Multidimensional Data Space Research Articles

DKPE-GraphSYN: a drug synergy prediction model based on joint dual kernel density estimation and positional encoding for graph representation.

Introduction: Synergistic medication, a crucial therapeutic strategy in cancer treatment, involves combining multiple drugs to enhance therapeutic effectiveness and mitigate side effects. Current research predominantly employs deep learning models for extracting features from cell line and cancer drug structure data. However, these methods often overlook the intricate nonlinear relationships within the data, neglecting the distribution characteristics and weighted probability densities of gene expression data in multi-dimensional space. It also fails to fully exploit the structural information of cancer drugs and the potential interactions between drug molecules. Methods: To overcome these challenges, we introduce an innovative end-to-end learning model specifically tailored for cancer drugs, named Dual Kernel Density and Positional Encoding (DKPE) for Graph Synergy Representation Network (DKPEGraphSYN). This model is engineered to refine the prediction of drug combination synergy effects in cancer. DKPE-GraphSYN utilizes Dual Kernel Density Estimation and Positional Encoding techniques to effectively capture the weighted probability density and spatial distribution information of gene expression, while exploring the interactions and potential relationships between cancer drug molecules via a graph neural network. Results: Experimental results show that our prediction model achieves significant performance enhancements in forecasting drug synergy effects on a comprehensive cancer drug and cell line synergy dataset, achieving an AUPR of 0.969 and an AUC of 0.976. Discussion: These results confirm our model's superior accuracy in predicting cancer drug combinations, providing a supportive method for clinical medication strategy in cancer.

Применение агентной модели интерактивной визуализации для создания средств визуального управления онтологическими данными

The paper presents an agent-based model of interactive visualization and proposes a method for its application in the development of the ontological data visual management tool. The process of user interaction with the visualization is represented as a state graph, where each node is a separate visualization. The initial state is known, and the final state is formed in the process of user interaction with the ontological data visual management tool. Each intermediate visualization from the state graph is transferred into the multidimensional Euclidean data space formed on the basis of the visualized ontology, which allows to calculate weighting coefficients on the state graph edges and to search the graph using known algorithms. Application of the model allows for a reduction in the labor intensity of executing user scripts due to the reduction of step-by-step visualization creation to the task of searching in the state space.

Robust reservoir identification by multi-well cluster analysis of wireline logging data

A novel clustering method is applied to well logs for improved rock type identification in hydrocarbon formations. For grouping the objects in the multi-dimensional data space, we propose a Most Frequent Value (MFV) based clustering technique applied to natural gamma ray, bulk density, sonic, photoelectric index, and resistivity logs. The MFV method is a robust estimator, which assists in finding the cluster centers more reliably than a more noise sensitive K-means clustering approach. The result of K-means cluster analysis highly depends on the choose of the initial centroids. To reduce the risk of inappropriately chosen starting values, we apply a histogram-based selection method to give the best position of the initial cluster centers. We assure the robustness of the solution by calculating the centroid as the MFV of the cluster elements and defining the overall deviation of cluster elements from the center by a weighted Euclidean (Steiner-) distance. The proposed workflow relies on a fully automated weighting of the cluster elements, which does not require a constraint on the statistical distribution of the observed variables. The processing of synthetic data shows high noise rejection capability and efficient cluster recognition even beside considerable amount of outlying and missing data; the accuracy is measured by the difference between the estimated and the exactly known distribution of cluster numbers. The clustering tool is first applied to single borehole data, then the procedure is extended to multi-well logging datasets to reconstruct the multi-dimensional spatial distributions of clusters revealing the lithological and petrophysical characteristics of the studied formations. A large in situ dataset acquired from several boreholes traversing Hungarian gas-bearing clastic reservoirs of Miocene age is analyzed. The accuracy of the field results is confirmed by core permeability measurements, independent well log analysis and a gradient metrics characterizing the noise rejection capability of the clustering method.

