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**1 - 4**of**4**### )−1ΣY. Keywords Maximum Likelihood Estimation · MLE · Cross Entropy Optimal Coordinates · Mahalanobis distance

"... Abstract One of the basic problems in data analysis lies in choosing the optimal rescaling (change of coordinate system) to study properties of a given data-set Y. The classical Mahalanobis approach has its basis in the classical normalization/rescaling formula Y 3 y → Σ−1/2Y ·(y−mY), where mY denot ..."

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Abstract One of the basic problems in data analysis lies in choosing the optimal rescaling (change of coordinate system) to study properties of a given data-set Y. The classical Mahalanobis approach has its basis in the classical normalization/rescaling formula Y 3 y → Σ−1/2Y ·(y−mY), where mY denotes the mean of Y and ΣY the covariance matrix. Based on the cross-entropy we generalize this approach and define the parameter which measures the fit of a given affine rescaling of Y compared to the Mahalanobis one. This allows in particular to find an optimal change of coordinate system which satisfies some additional conditions. In particular we show that in the case when we put origin of coordinate system in m the optimal choice is given by the transformation Y 3 y → Σ−1/2Y ·(y−mY), where Σ = ΣY (ΣY − (m−mY)(m−mY) T

### The memory centre

, 2014

"... Let x ∈ R be given. As we know the, amount of bits needed to binary code x with given accuracy (h ∈ R) is approximately mh(x) ≈ log2(max{1, |xh |}). We consider the problem where we should translate the origin a so that the mean amount of bits needed to code randomly chosen element from a realizati ..."

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Let x ∈ R be given. As we know the, amount of bits needed to binary code x with given accuracy (h ∈ R) is approximately mh(x) ≈ log2(max{1, |xh |}). We consider the problem where we should translate the origin a so that the mean amount of bits needed to code randomly chosen element from a realization of a random variable X is minimal. In other words, we want to find a ∈ R such that R 3 a → E(mh(X − a)) attains minimum. We show that under reasonable assumptions, the choice of a does not depend on h asymptotically. Consequently, we reduce the problem to finding minimum of the function R 3 a→ R ln(|x − a|)f(x)dx, where f is the density distribution of the random variable X. Moreover, we pro-vide constructive approach for determining a.

### 1k-means Approach to the Karhunen-Loéve Transform

"... Abstract—We present a simultaneous generalization of the well-known Karhunen-Loéve (PCA) and k-means algorithms. The basic idea lies in approximating the data with k affine subspaces of a given dimension n. In the case n = 0 we obtain the classical k-means, while for k = 1 we obtain PCA algorithm. ..."

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Abstract—We present a simultaneous generalization of the well-known Karhunen-Loéve (PCA) and k-means algorithms. The basic idea lies in approximating the data with k affine subspaces of a given dimension n. In the case n = 0 we obtain the classical k-means, while for k = 1 we obtain PCA algorithm. We show that for some data exploration problems this method gives better result then either of the classical approaches. Index Terms—Karhunen-Loéve Transform, PCA, k-Means, optimization, compression, data compression, image compression.

### A local Gaussian filter and adaptive morphology as tools for completing partially discontinuous curves

"... This paper presents a method for extraction and analysis of curve–type struc-tures which consist of disconnected components. Such structures are found in electron–microscopy (EM) images of metal nanograins, which are widely used in the field of nanosensor technology. The topography of metal nanograi ..."

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This paper presents a method for extraction and analysis of curve–type struc-tures which consist of disconnected components. Such structures are found in electron–microscopy (EM) images of metal nanograins, which are widely used in the field of nanosensor technology. The topography of metal nanograins in compound nanomaterials is cru-cial to nanosensor characteristics. The method of completing such templates consists of three steps. In the first step, a local Gaussian filter is used with different weights for each neighborhood. In the second step, an adaptive morphology operation is applied to detect the endpoints of curve segments and connect them. In the last step, pruning is employed to extract a curve which optimally fits the template.

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