Vertex component analysis: a geometric-based approach to unmix hyperspectral data

Hyperspectral remote sensing exploits the electromagnetic scattering patterns of the different materials at specific wavelengths [2, 3]. Hyperspectral sensors have been developed to sample the scattered portion of the electromagnetic spectrum extending from the visible region through the near-infrared and mid-infrared, in hundreds of narrow contiguous bands [4, 5]. The number and variety of potential civilian and military applications of hyperspectral remote sensing is enormous [6, 7]. Very often, the resolution cell corresponding to a single pixel in an image contains several substances (endmembers) [4]. In this situation, the scattered energy is a mixing of the endmember spectra. A challenging task underlying many hyperspectral imagery applications is then decomposing a mixed pixel into a collection of reflectance spectra, called endmember signatures, and the corresponding abundance fractions [8–10]. Depending on the mixing scales at each pixel, the observed mixture is either linear or nonlinear [11, 12]. Linear mixing model holds approximately when the mixing scale is macroscopic [13] and there is negligible interaction among distinct endmembers [3, 14]. If, however, the mixing scale is microscopic (or intimate mixtures) [15, 16] and the incident solar radiation is scattered by the scene through multiple bounces involving several endmembers [17], the linear model is no longer accurate. Linear spectral unmixing has been intensively researched in the last years [9, 10, 12, 18–21]. It considers that a mixed pixel is a linear combination of endmember signatures weighted by the correspondent abundance fractions. Under this model, and assuming that the number of substances and their reflectance spectra are known, hyperspectral unmixing is a linear problem for which many solutions have been proposed (e.g., maximum likelihood estimation [8], spectral signature matching [22], spectral angle mapper [23], subspace projection methods [24,25], and constrained least squares [26]). In most cases, the number of substances and their reflectances are not known and, then, hyperspectral unmixing falls into the class of blind source separation problems [27]. Independent component analysis (ICA) has recently been proposed as a tool to blindly unmix hyperspectral data [28–31]. ICA is based on the assumption of mutually independent sources (abundance fractions), which is not the case of hyperspectral data, since the sum of abundance fractions is constant, implying statistical dependence among them. This dependence compromises ICA applicability to hyperspectral images as shown in Refs. [21, 32]. In fact, ICA finds the endmember signatures by multiplying the spectral vectors with an unmixing matrix, which minimizes the mutual information among sources. If sources are independent, ICA provides the correct unmixing, since the minimum of the mutual information is obtained only when sources are independent. This is no longer true for dependent abundance fractions. Nevertheless, some endmembers may be approximately unmixed. These aspects are addressed in Ref. [33]. Under the linear mixing model, the observations from a scene are in a simplex whose vertices correspond to the endmembers. Several approaches [34–36] have exploited this geometric feature of hyperspectral mixtures [35]. Minimum volume transform (MVT) algorithm [36] determines the simplex of minimum volume containing the data. The method presented in Ref. [37] is also of MVT type but, by introducing the notion of bundles, it takes into account the endmember variability usually present in hyperspectral mixtures. The MVT type approaches are complex from the computational point of view. Usually, these algorithms find in the first place the convex hull defined by the observed data and then fit a minimum volume simplex to it. For example, the gift wrapping algorithm [38] computes the convex hull of n data points in a d-dimensional space with a computational complexity of O(nbd=2cþ1), where bxc is the highest integer lower or equal than x and n is the number of samples. The complexity of the method presented in Ref. [37] is even higher, since the temperature of the simulated annealing algorithm used shall follow a log( ) law [39] to assure convergence (in probability) to the desired solution. Aiming at a lower computational complexity, some algorithms such as the pixel purity index (PPI) [35] and the N-FINDR [40] still find the minimum volume simplex containing the data cloud, but they assume the presence of at least one pure pixel of each endmember in the data. This is a strong requisite that may not hold in some data sets. In any case, these algorithms find the set of most pure pixels in the data. PPI algorithm uses the minimum noise fraction (MNF) [41] as a preprocessing step to reduce dimensionality and to improve the signal-to-noise ratio (SNR). The algorithm then projects every spectral vector onto skewers (large number of random vectors) [35, 42,43]. The points corresponding to extremes, for each skewer direction, are stored. A cumulative account records the number of times each pixel (i.e., a given spectral vector) is found to be an extreme. The pixels with the highest scores are the purest ones. N-FINDR algorithm [40] is based on the fact that in p spectral dimensions, the p-volume defined by a simplex formed by the purest pixels is larger than any other volume defined by any other combination of pixels. This algorithm finds the set of pixels defining the largest volume by inflating a simplex inside the data. ORA SIS [44, 45] is a hyperspectral framework developed by the U.S. Naval Research Laboratory consisting of several algorithms organized in six modules: exemplar selector, adaptative learner, demixer, knowledge base or spectral library, and spatial postrocessor. The first step consists in flat-fielding the spectra. Next, the exemplar selection module is used to select spectral vectors that best represent the smaller convex cone containing the data. The other pixels are rejected when the spectral angle distance (SAD) is less than a given thresh old. The procedure finds the basis for a subspace of a lower dimension using a modified Gram–Schmidt orthogonalizati on. The selected vectors are then projected onto this subspace and a simplex is found by an MV T pro cess. ORA SIS is oriented to real-time target detection from uncrewed air vehicles using hyperspectral data [46]. In this chapter we develop a new algorithm to unmix linear mixtures of endmember spectra. First, the algorithm determines the number of endmembers and the signal subspace using a newly developed concept [47, 48]. Second, the algorithm extracts the most pure pixels present in the data. Unlike other methods, this algorithm is completely automatic and unsupervised. To estimate the number of endmembers and the signal subspace in hyperspectral linear mixtures, the proposed scheme begins by estimating sign al and noise correlation matrices. The latter is based on multiple regression theory. The signal subspace is then identified by selectin g the set of signal eigenvalue s that best represents the data, in the least-square sense [48,49 ], we note, however, that VCA works with projected and with unprojected data. The extraction of the end members exploits two facts: (1) the endmembers are the vertices of a simplex and (2) the affine transformation of a simplex is also a simplex. As PPI and N-FIND R algorithms, VCA also assumes the presence of pure pixels in the data. The algorithm iteratively projects data on to a direction orthogonal to the subspace spanned by the endmembers already determined. The new end member signature corresponds to the extreme of the projection. The algorithm iterates until all end members are exhausted. VCA performs much better than PPI and better than or comparable to N-FI NDR; yet it has a computational complexity between on e and two orders of magnitude lower than N-FINDR. The chapter is structure d as follows. Section 19.2 describes the fundamentals of the proposed method. Section 19.3 and Section 19.4 evaluate the proposed algorithm using simulated and real data, respectively. Section 19.5 presents some concluding remarks.