Neural network analysis of energy efficiency of the regional economy as a factor of Russia's sustainable development under conditions of big challenges

Subject. We consider the energy efficiency of Russia’s regional economy and its impact on sustainable development of the country in the face of big challenges. Objectives. The focus is on solving the multidimensional task of analyzing the development of energy efficiency of the economy of Russian regions, which relates to difficult-to-formalize tasks and harmonizes with modern requirements of competitiveness. Methods. We employ the cluster analysis based on neural networks, which are a relevant component of artificial intelligence. We also use the toolkit of artificial neural networks, i.e. Kohonen self-organizing maps. The said tools are free from model limitations and external interference in the functioning of the neural network, and enable to visualize the clustering results of multidimensional data space on the plane. Results. The cluster analysis of heterogeneous data enabled to distribute Russian regions across eight cluster formations. The considered indicators characterizing the energy efficiency of Russia’s regional economy had different effects on the creation of clusters. We obtained a significant unevenness of the distribution of Russian regions by cluster: the number of regions in clusters varied more than fourfold. We determined different levels of energy efficiency of the regions’ economy according to the studied indicators on cluster scale. This requires the application of different economic development strategies for regions of the Russian Federation in the focus of cluster formations. Conclusions. The paper shows the influence of big challenges on the development of energy efficiency of Russia. The findings indicate that to improve the sustainable and progressive development of Russia’s economy, innovative organizational and managerial methods are needed that generate a vector of orientation towards effective solution of urgent challenges facing the country.

Analysis of human capital development in Russia by means of physical culture and sports using neural network modeling

Subject. The article investigates the human capital in the regions of the Russian Federation in the context of health promotion and active longevity of the population of the country. Objectives. The purpose is to examine the state of human capital driven by the development of physical culture and sports in Russian regions, using neural network modeling. Methods. Neural networks are used as a method of studying the multidimensional statistical data. The method of data clustering based on neural network modeling is not affiliated with model constraints. This method is aimed at self–training of neural networks, i.e. self-organizing Kohonen maps, and is a promising means of visual representation of multidimensional data space. Results. Using the neural network modeling, we placed 85 Russian regions in six cluster formations. We assessed the influence of each studied indicator on cluster construction. The paper demonstrates a strong difference in the number of regions of the Russian Federation in clusters, presents average values of the considered indicators in clusters for 2021. Conclusions. The study shows uneven development of human capital in Russian regions from the point of view of physical culture and sports, which predetermines different strategies for the development of the regional sports sphere. The positive trend in the number of sports facilities, personnel, and population engaged in physical culture and sports is facilitated by a systems approach based on program documents, which focuses on strengthening the health of citizens and their active longevity.

Granular data representation under privacy protection: Tradeoff between data utility and privacy via information granularity

Information granules describe available experimental evidence at a more abstract level and facilitate the concise characterization of the structure of the numeric data. The utilization of unreliable data collectors poses a key challenge with respect to the privacy of the participants. In order to guarantee data confidentiality, we propose a granular data publishing method under differential privacy. In this study, a two-stage design of information granules considering the requirement of data privacy is proposed. The main idea is to protect individual sensitive pieces of data information against inference attacks and optimize the representation capabilities of information granules in a multi-dimensional data space. In the first stage, a clustering algorithm based on differential privacy is proposed to find some numeric prototypes in a numeric space, which considers sensitive data privacy protection. Second, this paper constructs spherical information granules involving the requirement of differential privacy. Their multi-dimensional and fuzzy granularity characteristics are considered. An improvement of the principle of justifiable granularity is utilized to optimize the bounds of information granules, and a new objective function is established to optimize the information granules. Our work is devoted to tradeoff data privacy and utility by constructing reasonable information granules. Unlike numerical data publishing, the publishing of interval information granules improves data utility in data protection. Experimental results demonstrate the superiority of the proposed granular description method based on differential privacy.

