Measures of fit in multiple correspondence analysis of crisp and fuzzy coded data

When continuous data are coded to categorical variables, two types of coding are possible: crisp coding in the form of indicator, or dummy, variables with values either 0 or 1; or fuzzy coding where each observation is transformed to a set of "degrees of membership" between 0 and 1, using co-called membership functions. It is well known that the correspondence analysis of crisp coded data, namely multiple correspondence analysis, yields principal inertias (eigenvalues) that considerably underestimate the quality of the solution in a low-dimensional space. Since the crisp data only code the categories to which each individual case belongs, an alternative measure of fit is simply to count how well these categories are predicted by the solution. Another approach is to consider multiple correspondence analysis equivalently as the analysis of the Burt matrix (i.e., the matrix of all two-way cross-tabulations of the categorical variables), and then perform a joint correspondence analysis to fit just the off-diagonal tables of the Burt matrix - the measure of fit is then computed as the quality of explaining these tables only. The correspondence analysis of fuzzy coded data, called "fuzzy multiple correspondence analysis", suffers from the same problem, albeit attenuated. Again, one can count how many correct predictions are made of the categories which have highest degree of membership. But here one can also defuzzify the results of the analysis to obtain estimated values of the original data, and then calculate a measure of fit in the familiar percentage form, thanks to the resultant orthogonal decomposition of variance. Furthermore, if one thinks of fuzzy multiple correspondence analysis as explaining the two-way associations between variables, a fuzzy Burt matrix can be computed and the same strategy as in the crisp case can be applied to analyse the off-diagonal part of this matrix. In this paper these alternative measures of fit are defined and applied to a data set of continuous meteorological variables, which are coded crisply and fuzzily into three categories. Measuring the fit is further discussed when the data set consists of a mixture of discrete and continuous variables.

An Automatic Approach for Learning and Tuning Gaussian Interval Type-2 Fuzzy Membership Functions Applied to Lung CAD Classification System

The potential of type-2 fuzzy sets for managing high levels of uncertainty in the subjective knowledge of experts or of numerical information has focused on control and pattern classification systems in recent years. One of the main challenges in designing a type-2 fuzzy logic system is how to estimate the parameters of type-2 fuzzy membership function (T2MF) and the Footprint of Uncertainty (FOU) from imperfect and noisy datasets. This paper presents an automatic approach for learning and tuning Gaussian interval type-2 membership functions (IT2MFs) with application to multi-dimensional pattern classification problems. T2MFs and their FOUs are tuned according to the uncertainties in the training dataset by a combination of genetic algorithm (GA) and crossvalidation techniques. In our GA-based approach, the structure of the chromosome has fewer genes than other GA methods and chromosome initialization is more precise. The proposed approach addresses the application of the interval type-2 fuzzy logic system (IT2FLS) for the problem of nodule classification in a lung Computer Aided Detection (CAD) system. The designed IT2FLS is compared with its type-1 fuzzy logic system (T1FLS) counterpart. The results demonstrate that the IT2FLS outperforms the T1FLS by more than 30% in terms of classification accuracy.

Fuzzy coding in constrained ordinations

Canonical correspondence analysis and redundancy analysis are two methods of constrained ordination regularly used in the analysis of ecological data when several response variables (for example, species abundances) are related linearly to several explanatory variables (for example, environmental variables, spatial positions of samples). In this report I demonstrate the advantages of the fuzzy coding of explanatory variables: first, nonlinear relationships can be diagnosed; second, more variance in the responses can be explained; and third, in the presence of categorical explanatory variables (for example, years, regions) the interpretation of the resulting triplot ordination is unified because all explanatory variables are measured at a categorical level.

The modal nature of structures in ontic structural realism

Ontic structural realism is the view that structures are what is real in the first place in the domain of fundamental physics. The structures are usually conceived as including a primitive modality. However, it has not been spelled out as yet what exactly that modality amounts to. This paper proposes to fill this lacuna by arguing that the fundamental physical structures possess a causal essence, being powers. Applying the debate about causal vs. categorical properties in analytic metaphysics to ontic structural realism, I show that the standard argument against categorical and for causal properties holds for structures as well. Structural realism, as a position in the metaphysics of science that is a form of scientific realism, is committed to causal structures. The metaphysics of causal structures is supported by physics, and it can provide for a complete and coherent view of the world that includes all domains of empirical science.

