Location adjustment for the minimum volume ellipsoid estimator.

Estimating multivariate location and scatter with both affine equivariance and positive breakdown has always been difficult. A well-known estimator which satisfies both properties is the Minimum Volume Ellipsoid Estimator (MVE). Computing the exact MVE is often not feasible, so one usually resorts to an approximate algorithm. In the regression setup, algorithms for positive-breakdown estimators like Least Median of Squares typically recompute the intercept at each step, to improve the result. This approach is called intercept adjustment. In this paper we show that a similar technique, called location adjustment, can be applied to the MVE. For this purpose we use the Minimum Volume Ball (MVB), in order to lower the MVE objective function. An exact algorithm for calculating the MVB is presented. As an alternative to MVB location adjustment we propose L-1 location adjustment, which does not necessarily lower the MVE objective function but yields more efficient estimates for the location part. Simulations compare the two types of location adjustment. We also obtain the maxbias curves of both L-1 and the MVB in the multivariate setting, revealing the superiority of L-1.Model;

Robust Wilks' Statistic based on RMCD for One-Way Multivariate Analysis of Variance (MANOVA)

The classical Wilks' statistic is the most using for testing the hypotheses of equal mean vectors of several multivariate normal populations for one-way MANOVA. It is extremely sensitive to the influence of outliers. Therefore, the robust Wilks' statistic based on reweighted minimum covariance determinant (RMCD) estimator with Hampel weighted function has been proposed. The distribution of the proposed statistic differs from the classical one. Mont Carlo simulations are used to evaluate the performance of the test statistic under the normal and contaminated distribution for the data set. Moreover, the type I error rate and power of test have been considered as statistical measures to comparison between the classical and the robust statistics. The results show that, the robust Wilks' statistic based on RMCD is closely to the classical Wilks' statistic in case of normal distribution for the data set while in case of contaminated distribution the method in question is the best. Keywords: One-Way Multivariate Analysis of Variance, Wilks' Statistic, Outliers, Robustness, Minimum Covariance Determinant Estimator.

Summary Conclusions: Computation of Minimum Volume Covering Ellipsoids*

We present a practical algorithm for computing the minimum volume n-dimensional ellipsoid that must contain m given points a₁,..., am â Rn. This convex constrained problem arises in a variety of applied computational settings, particularly in data mining and robust statistics. Its structure makes it particularly amenable to solution by interior-point methods, and it has been the subject of much theoretical complexity analysis. Here we focus on computation. We present a combined interior-point and active-set method for solving this problem. Our computational results demonstrate that our method solves very large problem instances (m = 30,000 and n = 30) to a high degree of accuracy in under 30 seconds on a personal computer.Singapore-MIT Alliance (SMA

Relatively Smooth Convex Optimization by First-Order Methods, and Applications

The usual approach to developing and analyzing first-order methods for smooth convex optimization assumes that the gradient of the objective function is uniformly smooth with some Lipschitz constant L. However, in many settings the differentiable convex function f(?) is not uniformly smooth-for example, in D-optimal design where f(x) := -ln det(HXHT) and X := Diag(x), or even the univariate setting with f(x) := -ln(x)+x2. In this paper we develop a notion of "relative smoothness" and relative strong convexity that is determined relative to a user-specified "reference function" h(?) (that should be computationally tractable for algorithms), and we show that many differentiable convex functions are relatively smooth with respect to a correspondingly fairly simple reference function h(?). We extend two standard algorithms-the primal gradient scheme and the dual averaging scheme-to our new setting, with associated computational guarantees. We apply our new approach to develop a new first-order method for the D-optimal design problem, with associated computational complexity analysis. Some of our results have a certain overlap with the recent work [H. H. Bauschke, J. Bolte, and M. Teboulle, Math. Oper. Res., 42 (2017), pp. 330-348]

Analysis of the Frank-Wolfe Method for Convex Composite Optimization involving a Logarithmically-Homogeneous Barrier

