Procrustes Analysis of Truncated Least Squares Multidimensional Scaling

Multidimensional Scaling (MDS) is an important class of techniques for embedding sets of patterns in Euclidean space. Most often it is used to visualize in mathbbR3 multidimensional data sets or data sets given by dissimilarity measures that are not distance metrics. Unfortunately, embedding n patterns with MDS involves processing O(n2) pairwise pattern dissimilarities, making MDS computationally demanding for large data sets. Especially in Least Squares MDS (LS-MDS) methods, that proceed by finding a minimum of a multimodal stress function, computational cost is a limiting factor. Several works therefore explored approximate MDS techniques that are less computationally expensive. These approximate methods were evaluated in terms of correlation between Euclidean distances in the embedding and the pattern dissimilarities or value of the stress function. We employ Procrustes Analysis to directly quantify differences between embeddings constructed with an approximate LS-MDS method and embeddings constructed with exact LS-MDS. We then compare our findings to the results of classical analysis, i.e. that based on stress value and correlation between Euclidean distances and pattern dissimilarities. Our results demonstrate that small changes in stress value or correlation coefficient can translate to large differences between embeddings. The differences can be attributed not only to the inevitable variability resulting from the multimodality of the stress function but also to the approximation errors. These results show that approximation may have larger impact on MDS than what was thus far revealed by analyses of stress value and correlation between Euclidean distances and pattern dissimilarities

An evaluation of the impact of shot and receiver lines spacing on seismic data quality – the Wierzbica 3D AGH seismic experiment

An attempt was made to describe the quality of the stacked seismic data semi-quantitatively with respect to the spacing of shot and receiver lines. The methods used included: signal-to-noise ratio calculation, seismic-to-well tie accuracy, wavelet extraction effectiveness and reliability of semi-automated interpretation of seismic attributes. This study was focused on the Ordovician-Silurian interval of the Lublin Basin, Poland, as it was considered as a main target for the exploration of unconventional hydrocarbon deposits. Our results reconfirm the obvious dependency between the density of the acquisition parameters and data quality. However, we also discovered that the seismic data quality is less affected by the shot line spacing than by comparable receiver line spacing. We attributed this issue to the fact of the higher irregularity of the shot points than receiver points, imposed by the terrain accessibility. We have also proven that the regularity of receiver and shot point distribution is crucial for the reliable interpretation of structural seismic attributes, since these were found to be highly sensitive to the acquisition geometry

Zastosowanie samoorganizujących sieci neuronowych Kohonena w klasyfikacji sejsmofacjalnej (rejon Ujkowice - Batycze) Application of Kohonen's Self Organizing Networks in seismofacies classification (the Ujkowice - Batycze area) /

Tyt. nagł.Bibliografia s. 449.Dostępny również w formie drukowanej.STRESZCZENIE: Artykuł przedstawia zastosowanie samoorganizujących sieci neuronowych Kohonena w klasyfikacji formy zapisu sejsmicznego. Klasyfikacja ta jest jednym z podstawowych elementów analizy sejsmofacjalnej, prowadzącej do wyciągnięcia znaczących wniosków poszukiwawczych. Istotnymi elementami takiej analizy są: wybór atrybutów sejsmicznych oraz użycie właściwego sposobu klasteryzacji. Do klasteryzacji użyto atrybutów AVA, które niosą ze sobą informacje o własnościach petrofizycznych skał. W celu zbadania rozkładu facji sejsmicznej na wybranym obszarze posłużono się dodatkowo innymi metodami wielowymiarowej analizy atrybutów sejsmicznych: klasyfikacją wybranego obszaru krossplotu "intercept-gradient" oraz klasteryzacją wykonaną metodą minimalizującą iloczyn odległości obiektów w wydzielanych grupach. Weryfikacji optymalnej metody klasyfikacji danych dokonano na podstawie obserwacji kształtów klastrów i ich charakterystyk. SŁOWA KLUCZOWE: atrybuty sesjmiczne, analiza sejsmofacjalna, krosskorelacja, samoorganizująca sieć neuronowa Kohonena. ABSTRACT: This paper presents the application of Kohonen's Self Organizing Networks in classification of seismic waveform. The classification is one of the basic elements of seismofacies analysis and it often leads to significant exploratory conclusions. Important elements of this kind of analysis are: selection of seismic attributes and usage of appropriate clustering method. There were used AVA attributes, which include information about petrophysical properties of rocks. There used two additional multi-dimensional methods to examine seismic facies distribution on selected area: classification of chosen crossplot intercept-gradient area and classification carried out by method which minimizes the product of objects distances in groups. Verification of optimal method for data classification was made based on observation of clusters shape and their characteristic due to insufficient information from wells. KEYWORDS: seismic attributes, seismofacies analysis, crosscorrelation Kohonen's Self Organizing Network

Natural Solvers in Problems of Searching for the Best Solution

In the paper we present a new method, which can be used as a natural solver for searching the best solution in the multidimensional and multimodal parameter space. The method is based on a well-known simulation technique, i.e., molecular dynamics. To show advantages and disadvantages ofthe particle method in comparison to the standard genetic algorithm, we analyse efficiency of the methods in finding the global minimum of multi-dimensional and multi-modal test-bed functions and we calculate the evaluation indices. We analyse also the ways the solution space is explored and the parameters of algorithms adjusted. The optimal heuristics are proposed. The tests carried out show that the choice of the most appriopriate optimization method depends on type of a problem considered. We show that the particle method is more efficient for finding the optimal solution for multi-modal problems with distinct global extreme, while the genetic algorithm is better for deceptive functions with several locals extreme, which are placed far away from the global optimum. This comes from the different ways in which the particle method and genetic algorithm explore the solution space. The particle method can be used for initial analysis of functions, which character is unknown

NATURAL SOLVERS IN PROBLEMS OF SEARCHING FOR THE BEST SOLUTION

In the paper we present a new method, which can be used as a natural solver for searching the best solution in the multidimensional and multimodal parameter space. The method is based ona well-known simulation techniąue, i.e., molecular dynamics. To show advantages and disadvanta- ges of the particie method in comparison to the standard genetic algorithm, we analyse efficiency of the methods in finding the global minimum of multi-dimensional and multi-modal test-bed functions and we calculate the evaluation indices. We analyse also the ways the solution space is explored and the parameters of algorithms adjusted. The optimal heuristics are proposed. The tests carried out show that the choice of the most appriopriate optimization method depends on type of a problem considered. We show that the particie method is morę efficient for finding the optimal solution for multi-modal problems with distinct global extreme, while the genetic algo­ rithm is better for deceptive functions with several locals extreme, which are placed far away from the global optimum. This comes from the different ways in which the particie method and genetic algorithm explore the solution space. The particie method can be used for initial analysis of functions, which character is unknown