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Optimization of the maximum likelihood estimator for determining the intrinsic dimensionality of high-dimensional data

Karbauskaitė R., Dzemyda G.

International Journal of Applied Mathematics and Computer Science

2015

Vol. 25, no. 4

895--913

One of the problems in the analysis of the set of images of a moving object is to evaluate the degree of freedom of motion and the angle of rotation. Here the intrinsic dimensionality of multidimensional data, characterizing the set of images, can be used. Usually, the image may be represented by a high-dimensional point whose dimensionality depends on the number of pixels in the image. The knowledge of the intrinsic dimensionality of a data set is very useful information in exploratory data analysis, because it is possible to reduce the dimensionality of the data without losing much information. In this paper, the maximum likelihood estimator (MLE) of the intrinsic dimensionality is explored experimentally. In contrast to the previous works, the radius of a hypersphere, which covers neighbours of the analysed points, is fixed instead of the number of the nearest neighbours in the MLE. A way of choosing the radius in this method is proposed. We explore which metric—Euclidean or geodesic—must be evaluated in the MLE algorithm in order to get the true estimate of the intrinsic dimensionality. The MLE method is examined using a number of artificial and real (images) data sets.

On the maximum likelihood estimator in the generalized beta regression model

Rydlewski J. P., Mielczarek D.

Opuscula Mathematica

2012

Vol. 32, no. 4

761-774

The subject of this article is to present the beta – regression model, where we assume that one parameter in the model is described as a combination of algebraically independent continuous functions. The proposed beta model is useful when the dependent variable is continuous and restricted to the bounded interval. The parameters are obtained by maximum likelihood estimation. We prove that estimators are consistent and asymptotically normal.

A note on the maximum likelihood estimator in the gamma regression model

Rydlewski J. P.

Opuscula Mathematica

2009

Vol. 29, no. 3

305-312

This paper considers a nonlinear regression model, in which the dependent variable has the gamma distribution. A model is considered in which the shape parameter of the random variable is the sum of continuous and algebraically independent functions. The paper proves that there is exactly one maximum likelihood estimator for the gamma regression model.

Rates of convergence for the maximum likelihood estimator in the convolution model

Majerski P.

Opuscula Mathematica

2006

Vol. 26, no. 1

99-108

Rates of convergence for the maximum likelihood estimator in the convolution model, obtained recently by S. van de Geer, are reconsidered and corrected.