Pattern Labelling of Business Communication Data

Systematic pattern recognition as well as the corresponding description of determined patterns entail numerous challenges in the application context of high-dimensional communication data. These can cause increased effort, especially with regard to machine-based processing concerning the determination of regularities in underlying datasets. Due to the increased expansion of dimensions in multidimensional data spaces, determined patterns are no longer interpretable by humans. Taking these challenges into account, this paper investigates to what extent pre-defined communication patterns can be interpreted for the application area of high-dimensional business communication data. An analytical perspective is considered by taking into account a holistic research approach and by subsequently applying selected Machine Learning methods from Association Rule Discovery, Topic Modelling and Decision Trees with regard to the overall goal of semi-automated pattern labelling. The results show that meaningful descriptions can be derived for the interpretation of pre-defined patterns.

Triboinformatics: machine learning algorithms and data topology methods for tribology

Friction and wear are very common phenomena found virtually everywhere. However, it is very difficult to predict tribological (i.e. related to friction and wear) structure–property relationships from fundamental physical principles. Consequently, tribology remains a data-driven, mostly empirical discipline. With the advent of new machine learning (ML) and artificial intelligence methods, it becomes possible to establish new correlations in tribological data to predict and control better the tribological behavior of novel materials. Hence, the new area of triboinformatics has emerged combining tribology with data science. This paper reviews ML algorithms used to establish correlations between the structures of metallic alloys and composite materials, tribological test conditions, friction and wear. This paper also discusses novel methods of surface roughness analysis involving the concept of data topology in multidimensional data space, as applied to macro- and nanoscale roughness. Other triboinformatic approaches are considered as well.

The benefits and dangers of using artificial intelligence in petrophysics

Artificial Intelligence, or AI, is a method of data analysis that learns from data, identify patterns and makes predictions with the minimal human intervention. AI is bringing many benefits to petrophysical evaluation. Using case studies, this paper describes several successful applications. The future of AI has even more potential. However, if used carelessly there are potentially grave consequences. A complex Middle East Carbonate field needed a bespoke shaly water saturation equation. AI was used to ‘evolve’ an ideal equation, together with field specific saturation and cementation exponents. One UKCS gas field had an ‘oil problem’. Here, AI was used to unlock the hidden fluid information in the NMR T1 and T2 spectra and successfully differentiate oil and gas zones in real time. A North Sea field with 30 wells had shear velocity data (Vs) in only 4 wells. Vs was required for reservoir modelling and well bore stability prediction. AI was used to predict Vs in all 30 wells. Incorporating high vertical resolution data, the Vs predictions were even better than the recorded logs. As it is not economic to take core data on every well, AI is used to discover the relationships between logs, core, litho-facies and permeability in multi-dimensional data space. As a consequence, all wells in a field were populated with these data to build a robust reservoir model. In addition, the AI predicted data upscaled correctly unlike many conventional techniques. AI gives impressive results when automatically log quality controlling (LQC) and repairing electrical logs for bad hole and sections of missing data. AI doesn’t require prior knowledge of the petrophysical response equations and is self-calibrating. There are no parameters to pick or cross-plots to make. There is very little user intervention and AI avoids the problem of ‘garbage in, garbage out’ (GIGO), by ignoring noise and outliers. AI programs work with an unlimited number of electrical logs, core and gas chromatography data; and don’t ‘fall-over’ if some of those inputs are missing. AI programs currently being developed include ones where their machine code evolves using similar rules used by life’s DNA code. These AI programs pose considerable dangers far beyond the oil industry as described in this paper. A ‘risk assessment’ is essential on all AI programs so that all hazards and risk factors, that could cause harm, are identified and mitigated.

Element Abundance Analysis of the Metal-rich Stellar Halo and High-velocity Thick Disk in the Galaxy

Abstract Based on the second Gaia data release (DR2) and APOGEE (DR16) spectroscopic surveys, we define two kinds of star samples: high-velocity thick disk (HVTD) with v ϕ > 90 km s−1 and metal-rich stellar halo (MRSH) with v ϕ < 90 km s−1. Due to high-resolution spectra data from APOGEE (DR16), we can accurately analyze the element abundance distribution of HVTD and MRSH. These element abundances constitute a multidimensional data space, and we introduce an algorithm method for processing multidimensional data to give the result of dimensionality reduction clustering. According to chemical property analysis, we derive that some HVTD stars could originate from the thin disk and some MRSH stars could originate from dwarf galaxies, but those stars that have similar chemical abundance characteristics in both samples may form in situ.