Biplots of fuzzy coded data

A biplot, which is the multivariate generalization of the two-variable scatterplot, can be used to visualize the results of many multivariate techniques, especially those that are based on the singular value decomposition. We consider data sets consisting of continuous-scale measurements, their fuzzy coding and the biplots that visualize them, using a fuzzy version of multiple correspondence analysis. Of special interest is the way quality of fit of the biplot is measured, since it is well-known that regular (i.e., crisp) multiple correspondence analysis seriously under-estimates this measure. We show how the results of fuzzy multiple correspondence analysis can be defuzzified to obtain estimated values of the original data, and prove that this implies an orthogonal decomposition of variance. This permits a measure of fit to be calculated in the familiar form of a percentage of explained variance, which is directly comparable to the corresponding fit measure used in principal component analysis of the original data. The approach is motivated initially by its application to a simulated data set, showing how the fuzzy approach can lead to diagnosing nonlinear relationships, and finally it is applied to a real set of meteorological data.

Multi-modal assessment of long-term erythropoietin treatment after neonatal hypoxic-ischemic injury in rat brain.

Erythropoietin (EPO) has been recognized as a neuroprotective agent. In animal models of neonatal brain injury, exogenous EPO has been shown to reduce lesion size, improve structure and function. Experimental studies have focused on short course treatment after injury. Timing, dose and length of treatment in preterm brain damage remain to be defined. We have evaluated the effects of high dose and long-term EPO treatment in hypoxic-ischemic (HI) injury in 3 days old (P3) rat pups using histopathology, magnetic resonance imaging (MRI) and spectroscopy (MRS) as well as functional assessment with somatosensory-evoked potentials (SEP). After HI, rat pups were assessed by MRI for initial damage and were randomized to receive EPO or vehicle. At the end of treatment period (P25) the size of resulting cortical damage and white matter (WM) microstructure integrity were assessed by MRI and cortical metabolism by MRS. Whisker elicited SEP were recorded to evaluate somatosensory function. Brains were collected for neuropathological assessment. The EPO treated animals did not show significant decrease of the HI induced cortical loss at P25. WM microstructure measured by diffusion tensor imaging was improved and SEP response in the injured cortex was recovered in the EPO treated animals compared to vehicle treated animals. In addition, the metabolic profile was less altered in the EPO group. Long-term treatment with high dose EPO after HI injury in the very immature rat brain induced recovery of WM microstructure and connectivity as well as somatosensory cortical function despite no effects on volume of cortical damage. This indicates that long-term high-dose EPO induces recovery of structural and functional connectivity despite persisting gross anatomical cortical alteration resulting from HI.

Application of fuzzy-rule-based postprocessing to correlation methods in pattern recognition

One of the most important problems in optical pattern recognition by correlation is the appearance of sidelobes in the correlation plane, which causes false alarms. We present a method that eliminate sidelobes of up to a given height if certain conditions are satisfied. The method can be applied to any generalized synthetic discriminant function filter and is capable of rejecting lateral peaks that are even higher than the central correlation. Satisfactory results were obtained in both computer simulations and optical implementation.

Fuzzy logic applied to the modeling of water dynamics in an Oxisol in northeastern Brazil

Modeling of water movement in non-saturated soil usually requires a large number of parameters and variables, such as initial soil water content, saturated water content and saturated hydraulic conductivity, which can be assessed relatively easily. Dimensional flow of water in the soil is usually modeled by a nonlinear partial differential equation, known as the Richards equation. Since this equation cannot be solved analytically in certain cases, one way to approach its solution is by numerical algorithms. The success of numerical models in describing the dynamics of water in the soil is closely related to the accuracy with which the water-physical parameters are determined. That has been a big challenge in the use of numerical models because these parameters are generally difficult to determine since they present great spatial variability in the soil. Therefore, it is necessary to develop and use methods that properly incorporate the uncertainties inherent to water displacement in soils. In this paper, a model based on fuzzy logic is used as an alternative to describe water flow in the vadose zone. This fuzzy model was developed to simulate the displacement of water in a non-vegetated crop soil during the period called the emergency phase. The principle of this model consists of a Mamdani fuzzy rule-based system in which the rules are based on the moisture content of adjacent soil layers. The performances of the results modeled by the fuzzy system were evaluated by the evolution of moisture profiles over time as compared to those obtained in the field. The results obtained through use of the fuzzy model provided satisfactory reproduction of soil moisture profiles.

