scholarly journals A construction of the minimum volume ellipsoid containing a set of points using BRKGA metaheuristic

2021 ◽  
Author(s):  
Antonio A. M. Raposo ◽  
Valeska Martins de Souza ◽  
Luís Roberto A. G. Filho
Keyword(s):  
Minimum Volume ◽  
Minimum Volume Ellipsoid ◽  
Set Of Points

scholarly journals Antioptimization of earthquake exitation and response

1998 ◽  
Vol 4 (1) ◽  
pp. 1-19 ◽  
Author(s):  
G. Zuccaro ◽  
I. Elishakoff ◽  
A. Baratta
Keyword(s):  
Historical Data ◽  
Dimensional Space ◽  
Stochastic Approach ◽  
Viable Alternative ◽  
Maximum Response ◽  
Minimum Volume ◽  
Earthquake Excitation ◽  
Novel Approach ◽  
Space Modeling ◽  
Minimum Volume Ellipsoid

The paper presents a novel approach to predict the response of earthquake-excited structures. The earthquake excitation is expanded in terms of series of deterministic functions. The coefficients of the series are represented as a point inN-dimensional space. Each available ccelerogram at a certain site is then represented as a point in the above space, modeling the available fragmentary historical data. The minimum volume ellipsoid, containing all points, is constructed. The ellipsoidal models of uncertainty, pertinent to earthquake excitation, are developed. The maximum response of a structure, subjected to the earthquake excitation, within ellipsoidal modeling of the latter, is determined. This procedure of determining least favorable response was termed in the literature (Elishakoff, 1991) as an antioptimization. It appears that under inherent uncertainty of earthquake excitation, antioptimization analysis is a viable alternative to stochastic approach.

scholarly journals The Arithmetic Tutte polynomial of two matrices associated to Trees

2018 ◽  
Vol 6 (1) ◽  
pp. 310-322
Author(s):  
R. B. Bapat ◽  
Sivaramakrishnan Sivasubramanian
Keyword(s):  
Analogous Result ◽  
Polynomial Matrix ◽  
Tutte Polynomial ◽  
Distance Matrix ◽  
Minimum Volume ◽  
Maximum Volume ◽  
Recent Interest ◽  
Minimum Volume Ellipsoid ◽  
Tutte Polynomials ◽  
Exponential Distance

Abstract Arithmetic matroids arising from a list A of integral vectors in Zn are of recent interest and the arithmetic Tutte polynomial MA(x, y) of A is a fundamental invariant with deep connections to several areas. In this work, we consider two lists of vectors coming from the rows of matrices associated to a tree T. Let T = (V, E) be a tree with |V| = n and let LT be the q-analogue of its Laplacian L in the variable q. Assign q = r for r ∈ ℤ with r/= 0, ±1 and treat the n rows of LT after this assignment as a list containing elements of ℤn. We give a formula for the arithmetic Tutte polynomial MLT (x, y) of this list and show that it depends only on n, r and is independent of the structure of T. An analogous result holds for another polynomial matrix associated to T: EDT, the n × n exponential distance matrix of T. More generally, we give formulae for the multivariate arithmetic Tutte polynomials associated to the list of row vectors of these two matriceswhich shows that even the multivariate arithmetic Tutte polynomial is independent of the tree T. As a corollary, we get the Ehrhart polynomials of the following zonotopes: - ZEDT obtained from the rows of EDT and - ZLT obtained from the rows of LT. Further, we explicitly find the maximum volume ellipsoid contained in the zonotopes ZEDT, ZLT and show that the volume of these ellipsoids are again tree independent for fixed n, q. A similar result holds for the minimum volume ellipsoid containing these zonotopes.

Experimental Data Have to Decide Which of the Nonprobabilistic Uncertainty Descriptions—Convex Modeling or Interval Analysis—to Utilize

2008 ◽  
Vol 75 (4) ◽  
Author(s):  
Xiaojun Wang ◽  
Isaac Elishakoff ◽  
Zhiping Qiu
Keyword(s):  
Experimental Data ◽  
Interval Analysis ◽  
Structural Response ◽  
Main Idea ◽  
Minimum Volume ◽  
Analytical Treatment ◽  
Numerical Examples ◽  
Minimum Volume Ellipsoid ◽  
Nonprobabilistic Uncertainty ◽  
Existing Data

