A Fuzzy Regression Technique Through Same-Points in Fuzzy Geometry

The rule representation opens a new dimension for expressing changes of fuzziness in fuzzy rule-based systems. In the case of standard Fuzzy rule representations, it is difficult to describe fuzzy functions in which crisp observations are required to have fuzzy conclusions, or in which an increase in the fuzziness of observations leads to reduced fuzziness in conclusions. These problems are mainly due to a lack of information. A fuzzy point rule determines the connection between a pair of fuzzy sets taken from the domain and the range of the rule. Expressing the fuzzy function through a set of fuzzy points and fuzzy interpolation between pairs of those points, each fuzzy point can be considered as a node point with given location and fuzziness. In common, sparse rule-base definitions, these node points are usually disjunctive on the domain, defining only single antecedent-consequent fuzziness connections at the location of the fuzzy points. However, this kind of information is insufficient when the goal is to express changes in the fuzziness of a given location in the domain. One solution to this problem is the double fuzzy point rule representation concept. Double fuzzy points are pairs of fuzzy points which share the same reference locations, but have different fuzziness properties. The existence of two different fuzziness values in a single location within the domain creates new possibilities for introducing fuzzy interpolation methods capable of interpolating not only between locations, but between changes in local fuzziness values as well. The main goal of this paper is to discuss how two-step Rule Interpolation methods can be adapted to be able to handle the double fuzzy point concept. To this end, an approach referred to as the Generalized Double Point Methodology (GDFPM) is proposed.

Imprecise points are regions in which one point should be placed. We study computing the largest and smallest possible values of various basic geometric measures on sets of imprecise points, such as the diameter, width, closest pair, smallest enclosing circle, and smallest enclosing bounding box. We give efficient algorithms for most of these problems, and identify the hardness of others.

There are various studies in which a variety of prediction tools have been introduced in time-series prediction literature. Non-probabilistic approaches which are based on fuzzy set theory, especially in recent years, have been put forward. Although these approaches including adaptive network fuzzy inference system, fuzzy functions approach, and fuzzy regression can be successfully utilized as a prediction tool, they have not been designed for prediction problem and they pass over the dependency structure of time-series observations. From this point forth, designing a prediction tool that considers the dependency structure of the observations of time series will procure to get predictions more accurately. Although the membership values, in the analysis process, are taken into account in almost all fuzzy methods, the non-membership and hesitation values are not considered. However, using as much information as possible on time series may be another positive factor that gives more accurate predictions. The primary aim of this study, for time-series prediction, is to introduce an intuitionistic fuzzy regression functions approach based on hesitation margin (I-FRF-HM). In the introduced intuitionistic fuzzy regression functions approach, two inference systems are separately constituted such that while one of them uses membership, other one uses non-membership values as inputs of inference system in addition with the crisp observations of time series. Predictions obtained from each system are converted into final predictions of whole inference system via an approach based on hesitation margin. Intuitionistic fuzzy C-means are utilized to get membership and non-membership values in the proposed model. The proposed I-FRF-HM has been applied to various real-world time series. The obtained findings are evaluated along with the results of some other time-series prediction models. The results show that the proposed I-FRF-HM has superior prediction performance to other prediction models.

Every organization needs adequate forecasts for planning the future. The accuracy of forecasts is influenced by both the quality of past data and the method selected to forecast the future. In this paper, we carry out a comparative study between the time series forecasts from (1) the Quick-prop neural network, (2) a fuzzy neural network (adaptive-network-based fuzzy inference system (ANFIS)), (3) a fuzzy regression and identification decision tree (ADRI), and (4) traditional time series methods (ARIMA models). We augment ANFIS by using fuzzy curves to identify the input variables that have the most influence on the output. This method identifies the significant input variables that lead to a considerable decrease in training time for ANFIS, while keeping the performance at least as good. We test the performance of ANFIS with the fuzzy curve pruning technique on empirical time series data (the national private consumption) from the Spanish economy. ANFIS produced the best performance on forecasting the empirical time series data compared to ADRI and ARIMA.

Assume that a set of imprecise points is given, where each point is specified by a region in which the point may lie. We study the problem of computing the smallest and largest possible tours and convex hulls, measured by length, and in the latter case also by area. Generally we assume the imprecision region to be a square, but we discuss the case where it is a segment or circle as well. We give polynomial time algorithms for several variants of this problem, ranging in running time from O(n) to O(n 13), and prove NP-hardness for some geometric problems on imprecise points.KeywordsLine SegmentConvex HullExtreme PointMinimum Span TreeTangent PointThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we introduce the outlier detection problem in a set of uncertain points. We study two variants of the problems based upon the definition of the outlier. For a given positive integer \(k(<n)\) and a set \(\mathfrak {R}\) of n regions as the imprecise points, the first type of the outlier detection problem that we study is to locate \(n-k\) points on distinct regions, such that the size of the smallest axis-aligned bounding box (AABB), the diameter or the smallest enclosing circle (SEC) of the resulting points gets minimized. The uncertainty regions we study are squares or disks, and the excluded k regions are considered as outliers.We also study the covering versions in which the objectives of the SEC and the AABB problems are to find the smallest circle or axis-aligned bounding box, respectively, that covers the area of at least \(n-k\) regions.In the second-type of outliers, the outliers are those k regions that mostly reduce the uncertainty-induced gap between the lower bound and the upper bound on the size of the output. We give polynomial time algorithms for several variants of the mentioned problems, ranging in running time from \(O(n \log n)\) to \(O(n^{5.5}\log n)\).