Automatic water detection from multidimensional hierarchical clustering for Sentinel-2 images and a comparison with Level 2A processors

Continuous monitoring of water surfaces is essential for water resource management. This study presents a nonparametric unsupervised automatic algorithm for the identification of inland water pixels from multispectral satellite data using multidimensional clustering and a high-performance subsampling approach for large scenes. Clustering analysis is a technique that is used to identify similar samples in a multidimensional data space. The spectral information and derived indices were used to characterize each scene pixel individually. A machine learning approach with random subsampling and generalization through a Naïve Bayes classifier was also proposed to make the application of complex algorithms to large scenes feasible. Accuracy was evaluated using an independent dataset that provides water bodies in 15 Sentinel-2 images over France acquired in different seasons and that covers a large range of water bodies and water colour types. The validation dataset covers a water surface of more than 1200 km2 (approximately 12 million pixels) including over 80,000 water bodies outlined using a semiautomatic active learning method, which were manually revised. The classification results were compared to the water pixel classification using three of the major Level 2A processors (MAJA, Sen2Cor and FMask) and two of the most common thresholding techniques: Otsu and Canny-edge. An input mask was used to remove coastal waters, clouds, shadows and snow pixels. Water pixels were identified automatically from the clustering process without the need for ancillary or pretrained data. Combinations using up to three water indices (Modified Normalized Difference Water Index-MNDWI, Normalized Difference Water Index-NDWI and Multiband Water Index-MBWI) and two reflectance bands (B8 and B12) were tested in the algorithm, and the best combination was NDWI-B12. Of all the methods, our method achieved the highest mean kappa score, 0.874, across all tested scenes, with a per-scene kappa ranging from 0.608 to 0.980, and the lowest mean standard deviation of 0.091. Standard Otsu's thresholding had the worst performance due to the lack of a bimodal histogram, and the Canny-edge variation achieved an overall kappa of 0.718 when used with the MNDWI. For water masks provided by generic processors, FMask outperformed MAJA and Sen2Cor and obtained an overall kappa of 0.764. In-depth analysis shows a quick drop in performance for all of the methods in identifying water bodies with a surface area below 0.5 ha, but the proposed approach outperformed the second best method by 34% in this size class.

Gas Turbine Engine Condition Monitoring Using Gaussian Mixture and Hidden Markov Models

This paper investigates the problem of condition monitoring of complex dynamic systems, specifically the detection, localisation and quantification of transient faults. A data driven approach is developed for fault detection where the multidimensional data sequence is viewed as a stochastic process whose behaviour can be described by a hidden Markov model with two hidden states — i.e. ‘healthy / nominal’ and ‘unhealthy / faulty’. The fault detection is performed by first clustering in a multidimensional data space to define normal operating behaviour using a Gaussian-Uniform mixture model. The health status of the system at each data point is then determined by evaluating the posterior probabilities of the hidden states of a hidden Markov model. This allows the temporal relationship between sequential data points to be incorporated into the fault detection scheme. The proposed scheme is robust to noise and requires minimal tuning. A real-world case study is performed based on the detection of transient faults in the variable stator vane actuator of a gas turbine engine to demonstrate the successful application of the scheme. The results are used to demonstrate the generation of simple and easily interpretable analytics that can be used to monitor the evolution of the fault across time.