This study shows that the type of the analytical treatment that should be adopted for nonprobabilistic analysis of uncertainty depends on the available experimental data. The main idea is based on the consideration that the maximum structural response predicted by the preferred theory ought to be minimal, and the minimum structural response predicted by the preferred theory ought to be maximal, to constitute a lower overestimation. Prior to the analysis, the existing data ought to be enclosed by the minimum-volume hyper-rectangle V1 that contains all experimental data. The experimental data also have to be enclosed by the minimum-volume ellipsoid V2. If V1 is smaller than V2 and the response calculated based on it R(V1) is smaller than R(V2), then one has to prefer interval analysis. However, if V1 is in excess of V2 and R(V1) is greater than R(V2), then the analyst ought to utilize convex modeling. If V1 equals V2 or these two quantities are in close vicinity, then two approaches can be utilized with nearly equal validity. Some numerical examples are given to illustrate the efficacy of the proposed methodology.

scholarly journals Modifikasi Penaksir Robust dalam Pelabelan Outlier Multivariat

2018 ◽  
Vol 14 (1) ◽  
pp. 46
Author(s):  
Erna Tri Herdiani
Keyword(s):  
Minimum Volume ◽  
Data Set ◽  
Minimum Volume Ellipsoid ◽  
Minimum Vector

Outlier adalah suatu observasi yang polanya tidak mengikuti mayoritas data. Outlier dalam kasus multivariat sangat sulit untuk dideteksi, khususnya ketika dimensi lebih dari 2. Kesulitan ini meningkat ketika data set berukuran besar, yakni jumlah variabel menjadi besar. Metode-metode pendeteksian outlier telah lama berkembang dan beberapa digunakan untuk pelabelan outlier sehingga data dapat dipisahkan antara data yang dicurigai sebagai outlier dan data set pada umumnya. Metode-metode tersebut adalah minimum volume ellipsoid disingkat MVE, minimun covariance determinant disingkat MCD, dan minimum vector variance disingkat MVV. Dari ketiga metode tersebut MVV memiliki waktu perhitungan yang paling cepat. Berdasarkan algoritma MVV, kriteria mengurutkan data menggunakan jarak mahalanobis, maka pada paper ini akan dimodifikasi kriteria pengurutan data dengan menghindari penulisan dalam bentuk invers dari matriks variansi kovariansi. Hasil yang diperoleh adalah metode MVV menjadi lebih cepat dengan menggunakan kriteria baru dengan kecermatan yang sama dengan MVV sebelumnya serta akan diaplikan untuk data real dan data simulasi.

scholarly journals Mahalanobis distances for ecological niche modelling and outlier detection: implications of sample size, error, and bias for selecting and parameterising a multivariate location and scatter method

PeerJ ◽  
2021 ◽  
Vol 9 ◽  
pp. e11436
Author(s):  
Thomas R. Etherington
Keyword(s):  
Ecological Niche ◽  
Minimum Volume ◽  
Minimum Covariance Determinant ◽  
Niche Modelling ◽  
Data Set ◽  
Sample Mean ◽  
Mahalanobis Distances ◽  
Scatter Method ◽  
Multivariate Location ◽  
Minimum Volume Ellipsoid

The Mahalanobis distance is a statistical technique that has been used in statistics and data science for data classification and outlier detection, and in ecology to quantify species-environment relationships in habitat and ecological niche models. Mahalanobis distances are based on the location and scatter of a multivariate normal distribution, and can measure how distant any point in space is from the centre of this kind of distribution. Three different methods for calculating the multivariate location and scatter are commonly used: the sample mean and variance-covariance, the minimum covariance determinant, and the minimum volume ellipsoid. The minimum covariance determinant and minimum volume ellipsoid were developed to be robust to outliers by minimising the multivariate location and scatter for a subset of the full sample, with the proportion of the full sample forming the subset being controlled by a user-defined parameter. This outlier robustness means the minimum covariance determinant and the minimum volume ellipsoid are highly relevant for ecological niche analyses, which are usually based on natural history observations that are likely to contain errors. However, natural history observations will also contain extreme bias, to which the minimum covariance determinant and the minimum volume ellipsoid will also be sensitive. To provide guidance for selecting and parameterising a multivariate location and scatter method, a series of virtual ecological niche modelling experiments were conducted to demonstrate the performance of each multivariate location and scatter method under different levels of sample size, errors, and bias. The results show that there is no optimal modelling approach, and that choices need to be made based on the individual data and question. The sample mean and variance-covariance method will perform best on very small sample sizes if the data are free of error and bias. At larger sample sizes the minimum covariance determinant and minimum volume ellipsoid methods perform as well or better, but only if they are appropriately parameterised. Modellers who are more concerned about the prevalence of errors should retain a smaller proportion of the full data set, while modellers more concerned about the prevalence of bias should retain a larger proportion of the full data set. I conclude that Mahalanobis distances are a useful niche modelling technique, but only for questions relating to the fundamental niche of a species where the assumption of multivariate normality is reasonable. Users of the minimum covariance determinant and minimum volume ellipsoid methods must also clearly report their parameterisations so that the results can be interpreted correctly.

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