Feature selection for clustering categorical data with an embedded modelling approach

Research on the problem of feature selection for clustering continues to develop. This is a challenging task, mainly due to the absence of class labels to guide the search for relevant features. Categorical feature selection for clustering has rarely been addressed in the literature, with most of the proposed approaches having focused on numerical data. In this work, we propose an approach to simultaneously cluster categorical data and select a subset of relevant features. Our approach is based on a modification of a finite mixture model (of multinomial distributions), where a set of latent variables indicate the relevance of each feature. To estimate the model parameters, we implement a variant of the expectation-maximization algorithm that simultaneously selects the subset of relevant features, using a minimum message length criterion. The proposed approach compares favourably with two baseline methods: a filter based on an entropy measure and a wrapper based on mutual information. The results obtained on synthetic data illustrate the ability of the proposed expectation-maximization method to recover ground truth. An application to real data, referred to official statistics, shows its usefulness.

Diagnosis of Tinea pedis and onychomycosis in patients from Portuguese National Institute of Health: a four-year study

ABSTRACT - Tinea pedis and onychomycosis are two rather diverse clinical manifestations of superficial fungal infections, and their etiologic agents may be dermatophytes, non-dermatophyte moulds or yeasts. This study was designed to statistically describe the data obtained as results of analysis conducted during a four year period on the frequency of Tinea pedis and onychomycosis and their etiologic agents. A questionnaire was distributed from 2006 to 2010 and answered by 186 patients, who were subjected to skin and/or nail sampling. Frequencies of the isolated fungal species were cross-linked with the data obtained with the questionnaire, seeking associations and predisposing factors. One hundred and sixty three fungal isolates were obtained, 24.2% of which composed by more than one fungal species. Most studies report the two pathologies as caused primarily by dermatophytes, followed by yeasts and lastly by non-dermatophytic moulds. Our study does not challenge this trend. We found a frequency of 15.6% of infections caused by dermatophytes (with a total of 42 isolates) of which T. rubrum was the most frequent species (41.4%). There was no significant association (p >0.05) among visible injury and the independent variables tested, namely age, gender, owning pet, education, swimming pools attendance, sports activity and clinical information. Unlike other studies, the variables considered did not show the expected influence on dermatomycosis of the lower limbs. It is hence necessary to conduct further studies to specifically identify which variables do in fact influence such infections.