МНОГОМЕРНЫЙ АНАЛИЗ ДАННЫХ НА ОСНОВЕ РЕЗУЛЬТАТОВ РАСЧЕТОВ ЯДЕРНОЙ ТРАНСМУТАЦИИ В ЦИРКОНИЕВЫХ СПЛАВАХ

The effects of nuclear transmutation in zirconium alloys are considered. The calculated scheme of transmutation of elements in alloys of the system [Zr + x% Nb + y% Mo] is determined. Numerical estimates of the molybdenum content in zirconium alloys of various compositions were carried out during irradiation in research (SM-3, BOR-60) and power (VVER-1000, PWR) reactors. The dynamics of the accumulation of molybdenum during irradiation in the SM reactor and the subsequent exposure of the samples for one year in standard zirconium alloys Zr + 1% Nb, Zr + 2.5% Nb and for the proposed initial compositions Zr + 1% Nb + 0.5% Mo were estimated Zr + 2.5% Nb + 0.5% Mo. Reactor irradiation leads to a significant accumulation of molybdenum in the Zr + 2.5% Nb alloy as a result of nuclear reactions: the molybdenum concentration increases to about 0.7 % with an increase in neutron fluence (E> 0.1 MeV) to 1×1023 cm-2 at irradiation in a research reactor SM. The dynamics of the effects of nuclear transmutation of zirconium upon irradiation in the active zone of research (SM-3, BOR-60) and power (VVER-1000, PWR) reactors is estimated. The Zr element transmutes no more than 10 % in 100 years of exposure. The accumulation of molybdenum over the same time can be up to 104 ppm (appm). The results of the intellectual analysis of the calculations performed with multidimensional analysis based on OLAP of the multidimensional data space of the database provide knowledge in the form of classifications and confirm the need to develop zirconium alloys with additional doping with molybdenum, which can provide optimal physicomechanical properties, in particular, radiation growth characteristics, as well as acceptable level of nuclear-physical and corrosive properties.

Detection of Power Quality Disturbance using a Multidimensional Approach in an Embedded System

Power Quality (PQ) has emerged as an important research field in recent years. The development and increasing use of high power converters and the increase of non linear loads with high power cause unwanted changes in the electrical signals (current and voltage). These changes are called electrical disturbances. To understand such disturbances and investigate their causes, they must be firstly detected. This work proposes a real time multidimensional approach for detecting PQ disturbances. The innovation of this work is the use of a deviation measure easily extracted from a multidimensional data space to quantify the PQ disturbances. The method was implemented in a FPGA (Field-programmable gate array) device using a LabVIEW interface. It performed well with high detection rates and low computational complexity for both simulated and real signals. The proposed deviation measure may be useful as a general power quality index.

A Novel Deep Recurrent Belief Network Model for Trend Prediction of Transformer DGA Data

Oil chromatography data together with its variation trend provide the key basis for the evaluation of the transformer health state. The existing studies on deep belief network (DBN) and support vector machine (SVM) have reported a few results in the field of oil chromatography data prediction. However, the above-mentioned methods are proposed for the classification problem, so there is no theoretical basis for applying the above-mentioned methods to the time-series prediction. The wrong usage limits the accuracy of the predicted results, which is observed as the obvious “time-shift” error in the prediction curve, leading to the predicted state inconsistent with the actual situation. To fill the gap, a deep recurrent belief network (DRBN) model for the transformer state prediction was proposed based on the time-series theory and oil chromatography data characteristics. In this model, the construction shortage of DBN in time-series prediction was analyzed as a prototype pattern to structure the stereoscopic mapping relation. The self-adaptive delay network based on the principle of autocorrelation, together with the corresponding error feedback network, realized the stereoscopic flow of data in multi-dimensional space and, thus, ensured the model's valid expression of the time-domain correlation. In addition, the cross-entropy loss function, based on the Kullback-Leibler divergence and the Weibull distribution with an obvious characteristic of DGA, was constructed to eliminate the uncertainty of initialization process and also effectively control the direction and step size of the error gradient. The examples in the field were used to verify the method to find that the model proposed in this paper can availably overcome the “time-shift” error, and its prediction accuracy can reach more than 95.16%.

