By Gintautas Dzemyda

This e-book highlights contemporary advancements in multidimensional info visualization, featuring either new equipment and variations on vintage suggestions. in the course of the e-book, a variety of purposes of multidimensional facts visualization are offered together with its makes use of in social sciences (economy, schooling, politics, psychology), environmetrics, and drugs (ophthalmology, recreation drugs, pharmacology, sleep medicine).

The booklet offers contemporary learn leads to optimization-based visualization. Evolutionary algorithms and a two-level optimization procedure, in keeping with combinatorial optimization and quadratic programming, are analyzed intimately. The functionality of those algorithms and the improvement of parallel models are discussed.

The usage of recent visualization recommendations to enhance the capabilies of synthetic neural networks (self-organizing maps, feed-forward networks) is additionally discussed.

The e-book comprises over a hundred distinct photographs proposing examples of the numerous diversified visualization options that the booklet presents.

This ebook is meant for scientists and researchers in any box of analysis the place advanced and multidimensional info needs to be represented visually.

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**Extra info for Multidimensional Data Visualization: Methods and Applications**

**Sample text**

2 Dimensionality Reduction 31 where t denotes the order number of iteration and η is an optimization step parameter. The coordinates of all m points Yi ∈ R2 , i = 1, . . , m, are recomputed at an iteration. 19) ∑ δi j . i< j In fact, the results of ES (Y ) minimization depend on η and on the initial coordinates of points Y1 ,Y2 , . . ,Ym . 4]. However, a larger value η can be used as well [53]. The analysis of a relative performance of different algorithms in reducing the dimensionality indicates that Sammon’s mapping is one of widely used visualization methods.

E. ∀i d(Xi , Xi1 ) < d(Xi , Xi2 ) < · · · < d(Xi , Xik ) =⇒ d(Yi ,Yi1 ) < d(Yi ,Yi2 ) < · · · < d(Yi ,Yik ), where: • d(Xi , Xi j ) is the distance between the points Xi and Xi j , (Xi , Xi j ∈ Rn ). • d(Yi ,Yi j ) is the distance between the points Yi and Yi j , (Yi ,Yi j ∈ Rd ). The measure, estimating how exactly the topology is preserved, is called a topology preservation measure. The measure, estimating how exactly the distances between all the points are preserved, is called a distance preservation measure.

M, is performed, where Ad is a 3. The transformation Yi = (Xi − X)A d-column matrix consisting of the eigenvectors corresponding to the highest d eigenvalues of matrix S. Here X¯ = (x¯1 , x¯2 , . . , x¯n ) is the vector of averages of the features. The difference between LDA and PCA is that, in addition, the known classes of objects are applied. The Iris data set, visualized by LDA, is presented in Fig. 20. 4 Multidimensional Scaling Multidimensional scaling (MDS) refers to a group of methods that are widely used for dimensionality reduction and visualization of multidimensional data [14].