We present and analyze a new generalized Frank-Wolfe method for the composite optimization problem $(P):{\min}_{x\in\mathbb{R}^n}\; f(\mathsf{A} x) + h(x)$, where $f$ is a $\theta$-logarithmically-homogeneous self-concordant barrier, $\mathsf{A}$ is a linear operator and the function $h$ has bounded domain but is possibly non-smooth. We show that our generalized Frank-Wolfe method requires $O((\delta_0 + \theta + R_h)\ln(\delta_0) + (\theta + R_h)^2/\varepsilon)$ iterations to produce an $\varepsilon$-approximate solution, where $\delta_0$ denotes the initial optimality gap and $R_h$ is the variation of $h$ on its domain. This result establishes certain intrinsic connections between $\theta$-logarithmically homogeneous barriers and the Frank-Wolfe method. When specialized to the $D$-optimal design problem, we essentially recover the complexity obtained by Khachiyan using the Frank-Wolfe method with exact line-search. We also study the (Fenchel) dual problem of $(P)$, and we show that our new method is equivalent to an adaptive-step-size mirror descent method applied to the dual problem. This enables us to provide iteration complexity bounds for the mirror descent method despite even though the dual objective function is non-Lipschitz and has unbounded domain. In addition, we present computational experiments that point to the potential usefulness of our generalized Frank-Wolfe method on Poisson image de-blurring problems with TV regularization, and on simulated PET problem instances.Comment: See Version 1 (v1) for the analysis of the Frank-Wolfe method with adaptive step-size applied to the H\"older smooth function

Perbandingan Penggunaan Penduga Robust Minimum Volume Ellipsoid (Mve) Dan Minimum Covariance Determinant (Mcd) Pada Analisis Diskriminan Kuadratik

"Analisis diskriminan (Discriminant Analysis) adalah metode analisis multivariat yang bertujuan untuk mencari fungsi pembeda pada dua atau lebih kelompok respon (Johnson & Wichern, 1992). Metode ini umum digunakan untuk pengklasifikasian suatu observasi kedalam kelompok yang saling bebas dan menyeluruh berdasarkan sejumlah variabel bebas. Namun, dalam analisis diskriminan klasik sangatlah sensitif terhadap kondisi data yang mengandung pencilan. Sifat tidak robust dari penduga diskriminan klasik ini akan menyebabkan fungsi pemisahan yang terbentuk juga tidak robust sehingga mempengaruhi hasil pengklasifikasian. Selain itu, dalam analisis diskriminan terdapat salah satu asumsi yakni homogenitas matriks kovariansi. Pelanggaran pada asumsi ini ditangani menggunakan analisis diskriminan kuadratik. Kombinasi antara fungsi diskriminan kuadratik dan fungsi diskriminan robust dapat mengatasi permasalahan perbedaan matrik kovariansi dan kondisi data yang mengandung pencilan. Dalam penelitian ini akan dibandingkan penggunaan penaksir robust dengan high breakdown point yakni penduga MVE dan MCD dalam analisis diskriminan kuadratik dengan melihat nilai APER, sensitivitas, dan spesifisitas yang dihasilkan. Untuk mengukur kinerja MVE dan MCD dalam analisis diskriminan kuadratik pada penelitian ini, dilakukan simulasi terhadap data bangkitan dengan 2 kelompok populasi yang berdistribusi normal p variat (p = 4, 6, dan 8) serta memiliki ukuran sampel berturut-turut pada masing-masing kelompok populasi sebesar n = 40, 100, dan 200 yang diberikan tingkat pencilan 5%, 10%, 20%, dan 25%. Berdasarkan hasil simulasi pada 100 kali pengulangan, penduga MCD menghasilkan nilai APER yang lebih kecil serta nilai sensitivitas, dan spesifisitas yang lebih besar dibandingkan penduga MVE maupun Klasik. Oleh karena itu, dapat disimpulkan bahwa penggunaan penduga MCD pada analisis diskriminan kuadratik lebih baik pada kondisi data yang mengandung pencilan

Location adjustment for the minimum volume ellipsoid estimator

Estimating multivariate location and scatter with both affine equivariance and positive breakdown has always been difficult. A well-known estimator which satisfies both properties is the Minimum Volume Ellipsoid Estimator (MVE). Computing the exact MVE is often not feasible, so one usually resorts to an approximate algorithm. In the regression setup, algorithms for positive-breakdown estimators like Least Median of Squares typically recompute the intercept at each step, to improve the result. This approach is called intercept adjustment. In this paper we show that a similar technique, called location adjustment, can be applied to the MVE. For this purpose we use the Minimum Volume Ball (MVB), in order to lower the MVE objective function. An exact algorithm for calculating the MVB is presented. As an alternative to MVB location adjustment we propose L-1 location adjustment, which does not necessarily lower the MVE objective function but yields more efficient estimates for the location part. Simulations compare the two types of location adjustment. We also obtain the maxbias curves of both L-1 and the MVB in the multivariate setting, revealing the superiority of L-1.status: publishe