Principal component analysis and cluster analysis for evaluating the natural and anthropogenic territory safety

This paper presents an approach to evaluating the natural and technogenic safety of the one of the largest regions in Siberia through the comprehensive analysis of territorial indicators in order to explore geographical variations and patterns in occurrence of emergencies by applying the data mining techniques – principal component analysis and cluster analysis – to data of the Territory Safety Passports. For data modeling, two principal components are selected and interpreted taking account of the contribution of the data attributes to the principal components. Data distribution on the principal components is analyzed at different levels of the territory detail: municipal areas and settlements. Two- and three- cluster structures are constructed in multidimensional data space; the main clusters features are investigated. The results of this analysis have allowed to identify the high-risk territories and rank them according to danger degree of occurrence of the natural and technogenic emergencies. This evaluation gives the basis for decision making and makes it possible for authorities to allocate the forces and means for territory protection more efficiently and develop a system of measures to prevent and mitigate the consequences of emergencies in the large region. The suggested in this work approach in terms of its stages, techniques and reasoning procedures can be considered as a model of comprehensive multidimensional analysis of the control objects in various areas.

A collaborative recommender system enhanced with particle swarm optimization technique

In a web environment, one of the most evolving application is those with recommendation system (RS). It is a subset of information filtering systems wherein, information about certain products or services or a person are categorized and are recommended for the concerned individual. Most of the authors designed collaborative movie recommendation system by using K-NN and K-means but due to a huge increase in movies and users quantity, the neighbour selection is getting more problematic. We propose a hybrid model based on movie recommender system which utilizes type division method and classified the types of the movie according to users which results reduce computation complexity. K-Means provides initial parameters to particle swarm optimization (PSO) so as to improve its performance. PSO provides initial seed and optimizes fuzzy c-means (FCM), for soft clustering of data items (users), instead of strict clustering behaviour in K-Means. For proposed model, we first adopted type division method to reduce the dense multidimensional data space. We looked up for techniques, which could give better results than K-Means and found FCM as the solution. Genetic algorithm (GA) has the limitation of unguided mutation. Hence, we used PSO. In this article experiment performed on Movielens dataset illustrated that the proposed model may deliver high performance related to veracity, and deliver more predictable and personalized recommendations. When compared to already existing methods and having 0.78 mean absolute error (MAE), our result is 3.503 % better with 0.75 as the MAE, showed that our approach gives improved results.

An all-sky support vector machine selection ofWISEYSO candidates

We explored the AllWISE catalogue of the Wide-field Infrared Survey Explorer mission and identified Young Stellar Object candidates. Reliable 2MASS and WISE photometric data combined with Planck dust opacity values were used to build our dataset and to find the best classification scheme. A sophisticated statistical method, the Support Vector Machine (SVM) is used to analyse the multi-dimensional data space and to remove source types identified as contaminants (extragalactic sources, main sequence stars, evolved stars and sources related to the interstellar medium). Objects listed in the SIMBAD database are used to identify the already known sources and to train our method. A new all-sky selection of 133,980 Class I/II YSO candidates is presented. The estimated contamination was found to be well below 1% based on comparison with our SIMBAD training set. We also compare our results to that of existing methods and catalogues. The SVM selection process successfully identified >90% of the Class I/II YSOs based on comparison with photometric and spectroscopic YSO catalogues. Our conclusion is that by using the SVM, our classification is able to identify more known YSOs of the training sample than other methods based on colour-colour and magnitude-colour selection. The distribution of the YSO candidates well correlates with that of the Planck Galactic Cold Clumps in the Taurus--Auriga--Perseus--California region.

Image Understanding Applications of Lattice Autoassociative Memories.

