regression, Lasso, and Elastic Net proposed by Tibshirani, R. Hastie, T., and Zou H in \cite{Tib1996, HastieZou2005} are used. The prediction of the lower and upper of the interval response (dependent) variable is carried out from their midpoints and ranges, which are estimated from the linear regression models with shrinkage generated in the midpoints and the ranges of the interval-valued predictors. Methods presented in this document are applied to three real data sets cardiologic interval data set, Prostate interval data set and US Murder interval data set to then compare their performance and facility of interpretation regarding the Center Method and the Center and Range Method. For this evaluation, the root-mean-squared error and the correlation coefficient are used. Besides, the reader may use all the methods presented herein and verify the results using the {\tt RSDA} package written in {\tt R} language, that can be downloaded and installed directly from {\tt CRAN} \cite{Rod2014}.Comment: 22 page