Análise técnica dos principais índices bolsistas europeus

Mestrado em Controlo e Gestão e dos Negócios

Categorical data clustering using a minimum message length criterion

Research on cluster analysis for categorical data continues to develop, new clustering algorithms being proposed. However, in this context, the determination of the number of clusters is rarely addressed. We propose a new approach in which clustering and the estimation of the number of clusters is done simultaneously for categorical data. We assume that the data originate from a finite mixture of multinomial distributions and use a minimum message length criterion (MML) to select the number of clusters (Wallace and Bolton, 1986). For this purpose, we implement an EM-type algorithm (Silvestre et al., 2008) based on the (Figueiredo and Jain, 2002) approach. The novelty of the approach rests on the integration of the model estimation and selection of the number of clusters in a single algorithm, rather than selecting this number based on a set of pre-estimated candidate models. The performance of our approach is compared with the use of Bayesian Information Criterion (BIC) (Schwarz, 1978) and Integrated Completed Likelihood (ICL) (Biernacki et al., 2000) using synthetic data. The obtained results illustrate the capacity of the proposed algorithm to attain the true number of cluster while outperforming BIC and ICL since it is faster, which is especially relevant when dealing with large data sets.

Determining the number of clusters in categorical data

Cluster analysis for categorical data has been an active area of research. A well-known problem in this area is the determination of the number of clusters, which is unknown and must be inferred from the data. In order to estimate the number of clusters, one often resorts to information criteria, such as BIC (Bayesian information criterion), MML (minimum message length, proposed by Wallace and Boulton, 1968), and ICL (integrated classification likelihood). In this work, we adopt the approach developed by Figueiredo and Jain (2002) for clustering continuous data. They use an MML criterion to select the number of clusters and a variant of the EM algorithm to estimate the model parameters. This EM variant seamlessly integrates model estimation and selection in a single algorithm. For clustering categorical data, we assume a finite mixture of multinomial distributions and implement a new EM algorithm, following a previous version (Silvestre et al., 2008). Results obtained with synthetic datasets are encouraging. The main advantage of the proposed approach, when compared to the above referred criteria, is the speed of execution, which is especially relevant when dealing with large data sets.

El conservadurismo, el secreto y las opciones de financiación de las pyme portuguesas

Doutoramento em Economia Financeira e Contabilidade

Enhancing the selection of a model-based clustering with external categorical variables

In cluster analysis, it can be useful to interpret the partition built from the data in the light of external categorical variables which are not directly involved to cluster the data. An approach is proposed in the model-based clustering context to select a number of clusters which both fits the data well and takes advantage of the potential illustrative ability of the external variables. This approach makes use of the integrated joint likelihood of the data and the partitions at hand, namely the model-based partition and the partitions associated to the external variables. It is noteworthy that each mixture model is fitted by the maximum likelihood methodology to the data, excluding the external variables which are used to select a relevant mixture model only. Numerical experiments illustrate the promising behaviour of the derived criterion. © 2014 Springer-Verlag Berlin Heidelberg.

Approximate entropy normalized measures for analyzing social neurobiological systems

When considering time series data of variables describing agent interactions in social neurobiological systems, measures of regularity can provide a global understanding of such system behaviors. Approximate entropy (ApEn) was introduced as a nonlinear measure to assess the complexity of a system behavior by quantifying the regularity of the generated time series. However, ApEn is not reliable when assessing and comparing the regularity of data series with short or inconsistent lengths, which often occur in studies of social neurobiological systems, particularly in dyadic human movement systems. Here, the authors present two normalized, nonmodified measures of regularity derived from the original ApEn, which are less dependent on time series length. The validity of the suggested measures was tested in well-established series (random and sine) prior to their empirical application, describing the dyadic behavior of athletes in team games. The authors consider one of the ApEn normalized measures to generate the 95th percentile envelopes that can be used to test whether a particular social neurobiological system is highly complex (i.e., generates highly unpredictable time series). Results demonstrated that suggested measures may be considered as valid instruments for measuring and comparing complexity in systems that produce time series with inconsistent lengths.

ALSMon: sistema para monitorizar ALS

Trabalho Final de Mestrado para obtenção do grau de Mestre em Engenharia Informática e de Computadores

Enhancing the selection of a model-based clustering with external categorical variables

In cluster analysis, it can be useful to interpret the partition built from the data in the light of external categorical variables which are not directly involved to cluster the data. An approach is proposed in the model-based clustering context to select a number of clusters which both fits the data well and takes advantage of the potential illustrative ability of the external variables. This approach makes use of the integrated joint likelihood of the data and the partitions at hand, namely the model-based partition and the partitions associated to the external variables. It is noteworthy that each mixture model is fitted by the maximum likelihood methodology to the data, excluding the external variables which are used to select a relevant mixture model only. Numerical experiments illustrate the promising behaviour of the derived criterion.