Multivariate mathematical morphology (MMM) aims to extend the mathematical morphology from gray scale images to images whose pixels are high-dimensional vectors, such as remote sensing hyperspectral images and functional magnetic resonance images (fMRIs). Defining an ordering over the multidimensional image data space is a fundamental issue MMM, to ensure that ensuing morphological operators and filters are mathematically consistent. Recent approaches use the outputs of two-class classifiers to build such reduced orderings. This paper presents the applications of MMM built on reduced supervised orderings based on lattice autoassociative memories (LAAMs) recall error measured by the Chebyshev distance. Foreground supervised orderings use one set of training data from a foreground class, whereas background/foreground supervised orderings use two training data sets, one for each relevant class. The first case study refers to the realization of the thematic segmentation of the hyperspectral images using spatial-spectral information. Spectral classification is enhanced by a spatial processing consisting in the spatial correction guided by a watershed segmentation computed by the LAAM-based morphological operators. The approach improves the state-of-the-art hyperspectral spatial-spectral thematic map building approaches. The second case study is the analysis of resting state fMRI data, working on a data set of healthy controls, schizophrenia patients with and without auditory hallucinations. We perform two experiments: 1) the localization of differences in brain functional networks on population-dependent templates and 2) the classification of subjects into each possible pair of cases. In this data set, we find that the LAAM-based morphological features improve over the conventional correlation-based graph measure features often employed in fMRI data classification.

The F distribution and its relationship to the chi squared and t distributions

Many fundamental multivariate methods use the F distribution and its associated tests and critical values: it is the basis of the many common statistical tests in chemometrics, for example, for detecting outliers or whether an observation belongs to a predefined class. We have seen that the t distribution is appropriate for estimating critical values or confidence limits when a population has an underlying normal distribution, but the sample size is small. This is primarily a consequence of the difficulty of determining a population standard deviation, and using a method that more often than not underestimates it, the apparent distribution from the mean is distorted. When we discussed the chi squared distribution 1, we noted that this represented the distribution of squared Mahalanobis distances from the mean, and in particular that if more than one variable is measured, there is no specific positive or negative direction, and as such, using squared distances (which are independent of direction) was essential. Hence, the chi-square distribution naturally extends from univariate to multivariate data. The F distribution can be regarded as the equivalent extension of the t distribution when there is more than one variable but small sample sizes. There are numerous ways of introducing this distribution in the literature, which is widely employed in many diverse areas. In this and the next article, we focus primarily on the distribution of data in multidimensional space: the F distribution is often introduced in the context of analysis of variance. We will come across this distribution and its associated statistic in other contexts in later articles. The F is named after R. A. Fisher, who was a pioneering statistician most active in the 1920s and 1930s, and who worked in agricultural science in the UK. Many of the fundamental multivariate methods such as several approaches for one class classification or class modelling, including SIMCA and multivariate statistical process control, use the F distribution and its associated tests and critical values. It is the basis of the many common statistical tests in chemometrics, for example, for detecting outliers or whether an observation belongs to a predefined class. However, in order to understand it, it is necessary to also understand its relationship to other distributions. If there are a large number of observations (i.e. ν2 is large), then the shape of the F distribution is very similar to the chi squared distribution with ν1 degrees of freedom as illustrated in Figure 2, although there is a shift in position (in fact, chi squared equals ν1 F, and for 1 degree of freedom, they are both the same as ν1 = 1). Note that if both ν1 and ν2 are large, the F distribution also resembles the normal distribution, with a mean of 1. Figure 3 illustrates several different F distributions. We can note several things. If ν1 and ν2 are large, then we can see from the equations earlier that the mean is approximately equal to 1 and the variance to 4/ν2, so the larger ν2, the sharper the Gaussian (and more symmetric). This is illustrated in Figure 4 in the case where ν1 = 1000. Such situations, whilst rare in traditional statistical applications, may often be encountered in chemometrics where there may be a large number of variables, although would still require large sample sizes. Note that the mode changes position as the distribution becomes more symmetric as ν2 increases. In traditional statistical texts, it is usual to present F distribution tables. Because there are rather many possible F distributions, these are usually presented as critical values. A critical value of p = 0.01 gives the value of the F statistic that is expected to be exceeded by only 1% of the data, or in some cases, this can be called the 99% confidence limit. These tables are self-evident and are given in Tables 1 and 2 for two critical values. Note that F tables can be presented for different critical values. There are several more comprehensive tables available on the web 3, 4 although it is recommended that p values are calculated in Excel or any other common environment. Note that the tables later are presented for the one-tailed F cdf in this article. In some contexts, it is appropriate to look at two-tailed F tests, but we will not at this phase be concerned with this.