Krzysztof Domino

Krzysztof Domino, Piotr Gawron, Łukasz Pawela

In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the $d^\text{th}$ order cumulant can be presented in the form of an $d$-dimensional tensor, the algorithm is presented using tensor operations. The algorithm provided in the paper takes advantage of super-symmetry of cumulant and moment tensors. We show that the proposed algorithm considerably reduces the computational complexity and the computational memory requirement of cumulant calculation as compared with existing algorithms. For the sizes of interest, the reduction is of the order of $d!$ compared to the naive algorithm.

Krzysztof Domino

The basic goal of computer engineering is the analysis of data. Such data are often large data sets distributed according to various distribution models. In this manuscript we focus on the analysis of non-Gaussian distributed data. In the case of univariate data analysis we discuss stochastic processes with auto-correlated increments and univariate distributions derived from specific stochastic processes, i.e. Levy and Tsallis distributions. Deep investigation of multivariate non-Gaussian distributions requires the copula approach. A copula is an component of multivariate distribution that models the mutual interdependence between marginals. There are many copula families characterised by various measures of the dependence between marginals. Importantly, one of those are `tail' dependencies that model the simultaneous appearance of extreme values in many marginals. Those extreme events may reflect a crisis given financial data, outliers in machine learning, or a traffic congestion. Next we discuss higher order multivariate cumulants that are non-zero if multivariate distribution is non-Gaussian. However, the relation between cumulants and copulas is not straight forward and rather complicated. We discuss the application of those cumulants to extract information about non-Gaussian multivariate distributions, such that information about non-Gaussian copulas. The use of higher order multivariate cumulants in computer science is inspired by financial data analysis, especially by the safe investment portfolio evaluation. There are many other applications of higher order multivariate cumulants in data engineering, especially in: signal processing, non-linear system identification, blind sources separation, and direction finding algorithms of multi-source signals.

Agnieszka Anna Tomaka, Krzysztof Domino, Dariusz Pojda et al.

A computational method for quantitative analysis of temporomandibular joint (TMJ) configuration using occlusal positioning splints is proposed and demonstrated. The method models a positioning splint as a physical realization of a predefined rigid transformation of the mandible, derived from multimodal data, including CBCT, facial motion acquisition, and dental scans integrated within a common coordinate system. Splints corresponding to selected mandibular positions are designed and fabricated, and their positioning accuracy is evaluated using repeated scans of plaster models. Discrepancies are represented as error transformations and analyzed statistically in the space of rigid motions. The estimated transformations are propagated to segmented TMJ structures, enabling simulation-based evaluation of joint space changes. Transformation-based error analysis and surface distance metrics are used to quantify differences between planned and achieved configurations. The method enables indirect assessment of TMJ configuration using a single anatomical model and transformation data, reducing the need for repeated imaging across multiple mandibular positions. This study is intended as a methodological demonstration, supported by a clear step-by-step graphical presentation, and does not aim to provide clinical validation.

Agnieszka A. Tomaka, Leszek Luchowski, Dariusz Pojda et al.

The purpose of this chapter is to discuss methods of acquisition, visualization and analysis of the dynamics of a complex biomedical system, illustrated by the human stomatognathic system. The stomatognathic system consists of the teeth and the skull bones with the maxilla and the mandible. Its dynamics can be described by the change of mutual position of the lower/mandibular part versus the upper/maxillary one due to the physiological motion of opening, chewing and swallowing. In order to analyse the dynamics of the stomatognathic system its morphology and motion has to be digitized, which is done using static and dynamic multimodal imagery like CBCT and 3D scans data and temporal measurements of motion. The integration of multimodal data incorporates different direct and indirect methods of registration - aligning of all the data in the same coordinate system. The integrated sets of data form 4D multimodal data which can be further visualized, modeled, and subjected to multivariate time series analysis. Example results are shown. Although there is no direct method of imaging the TMJ motion, the integration of multimodal data forms an adequate tool. As medical imaging becomes ever more diverse and ever more accessible, organizing the imagery and measurements into unified, comprehensive records can deliver to the doctor the most information in the most accessible form, creating a new quality in data simulation, analysis and interpretation.

Przemysław Głomb, Krzysztof Domino, Michał Romaszewski et al.

In the small target detection problem a pattern to be located is on the order of magnitude less numerous than other patterns present in the dataset. This applies both to the case of supervised detection, where the known template is expected to match in just a few areas and unsupervised anomaly detection, as anomalies are rare by definition. This problem is frequently related to the imaging applications, i.e. detection within the scene acquired by a camera. To maximize available data about the scene, hyperspectral cameras are used; at each pixel, they record spectral data in hundreds of narrow bands. The typical feature of hyperspectral imaging is that characteristic properties of target materials are visible in the small number of bands, where light of certain wavelength interacts with characteristic molecules. A target-independent band selection method based on statistical principles is a versatile tool for solving this problem in different practical applications. Combination of a regular background and a rare standing out anomaly will produce a distortion in the joint distribution of hyperspectral pixels. Higher Order Cumulants Tensors are a natural `window' into this distribution, allowing to measure properties and suggest candidate bands for removal. While there have been attempts at producing band selection algorithms based on the 3 rd cumulant's tensor i.e. the joint skewness, the literature lacks a systematic analysis of how the order of the cumulant tensor used affects effectiveness of band selection in detection applications. In this paper we present an analysis of a general algorithm for band selection based on higher order cumulants. We discuss its usability related to the observed breaking points in performance, depending both on method order and the desired number of bands. Finally we perform experiments and evaluate these methods in a hyperspectral detection scenario.

Krzysztof Domino, Piotr Gawron

High order cumulant tensors carry information about statistics of non-normally distributed multivariate data. In this work we present a new efficient algorithm for calculation of cumulants of arbitrary order in a sliding window for data streams. We showed that this algorithms enables speedups of cumulants updates compared to current algorithms. This algorithm can be used for processing on-line high-frequency multivariate data and can find applications in, e.g., on-line signal filtering and classification of data streams. To present an application of this algorithm, we propose an estimator of non-Gaussianity of a data stream based on the norms of high-order cumulant tensors. We show how to detect the transition from Gaussian distributed data to non-Gaussian ones in a~data stream. In order to achieve high implementation efficiency of operations on super-symmetric tensors, such as cumulant tensors, we employ the block structure to store and calculate only one hyper-pyramid part of such tensors.

Krzysztof Domino

The cumulant analysis plays an important role in non Gaussian distributed data analysis. The shares' prices returns are good example of such data. The purpose of this research is to develop the cumulant based algorithm and use it to determine eigenvectors that represent investment portfolios with low variability. Such algorithm is based on the Alternating Least Square method and involves the simultaneous minimisation 2'nd -- 6'th cumulants of the multidimensional random variable (percentage shares' returns of many companies). Then the algorithm was tested during the recent crash on the Warsaw Stock Exchange. To determine incoming crash and provide enter and exit signal for the investment strategy the Hurst exponent was calculated using the local DFA. It was shown that introduced algorithm is on average better that benchmark and other portfolio determination methods, but only within examination window determined by low values of the Hurst exponent. Remark that the algorithm of is based on cumulant tensors up to the 6'th order calculated for a multidimensional random variable, what is the novel idea. It can be expected that the algorithm would be useful in the financial data analysis on the world wide scale as well as in the analysis of other types of non Gaussian distributed data.