would be an excellent book for undergraduate, graduate and beyond….The style of writing is easy to read and the author does a good job of adding humor in places. The integration of basic programming in R with the data that is collected for any experiment provides a powerful platform for analysis of data…. having the understanding of data analysis that this book offers will really help researchers examine their data and consider its value from multiple perspectives – and this applies to people who have small AND large data sets alike! This book also helps people use a free and basic software system for processing and plotting simple to complex functions. Michelle Pantoya, Texas Tech University Measurements of quantities that vary in a continuous fashion, e.g., the pressure of a gas, cannot be measured exactly and there will always be some uncertainty with these measured values, so it is vital for researchers to be able to quantify this data. Uncertainty Analysis of Experimental Data with R covers methods for evaluation of uncertainties in experimental data, as well as predictions made using these data, with implementation in R. The books discusses both basic and more complex methods including linear regression, nonlinear regression, and kernel smoothing curve fits, as well as Taylor Series, Monte Carlo and Bayesian approaches. Features: 1. Extensive use of modern open source software (R). 2. Many code examples are provided. 3. The uncertainty analyses conform to accepted professional standards (ASME). 4. The book is self-contained and includes all necessary material including chapters on statistics and programming in R. Benjamin D. Shaw is a professor in the Mechanical and Aerospace Engineering Department at the University of California, Davis. His research interests are primarily in experimental and theoretical aspects of combustion. Along with other courses, he has taught undergraduate and graduate courses on engineering experimentation and uncertainty analysis. He has published widely in archival journals and became an ASME Fellow in 2003.

Mixed-fuzzy functions are proposed as an alternate to fuzzy rule base formation in the structure identification of system models and reasoning with them. These mixed-fuzzy functions can be determined by any function identification method such as least squares' estimates, LSE, maximum likelihood estimates, MLE, support vector machines, SVM, etc. For this purpose, working knowledge of a fuzzy clustering algorithm such as FCM or its variations, such as Improved fuzzy clustering method (IFCM), would be sufficient to obtain membership values of input vectors. The membership values together with scalar input variables are then used by the LSE technique to determine "mixed fuzzy functions" for each cluster identified by FCM and/IFCM. These functions are different from "fuzzy rule base" approaches as well as "fuzzy regression" approaches. In mixed fuzzy functions, various transformations of the membership values are included as new variables in addition to original selected scalar input variables; and at times, a logistic transformation of non-scalar original selected input variables may also be included as a new variable.

This paper describes the theory and algorithms of fuzzy distance transform (FDT). Fuzzy distance is defined as the length of the shortest path between two points. The length of a path in a fuzzy subset is defined as the integration of fuzzy membership values of the points along the path. The shortest path between two points is the one with the minimum length among all (infinitely many) paths between the two points. It is demonstrated that, unlike in the binary case, the shortest path in a fuzzy subset is not necessarily a straight-line segment. The support of a fuzzy subset is the set of points with nonzero membership values. It is shown that, for any fuzzy subset, fuzzy distance is a metric for the interior of its support. FDT is defined as the process on a fuzzy subset that assigns at each point the smallest fuzzy distance from the boundary of the support. The theoretical framework of FDT in continuous space is extended to digital spaces and a dynamic programming-based algorithm is presented for its computation. Several potential medical imaging applications are presented including the quantification of blood vessels and trabecular bone thickness in the regime of limited special resolution.

During the Vth Seminar on Fuzzy Set Theory (Linz, Austria, September 1983) U. Cerruti and U. Hohle stated the following open problem from their work on fuzzy equalities. “16 (X,E) is a set with [0,1]-valued fuzzy equality E, and (P(X),E) is the set of fuzzy points with the induced fuzzy equality E, what is the precise relationship between X and P(X) and between E and E?” In particular they were unable to show that P(X) in general is “larger” than X and wondered if each fuzzy point ρ was of the type ρ(z) = E(x, z) for some x e X. During the seminar we solved this question in the negative by the example following Corollary 2.3. based on [3] and conjectured that it could be generalized to arbitrary pseudometric spaces. In this paper we prove this conjecture. We show that for a large class of fuzzy equalities the relationship between X and P(X) is as between a pseudometric space and its completion and between E and [Etilde] is as between a uniformly continuous map and its continuous extension. A first proof i...