Espectros Vis-NIR do Solo e Fuzzy K-Médias Aplicados na Delimitação de Unidades de Mapeamento de Solos em Topossequências

RESUMO O conhecimento dos solos é cada vez mais importante para que o uso dele seja realizado corretamente na agropecuária, no crescimento urbano, na conservação dos recursos naturais, entre outros. Entretanto, verifica-se carência de profissionais qualificados para a caracterização e os mapeamentos pedológicos, particularmente em escalas de maior detalhamento. Essa carência, aliada aos avanços das ferramentas computacionais e do sensoriamento remoto, promoveu o surgimento do Mapeamento Digital de Solos (MDS), que busca auxiliar e agilizar as atividades de levantamento pedológico. Assim, este trabalho objetivou desenvolver uma metodologia de delimitaçao de unidades de solos em topossequências por meio do comportamento espectral dos solos no comprimento de onda do Visível-Infravermelho Próximo (Vis-NIR). A metodologia espectral consistiu na obtenção das curvas espectrais dos solos por meio do espectrorradiômetro FieldSpecPro e da redução do número de informações espectrais por meio da análise de Componentes Principais, seguida de agrupamento das amostras mediante método fuzzy k-médias. Foram selecionadas cinco topossequências com pontos equidistantes de 30 m para caracterizar as classes de solos e amostragens. Foram descritas oito classes de solos distintas, que tiveram caracterização detalhada e classificação em perfis pedológicos. No restante dos pontos, a caracterização das classes de solos foi feita com base na classificação dos solos realizada nos perfis pedológicos, com coleta de amostras por meio de tradagens nas profundidades de 0,00-0,20 e 0,80-1,00 m, perfazendo o total de 162 amostras ao longo das cinco topossequências. As amostras foram analisadas pelas metodologias convencional e espectral, para que os resultados pudessem ser comparados e avaliados. Dessa forma, foram realizadas análises morfológicas, físicas (textura) e químicas nas amostras de solo. Das cinco topossequências estudadas, os resultados foram satisfatoriamente semelhantes; alguns solos não foram perfeitamente individualizados pela metodologia espectral, em razão da grande semelhança em seus comportamentos espectrais, como demonstrado pelo Latossolo Vermelho Férrico e Nitossolo Vermelho Férrico. A metodologia espectral foi capaz de diferenciar solos com resposta espectral distinta e estabelecer limites nas topossequências, apresentando grande potencial para ser implementada em levantamentos pedológicos.

Identification of different heart tissues from MRI C-SENC images using an unsupervised multi-stage fuzzy clustering technique.

PURPOSE: To objectively characterize different heart tissues from functional and viability images provided by composite-strain-encoding (C-SENC) MRI. MATERIALS AND METHODS: C-SENC is a new MRI technique for simultaneously acquiring cardiac functional and viability images. In this work, an unsupervised multi-stage fuzzy clustering method is proposed to identify different heart tissues in the C-SENC images. The method is based on sequential application of the fuzzy c-means (FCM) and iterative self-organizing data (ISODATA) clustering algorithms. The proposed method is tested on simulated heart images and on images from nine patients with and without myocardial infarction (MI). The resulting clustered images are compared with MRI delayed-enhancement (DE) viability images for determining MI. Also, Bland-Altman analysis is conducted between the two methods. RESULTS: Normal myocardium, infarcted myocardium, and blood are correctly identified using the proposed method. The clustered images correctly identified 90 +/- 4% of the pixels defined as infarct in the DE images. In addition, 89 +/- 5% of the pixels defined as infarct in the clustered images were also defined as infarct in DE images. The Bland-Altman results show no bias between the two methods in identifying MI. CONCLUSION: The proposed technique allows for objectively identifying divergent heart tissues, which would be potentially important for clinical decision-making in patients with MI.

The contributions of sensory dominance and attentional bias to cross-modal enhancement of visual cortex excitability.