An elliptic curve can be defined as a smooth projective curve of degree 3 in the projective plane with a distinguished point chosen on it. The set of points on the curve can thus be endowed with a natural additive group structure. The most concrete description of an elliptic curve comes from its affine equation, written as $$y^2=x^3+ax+b,\quad \text{with}\ 4a^3+27b^2\not=0.$$ The theory of elliptic curves is a marvelous mixture of elementary mathematics and profound, advanced mathematics, a mixture which moreover lies on the crossroads of multiple theories: arithmetic, algebraic geometry, group representations, complex analysis, etc. Here, we will provide an introduction to the subject and prove the main Diophantine theorems: the group of rational points is finitely generated (the Mordell-Weil theorem) and the set of integral points is finite (Siegel’s theorem). Finally, we will evoke the famous theorem of Wiles—whose proof resulted in the proof of Fermat’s last theorem—and the Birch & Swinnerton-Dyer conjecture.

Background Dietary antioxidants have been found to improve various diseases, including digestive, cardiovascular, and urinary disorders. However, the relationship between the composite dietary antioxidant index (CDAI) and constipation remains unclear. This study aims to investigate the potential link between CDAI and constipation among adults in the United States. Methods This study utilized data from the National Health and Nutrition Examination Survey (NHANES) collected between 2005 and 2010. Dietary data from the respondents were obtained through two 24-hour dietary recall interviews. Multiple statistical methods, including multivariable logistic regression, subgroup analysis, and smooth curve fitting analysis, were used to explore the association between CDAI and chronic constipation. Results A total of 10,904 participants were included in the study, of whom 1,184 were identified as having chronic constipation. After adjusting for potential confounders, multivariable logistic regression analysis showed that higher CDAI was significantly associated with a lower risk of constipation (OR = 0.958 [0.929, 0.987]). Compared to the lowest quartile, the highest quartile of CDAI was associated with a significantly reduced prevalence of constipation (OR = 0.704 [0.535, 0.927]). Subgroup analysis indicated that differences in gender, alcohol intake, and smoking status might influence the association between CDAI and constipation. Smooth curve analysis revealed an "n" shaped relationship between CDAI and constipation among non-alcohol consumers, with a turning point at a CDAI value of 1.08. Conclusion An elevated CDAI is negatively correlated with the incidence of chronic constipation, suggesting that increasing dietary antioxidant intake may reduce constipation prevalence. These findings underscore the importance of dietary antioxidants in maintaining gut health and provide comprehensive guidance for clinical and public health practices.

: This note describes software implementing the SMART(Smooth Multiple Additive Regression Technique) algorithm. SMART generalizes the projection pursuit method to classification and multiple response regression. SMART also provides a more efficient algorithm for single response projection pursuit regression. Originator-supplied keywords include: Multiple response regression, Non parametric regression, Classification, and Discriminant analysis.

Let C be a smooth non-degenerate integral curve of degree d and genus g in P3 over an algebraically closed field of characteristic zero. For each point P in P3 let VP be the linear system on C induced by the hyperplanes through P . By VP one maps C onto a plane curve CP , such a map can be seen as a projection of C from P . If P is not the vertex of a cone of bisecant lines, then CP will have only finitely many singular points; or to put it slightly different: The secant scheme SP = (VP ) 1 2 parametrizing divisors in the second symmetric product C2 that fail to impose independent conditions on VP will be finite. Hence each such point P gives rise to a partition {a1 ≥ a2 ≥ . . . ≥ ak} of ∆(d, g) = 1 2 (d− 1)(d− 2)− g, where the ai are the local multiplicities of the scheme SP . If P is the vertex of a cone of bisecant lines (for example if P is a point of C), we set a1 = ∞. It is clear that the set of points P with a1 ≥ 2 is the surface F of stationary bisecant lines (including some tangent lines); a generic point P on F gives a tacnodial CP . We give two results valid for all curves C. The first one describes the set of points P with a1 ≥ 3. The second result describes the set of points with a1 ≥ 4.

Aiming at the problem of insufficient accuracy of traditional prediction methods for the bearing capacity of CFG pile composite foundation, an intelligent prediction model based on BP neural network is proposed. By constructing a four layer network architecture, the input layer, transition layer, hidden layer, and output layer are organically combined to achieve nonlinear mapping of geological parameter depth effects. Using Min Max standardization and smooth spline regression dimensionality reduction techniques to handle the equivalent transformation of heterogeneous foundation parameters and reduce data dimensionality redundancy. Design a three-level optimization strategy for Global Search global optimization and adaptive BP training to overcome the shortcomings of traditional algorithms that are prone to falling into local optima. The experimental results show that the root mean square error of the model in the test set in North China is 21.3 kPa, with a coefficient of determination of 0.92, which is 53 % higher than the prediction accuracy of the standard formula method. The relative error rate of cross regional validation is controlled at 8.7 %, reflecting the generalization ability to different geological conditions. Parameter sensitivity analysis reveals that pile length and natural porosity ratio are the core influencing factors, with a weight ratio of over 40 %, verifying the physical rationality of the model parameters. The study also pointed out that although the design of transition layers enhances physical interpretability, it increases network complexity; The encoding method of discrete variables such as construction technology suffers from information loss, which affects the accuracy of the model in characterizing process differences. In the future, the interpretability and robustness of the model can be further improved by introducing physical constraint regularization, multi-source data fusion technology, and emerging deep learning architectures.