Approaching or looming sounds (L-sounds) have been shown to selectively increase visual cortex excitability [Romei, V., Murray, M. M., Cappe, C., & Thut, G. Preperceptual and stimulus-selective enhancement of low-level human visual cortex excitability by sounds. Current Biology, 19, 1799-1805, 2009]. These cross-modal effects start at an early, preperceptual stage of sound processing and persist with increasing sound duration. Here, we identified individual factors contributing to cross-modal effects on visual cortex excitability and studied the persistence of effects after sound offset. To this end, we probed the impact of different L-sound velocities on phosphene perception postsound as a function of individual auditory versus visual preference/dominance using single-pulse TMS over the occipital pole. We found that the boosting of phosphene perception by L-sounds continued for several tens of milliseconds after the end of the L-sound and was temporally sensitive to different L-sound profiles (velocities). In addition, we found that this depended on an individual's preferred sensory modality (auditory vs. visual) as determined through a divided attention task (attentional preference), but not on their simple threshold detection level per sensory modality. Whereas individuals with "visual preference" showed enhanced phosphene perception irrespective of L-sound velocity, those with "auditory preference" showed differential peaks in phosphene perception whose delays after sound-offset followed the different L-sound velocity profiles. These novel findings suggest that looming signals modulate visual cortex excitability beyond sound duration possibly to support prompt identification and reaction to potentially dangerous approaching objects. The observed interindividual differences favor the idea that unlike early effects this late L-sound impact on visual cortex excitability is influenced by cross-modal attentional mechanisms rather than low-level sensory processes.

ConnectomeViewer - Multi-Modal Multi-Level Network Visualization and Analysis.

Introduction: The field of Connectomic research is growing rapidly, resulting from methodological advances in structural neuroimaging on many spatial scales. Especially progress in Diffusion MRI data acquisition and processing made available macroscopic structural connectivity maps in vivo through Connectome Mapping Pipelines (Hagmann et al, 2008) into so-called Connectomes (Hagmann 2005, Sporns et al, 2005). They exhibit both spatial and topological information that constrain functional imaging studies and are relevant in their interpretation. The need for a special-purpose software tool for both clinical researchers and neuroscientists to support investigations of such connectome data has grown. Methods: We developed the ConnectomeViewer, a powerful, extensible software tool for visualization and analysis in connectomic research. It uses the novel defined container-like Connectome File Format, specifying networks (GraphML), surfaces (Gifti), volumes (Nifti), track data (TrackVis) and metadata. Usage of Python as programming language allows it to by cross-platform and have access to a multitude of scientific libraries. Results: Using a flexible plugin architecture, it is possible to enhance functionality for specific purposes easily. Following features are already implemented: * Ready usage of libraries, e.g. for complex network analysis (NetworkX) and data plotting (Matplotlib). More brain connectivity measures will be implemented in a future release (Rubinov et al, 2009). * 3D View of networks with node positioning based on corresponding ROI surface patch. Other layouts possible. * Picking functionality to select nodes, select edges, get more node information (ConnectomeWiki), toggle surface representations * Interactive thresholding and modality selection of edge properties using filters * Arbitrary metadata can be stored for networks, thereby allowing e.g. group-based analysis or meta-analysis. * Python Shell for scripting. Application data is exposed and can be modified or used for further post-processing. * Visualization pipelines using filters and modules can be composed with Mayavi (Ramachandran et al, 2008). * Interface to TrackVis to visualize track data. Selected nodes are converted to ROIs for fiber filtering The Connectome Mapping Pipeline (Hagmann et al, 2008) processed 20 healthy subjects into an average Connectome dataset. The Figures show the ConnectomeViewer user interface using this dataset. Connections are shown that occur in all 20 subjects. The dataset is freely available from the homepage (connectomeviewer.org). Conclusions: The ConnectomeViewer is a cross-platform, open-source software tool that provides extensive visualization and analysis capabilities for connectomic research. It has a modular architecture, integrates relevant datatypes and is completely scriptable. Visit www.connectomics.org to get involved as user or developer.

A fuzzy-based decision model application on strategic management

In this article, the objective is to demonstrate the effects of different decision styles on strategic decisions and likewise, on an organization. The technique that was presented in the study is based on the transformation of linguistic variables to numerical value intervals. In this model, the study benefits from fuzzy logic methodology and fuzzy numbers. This fuzzy methodology approach allows us to examine the relations between decision making styles and strategic management processes when there is uncertainty. The purpose is to provide results to companies that may help them to exercise the most appropriate decision making style for its different strategic management processes. The study is leaving more research topics for further studies that may be applied to other decision making areas within the strategic management process.